Line Graphs and Line Digraphs (Developments in Mathematics, 68) 3030813843, 9783030813840

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Developments in Mathematics

Lowell W. Beineke Jay S. Bagga

Line Graphs and Line Digraphs

Developments in Mathematics Volume 68

Series Editors Krishnaswami Alladi, Department of Mathematics, University of Florida, Gainesville, FL, USA Pham Huu Tiep, Department of Mathematics, Rutgers University, Piscataway, NJ, USA Loring W. Tu, Department of Mathematics, Tufts University, Medford, MA, USA

Aims and Scope The Developments in Mathematics (DEVM) book series is devoted to publishing well-written monographs within the broad spectrum of pure and applied mathematics. Ideally, each book should be self-contained and fairly comprehensive in treating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential relationship to other fields are most welcome. High-quality edited volumes conveying current state-of-the-art research will occasionally also be considered for publication. The DEVM series appeals to a variety of audiences including researchers, postdocs, and advanced graduate students.

More information about this series at http://www.springer.com/series/5834

Lowell W. Beineke • Jay S. Bagga

Line Graphs and Line Digraphs

Lowell W. Beineke Department of Mathematical Sciences Purdue University Fort Wayne Fort Wayne, IN, USA

Jay S. Bagga Department of Computer Science Ball State University Muncie, IN, USA

ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-030-81384-0 ISBN 978-3-030-81386-4 (eBook) https://doi.org/10.1007/978-3-030-81386-4 Mathematics Subject Classification: 05C76, 05C20, 05C75, 05C50, 05C45 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland.

We dedicate this book to our wives Judith Beineke Daljit Bagga

Foreword

For a subject that became one of the most studied and popular areas of discrete mathematics, graph theory has had a rather curious history. It is considered to have begun with the Königsberg Bridge Problem (Does there exist a walking route about the eighteenth century city of Königsberg that crosses each of its seven bridges exactly once?) and its solution in 1736 by the famous Swiss mathematician Leonhard Euler. The paper by Euler containing the solution indirectly suggests a structure that was later to be called a graph (a term coined by the British mathematician James Joseph Sylvester). For the next century and a half, similar structures appeared, often indirectly again, through questions, problems, and puzzles that were more of a recreational nature. Even then, it was often only later that it was seen that the problems could be looked at more mathematically with the aid of graphs. These graphs repeatedly provided a more visual way of looking at the problems and often suggested other questions. As time passed, graphs themselves became the subject of study, in a number of cases inspired by attempts to solve one of the famous problems of the time, the Four Color Problem (Can the countries of every map be colored with one of four colors so that two countries with a common boundary are colored differently?). This problem gave rise to research papers on graphs, including an important one in 1890 by the British mathematician Percy John Heawood. One year later, a paper of a strictly theoretical nature, dealing with regular graphs, was written by the Danish mathematician Julius Petersen. Graphs have been shown to have a wide range of applications and have appeared in diverse contexts, even with sporting events. For example, suppose that there are several tennis players who are to participate in a tennis competition. Certain pairs of these individuals are scheduled to be involved in a match. This situation gives rise quite naturally to a graph G. The participants will be the vertices of G and there is an edge between two vertices if the corresponding participants are to have a match in the competition. The graph G then gives a more visual picture of the entire competition. At the conclusion of the competition, the graph G could be converted into a directed graph (digraph) D by assigning a direction to an edge from vertex u to vertex v if u defeated v in their match. The digraph D not only gives us a picture of the entire competition but provides us with the outcomes and which participants vii

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Foreword

did well and which didn’t. Such a digraph often suggests other questions. As with graphs, digraphs have become the object of mathematical study. In the tennis competition example just described, another question occurs with regard to the graph G. Can the graph G be used to provide a possible time schedule of all matches to take place in the competition? One observation can be made immediately. Two matches cannot be scheduled to take place at the same time if the matches involve the same participant. To help us answer the scheduling question, there is another structure (another graph) that can be analyzed—one in which the basic objects are the matches (the edges of G). In this new graph, to be denoted by L(G), the vertices are the edges of G and there is an edge between two vertices of L(G) if the corresponding edges of G have a vertex of G in common. The graph L(G) is called the line graph of G. If, instead of beginning with a graph G, we begin with a digraph D, then a similar structure, the line digraph L(D) of D, can be constructed and studied. The concept of line graphs is considered to have originated by the American mathematician and mathematics educator Hassler Whitney in 1932, although it appears only indirectly in his work. The growing interest in this concept became even more evident when it appeared in the 1962 textbook on graph theory by Oystein Ore—the first book on graph theory written in English. In his book, Ore looked at a line graph as emanating from the edge adjacency matrix of some graph G, which in turn could be looked at in terms of the vertex adjacency matrix of a new graph, namely the line graph L(G). Ore called these new graphs interchange graphs. Indeed, line graphs have been referred to by many names, but it was the term line graph that became the standard, a term coined by the famous graph theorist Frank Harary, who was known to many as Mr. Graph Theory. Over the years, the study of graphs that are the line graph of some graph has grown in interest; numerous interesting questions and problems developed and were studied by many in all parts of the world. The most fundamental of these questions was: Is a given graph the line graph of some graph? This question has been answered in different ways. One especially unique way of answering this question was provided by Lowell Beineke (one of the authors of the book Line Graphs and Line Digraphs) in a theorem that became one of the classic theorems of graph theory. He showed that if a given graph contained any of nine certain substructures, then it was impossible for this graph to be the line graph of some graph; while if a graph failed to contain any of these nine substructures, then it must be the line graph of some graph. In the book Line Graphs and Line Digraphs, the authors present the primary results dealing with line graphs and line digraphs. Two principal questions addressed in the book are: If a graph G possesses a particular property, then what does this tell us about its line graph L(G)? If the line graph L(G) of a graph G possesses a particular property, what does this tell us about G? Corresponding questions are also addressed for digraphs and line digraphs. The definition of a line graph itself has suggested other possible related graphs emanating from a given graph, resulting in generalizations of line graphs. The authors discuss these generalizations as well.

Foreword

ix

With all of the research and resulting information that have been obtained on line graphs and line digraphs for nearly a century, it is both timely and fortunate that a book has been written on these topics—and written by two mathematicians not only known for their research on the aforementioned topics but also known for their writing styles and clarity of writing: Lowell Beineke and Jay Bagga. These two highly respected mathematicians have not only done much research on line graphs and line digraphs but have an added connection to these topics, for the person who originated the term line graph, Frank Harary, was the academic father of Lowell Beineke and the academic grandfather of Jay Bagga. The authors have superbly presented in a clear and interesting manner how this class of graphs and digraphs has caught both the attention of and interest in and fascinated many for so many years. Kalamazoo, MI, USA April 2021

Gary Chartrand

Preface

In the past century, the field of graph theory has expanded enormously, proceeding from König’s foundational book in 1936 to numerous books today, some general, some specialized. One important area that appears to have been overlooked is that of line graphs, and this book is our response to that need. The precise origins of the idea of focusing on the edges of a graph (by whatever name) more than on the vertices are difficult to pinpoint, and are likely to have occurred at different times. One giant step in this direction was taken in 1880 when the Scottish mathematical physicist P. G. Tait converted the four color problem (in the terminology of graph theory) from coloring the vertices of a graph to an equivalent problem of coloring the edges of a different graph. Another breakthrough in coloring the edges of graphs came in a 1916 paper by Dénes König, but it was apparently the work of Hassler Whitney on edge-isomorphisms in papers published in 1932 and 1933 that led to the study of line graphs as we know them today. To the best of our knowledge, the next paper on line graphs was in Hungarian and appeared a decade later, in 1943. It was written by J. Krausz and included a characterization of those graphs that are line graphs, and this was his only known publication. An even longer period of time passed before another landmark appeared: in 1960, Frank Harary and R. Z. Norman introduced line digraphs to the world. By this time, graph theory had emerged as a subject in its own right, and line graphs and line digraphs had come of interest. Among the milestones on this journey are the following: The first doctoral dissertation on line graphs that we are aware of was that of Gary Chartrand at Michigan State University in 1964; the first chapter on line graphs in a book appeared in Harary’s Graph Theory, published in 1969, and the first book on a topic that is ostensibly about line graphs was Edge-colourings of Graphs by Stanley Fiorini and Robin Wilson in 1977. The idea of writing this book actually originated with Badri Varma while he was a professor at the University of Wisconsin-Fox Valley. Together, the three of us worked on an initial list of chapters. Varma’s diligent work in creating a comprehensive bibliography and his efforts in gathering reprints of many papers were crucial in our later work. After a couple of years, Varma decided that he would not be able to continue with this project due to his pursuit of other creative interests. xi

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However, his imprint on our work is enormous, and we value his assistance, motivation, and inspiration. We shift now to our individual stories that led us to writing this book (we use our initials to indicate which of us is the author). LB: In his graph theory course at the University of Michigan, Frank Harary asked the class to answer the question of which graphs had planar line graphs. Unbeknownst to him at the time, the solution had recently been published by Jiˇrí Sedláˇcek. Although I was a bit disappointed in not being the first to find the answer, I believe that this was the beginning of my fascination with line graphs. A major contribution to my story came in a 1965 paper by Arnoud van Rooij and Herb Wilf. It contained a second characterization of line graphs and led me to the third characterization, the discovery of the nine key forbidden subgraphs for line graphs. A subsequent delight occurred when Robin Whitty featured this result as his online Theorem of the Day (#48). His article included a version of this limerick: The mathematician named Lowell Beineke Determined the graphs line, to be, Those it’s forbidden In G to find hidden A graph from his set of size nine—the key. JB: Interestingly, I too was introduced to graph theory by Frank Harary, but through his now-classic book Graph Theory, while I was a graduate student in Mumbai. This continued with a course in graph theory at Purdue University. I became totally immersed in the subject after discovering Professor Beineke’s (my graph theory guru) work and attending his lectures. In the course of my doctoral work in bipartite tournaments under his direction, I discovered his work on the characterization of line graphs. The chapter “Line Graphs and Line Digraphs” (by Hemminger and Beineke) in the book Selected Topics in Graph Theory by Beineke and Wilson provided ample detailed material in this regard. This led to my collaboration with Beineke and Varma in which we began research in super line graphs, and this resulted in several publications. The 17 chapters of the book are divided in three parts. If you are new to line graphs, we recommend that you begin your study with the first three chapters. The remaining chapters in Part I can be read in any order. The three chapters of Part II cover line digraphs and their properties. The five chapters of Part III discuss a variety of generalizations of line graphs, and this is an active area of research. In addition to being of considerable value to individuals both for its content and its research methodology, the book can be used as a guided study in a graduate level class. A basic introduction to graph theory will be sufficient but is not necessary as background. Almost any book contains errors and we expect this does too. We appreciate your notifying us of any errors that you may find. As appropriate, reported errata will eventually appear on the book’s website. We are grateful to the more than 150 mathematicians involved in the discovery and proof of results that added so much to the theory of line graphs and line digraphs and their generalizations. We have chosen to cite one representative paper from each

Preface

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of the 17 chapters. For obvious reasons, none of our own papers is on the list, and only Harary is mentioned twice. Many of the selected papers were chosen for their historical significance as well as for their contribution to the subject. 1. Fundamentals of Line Graphs: “Démonstration nouvelle d’un théorème de Whitney sur les réseaux” by J. Krausz. This was arguably the first paper that viewed line graphs as objects to be analyzed in their own right, with a variety of properties. 2. Line Graph Isomorphisms: “2-isomorphic graphs” by H. Whitney. Whitney was the first to develop a theory of mappings of edges that preserved the adjacency. 3. Characterizations of Line Graphs: “The interchange graph of a finite graph” by A. C. M. van Rooij and H. S. Wilf. This paper describes the conditions under which pairs of triangles can appear in line graphs. 4. Spectral Properties of Line Graphs: Über Teiler, Faktoren und characteristische Polynome von Graphen II” by H. Sachs. The neat result of Sachs’s is the fact that −2 is a sharp lower bound on the eigenvalues of line graphs. 5. Planarity of Line Graphs: “Some properties of interchange graphs” by J. Sedláˇcek. This paper gives elementary necessary and sufficient conditions for a graph to have a planar line graph, the beginning of a theory of planarity and line graphs. 6. Connectivity of Line Graphs: “On the line-connectivity of line-graphs” by T. Zamfirescu. Among the various connectivity theorems involving line graphs, the author establishes connections between the edge-connectivity of a graph and that of its line graph. 7. Traversability in Line Graphs: “On Hamiltonian line-graphs” by G. Chartrand. One of the main topics in this paper is determination of the minimum number of iterations of a graph that will yield a Hamiltonian line graph. 8. Colorability in Line Graphs: “On an estimate of the chromatic class of a pgraph (Russian)” by V. G. Vizing. This paper establishes one of the most important theorems on line graphs, indeed of chromatic graph theory, namely that the chromatic number of a line graph always has one of two possible values. 9. Distance and Transitivity in Line Graphs: “Centers in line graphs” by M. Knor, L'. Niepel, and L'Šoltés. Sharp bounds on the radius and diameter of a line graph are among the interesting results in this paper. 10. Fundamentals of Line Digraphs: “Some properties of line digraphs” by F. Harary and R. Z. Norman. In this, the first article on line digraphs, necessary and sufficient conditions are found for a digraph to be a line digraph. 11. Characterizations of Line Digraphs: “Sur une certaine correspondance entre graphs” by C. Heuchenne. The author provides another characterization of line digraphs, this one an elementary local condition involving arcs on small sets of vertices. 12. Iterated Line Digraphs: “Digraphs with periodic line digraphs” by R. L. Hemminger. The author’s main result involves finding those digraphs for which the iterated line digraphs eventually become periodic.

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13. Total Graphs and Total Digraphs: “A characterization of total graphs” by M. Behzad. This paper introduces a graph operation that combines a graph and its line graph using both adjacency and incidence and then gives a characterization of these graphs. 14. Path Graphs and Path Digraphs: “Path graphs” by H. J. Broersma and C. Hoede. The vertices of the path graph of a graph are the paths of length 2, with adjacency appropriately defined. These graphs turn out to be more complex than line graphs. 15. Super Line Graphs and Super Line Digraphs: “Path-comprehensive and vertexpancyclic properties of super line graphs L2 (G)” by X. Li, H. Li, and H. Zhang. Among the conclusions in this paper are results that Hamiltonian cycles are abundant in the extensions of line graphs to pairs of edges in a graph. 16. Line Graphs of Signed Graphs: “On the notion of balance of a signed graph” by F. Harary. This was one of the earliest papers to be motivated by the social sciences, that being an application of "loves" and "hates" in the field of social psychology. 17. The Krausz Dimension of a Graph: “Computational Complexity of the Krausz dimension of graphs” by P. Hlinˇený and J. Kratochvíl. One of the key properties of line graphs is that the edges can be partitioned into complete subgraphs so that each vertex in at most two of them. The idea of this paper is to investigate graphs in which the number 2 is allowed to increase. In addition to the multitude of academicians to whom we are grateful for their contributions to the theory of line graphs and line digraphs, there are many others to whom we want to express our thanks. We are also grateful to our hosts at several universities: our home institutions of course, Ball State University and Purdue University Fort Wayne (formerly Indiana University Purdue University Fort Wayne, including while JB was on sabbatical), Oxford University (UK) (where LB was on sabbatical), and Kalasalingam University (India) (where both JB and LB were visiting mathematicians). This project has consumed a substantial portion of time, including time away from our spouses, children, and grandchildren. Our wives Judith and Daljit have been most understanding, patient, and supportive, and we are especially grateful to them, which is why we have dedicated this book to them. The present form of our book is the result of diligent support, constant encouragement, and frequent advice from Dr. Remi Lodh, mathematics editor at Springer, and we want to convey our special appreciation to him and other members of the editorial and production staff there. Naturally, writing this tome has been on balance a most rewarding experience for us, and of course we hope that you as a reader will gain from it both in knowledge and pleasure. Fort Wayne, IN, USA Muncie, IN, USA September 2021

Lowell W. Beineke Jay S. Bagga

Contents

Part I

Line Graphs

1

Fundamentals of Line Graphs . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Line Graphs of Common Families . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Degree Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Iterated Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3 3 3 5 9 12

2

Line Graph Isomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Edge-Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

17 17 17 20

3

Characterizations of Line Graphs . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Primary Characterizations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Bipartite Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Complements of Line Graphs.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

25 25 26 30 36 39

4

Spectral Properties of Line Graphs . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Eigenvalues of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 The Spectrum of a Line Graph . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 An Extension of Line Graphs .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Graphs Whose Spectrum Is Bounded Below by −2 .. . . . . . . . . . . . . . .

51 51 51 54 55 58

5

Planarity of Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Graphs Having Planar Line Graphs . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Regular Planar Line Graphs . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.4 Outerplanar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

61 61 61 67 74 xv

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Contents

5.5 5.6

Crossing Numbers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Planarity of Iterated Line Graphs .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

76 81

6

Connectivity of Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Vertex- and Edge-Connectivity . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Connectivity of Iterated Line Graphs . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

87 87 87 88 92

7

Traversability in Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 7.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 97 7.2 Eulerian Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 98 7.3 Hamiltonian Graphs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99 7.4 Connectivity and Hamiltonian Cycles . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 102 7.5 The Hamiltonian Index.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105

8

Colorability in Line Graphs .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Line Graphs of Planar Cubic Graphs .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.3 Quartic Line Graphs.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.4 König’s and Vizing’s Theorems . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.5 Extensions and Variations.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

109 109 110 112 117 123

9

Distance and Transitivity in Line Graphs . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Distance .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Transitive Orientations of Line Graphs . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

127 127 127 133

Part II

Line Digraphs

10 Fundamentals of Line Digraphs . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.3 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 10.4 Digraphs with Isomorphic Line Digraphs .. . . . . .. . . . . . . . . . . . . . . . . . . . 10.5 Connected Digraphs Isomorphic to Their Line Digraphs . . . . . . . . . . 10.6 Spectra of Line Digraphs . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

145 145 146 148 151 154 156

11 Characterizations of Line Digraphs . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.2 Background .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.3 Characterization of Line Digraphs of Multidigraphs . . . . . . . . . . . . . . . 11.4 Families of Line Digraphs . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 11.5 Excluded Induced Subdigraphs .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

159 159 159 161 164 167

12 Iterated Line Digraphs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 173 12.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 173 12.2 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 173

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12.3 Periodic Line Digraphs.. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 177 12.4 Characterization of Second Order Iterated Line Digraphs . . . . . . . . . 184 12.5 Characterizations of Families of Second-Order Line Digraphs .. . . 193 Part III

Generalizations

13 Total Graphs and Total Digraphs . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.2 Basics of Total Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.3 Special Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.4 Ordinary Graphs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.5 Planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.6 Traversability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 13.7 Total Digraphs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

203 203 204 207 208 210 212 213

14 Path Graphs and Path Digraphs .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.2 Definition and Basic Properties .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.3 Characterization of Path Graphs .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.4 Graphs with Isomorphic Path Graphs . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.5 Path Digraphs .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 14.6 Clique Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

219 219 220 223 225 227 230

15 Super Line Graphs and Super Line Digraphs. . . . . . .. . . . . . . . . . . . . . . . . . . . 15.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.2 Definition and Basic Properties .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.3 Independence Number . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.4 Degrees of Index-2 Super Line Graphs . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.5 Paths and Cycles in Super Line Graphs . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.6 The Line Completion Number .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.7 Super Line Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15.8 Super Line Multigraphs .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

233 233 234 236 237 241 244 252 254

16 Line Graphs of Signed Graphs . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.2 Line Graphs of Signed Graphs . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16.3 Boolean Signed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

257 257 257 262

17 The Krausz Dimension of a Graph . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.2 Some Properties of the Krausz Dimension .. . . . .. . . . . . . . . . . . . . . . . . . . 17.3 Some Interesting Families . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 17.4 The Krausz Dimension and Graph Operations... . . . . . . . . . . . . . . . . . . . 17.5 Krausz Critical Graphs .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

269 269 269 271 276 278

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Contents

Glossary . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 283 References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 285 Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293 Index of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 297

Part I

Line Graphs

Chapter 1

Fundamentals of Line Graphs

1.1 Introduction Among the many graph transformations that mathematicians have studied, the line graph is without doubt the most interesting. Informally, the line graph L(G) of a graph G is the result of taking the edges of a graph as the vertices of a new graph and joining two of these vertices by an edge if the corresponding edges in G have a common vertex. In Sect. 1.2, we define the line graph formally, and after looking at some examples, we examine some of the elementary properties of line graphs, in particular those involving the number of edges and the degrees of the vertices in a line graph. This is followed by determining the line graphs of the graphs in several families of graphs, starting with paths and cycles, and then the more interesting families of complete graphs and complete bipartite graphs. We then return to consideration of the degrees of the vertices in a line graph. One of the topics considered here is the maximum and minimum degrees among the vertices of the line graph and a second is that of the family of graphs for which the degrees of all of the vertices are equal. Since the line graph of a graph is another graph, the process can of course be repeated, and this will generally yield a sequence of iterated line graphs. The chapter concludes with results on the number of vertices and the degrees in the sequence. The topic of iterated line graphs will reappear in some of the later chapters.

1.2 Basic Concepts The concept that has come to be known as the line graph first appeared in the context of the “2-isomorphism” of graphs in a 1933 paper by Hassler Whitney [174]. A decade later, Krausz [122] began the study of line graphs in their own right, providing the first characterization of them as a family of graphs. It still took © Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_1

3

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1 Fundamentals of Line Graphs

some time for the concept to develop coherence, with the next paper on record being by Harary and Norman [93] in 1960 on directed graphs, and this may well have been the impetus towards usage of the term line graph (motivated by Harary calling the ‘vertices’ and ‘edges’ of a graph ‘points’ and ‘lines’). We believe that the first comprehensive collection of results in this area was the chapter ‘Line Graphs’ in Harary’s 1969 book Graph Theory [91], and it is very likely that the name ‘Line Graphs’ and even more significantly, continued research in this area, was stimulated by this book. Harary began his chapter with these sentences: “The concept of the line graph of a given graph is so natural that it has been independently discovered by many authors. Of course, each gave it a different name:” and he provided this list: interchange graph, edge-to-vertex dual, covering graph, derivative, derived graph, adjoint, and conjugate (and that was only in English). (Curiously, even though Harary’s first paper on this subject was on digraphs, line digraphs were relegated to exercises in his book. The chapter by Hemminger and Beineke in the book Selected Topics in Graph Theory [103] went one step better and included both graphs and digraphs in their title.) Given a graph G with at least one edge, the line graph L(G) is that graph whose vertices are the edges of G, with two of these vertices being adjacent if the corresponding edges are adjacent in G. The operation of going from adjacent edges in G to adjacent vertices in L(G) is indicated in Fig. 1.1. A simple example of a line graph is shown in Fig. 1.2. We observe that the line graph of the path Pn of n vertices (n > 1) is the path Pn−1 . Consequently, if G is a non-trivial connected graph, then L(G) is also connected. Furthermore, if H is a non-null subgraph (that is, has at least one edge) of G, then L(H ) is an induced subgraph of L(G). If F is a graph whose line graph is G, that is, L(F ) = G, then F is called a root graph (or simply a root) of G. For instance, K3 and K1,3 are both roots of K3 . We

f

e

L

Fig. 1.1 The operation of forming the line graph

G:

Fig. 1.2 A graph and its line graph

L(G):

e

f

1.3 Line Graphs of Common Families

5

note also it is not hard to show that the only connected root of K4 is K1,4 . In fact, in Chap. 2 we will show that K3 is the only connected graph with two non-isomorphic roots. Our first theorem gives some elementary properties involving numbers in line graphs. Theorem 1.1 Let G be a non-null graph with n vertices and m edges. Then  (a) L(G) has m vertices and 12 (deg v)2 − m edges. (b) The degree of a vertex e = vw in L(G) is deg e = deg v + deg w − 2. Proof (a) By definition, L(G) has as many vertices as G has edges. It also follows from the definition that each edge of L(G) arises from precisely one pair   of edges at some vertex of G. Hence, each vertex v of degree d contributes d2 different edges to L(G). Summing and using the fact that the sum of the degrees in G is 2m yields the result. (b) This follows from the fact that, given an edge vw, there are deg v − 1 other edges at vertex v in G and deg w − 1 other edges at w, each of which gives rise to an edge at vw in L(G). Since every edge at vw in L(G) arises exactly once in this way, the result follows.   We find it useful to call the value in (b) of the theorem the line-degree of the edge vw in graph G. Corollary 1.1 If G is an r-regular graph of order n, then L(G) is 2(r − 1)-regular and has nr 2 vertices. Clearly, the edges at any vertex v of a graph G generate a complete subgraph of the line graph L(G). Furthermore, as we shall show, this is the only way that complete subgraphs of order greater than 3 can arise. First we note that a copy of K3 can arise in L(G) from either K3 or K1,3 (and only in these ways). But one cannot get K4 in L(G) by adding an edge to a 3-cycle of G, so the only way to get K4 in L(G) is from K1,4 in G. Clearly the same principle also holds for any complete subgraph of order greater than 4 in a line graph.

1.3 Line Graphs of Common Families The line graphs of some elementary families of graphs are straightforward to find: ∼ Pn−1 for n ≥ 2. (a) Paths: L(Pn ) = (b) Cycles: L(Cn ) ∼ = Cn . (c) Stars: L(K1,s ) ∼ = Ks . Two of the most important families of graphs are the complete graphs Kn and the complete bipartite graphs Kr,s . Their line graphs also turn out to have some interesting and significant properties. We introduce our graphs with a set-theoretic

6

1 Fundamentals of Line Graphs

T5 :

T4 :

Fig. 1.3 The octahedron and its successor in the family

T2,3 :

T3,3 :

Fig. 1.4 The triangular prism and and its successor in the family

foundational approach to two families of graphs derived from the 2-element subsets of a set S and the ordered pairs of S, and we begin   with the subsets. Let Tn be the graph whose vertices are the n2 pairs of elements from a set of n elements, with two vertices adjacent if the corresponding sets have a non-empty intersection. For example, T4 is the graph of the octahedron, shown in Fig. 1.3, while T5 , also shown in shown in Fig. 1.3, has order 10 and is regular of degree 6; it happens to be the complement of the Petersen graph (whereas the complement of T4 is 3K2 ). We now consider the ordered pairs graphs, generalizing the definition to a pair of sets that are not necessarily of the same size. Let Tr,s be the graph whose vertices are the rs ordered pairs of elements from a set R of r elements and a set S of s elements, with two vertices adjacent if the corresponding pairs have a common entry. For instance, Fig. 1.4 shows T2,3 and T3,3 , which are isomorphic to the triangular prism K2 × C3 and C3 × C3 respectively. It can be easily verified that Tr,s ∼ = Kr × Ks , the Cartesian product of complete graphs. One of the families of graphs that is of particular interest in algebraic graph theory consists of regular graphs with additional requirements having to do with sets of vertices that are common neighbors of a pair of vertices. A graph is strongly regular if it is regular (say of degree δ) and all pairs of adjacent vertices have the same number of common neighbors (say λ) and all pairs of non-adjacent vertices also have the same number of common neighbors (say μ).

1.3 Line Graphs of Common Families

7

We consider first the previously mentioned graphs of 2-element subsets. It follows from the  definition of Tn that it is actually the line graph of Kn . Hence, it has order n2 and is regular of degree 2n − 4. Beyond this, it has the property that every pair of adjacent vertices have n − 2 common neighbors and every pair of non-adjacent vertices have four common neighbors. An interesting question is whether there are other graphs having the same parameters as Tn . It turns out that nearly all of them are unique, with the only exception being for n = 8. It was shown by Chang [52] in 1959 that L(K8 ) has three companion graphs, that is, there are three graphs with the same parameters as L(K8 ): n = 28, δ = 12, λ = 6, and μ = 4. Chang showed that each of the three can be obtained from T8 by switching adjacency and non-adjacency at eight vertices. In one, the eight vertices are the ends of four independent edges; in a second, they are the vertices of an 8cycle, and in the third, the vertices of a 3-cycle and a 5-cycle. We summarize these facts in the following theorem, omitting the proofs of most of the details. Theorem 1.2 The line graph of the complete graph of order n, L(Kn ) is the strongly regular graph Tn , which   (i) has order n2 , (ii) is regular of degree 2n − 4, (iii) has 12 n(n − 1)(n − 2) pairs of adjacent vertices, all with n − 2 common neighbors, and (iv) has 18 n(n − 1)(n − 2)(n − 3) pairs of non-adjacent vertices, all with four common neighbors. Furthermore, these are the only graphs with these parameters except for three graphs that have the same parameters as L(K8 ). We move on now to the graphs of ordered pairs of numbers. It follows from the definition of Tr,s that it is actually the line graph L(Kr,s ). Hence, it has order rs and is regular of degree r + s − 2 unless s = r. Therefore, for now we restrict our attention to that case, which does turn out to be strongly regular. Thus, L(Kr,r ) is the strongly regular graph Tr,r of order r 2 and parameters δ = 2r − 2, λ = r − 2, and μ = 2. As it happened, the same year in which Chang published his result involving the strong regularity of line graphs of complete graphs, Shrikhande [159] published the analogous result for complete bipartite graphs. He showed that once again the root graph with eight vertices is special. For all r ≥ 1, L(Kr,r ) is a strongly regular graph of order r 2 , degree δ = 2r − 2, λ = r − 2, and μ = 2. Furthermore, except for r = 4, there are no other graphs with the same parameters. Shrikhande showed that there is only one exceptional graph, which is shown in Fig. 1.5. Moon [139] investigated the parameters for complete bipartite graphs to the cases in which the two partite sets have different sizes: that is, L(Kr,s ) with 1 ≤ r < s. It has rs vertices, all of degree r + s − 2. However, not all pairs of adjacent vertices have the same number of common neighbors, and so L(Kr,s ) is not strongly regular. Moon proved that in all cases except two, which were left undetermined at the time,

8

1 Fundamentals of Line Graphs

Fig. 1.5 The Shrikhande graph

there are no other graphs with these parameters. These two cases were r = 3 and s = 4 and r = 4 and s = 5, and Hoffman [107] was able to settle those. The following theorem summarizes the results for the line graphs of all complete bipartite graphs. Theorem 1.3 The line graph of the complete bipartite graph L(Kr,s ) has the following properties: (a) For s = r, it is the strongly regular graph Tr,r , which (i) has order r 2 , (ii) is regular of degree 2r − 2, (iii) has r 2 (r − 1) pairs of adjacent vertices, all with r − 2 common neighbors, and (iv) has 12 r 2 (r − 1)2 pairs of non-adjacent vertices, all with 2 common neighbors. Furthermore, these are the only graphs with these parameters except for one graph that has the same parameters as L(K4,4 ). (b) For 1 ≤ r < s, it is the graph Tr,s , which (i) has order rs, (ii) is regular   of degree r + s − 2, (iii) has r 2s − 2 pairs of adjacent vertices with s − 2 common neighbors and  s 2r − 2 pairs of adjacent vertices with r − 2 common neighbors, (iv) all pairs of non-adjacent vertices have just two common neighbors. Furthermore, there are no other graphs with these parameters. We conclude this section by adding to the list of paths, cycles, and stars with which we began the section these two families: (d) Complete graphs: L(Kn ) ∼ = Tn for n ≥ 2. (e) Complete bipartite graphs: L(Kr,s ) ∼ = Tr,s for r, s ≥ 1.

1.4 Degree Sequences

9

1.4 Degree Sequences As we have seen, the degree of a vertex e in the line graph L(G) of a graph G is deg v + deg w − 2 where e = vw in G. Two of the parameters of interest in a graph G are the largest and smallest degrees among its vertices; we denote these by δ(G) for the minimum and Δ(G) for the maximum. The following observation was made by Chartrand and Stewart [58]; it readily follows from the preceding fact about the degrees of vertices in line graphs. Theorem 1.4 Let G be a graph with at least one edge. (a) δ(L(G)) ≥ 2δ(G) − 2 with equality if and only if G has two adjacent vertices of degree δ(G). (b) Δ(L(G)) ≤ 2Δ(G) − 2 with equality if and only if G has two adjacent vertices of degree Δ(G). It follows that if G is a connected nontrivial graph with m edges (m ≥ 2), the degrees of the vertices in L(G) constitute a collection of m positive integers. A natural question is which such collections of numbers less than m are the degrees of some line graph? This is an interesting question that remains open in general and one on which not a great deal is known. The pioneering work was done by Bauer in his 1978 doctoral thesis at Stevens Institute of Technology. Here we look at some of the results from Bauer [18, 19] on regular line graphs and those in which at least one vertex of the line graph is adjacent to all of the others. For convenience, since connected graphs with fewer than four edges are few in number, we consider connected line graphs L(G) of order m ≥ 4 having degree sequence π = (d1 , d2 , . . . , dm ) with d1 = m − 1 ≥ d2 ≥ . . . ≥ dm ≥ 1. When a graph has multiple vertices with the same degree, we find it useful to adopt the practice of using exponents in parentheses to denote this. For instance, π = (3, 3, 3, 3, 2, 2, 2) might be written as π = (3(4) , 2(3)). Regular Line Graphs Let L(G) be a connected d-regular line graph of order m. Clearly the sequence consisting of m entries all of which are m−1 constitutes the degrees of the complete graph Km and this is the line graph of K1,m . Having handled this case, from here on we assume that L(G) is not complete, that is, 2 ≤ d ≤ m − 2. Since an edge e = vw in the root graph G has degree deg v + deg w − 2 = d, there are clearly only two types of root graph G possible: those in which deg v = deg w for every edge and those in which deg v = deg w. It follows that in the first instance G must be regular, and in the second it must be bipartite and bi-regular (that is, all vertices in each partite set must have the same degree). We now look at the first of these and assume that L(G) has a root graph G that is a connected l-regular graph with n vertices and m edges. Then since the linedegree of an edge of G is 2l − 2, the degree of regularity d of L(G) must be even, say d = 2k. Consequently, the degree-sum in G is 2m = n( 12 d + 1). Rewriting this, we have n(d + 2) = 4m; in other words, 4m is a multiple of d + 2. These

10

1 Fundamentals of Line Graphs

are thus necessary conditions for a d-regular line graph (with d even) of order m to have a root graph that is regular. We now show that they are also sufficient. Let m and d be positive integers with d even and 4m a multiple of d + 2, say d = 2k and p(d + 2) = 4m. Thus p(k + 1) = 2m, and from this it follows that there is a (k + 1)-regular graph with p vertices and m edges. By definition, its line graph has m vertices each of degree d. We now assume that L(G) is a d-regular line graph of order m which has an r ×s bipartite root graph G with 2 ≤ r < s and each of r vertices having degree a and each of the other s vertices having degree b. Clearly the number m of edges in G is m = ra = sb, and each edge vw has the same line-degree d = deg v + deg w − 2 = a +b −2. Thus, its line graph L(G) is a d-regular graph of order m. Now let m and d be fixed and assume that integers a and b (with a, b ≥ 2) are such that a +b = d +2 and there are integers r and s for which ra = sb = m. It is readily seen that there is an r × s bipartite graph G with the vertices in the respective partite sets having degrees a and b, and its line graph is d-regular and of order m. This establishes the following theorem characterizing regular connected line graphs that are not complete: Theorem 1.5 For positive integers d and m with 2 ≤ d ≤ m − 2, there exists a d-regular line graph of order m if and only if at least one of the following two sets of conditions holds: (a) d is even and 4m is a multiple of d + 2; (b) there exist integers a, b, r, and s each at least 2 for which a + b = d + 2 and ra = sb = m. Examples illustrating the theorem are easy to come by. For the first type, we let d = 4 and m = 9, from which it follows that a graph that with six vertices and nine edges will have a 4-regular line graph of order 9. There are in fact two such root graphs, K3,3 and the triangular prism K2 × K3 . For an example of the second type, we start with d = 5 and m = 12 and let a = 4 and b = 3. The line graph of K3,4 is then a 5-regular line graph of order 12. Line Graphs with a Spanning Star We next consider the case in which d1 = m − 1, and we let e = vw be an edge in a graph G adjacent to all of the other edges. It follows that the end vertices of each of the other edges are adjacent either (a) only to v, (b) only to w, or (c) to both v and w (see Fig. 1.6). Further, let A, B, and C be the sets of edges having the respective vertices (not including e). Now we assume that L(G) is not a complete graph, that is, G is not a star. We also assume there is at least one other edge f adjacent to all of the other edges. If f is in A, it follows that B = C = ∅, in which case G is the star K1,m and the degree sequence of L(G) is ((m − 1)(m) ). So we assume next that this is not the case, so that f is in C, say f = uw. From Fig. 1.6 we can see that for f to be adjacent to all other edges, there cannot be an edge in C other than uv and uw and also that A must be empty. Thus, G must consist

1.4 Degree Sequences Fig. 1.6 A graph with one edge adjacent to all others

11 B

C

A

v Fig. 1.7 A graph with two edges adjacent to all others

w B

C u

v

w

of a triangle added to a star as in Fig. 1.7, that is, K1,m−1 with one additional edge. The degree sequence of this line graph is therefore ((m − 1)(2), (m − 2)(m−3) , 2). We observe that it also follows from this argument that there cannot be a third edge of G of line-degree m − 1 without G being the star K1,m . Now we assume that there is only the one edge vw in L(G) of line-degree m − 1, as in Fig. 1.6. Let a, b, and 2c be the number of edges in A, B, and C respectively, so a + b + 2c = m − 1. Also, to avoid stars either on their own or with one edge added, we eliminate the cases where either a or b is 0 and c is either 0 or 1. Case 1. c = 0, a + b = m − 1, and a ≥ b ≥ 1. Then π = (m − 1, a (a), b (b)). Case 2. c = 1, a + b = m − 3, and a ≥ b ≥ 1. Then π = (m − 1, a + 2, (a + 1)(a) , b + 2, (b + 1)(b) ) in some order. Case 3. c ≥ 2 and a ≥ b ≥ 0. Then π = (m − 1, (a + c + 1)(c) , (a + c)(a), (b + c + 1)(c) , (b + c)(b)) in some order. We summarize these results in the following theorem. Note that here the sequences are not necessarily in non-increasing order. Theorem 1.6 A sequence of positive integers (d1 , d2 , . . . , dm ) with m ≥ 4 and d1 = m − 1 is line-graphical if and only if it is one of these types: • • • • •

((m − 1)(m) ), ((m − 1)(2), (m − 2)(m−3) , 2), (m − 1, a (a), b (b)) with a + b = m − 1, (m − 1, a + 2, (a + 1)(a), b + 2, (b + 1)(b) ) with a + b = m − 3, (m − 1, (a + c + 1)(c) , (a + c)(a), (b + c + 1)(c) , (b + c)(b) ) with c ≥ 2 and a + b + 2c = m − 1.

12

1 Fundamentals of Line Graphs

We conclude this section by observing that beyond these two families, those line graphs that are regular or have at least one vertex adjacent to all of the others (in which case either all of the vertices have this property or only one or two do), there are not many results known about line-graphical sequences.

1.5 Iterated Line Graphs Obviously one can repeat the operation of taking the line graph of a graph. Formally, the iterated line graphs of a graph G are defined by L1 (G) = L(G) and Lk+1 (G) = L(Lk (G)) for k ≥ 1. Figure 1.8 shows the first four iterates of one graph. Notice that in this example the line graph L(G) is smaller than G itself (fewer vertices and no more edges). However, the iterated line graphs appear to get larger. For convenience here, a graph is called prolific if it is connected, has a vertex of degree at least 3, and is not K1,3 , in other words, if it is connected and not a path, cycle, or claw. For connected graphs, the first observation holds only for trees, and since (as observed earlier) the number of edges in a line graph is the sum of the binomial coefficients of the degrees of the vertices of the original graph G, this will be less than the number m of edges in G if and only if G is a path and will be equal to m if and only if G has only one vertex of degree greater than 2 and that degree is 3 (as in this example). Following the lines of an argument provided by van Rooij and Wilf [168], we give a formal proof of this result, one that contains two interesting facts. The first is that for any connected graph other than K1,3 having maximum degree at least 3, the third line graph has a pair of edge-disjoint cycles. The second is that if a graph has a pair of such cycles at distance d from each other, then its line graph has a pair of such cycles at distance d + 1. We illustrate this fact with the graph G in Fig. 1.9.

Fig. 1.8 A graph and four iterated line graphs G:

L3 (G) :

L2 (G) :

L(G) :

L4 (G) :

1.5 Iterated Line Graphs

13

Fig. 1.9 Cycles further apart in a line graph G:

L(G):

Theorem 1.7 Let G be a prolific graph, and let nk be the number of vertices in Lk (G). Then lim nk = ∞. k→∞

Proof Let G be a prolific graph. We first show that for some k, Lk (G) has a pair of edge-disjoint cycles. This will be done in several cases, depending on the maximum degree Δ in G. First, if Δ ≥ 5, then L(G) contains the complete graph K5 , and this clearly has a pair of edge-disjoint cycles. Next, if Δ = 4, then L(G) contains the complete graph K4 and its line graph L2 (G) has such a pair of cycles. Now, if Δ = 3 and G has two (or more) vertices of degree 2, then L(G) clearly has such a pair, so we assume that G has only one vertex of degree 3, all others are of degree 1 or 2, and G is not K1,3 . This means that the first graph in Fig. 1.8 is a subgraph of G, and hence the third of its iterated line graphs has the desired pair of cycles. It now follows from the observation made before the statement of the theorem (and illustrated in Fig. 1.9) that the iterated line graphs in the sequence Lk (G) have paths that get arbitrarily long, proving the theorem.   Corollary 1.2 If G is a connected graph other than K1,3 for which Lk (G) ∼ = L(G) for some k, then L(G) ∼ = G and G is a cycle. We now turn to small degrees in graphs and their line graphs, in particular degrees 1 and 2, and we assume as before that the graphs are prolific. Recall that the degree of a vertex e in L(G) is deg v + deg w − 2 if e is the edge vw in G. Assuming that e is not an isolated edge, we see that deg e = 1 in L(G) if and only if either v or w has degree 1 and the other has degree 2. Furthermore, deg e = 2 if and only if either v or w has degree 1 and the other has degree 3 or both have degree 2. We will examine what happens in the line graphs of graphs with such vertices, having already assumed that the graph is not a cycle. Looking first at the simplest case, we assume that the graph G has vertices of degree 1 but none of degree 2. We note that then L(G) has no vertices of degree 1. Since a single edge and a claw are not allowed, the smallest possible graph is the H-graph shown in Fig. 1.10, along with its iterated line graphs L(H ), L2 (H ), and L3 (H ). We observe that L(H ) has four degree-2 vertices, while L2 (H ) has two

14

1 Fundamentals of Line Graphs

H:

L2 (H):

L(H):

L3 (H):

Fig. 1.10 The coming and going of vertices of degree 2

and L3 (H ) has none. This behavior will take place whenever the graph G with no degree-2 vertices has a vertex of degree 3 adjacent to a pair of degree-1 vertices. Furthermore, if G is such that the neighbor of each of its degree-1 vertices has degree greater than 3, then L(G) will have no vertices of degree 2. In addition, if G has at least one pair of adjacent degree-1 and degree-3 vertices (and still no degree-2 vertices), then by the same argument as for the graph H , L(G) will have a vertex of degree 2 but L2 (G) will not. Therefore we have the following result. Lemma 1.1 If G is a connected graph that is not a claw and has no vertices of degree 2, then L3 (G) has no vertices of degree 1 or 2. We now consider graphs having vertices of degree 2. Such a vertex will lie on a path P , all of whose internal vertices are also of degree 2 and whose end vertices are not. Since we have assumed that G is not a path, there are two possible types: either P will end in one vertex of degree 1 and one of degree at least 3 or both ends will have degree at least 3 as is illustrated in Fig. 1.11, where we show only the cases where the ends have degree 1 or 3. As informal objects, these graphs resemble chains, with the first of the two types attached at one end and the second attached at both ends. We therefore define the first to have a type-1 chain and the second a type-2 chain. We denote the corresponding graphs Q (l) and Q

(l), with l being the length of the path P (that is, l is 1 more than the number of 2s). Clearly, the line graph of a chain of either type has one fewer vertex of degree 2. Hence, Ll (Q

(l)) has no vertices of degree 2. Similarly, the iterates of Q (l) lose the vertices of degree 2 on the path, but because of its degree-1 vertex, the resulting line graph does have a vertex of degree 2. However, the next iterated line graph does not. Consequently, eventually an iterated line graph of any prolific graph has minimum

1.5 Iterated Line Graphs

15

Fig. 1.11 Maximal paths of degree-2 vertices

degree at least 3. Therefore, every prolific graph G has an iterated line graph Lk (G) with minimum degree d ≥ 3. Hence, the degree of any vertex e in Lk+1 (G) must satisfy deg e ≥ 2d − 2 > d. Therefore, the sequence of the minimum degrees of G is eventually strictly increasing, and this implies the following result. We note that it obviously has Theorem 1.7 as a corollary. Theorem 1.8 If G is a connected graph that is not a cycle, a path, or a claw, then lim δ(Lk (G)) = ∞.

k→∞

We conclude this chapter with a nice result on the extreme degrees, both minimum and maximum, of iterated line graphs. It was actually conjectured in two parts by Niepel et al. [144], and the two results were proved in papers 4 years apart by Hartke and Higgins [97, 98]. Combined, their theorems state that eventually every iterated line graph of a connected graph other than a path has equality holding in the inequalities in Theorem 1.4. Theorem 1.9 If G is a connected graph that is not a path, then there exists a positive integer k0 such that for all k ≥ k0 , δ(Lk+1 (G)) = 2δ(Lk (G)) − 2 and Δ(Lk+1 (G)) = 2Δ(Lk (G)) − 2.

Chapter 2

Line Graph Isomorphisms

2.1 Introduction The study of line graphs emerged from the field of group theory through the inspiration of Hassler Whitney. Consider first a one-to-one function from the edges of the complete graph K3 and the complete bipartite graph K1,3 . Whitney showed that this mapping not only preserves adjacency of edges, but these are the only connected non-isomorphic graphs with this property. Whitney further proved that there are three—and only three—connected graphs having automorphisms in which there are edges of triangles and Y-graphs that get interchanged. With Whitney’s theorem the primary result of Sect. 2.2, automorphism groups connected with line graphs are explored in Sect. 2.3. The chapter concludes with an interesting result of Frank Harary and Ed Palmer on those graphs that are not only isomorphic to their line graph but with their vertex-sets being preserved.

2.2 Edge-Isomorphisms As noted above, Whitney [173] proved that the two graphs K1,3 and K3 are the only two connected non-isomorphic graphs with isomorphic line graphs. Our main goal now is the determination of when a vertex-isomorphism induces an edgeisomorphism. Our proofs of the following results are based on those given by Jung [114]. We begin with some definitions and a result on isomorphisms and the sets of edges at the vertices. The set of edges at a vertex v is denoted S(v) and any non-empty subset of S(v) is called a star of G. Given graphs G and H , a mapping φ : E(G) → E(H ) is called star-preserving if, for every star S in G, φ(S) is a star in H .

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_2

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2 Line Graph Isomorphisms

Lemma 2.1 Let G and H be connected graphs and let φ be a bijection from E(G) onto E(H ). Then φ is an isomorphism of G onto H if and only if both φ and φ −1 are star-preserving. Proof Assume that G and H are connected graphs and φ is a star-preserving map from G to H . If v is a vertex of G of degree at least 2, then there must be just one vertex v in H for which φ(S(v)) ⊂ φ(S(v )). It follows that φ determines a function  φ : {v ∈ V (G) : deg v ≥ 2} → {v ∈ V (H ) : deg v ≥ 2}. Since −1 φ has the same properties as φ, it follows that  φ must be onto and therefore φ(S(v)) = S( φ(v)). We now assume that G has at least three vertices since otherwise the result is trivial. Consider an edge vw in G with deg w = 1. Then deg v > 1, and so φ(vw) ∈ φ(S(v), which is S( φ(v)). It follows from the above that φ(vw) = v w , where

v =  φ(v) and deg w = 1. Now extend  φ by taking  φ(w) to be w . Therefore, φ determines a function  φ : V (G) → V (H ) for which φ(S(v)) = S( φ(v)) for all vertices v in G. However, S(v) = S(w) if and only if v = w since G has at least three vertices, and so  φ(v) is a bijection. This proves that the stated condition is sufficient for edge-isomorphism of graphs. Since it is obviously necessary, the proof is complete.   We now turn to the topic of mappings between the edge sets of two connected graphs that are adjacency-preserving but not star-preserving. As we pointed out in Chap. 1, the two graphs K1,3 and K3 are non-isomorphic graphs with isomorphic line graphs, and so no one-to-one mapping from E(K1,3 ) to E(K3 ) can be starpreserving (see Fig. 2.1). Now consider two copies of the graph of order 4 consisting of a triangle with an added end edge (which we denote K2 · K3 ), with edge labels as in G1 and H1 in Fig. 2.2, and let φ be the function that assigns to each edge of G1 the edge of H1 having the same label (color). Then this function is edge-preserving but not starpreserving. Whitney proved that there are only two other connected graphs like this. Theorem 2.1 Let G and H be two graphs of order at least 4 with edge-labelings for which there is a correspondence that preserves edge-adjacency but does not preserve stars. Then G and H are isomorphic to one of the three pairs of graphs in Fig. 2.2.

Fig. 2.1 L(K1,3 ) = L(K3 ) = K3

2.2 Edge-Isomorphisms

19

a G1 :

H1 :

a c

b

G2 :

a

e b

c

G3 :

e b

f c

d

d

d

a

a

a

c

b

H2 :

d

c

b e d

H3 :

c

b e

f d

Fig. 2.2 Automorphisms of three graphs

Proof Assume that G and H are graphs with at least four vertices for which there is an edge correspondence φ that preserves the adjacency of edges but is not starpreserving. Without loss of generality, we may assume that G has a vertex v for which φ(S(v)) is not a star. It follows that deg v = 3 since two adjacent edges form a star and the only way that four edges can be mutually adjacent is in a star. Hence, φ(S(v)) must consist of the edges of a triangle. Let S(v) = {a, b, c}. Since the graph G is connected, there must be another edge d adjacent to one of the edges of the star S(v). However, since its image φ(d) is adjacent to two edges in H , d itself must be adjacent to two edges, say to b and c. This gives the first pair of graphs G1 and H1 in the figure. If there are two such edges, we get the pair G2 and H2 ; and if three, the pair G3 and H3 . As noted earlier in the proof, there cannot be another edge of G incident to v; also, any edge adjacent to a, b, or c has to be adjacent to two of them, so G cannot have any more edges. Hence, these three pairs of graphs, all of order 4, are the only possibilities. In Fig. 2.2, we observe that two colors are adjacent in Hi if and only if they are adjacent in Gi , for each i ∈ {1, 2, 3}.   We note that the two graphs in each of the three pairs are isomorphic to one another. We summarize the main consequences of Whitney’s results in two theorems. Theorem 2.2 The only two connected graphs with isomorphic line graphs are K3 and K1,3 . Theorem 2.3 If G is a connected graph for which L(G) has more automorphisms than G itself has, then G is K1,3 with one, two, or three added edges.

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2 Line Graph Isomorphisms

2.3 Groups Having looked at individual isomorphisms (edge- as well as vertex-), we now look at sets of isomorphisms. Just as the set of all (vertex-)isomorphisms from a graph G onto itself forms a group Γ (G), so does the set of all edge-isomorphisms, which we denote Γ (G). More generally, we let Γ (G, H ) denote the set of all isomorphisms from graph G into graph H (so Γ (G, G) = Γ (G)), and similarly Γ (G, H ) denotes the set of all edge-isomorphisms from G into H . Clearly, given a vertex-isomorphism, there is a corresponding natural edge-isomorphism. Formally, given an isomorphism φ ∈ Γ (G, H ), define φ ∗ : E(G) → E(H ) by φ ∗ (vw) = φ(v)φ(w). Now let Γ ∗ (G, H ) = {φ ∗ : φ ∈ Γ (G, H )}. The following theorem gives some of the connections between the three groups of graph isomorphisms, Γ (G, H ), Γ (G, H ), and Γ ∗ (G, H ). Theorem 2.4 The following hold for any two graphs G and H : (a) Γ ∗ (G, H ) is a subgroup of Γ (G, H ). (b) If G and H are connected and not one of the graphs in Fig. 2.2, then Γ ∗ (G, H ) = Γ (G, H ). (c) The mapping T : Γ (G, H ) → Γ ∗ (G, H ) for which T (φ) = φ ∗ is one-to-one if and only if G has no isolated edges and at most one isolated vertex. As this theorem implies, Whitney’s theorem tells us when Γ ∗ (G, H ) equals and that with only a few exceptions, the natural mapping T from Γ (G, H ) onto Γ ∗ (G, H ) is one-to-one. Thus, the three groups Γ (G, H ), Γ (G, H ), and Γ ∗ (G, H ) are closely related. This is even more so when G = H , in which case we denote the groups by Γ (G), Γ (G), and Γ ∗ (G). Then the mapping T is a homomorphism since for any elements φ and ψ of Γ (G) and all edges vw in G, Γ (G, H );

(φψ)∗ (vw) = (φψ)(v)(φψ)(w) = φ(ψ(v))φ(ψ(w)) = φ ∗ (ψ(v)ψ(w)) = φ ∗ ψ ∗ (vw).

Since both isomorphisms and edge-isomorphisms of graphs preserve connected components, the next theorem follows from these observations and Theorems 2.2 and 2.3. Theorem 2.5 The following hold for any graph G: (a) Γ ∗ (G) is a subgroup of Γ (G). (b) Γ ∗ (G) = Γ (G) if and only if G does not have a pair of components isomorphic to K1,3 and K3 nor a component isomorphic to one of the three graphs in Fig. 2.2. (c) Γ (G) ∼ = Γ ∗ (G) if and only if G has no isolated edges and at most one isolated vertex. (d) If G is connected, then Γ (G) ∼ = Γ (G) if and only if G is not isomorphic to one of the three graphs in Fig. 2.2.

2.3 Groups

21

Although this theorem, or at least parts of it, have been discovered many times, Sabidussi [153] was the first to publish the connections between the groups. We restate the main result, albeit in a somewhat different form. Theorem 2.6 If G is a connected graph of order at least 3 and is not one of the three special graphs in Fig. 2.2, then Γ (G) ∼ = Γ (L(G)). As we have observed, there is just one pair of non-isomorphic connected graphs with isomorphic line graphs and only three graphs, all of order 4, for which the automorphism group is smaller than the automorphism group of its line graph; that is, as automorphism groups, Γ (G)  Γ (L(G)). It follows that if G is any connected graph of order at least 5, then Γ (G) ∼ = Γ (L(G)). Harary and Palmer [94] considered the question of when G and L(G) are not only isomorphic but identical in that the elements of G and the elements of L(G) are always permuted in exactly the same way. More precisely, G and L(G) are said to be group-identical if for some isomorphism φ : Γ (G) → Γ (L(G)) there is a matching f between V (G) and V (L(G)) (that is, between V (G) and E(G)) for which φ acts consistently, that is, for all automorphisms α ∈ Γ (G) and all vertices v in G, f (α(v)) = φ(α)(f (v)). Figure 2.3 shows this as commutative operations. The first thing to note is that for a connected graph G to be group-identical to its line graph is that it must have the same number of edges as vertices. This means that, being connected, G must have a cycle and can have only one; that is, G must be unicyclic. We now consider an example. Figure 2.4 shows a graph G and its line graph, both having their vertices labeled.

V

f

E

α

φ (α)

V

f

E

Fig. 2.3 A commutative diagram w

G:

u

v

b x

z

y

L(G) :

c

a

d

f

e

Fig. 2.4 A graph and its line graph having isomorphic but not group-identical groups

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2 Line Graph Isomorphisms

With the standard group notation for permutations, we see that in addition to the identity permutation, G has the three permutations (uy)(vx)(w)(z), (wz)(u)(v) (x)(y), and (uy)(vx)(wz) while L(G) has (ad)(bc)(ef ), (bf )(ce)(a)(d), and (ad)(be)(cf ). As both groups are the Klein-4 group, they are isomorphic. However, G has a permutation with four fixed elements while L(G) does not, so the two graphs are not group-identical. It turns out that the number of fixed elements in permutations feature prominently in the characterization of Harary and Palmer of when a graph is group-identical to its line graph. Since every group-identical graph has exactly one cycle, it can be considered as being embedded in the plane with the cycle being a regular polygon with a tree (perhaps trivial) at each vertex. Since the automorphism group of a polygon is a dihedral group, the only possible automorphisms of a unicyclic graph are rotations and reflections, and, as shown by Harary and Palmer [94], it is reflections that are important to characterizing group-identical line graphs. It follows from what was shown earlier that the only connected unicyclic graph that is not isomorphic to its line graph is the graph K2 · K3 (K2 and K3 with one common vertex). Interestingly, that is the only graph in this family with its cycle odd that does not have the group-identical property. As the Harary-Palmer theorem states, the even case is more complicated. Theorem 2.7 A connected unicyclic graph G is group-identical to its line graph if and only if one of the following holds: • its cycle has odd length and G = K2 · K3 ; • its cycle has even length and the number of reflections that have exactly two fixed vertices equals the number of reflections that have exactly two fixed edges. Our consideration of unicyclic graphs can be simplified by having different trees at the vertices be paths of different lengths, effectively providing the basis of a canonical representation. Apparently a large fraction of those graphs with a nontrivial permutation are not group-identical. An example in which they are identical is the graph G in Fig. 2.5, where the cycle is an icosagon and the trees in each quadrant are, in order, P1 , P2 , P3 , P3 , P2 , P1 . With the vertices labeled as indicated in the figure, each of the two reflections through the pairs {v1 , v11 } and {v6 , v16 } fixes only those two vertices. Similarly, in L(G), with its vertices labeled as indicated in the figure, each of the two reflections through the pairs {e1 , e11 } and {e6 , e16 } fixes only those two vertices. Hence, by the theorem, G and L(G) are group-identical.

2.3 Groups

23

Fig. 2.5 The Harary and Palmer example

T15

T16

T14

T1 v21 v1 v2

v22 v3

T13 v16

G: T12

T10

T9

T6 T7

T8

e1 e2

e16

e11

T3 v25

v24 v4 T4 v5 v26 v6 v7 T5

v11

T11

L(G):

T2 v23

e3

e4 e5 e6 e7

Chapter 3

Characterizations of Line Graphs

3.1 Introduction In seeking a characterization of a family of graphs, it makes sense to look at basic effects of the definition or a key property of the family. In the case of a line graph, it is readily seen that the set of edges at a vertex of a graph G produces a complete subgraph in the line graph L(G). Furthermore, every edge of L(G) arises from some vertex in this way. J. Krausz noted not only this fact but also the fact that a vertex in a line graph is incident with either one or two of these complete subgraphs. Using these ideas, he produced the first of the three contrasting characterizations of line graphs. Another basic fact about line graphs is that the star K1,3 is not the line graph of any graph, and hence it cannot be an induced subgraph of any line graph. The second characterization was discovered by A. C. M. van Rooij and H. S. Wilf, who observed regarding induced subgraphs that even if a graph G does not contain K1,3 , if it contains K1,1,2 (that is, K4 with one edge missing) with some special properties, then G itself still cannot be a line graph. This was subsequently used to find a complete list of minimal graphs that cannot be line graphs, the third characterization. These results are proved in Sect. 3.2. Refinements of these characterizations appear in Sect. 3.3, including the fact that only eight of the nine forbidden graphs need to be considered for connected graphs with more than six vertices. Line graphs of some special families of graphs, in particular trees and other bipartite graphs, are characterized in Sect. 3.4.

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_3

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3 Characterizations of Line Graphs

3.2 Primary Characterizations Many mathematicians find characterizations of objects to be among the most interesting theorems, and this appears to be the case among the various types of graphs. We certainly find this to be so for line graphs, for which there are three primary characterizations. The three graphs shown in Fig. 3.1 will play a key role in the proof of the main theorem in this area. Each of them is a line graph, with F1 being the line graph of the graph obtained by adding one edge to K1,3 , F2 the line graph of F1 , and F3 the line graph of K4 . Some definitions needed for one of the characterizations in this theorem are the following. A clique in a graph G is a maximal complete subgraph of G. A triangle in G is called odd if some vertex of G is adjacent to an odd number of vertices of the triangle, and is called even otherwise. The conditions in the line graph characterization theorem were found as follows: (2) by Krausz [122], (3) van Rooij and Wilf [168], and (4) Beineke [29, 30]. Theorem 3.1 The following statements are equivalent for a graph G: (1) G is the line graph of some graph. (2) The edges of G can be partitioned into complete subgraphs in such a way that no vertex belongs to more than two of the subgraphs. (3) The graph K1,3 is not an induced subgraph of G, and if uvw and vwx are both odd triangles, then u and x are adjacent vertices. (4) None of the nine graphs in Fig. 3.2 is an induced subgraph of G.

Fig. 3.1 Three important line graphs F1 :

F2 :

F3 :

3.2 Primary Characterizations

27

L1 :

L2 :

L3 :

L4 :

L5 :

L6 :

L7 :

L8 :

L9 :

Fig. 3.2 The nine line-forbidden graphs

Proof Our proof proceeds through showing this sequence of implications: (1) ⇒ (2), (2) ⇒ (1), (2) ⇒ (4), (4) ⇒ (3), and (3) ⇒ (2). As usual, we assume throughout that G is connected. (1) ⇒ (2): Assume that G is the line graph of H . At each vertex of H , the edges determine a complete subgraph of G, and every edge of G lies in exactly one of these. Since each edge of H has two ends, the corresponding vertex in G is in at most two of these complete subgraphs. (2) ⇒ (1): Let S be the family of complete subgraphs given in (2) together with the family of single-vertex graphs that are in just one of the complete subgraphs. Note that each vertex of G is in exactly two of the members of S . Define the graph H to have S as its vertex-set with two vertices being adjacent whenever the corresponding subgraphs have a common vertex. We now show that G is the line graph of H . There is certainly a one-to-one correspondence φ from the edges of H to the vertices of G: for each edge e in H , let φ(e) be the vertex of G that is in the two subgraphs in S which are joined by e in H . What remains to be shown is that adjacency and non-adjacency are preserved between L(H ) and G. Let e and f be two edges in H . If e ∼ f in L(H ), then e joins (say) A and B from S , and f joins B and C. It follows that then φ(e) is adjacent to φ(f ) in G since B is a complete subgraph. Hence, φ preserves adjacency. On the

28

3 Characterizations of Line Graphs

other hand, if e and f are not adjacent in L(H ), then no member of S can contain both φ(e) and φ(f ), so φ(e) cannot be adjacent to φ(f ). Hence, φ also preserves non-adjacency. (2) ⇒ (4): It is easily seen that when any of the graphs L1 , L2 , · · · , L9 has its edges partitioned into complete subgraphs, some vertex is in more than two of the subgraphs. Therefore (since (1) implies (2)), none of these nine graphs can be a line graph. Because every induced subgraph of a line graph must itself be a line graph, the implication follows. (4) ⇒ (3): We show the contrapositive, namely, that if G does not contain K1,3 as an induced subgraph and does not satisfy (3), then it must contain one of the other eight graphs in Fig. 3.2 as an induced subgraph. That is, G must have two odd triangles uvw and vwx with u ∼ v. We consider separately the two cases of whether or not some vertex is adjacent to an odd number of vertices in both triangles. Case 1 There is a vertex y adjacent to an odd number of vertices in the set u, v, w, and to an odd number in v, w, x. If y is adjacent to all three vertices in one of the sets, then it must be adjacent to all four, and so L3 is induced. On the other hand, if y is adjacent to only one vertex in each set, this cannot be either v or w since then K1,3 would be induced, contrary to hypothesis. Consequently, y must be adjacent to u and x, and L2 is induced. Case 2 No vertex is adjacent to an odd number in each set. Assume that y is adjacent to an odd number of u, v, and w, and that z is adjacent to an odd number of v, w, and x. We note two facts that follow: (α) If either y or z is adjacent to v or to w, then it is also adjacent to u or to x since otherwise K1,3 is an induced subgraph. (β) Neither y nor z can be adjacent to both u and x since it would then be adjacent to an odd number of vertices in both triangles. There are now three further possibilities to consider, depending on the number of vertices of the corresponding triangle that each is adjacent to. Case 2(a) Both y and z are adjacent to just one vertex of the corresponding triangle. This is the most intricate of the three possibilities, and involves a number of further subcases: y ∼ u and z ∼ x. This implies that either L4 or L7 is an induced subgraph (because of (β)), depending on whether y ∼ z or not. y ∼ w and z ∼ x. It follows from (α) and (β) that y ∼ x and z ∼ u. If y ∼ z, then the induced subgraph v, x, y, z is K1,3 , while, if y ∼ z, then graph L8 is obtained. y ∼ w and z ∼ v. Necessarily y ∼ x and z ∼ u, so that if y ∼ z, a graph isomorphic to L8 is obtained, while if y ∼ z, then graph L3 is induced. y ∼ w and z ∼ w. Again y ∼ x and z ∼ u, so that, if y ∼ z, L9 is obtained, and, if y ∼ z, then K1,3 is an induced subgraph. Up to interchanging the roles of vertices, this exhausts the possibilities of this case.

3.2 Primary Characterizations

29

Case 2(b) Both y and z are adjacent to all three vertices of the corresponding triangle. It follows that y ∼ x and z ∼ u. Hence, if y ∼ z, then G has an induced subgraph isomorphic to L3 , while, if y ∼ z, then L6 occurs. Case 2(c) One of the vertices, say y, is adjacent to all three vertices of the corresponding triangle, and the other, z, to just one. Assume that y is adjacent to u, v, and w, and hence not to x. There are then two possibilities to consider, depending on which vertex of triangle vwx is adjacent to z. If z ∼ x, then L2 or L5 is obtained according as y is or is not adjacent to z. If z ∼ v or w, then either L3 or K1,3 is induced, depending on whether z is adjacent to both u and y or not. This exhausts all of the possibilities, thereby establishing this implication. (3) ⇒ (2): Assume that G satisfies (3). We first show that if, in addition, G has two even triangles with a common edge, then it must be isomorphic to one of the graphs F1 , F2 , or F3 in Fig. 3.1. To this end, let uvw and vwx be even triangles, whence u ∼ x. If G is not just F1 , then it must have a fifth vertex y adjacent to one of the others. Since both triangles were taken to be even, we may asume that y is adjacent either to just v and w or to the three vertices u, v, and x. The former cannot occur since then K1,3 would be induced as u, v, x, y. Hence, the latter must hold, and so we have graph F2 . If G has a sixth vertex z, then this same argument implies that z is adjacent to u, x, and either v or w. If z ∼ v, then y ∼ z implies that the induced subgraph v, w, y, z is K1,3 , while y ∼ z implies that uvy and vyx are odd triangles with u ∼ x. Since both of these violate our hypotheses, we must have z ∼ w. Also, z ∼ y since otherwise uvy and vyx are again odd triangles. It follows that graph F3 is obtained. This is as far as one can go since if there were a seventh vertex in G, it would have to be adjacent to precisely the same vertices as z and v, and then K1,3 would again be induced. Thus, the claim is established. That each of the three graphs in Fig. 3.1 can have its edge-set partitioned to satisfy (2) is readily verifiable. Now assume that G does not have a pair of even triangles with a common edge. Let S be the set of cliques that are not even triangles, and let T be the set of edges that lie on just one triangle and that triangle even. We now show that the graphs in S ∪ T determine a partition of the edge-set of G. Clearly, each edge of G lies in at least one member of the set. If an edge vw is in two members of S ∪ T , then both must be cliques that are not even triangles. There are therefore vertices u and x each in just one of these two cliques, and hence not adjacent to one another. But then uvw and vwx are odd triangles, each either being in a clique with at least four vertices or being itself a clique that is an odd triangle. Since this violates (3), it follows that the edges are partitioned into the complete subgraphs in S ∪ T . By considering three cases, we next show that no vertex lies in more than two of the graphs in S ∪ T . First, assume that v is a vertex in exactly one member of T , say edge vw of the even triangle uvw. Since v is on only one member of T , edge uv must be on an odd triangle tuv. Any vertex adjacent to v must also be adjacent to u since triangle uvw is even. Furthermore, any two such vertices y and z must be adjacent since both of the triangles uvy and uvz have an edge in common with triangle uvw and are thus

30

3 Characterizations of Line Graphs

odd. It follows that v lies in precisely one member of S and so is in exactly two members of S ∪ T . Next, assume that v is on two members of T . If these are edges vw and vw of different even triangles uvw and u vw , then u must be adjacent to u or w . But this means that uv is on two even triangles, contradicting our definition of T . Hence, both member of T that contain v must be on the same even triangle uvw. This means that v cannot be on any other edge of G since that would imply that uvw is an odd triangle or that uv or vw is on more than one triangle. Hence, v lies on only the two members of T and none of S . Finally, suppose that v lies on three members of S . Let x, y, and z respectively be other vertices in these cliques. Because (as we have shown) no edge lies in more than one clique in S , none of these vertices lies in either of the other two cliques. Furthermore, two of these vertices, say x and y, are adjacent, since otherwise K1,3 would be an induced subgraph. This implies that triangle vxy is even, since if not, it would be in a member of S containing both x and y. Of course the same argument can be used to show that triangle vyz is even. This contradicts our assumption that no two even triangles have a common edge. Therefore, no vertex can be in more than two members of S . Consequently, the edge-set of every graph G that satisfies (3) has a partition as described in (2), and this completes the proof.   This theorem is arguably the most important theorem in the theory of line graphs, and we shall adopt the notation of L for this family of nine graphs and L1 − L9 for the graphs themselves as in Fig. 3.2.

3.3 Refinements We begin this discussion by focusing our attention on the line-forbidden graph L9 , the wheel W5 with five rim vertices. Assume that G is a connected graph that contains only L9 as an induced subgraph from the set of nine forbidden graphs. Assume that L9 is labelled as in Fig. 3.3, and suppose that G contains another vertex u. If u ∼ v (the hub vertex), then it must be adjacent to at least one of each Fig. 3.3 The graph L9 with a labeling

w1

w5

w2 v

w3

w4

3.3 Refinements

31 x

v

v x

(a)

(b)

(c)

Fig. 3.4 Extensions of L9

pair of rim vertices at distance 2 in order not to have K1,3 as an induced subgraph in G. From this observation it follows that u must be adjacent to three consecutive rim vertices, and thus G must have L3 as an induced subgraph, in contradiction to our assumption. Hence, u can be adjacent only to rim vertices. Assume now that u ∼ w1 . Again in order for G not to have K1,3 as an induced subgraph, u must be adjacent to either w5 or w2 , say w2 . Our argument can be extended to deduce that u must have all of its neighbors consecutive on the rim and that there must be at least two of them. If the neighbors of u are w1 , w2 , and possibly w3 , then the induced subgraph G − {w3 } is isomorphic to L8 , while if u is adjacent to all except possibly w5 , then G − {w3 , w5 } is isomorphic to L2 (see Fig. 3.4b). Therefore, as first observed by Šoltés [162], the only graph that is not a line graph and does not have any of L1 − L8 as an induced subgraph is L9 itself. One consequence of this is that a connected graph with at least seven vertices is a line graph if and only if it does not contain any of L1 − L8 as an induced subgraph. Continuing in this vein, Šoltés proceeded to prove a similar result for L8 , although in this case there is one further exception. Before going on to discuss that result, we observe that L8 and L9 are special in this context. That is, for each of the other seven line-forbidden graphs Li , there are infinitely many connected nonline graphs containing none of the other eight graphs in L as an induced subgraph. Examples of such graphs are shown in Fig. 3.5, where for i ≤ 7, Fi illustrates this for Li . We observe that F3 is Kn − e, which includes L3 = K5 − e, while F6 is obtained by identifying an edge of L6 with Kr . For the remaining graphs, a path Pr is appended as shown. In an extension of Šoltés’s theorem on the graph L8 , three more graphs play key roles, as we shall see in the proof of his theorem. These are the graphs M1 , M2 , and M3 shown in Fig. 3.6. Theorem 3.2 Let G be a connected graph that is not a line graph. (a) If G does not contain any of L1 , L2 , . . . , L8 as an induced subgraph, then G is L9 . (b) If G does not contain any of L1 , L2 , . . . , L7 or L9 as an induced subgraph, then G is L8 or M1 . (c) If G does not contain any of L1 , L2 , . . . , L7 as an induced subgraph, then G is L8 , L9 , M1 , M2 , or M3 .

32

3 Characterizations of Line Graphs

F1 :

F3 :

F2 :

F5 :

F4 :

F6 :

F7 :

Fig. 3.5 Infinite families

Proof (a) This statement was verified in the discussion leading up to the statement of the theorem. (b) Let G be a graph that contains L8 and only L8 from the set of line-forbidden graphs L . Assume that L8 has the labelling shown in Fig. 3.7, and assume that x is a vertex of G adjacent to some vertex in this copy of L8 , and let H denote the subgraph induced by these seven vertices. There are a number of cases to consider. Case 1 x ∼ v1 and x ∼ v2 . In order for there not to be a copy of K1,3 , it must be that x is also adjacent to at least one member of each of these pairs of vertices: u1 or w1 , u2 or w2 , and w1 or w2 . Without loss of generality, we assume x ∼ w1 , whence x ∼ u2 or x ∼ w2 . However, in either case, there is an induced copy of L3 (see Fig. 3.8a). Case 2 x ∼ v1 and x ∼ v2 . In order for there not to be a copy of K1,3 , it must be that x is adjacent to u1 and also to w1 or w2 . We consider these two as separate subcases.

3.3 Refinements

33

Fig. 3.6 Graphs M1 , M2 , M3 of Theorem 3.2

M1 :

M2 :

M3 :

Fig. 3.7 The graph L8 with a labeling

u1

w2

v1

v2

w1

u2

Case 2(a) x ∼ u1 and x ∼ w2 . Then, as we just noted, x ∼ w1 , and this forces L9 to be an induced subgraph (whether or not x ∼ u2 ) (see Fig. 3.8b). Case 2(b) x ∼ u1 and x ∼ w2 . What occurs here further depends on what adjacencies x has with u2 and w1 . If x ∼ u2 , then L5 is induced as H − w1 whether or not w1 is adjacent to w2 . On the other hand, if x ∼ u2 and x ∼ w1 , then in the induced subgraph H − u1 , w2 is isomorphic to L2 , while if x ∼ u2 and x ∼ w1 , then in H − u1 , w2 is isomorphic to L2 (see Fig. 3.8c). Case 3 x ∼  v1 and x ∼ v2 . We again consider several subcases, depending on whether x is adjacent to either of w1 and w2 . Case 3(a) x ∼ w1 and x ∼ w2 . Then whether or not x is adjacent to u1 or u2 , the subgraph H − u1 , u2 is isomorphic to L2 (see Fig. 3.8d).

34

3 Characterizations of Line Graphs u1

w2

v1

v2

x

u1

w2

v1

v2

x w1

u2

w1

(a)

u2 (b)

u1

w2

u1

w2

v1

v2

v1

v2

x

x w1

u2

w1

(c)

u2 (d)

u1

w2

v1

v2

x w1

u2 (e)

Fig. 3.8 Graphs used in the proof of Theorem 3.2(b)

Case 3(b) x ∼ w1 and x ∼ w2 . We assume that x ∼ u1 . Now H − w1 is either L7 or L4 according as x is or is not adjacent to u2 (see Fig. 3.8e). Case 3(c) x ∼ w1 and x ∼ w2 . Then we must have x ∼ u1 . If also x ∼ u2 , then H − w2 is L7 (see Fig. 3.8e). However, if x ∼ u2 , the result is the graph M1 in Fig. 3.6. This exhausts all of the possibilities, thereby completing the proof that the only graphs that are not line graphs and have only L8 from L as an induced subgraph are L8 itself and M1 . (c) Assume now that J is not a line graph and that it has none of L1 − L7 as an induced subgraph. Of course, L8 and L9 are themselves such graphs. Furthermore, from the proofs of parts (a) and (b), M1 and M2 (with a little checking) are the only such graphs with seven vertices. Suppose now that J has order 8. Then it must be obtainable from one of these two graphs by adding an eighth vertex y. We consider them separately.

3.3 Refinements

35

Fig. 3.9 Graphs used in the proof of Theorem 3.2(c)

x

x y

y (a)

(b) x

x

y

v

v

(c)

(d)

y

Case 1 J does not contain L9 as an induced subgraph. It follows then that, with L8 as the starting point (assume it to be labelled as before), y is just like x, and there are now some subcases to consider. Case 1(a) y is adjacent to u1 and w2 . In order not to have a copy of K1,3 induced, we must have y ∼ x. But then J − {u2 , v2 } is an induced copy of L5 , contradicting our hypotheses (see Fig. 3.9a). Case 1(b) y is adjacent to u2 and w1 . In this case J − {u1 , u2 } is an induced copy of either L7 or L4 according as y is or is not adjacent to x (see Fig. 3.9b). Case 2 J contains L9 as an induced subgraph. This means that J − {x} and J − {y} are both isomorphic to M2 . We assume (as before) that x ∼ w1 and x ∼ w2 , in which case there are essentially three possibilities for y’s neighbors. Case 2(a) y ∼ w1 and x ∼ w2 . It follows that y ∼ x, whence J − {w3 , w5 }is a copy of L5 , so this cannot be (see Fig. 3.9c). Case 2(b) y ∼ w3 and x ∼ w4 . In this case J − {u1 , u2 } is an induced copy of either L7 or L4 according as y is or is not adjacent to x (see Fig. 3.9d).

36

3 Characterizations of Line Graphs

Case 2(c) y ∼ w2 and x ∼ w3 . In order that there not be an induced K1,3 at w2 , it must be that y ∼ x, and hence we have the graph M3 . It follows that the only possible 8-vertex graph that satisfies the hypotheses is M3 . Furthermore, there is no such graph with a ninth vertex z since its position relative to both x and y would have to be the same as the relative positions of x and y themselves, and this is impossible. This completes the proof of (c), and with that the proof of the theorem.   The essence of this theorem is captured in the following corollary. Corollary 3.1 Let G be a connected graph. (a) If |G| ≥ 7, then G is a line graph if and only if it does not contain any of L1 , L2 , . . . , L8 as an induced subgraph. (b) If |G| ≥ 9, then G is a line graph if and only if it does not contain any of L1 , L2 , . . . , L7 as an induced subgraph.

3.4 Bipartite Graphs As we observed in Sect. 3.2, when testing whether a graph G is a line graph using Krausz’s characterization, and when finding a root graph if it is, cliques play a key role. In fact, the only clique that might not arise from the edges at a vertex is one of order 3, that is, an even triangle. Thus, it is natural to ask when there are no even triangles in a line graph. The answer is quite simple: when the root graph is trianglefree. In other words, a graph G is the line graph of a triangle-free graph if and only if its cliques form a Krausz partition, and then the intersection graph of the set of cliques is a root graph of G except possibly that some end vertices may need to be added. Hence, for these graphs, the ‘clique graph transformation’ is essentially the inverse of the line graph transformation, an observation first made by Hedetniemi and Slater [99]. To make this more precise, it is useful to have the concept of the clique graph K(G) of a graph G to be the intersection graph of the sets of vertices of the cliques of G. Clique graphs will be revisited in Chap. 14 as a generalization of line graphs on their own, and the principle result there is their characterization as a family of graphs. Theorem 3.3 If G is a connected triangle-free graph of order at least 3, then K(L(G)) is isomorphic to the graph obtained by deleting any end vertices that G might have. For example, consider the graph G in Fig. 3.10. Its cliques are the seven sets of vertices listed. Denoting these (as indicated) by t, u, . . . , z, we find that the corresponding clique graph K(G) is that shown in the figure. By adding one more vertex, s, adjacent to t and corresponding to a being in only one clique, we get the root graph of G.

3.4 Bipartite Graphs

37 a

f

e

b

g

G:

h i j

t= u= v= w= x= y= z=

a, b b, c, g c, d, i, j d, e a, e, f f , g, h h, i

c

d (a) t

y

x

u

K(G) : z

w

(b)

v

s

Fig. 3.10 The clique graph of a graph

The following theorem gives several characterizations of the line graphs of triangle-free graphs, the nicest being the last, which says that there are only two forbidden induced subgraphs (see Fig. 3.11) for this family. Theorem 3.4 The following statements are equivalent for a connected graph G with at least four vertices: (1) G is the line graph of a triangle-free graph. (2) The subgraph induced by the common neighbors of the ends of each edge is complete, and that induced by the neighbors of each vertex is either complete or spanned by two complete graphs. (3) Two cliques of G have at most one common vertex, and the clique graph of G is triangle-free. (4) G does not contain either K1,3 or K1,1,2 as an induced subgraph.

38

3 Characterizations of Line Graphs

K1,3 :

K1,1,2 :

Fig. 3.11 Forbidden subgraphs for the line graph of a triangle-free graph

Proof The result follows by establishing this sequence of implications: (1) ⇒ (2), (2) ⇒ (3), (3) ⇒ (4), and (4) ⇒ (1). (1) ⇒ (2): Assume that G is the line graph of the triangle-free graph H . First consider a vertex v of G. If it corresponds to an end-edge of H , then its neighbors are all adjacent to one another, while if it corresponds to a non-end-edge, then its neighbors generate two complete graphs, one for each end of the edge. Thus, the second property in (2) holds. For the first property, we suppose that two adjacent vertices v and w in G have two common neighbors u and x that are not adjacent. Then K1,1,2 is induced by these four vertices in G, and so H contains the ‘musical triangle’ graph, K2 · K3 (K3 with an end-edge attached), and hence is not trianglefree. (2) ⇒ (3): Assume that (2) holds. It follows from the first condition of (2) that two different cliques share at most one vertex. Suppose that the clique graph K(G) contains a triangle, say on the three cliques A, B, and C. It follows from the second condition of (2) that no vertex of G can be on three cliques, and so A, B, and C meet pairwise in three different vertices, say A and B meet at u, B and C at v, and A and C at w. But then u, v, and w form a triangle, and so lie in a clique that meets each of the other cliques in more than one vertex, a contradiction. Hence K(G) must be triangle-free. (3) ⇒ (4): Assume that (3) holds. Then K1,3 cannot be an induced subgraph of G since if it were, its edges would constitute cliques generating a triangle. Furthermore, neither can K1,1,2 be an induced subgraph of G since if it were, the two cliques containing its triangles would have two vertices in common. Hence, (4) must hold. (4) ⇒ (1): Finally, assume that neither K1,3 nor K1,1,2 is an induced subgraph of G. Then, since the latter graph is an induced subgraph of each of the graphs L2 , L3 , . . . , L9 , G must be a line graph, say of H . However, H cannot contain a triangle, since if it did, because it is connected and has more than three vertices, it would contain K2 · K3 and so G would have K1,1,2 as an induced subgraph, contradicting our assumption.   In this theorem, (3) and (4) are due to Hedetniemi and Slater [99], while (2) is an unpublished result of A. R. Rao. By restricting the triangle-free graphs to

3.5 Complements of Line Graphs

39

have no odd cycles, we get the following characterizations of the line graphs of bipartite graphs. The excluded subgraph criterion was first found by Chartrand [53]. In practice, a combination of the two criteria would seem to be the most efficient: determine that the graph is the line graph of a triangle-free graph, find the root graph, and then determine whether or not it is bipartite. Corollary 3.2 The following statements are equivalent for a connected graph G with at least four vertices: (1) G is the line graph of a bipartite graph. (2) Two cliques of G have at most one common vertex, and the clique graph of G is bipartite. (3) G does not contain an induced subgraph isomorphic to K1,3 , K1,1,2, or an odd cycle of length greater than 3. The theorem can be further specialized to the line graphs of trees. Corollary 3.3 The following statements are equivalent for a connected graph G: (1) G is the line graph of a tree. (2) Two cliques of G have at most one common vertex, and the clique graph of G is a tree. (3) No vertex is in more than two blocks, and every block is complete. (4) G does not contain K1,3 as an induced subgraph, and every block of G is complete.

3.5 Complements of Line Graphs In this section we investigate properties of the complements of line graphs, starting with those graphs whose line graph and complement are isomorphic, followed by a theorem characterizing those triangle-free graphs whose complement is a line graph. The remainder of the chapter is devoted to finding those graphs that are both a line graph and the complement of a line graph. We recall that one of the classical early results on line graphs is that a connected graph is isomorphic to its line graph if and only if it is a cycle. Aigner [3] considered a similar question: For which graphs are the line graph and the complement isomorphic? In answering this question, we first determine how many edges such a graph with n vertices must have and then show that such a graph must be connected. Lemma 3.1 If G is a graph of order n whose complement is isomorphic to its line graph, then G must have n edges and L(G) must have n vertices and n(n−3) edges. 2 Proof Let G be such that its complement and its line graph are isomorphic: G ∼ = L(G). Then L(G) must all have the same number nof vertices as G, and L(G) must have the same number of edges as G, which is thus n2 −n, which equals n(n−3)  2 . 

40

3 Characterizations of Line Graphs

Lemma 3.2 If the complement of graph G is isomorphic to its line graph, then G is connected. Proof Clearly, if G is disconnected, then its complement is connected, so L(G) must also be connected. This implies that G is either connected itself or it has at least two components, at most one of which is nontrivial. Suppose G is of the latter type. We first note that G cannot have two isolated vertices. For if it did, then the non-trivial component F must have at least three cycles since it has n edges and at most n − 2 vertices. However, it must have at least two edges e and f adjacent to all of the other edges since G has at least two vertices of degree n − 1. Then F must be as shown in Fig. 3.12(a), which is a contradiction. Therefore G must be of the form F + K1 with F having an edge g = vw adjacent to all of the other edges. Furthermore, since F must have n − 1 vertices and n edges, it must consist of the diamond K1,1,2 (two triangles with a common edge) together with possible end edges at v and w, as shown in Fig. 3.12(b). Now suppose that there are r end edges at v and s end edges at w, with 0 ≤ r ≤ s. Then it is easy to see that the minimum degree of G is r + 1, while the minimum degree of L(G) is at least r + 3. This contradiction proves that G cannot be disconnected.   In fact, as is stated in the following theorem, there are only two graphs with the property under consideration. They are shown in Fig. 3.13. Theorem 3.5 If G is a graph whose complement is isomorphic to its line graph, then G is either a 5-cycle or the graph that is the result of attaching one edge at each vertex of a triangle.

e v

g

f (a) Fig. 3.12 Disconnected candidates

Fig. 3.13 The two graphs whose line graph is its complement

(b)

w

3.5 Complements of Line Graphs

41

Proof Assume now that G is a connected graph for which its complement and its line graph are isomorphic: G ∼ = L(G). It follows that G has exactly one cycle since it has the same number of edges as vertices. If G is itself a cycle, then L(G) must be a cycle and G must be the same. The only such cycle is C5 . We therefore turn to the case where that G is a connected graph that is not a cycle and we assume that its only cycle C has length k. Then G contains an end vertex and so G has a vertex of degree n − 2. However, if k ≥ 5, then each edge of G is disjoint from at least two other edges, and so each vertex of L(G) has degree less than n − 2. Hence, k < 5. Suppose k = 4. By the same argument as before, G cannot then have a vertex at distance 2 from the cycle C. Hence, every vertex not on C must be adjacent to C, and neither of the two opposite edges of C can have degree n−2 in L(G). Therefore, G can have only one or two vertices not on C. If just one, then n = 5 and G has five edges while L(G) has six. If two, then both are at the same vertex of C, and so G has a vertex of degree 1 while L(G) has no such vertex. Therefore, k = 4. Assume that k = 3. We begin with the cycle C of length 3 and observe that there cannot be a vertex at distance 2 from C. For, if there were such a vertex, then it and the three vertices of C give an induced claw K1,3 in G, which contradicts this being a line graph. We now consider the maximum degree of the vertices in G and L(G). Clearly, the maximum degree in G is n−2. Because the vertices of maximum degree in L(G) must come from the edges of C, there are at most three in number. Now, not all of the end vertices can be adjacent to the same vertex of C because if they were, then the adjacent edges on C would be adjacent to all of the other edges of G and then L(G) would have vertices of degree n − 1. Therefore, in order to have the number of end vertices in G match the number of vertices of maximum degree in L(G), there must be one end vertex adjacent to each vertex of C; that is, G must be the graph of order 6 shown in Fig. 3.13.   We turn now to the more general problem of finding those graphs whose complement is the line graph of some graph. We begin with trees and consider the tree H = K3,3 − C4 shown in Fig. 3.14. Its complement is also shown in the figure, and it has the third graph, the bow-tie graph B, as its root graph; that is, L(B) ∼ = H. Hence, the tree H is one answer to our question. (As an aside, we observe that B is itself a line graph and in fact the line graph of H , so H ∼ = L2 (H ).) A second example is the tree T = K3,4 − C6 , the details of which we leave to the reader, with the graphs shown in Fig. 3.15. This problem was first investigated by Cvetkovi´c and Simi´c [67], where the focus is on bipartite graphs whose complements are line graphs. We observe that if

H:

H = L(B):

Fig. 3.14 A tree whose complement is a line graph

B:

42

T:

3 Characterizations of Line Graphs

T = L(G):

G:

Fig. 3.15 A larger tree whose complement is a line graph

a graph G has this property, then so do all of its induced subgraphs. Hence, where feasible, maximal graphs in a given family are focused upon, and, as trees, both of the graphs H and T are maximal. The only other trees that qualify are the stars K1,s , and they do so in an obvious way since the complement of K1,s is the disjoint union Ks−1 + K1 and this is the line graph of K1,s−1 + K2 . Theorem 3.6 A tree is the complement of a line graph if and only if it is a star or an induced subtree of either K3,3 − C4 or K3,4 − C6 . The extension of this to forests appeared in [67], with the only additional graphs being null graphs or stars together with one or two isolated vertices. We next consider only non-null graphs, and assume that G is a graph with at least three components, at least two of which have edges. Using the forbidden subgraphs L1 and L2 , one can show that the only possibilities are 2K2 + K1 or 3K2 . Therefore we move on to consider those graphs with at most two components, at least one of which has more than one edge. The following theorem completes the solution for bipartite graphs. Theorem 3.7 Let G be a bipartite graph with either one or two components, at least one with a vertex of degree greater than 1. (a) If G is connected, then its complement is a line graph only if it is complete bipartite or an induced subgraph of one of these graphs: K3,3 − C4 ; K3,s − C6 (s ≥ 3); Kr,s − tK2 (t ≥ 0). (b) If G is disconnected, then its complement is a line graph only if it is an induced subgraph of one of these graphs: (K1 + Kr,s ) − tK2 (t ≥ 0). Going beyond bipartite graphs, Cvetkovi´c and Simi´c proved one additional theorem, one that has a nice and perhaps surprising conclusion. Theorem 3.8 If a triangle-free graph is not bipartite, then it is an induced subgraph of the Petersen graph.

3.5 Complements of Line Graphs

43

The main objective of this section is to find those graphs that are both a line graph and the complement of a line graph. Clearly, complete graphs and null graphs have this property, and so we eliminate those from further consideration and define a graph to be a co-line graph if it is neither complete nor null and both it and its complement are line graphs. We begin with an ‘excluded subgraph’ characterization. We observe that if a graph F is an induced subgraph of G, then its complement F is an induced subgraph of G. Consequently, we have the following result, which, although simple, is included for the sake of completeness. Theorem 3.9 A graph G is the complement of a line graph if and only if it does not contain any of the nine graphs in Fig. 3.16 as an induced subgraph. Theorem 3.10 A graph G is a co-line graph if and only if it does not contain any of the ten graphs in Fig. 3.17 as an induced subgraph. Proof It follows from earlier results that a graph is a co-line graph if and only if it does not contain any of the nine graphs L1 , L2 , . . . , L9 or any of their complements as an induced subgraph. However, this set of eighteen graphs is not minimal. We

L1 :

L2 :

L3 :

L4 :

L5 :

L6 :

L7 :

L8 :

L9 :

Fig. 3.16 Forbidden subgraphs for the complement of a line graph

44

3 Characterizations of Line Graphs

L1 :

L1 :

L2 :

L2 :

L3 :

L3 :

L6 :

L6 :

L9 :

L9 :

Fig. 3.17 Forbidden subgraphs for a co-line graph

observe that the complements of L4 , L5 , L7 , and L9 all have the claw K1,3 as an induced subgraph. This leaves the ten graphs in Fig. 3.17, and it is a routine matter to verify that none of these contains any of the others as an induced subgraph, so this is a minimal set.  

3.5 Complements of Line Graphs

45

We now determine precisely which graphs are co-line graphs of any given order. As we will see, there are such graphs only for the orders from 3 to 9. The numbers of a given order are given in a corollary and a figure. For six or fewer vertices, the set was found by elimination from a set of all graphs of that order, but of course it could also be done by computer. Corollary 3.4 There are two co-line graphs of order 3, shown in Fig. 3.18. Corollary 3.5 There are seven co-line graphs of order 4, shown in Fig. 3.19. Corollary 3.6 There are twelve co-line graphs of order 5, shown in Fig. 3.20. Corollary 3.7 There are ten co-line graphs of order 6, shown in Fig. 3.21. Since none of the ten co-line forbidden subgraphs in Fig. 3.17 has more than six vertices, there is a recursive criterion for co-line graphs of order 7 or more: Lemma 3.3 For n ≥ 7, a graph of order n is a co-line graph if and only if each of its induced subgraphs of order n − 1 is. Corollary 3.8 There are four co-line graphs of order 7, shown in Fig. 3.22. Proof Because the complement of a forbidden subgraph is forbidden, only graphs with seven vertices and up to ten edges need to be considered. Assume that G is such a graph. Because L3 is forbidden, G cannot have more than three components. Suppose that it has three. Since L1 is excluded, G cannot have a triangle. Since one

Fig. 3.18 The co-line graphs of order 3

Fig. 3.19 The co-line graphs of order 4

46

Fig. 3.20 The co-line graphs of order 5

Fig. 3.21 The co-line graphs of order 6

3 Characterizations of Line Graphs

3.5 Complements of Line Graphs

47

Fig. 3.22 The co-line graphs of order 7

of the components must have at least three vertices and L2 is excluded, two of the three components must be trivial and the third must be a triangle-free graph with five vertices. We see from Fig. 3.20 that this component must be a path or a cycle, and therefore G must contain L3 as an induced subgraph. Hence, G cannot have three components. Now assume that G has two components. Again, because of L1 and L2 , it must be triangle-free and one component must be trivial. From the connected graphs in Fig. 3.21, we see that the other component must be a 6-cycle, and it is straightforward to verify that both K1 + C6 and its complement are line graphs. The remaining case is that G be connected. We claim that G must have a triangle. Suppose it does not. Then G must have a vertex v for which G − v is connected. Then from Fig. 3.21 we can deduce that G−v is a 6-cycle C. However, to then avoid G having an induced claw, the seventh vertex must be adjacent to two consecutive vertices of C, so G must indeed have a triangle. Furthermore, to avoid G having L1 as an induced subgraph, every vertex must be adjacent to at least one vertex on each triangle. We will see that this is an important observation. We next show that, keeping in mind that G has at most ten edges, it cannot have two triangles that share an edge. Suppose that G has K4 is a subgraph. Then each of the other three vertices must be adjacent to at least two of its vertices, and for this, G would have to have twelve edges. So suppose that uvw and vwx are triangles of G with u and x not adjacent. Because of the restriction on the total number of edges, at least one of the other vertices, say y, is adjacent to only one vertex on these two triangles. Since every triangle must have a vertex adjacent to y, that vertex would have to be either v or w. However, that means that G would have an induced claw. Therefore, no two triangles can have a common edge. If T is a triangle of G, then each of the other four vertices must be adjacent to exactly one of its three vertices, so two must be at the same vertex of T . Furthermore, they must be adjacent to each other since otherwise G has a claw. Thus, G has two triangles uvw and wxy (that is, they form a bow-tie graph). Neither of the other

48

3 Characterizations of Line Graphs

Fig. 3.23 The co-line graph of order 8

two vertices can be adjacent to w, and since G has at most ten edges, it follows that each of them is adjacent to a different one of each triangle; that is, up to their labels, one is adjacent to u and x and the other to v and y. There is only one graph that satisfies these conditions, the one with ten edges shown in Fig. 3.22, and it is a co-line graph. This completes the proof that there are just four co-line graphs with seven vertices.   Corollary 3.9 There is only one co-line graph of order 8, shown in Fig. 3.23. Proof Assume that G is a co-line graph with eight vertices and at most fourteen edges. Suppose that G is disconnected. Then, as in the 7-vertex case, it must be triangle-free and have just two components, one of which is trivial. It is then clear from the co-line graphs with seven vertices that no such graph exists. Therefore, G must be connected. Consequently, its eight induced subgraphs with seven vertices must each be one of the four co-line graphs of order 7. The one with an isolated vertex cannot play this role of since then G would have a vertex of degree 1, and that is incompatible with the eligible 7-vertex co-line graphs. Hence, each of the eight vertex-deleted subgraphs of G must be one of the two with ten or eleven edges. Since neither of these has two triangles with a common edge, neither can G. Furthermore, since none of the 7-vertex graphs has a vertex of degree 5, G cannot have one of degree 2 or 5. Now not all eight subgraphs can be the same since that would force G to be regular, and since both of the 7-vertex candidates have vertices of degree 2 and 4, that is not possible. Consider a copy of the one with ten edges. Then the eighth vertex must be adjacent to both of the vertices of degree 2. Furthermore, in order for G to be claw-free, it must also be adjacent to a neighbor of each of these and those neighbors cannot themselves be adjacent since that would violate the property of there being no triangles with a common edge. The result has fourteen edges, and thus, up to isomorphism, there is only one eligible graph of order 8 possible. It is in fact self-complementary, and can be verified to be a co-line graph.   Corollary 3.10 There is only one co-line graph of order 9; it is shown in Fig. 3.24. Proof Assume that G is a co-line graph of order 9. The only way in which this is possible is for all of its induced 8-vertex subgraphs to be isomorphic to the co-line graph of that order. This requires the addition of the ninth vertex to be adjacent to

3.5 Complements of Line Graphs

49

Fig. 3.24 The co-line graph of order 9

the four vertices of degree 3 (and only these). The result is the graph in Fig. 3.24, and it can be verified to be a co-line self-complementary graph.   Recalling that the definition of co-line included the requirement that a graph be neither complete nor null, we observe that there are no co-line graphs of order 10 since the existence of such a graph would require that there be a co-line graph of order 9 with a vertex of degree 5. We summarize some of our results in the following theorem of Beineke [31]. Theorem 3.11 There are precisely 37 co-line graphs and the largest has nine vertices. Having determined which graphs G have both G and G line graphs, we can classify those for which neither is a line graph, and those for which one is and the other is not. Based on earlier results, it is the case that neither a graph or its complement is a line graph if and only if G contains some Li (for i = 1, 2, . . . , 9) as an induced subgraph and G contains some Li (for i = 1, 2, . . . , 9) or vice versa. Of course, from these results one can determine whether it is the case that G or G is a line graph and the other is not.

Chapter 4

Spectral Properties of Line Graphs

4.1 Introduction One of the most important ways in which graphs are used in computation is to encode their structure as a matrix, and the primary purpose of this chapter is to make connections between matrices and line graphs. A classic example of this is the well-known result of Horst Sachs: every eigenvalue of the adjacency matrix of the line graph of a graph is at least −2. In the next section, we review some of the basic material on graphs and matrices before moving on to linear algebra aspects of line graphs themselves in Sect. 4.3. (It is assumed that readers have some familiarity with matrices, and concepts that are needed and not described here can be found in any elementary book on linear algebra.)

4.2 Eigenvalues of Graphs We begin with some definitions and examples. Let G be a graph with vertices v1 , v2 , . . . , vn . The adjacency matrix A = A(G) = (aij ) of G is the n × n matrix in which the (i, j ) entry aij is 1 if vertices vi and vj are adjacent and is 0 otherwise. A simple example is shown in Fig. 4.1. One of the basic theorems on adjacency matrices involves the number of walks from one vertex to another. Clearly, every adjacency matrix is symmetric, and the ith row and the ith column sums equal the degree of the vertex vi . A straightforward induction proof shows that the (i, j ) entry of the kth power of A is the number of walks of length k from vertex vi to vertex vj . One of the most appreciated results is the classic 1847-theorem due to Kirchhoff [117] (of course it applies to line graphs). It gives the number of spanning trees in terms of the determinant of a matrix closely related to the adjacency matrix.

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_4

51

52

4 Spectral Properties of Line Graphs

Our interest here is primarily in the eigenvalues of the adjacency matrix of a graph, and we provide some of the basic definitions involving this material. Because the foundation of this material is the adjacency matrix, we assume here that the entries in our matrices are nonnegative. For a square n × n matrix M, its characteristic polynomial φ(M; x) is the determinant of the matrix xI − M, its eigenvalues are the roots of the characteristic polynomial, its spectrum is the list of the eigenvalues with their multiplicities, frequently written in non-increasing order as λ1 ≥ λ2 ≥ . . . ≥ λn , and its eigenvectors are the non-zero column vectors x for which Mx = λx for some eigenvalue λ. The matrix terminology carries over to graphs through the adjacency matrix. Thus, if G is a graph of order n and A is its adjacency matrix, then the characteristic polynomial, the eigenvalues, the spectrum, and the eigenvectors of G are those of A. We note that the order given to the vertices of a graph affects the appearance of the adjacency matrix but does not affect the related objects. The characteristic polynomial of Q3 shown in Fig. 4.1 is seen to be φ(Q3 ; x) = x 8 − 12x 6 + 30x 4 − 28x 2 + 9 = (x − 3)(x + 3)(x − 1)3 (x + 1)3 , with the spectrum 3,1,1,1,−1,−1,−1,−3. Some examples of the eigenvalues of line graphs are shown in Table 4.1 [70]. Before focusing on line graphs, we state some results on eigenvalues of graphs in general (see Doob [70] for details). Some notation that we use for convenience

v7

v4

v3

v8

v5

v2

Q3 :

(Q3 ) =

0 0 0 0 1 1 1 0

0 0 0 0 1 1 0 1

0 0 0 0 1 0 1 1

0 0 0 0 0 1 1 1

1 1 1 0 0 0 0 0

1 1 0 1 0 0 0 0

1 0 1 1 0 0 0 0

0 1 1 1 0 0 0 0

v6

v1 Fig. 4.1 Q3 and its adjacency matrix

Table 4.1 Some line graphs and their spectra Line graph Pn Cn (n even) Cn (n odd) L(K4 ) = K2,2,2 L(Kn ) L(Kr,s )

Eigenvalues kπ 2 cos( n+1 ), k = 1, 2, · · · , n 2kπ 2 cos( n ), k = 1, 2, · · · , n 2 cos( 2kπ n ), k = 1, 2, · · · , n 4, 0, −2 2n − 4, n − 4, −2 r + s − 2, r − 2, s − 2, −2

Respective multiplicities 1, 1, · · · , 1 2, 1, 1, · · · , 1, 2 1, 1, · · · , 1, 2 1, 3, 2 1, n − 1, n(n − 3)/2 1, s − 1, r − 1, (r − 1)(s − 1)

4.2 Eigenvalues of Graphs

53

in this chapter is to let λ denote the minimum eigenvalue of a graph G and to let Λ denote the maximum eigenvalue (analogous to δ and Δ denoting the minimum and maximum vertex degrees). Further, Ir and Jr denote respectively the identity r × r matrix and the universal (meaning all 1s) r × r matrix. Of course, these theorems all hold for line graphs. Theorem 4.1 Let G be a nontrivial connected graph with eigenvalues λ1 ≥ λ2 ≥ . . . ≥ λn . (a) The sum of the eigenvalues is 0. (b) If G is bipartite, then λi = −λn+1−i . It is interesting to note that (b) in this theorem was first discovered in a chemistry context in the study of molecules by Coulson and Rushbrooke [63]. The next theorem is the result of applying a rather deep result on nonnegative matrices (known as the Perron-Frobenius theorem) to graphs (see [157]). Theorem 4.2 If G is a nontrivial connected graph with maximum eigenvalue Λ, then the following hold: (a) (b) (c) (d) (e)

Λ is a simple root of the characteristic polynomial. Λ has an eigenvector with all entries positive. If λi is any other eigenvalue of G, then −Λ ≤ λi < Λ. If e is an edge of G, then the largest eigenvalue of G − e is less than Λ. If G has maximum degree Δ and average degree d, then either G is regular and Λ = Δ or G is not regular and d < Λ < Δ.

Since this theorem does not involve line graphs per se, we do not prove it in its entirety, but we do give a proof of part (e). Proof (e) Let G be a graph with maximum degree Δ and maximum eigenvalue Λ. Further, let v = [c1 , c2 , . . . , cn ]T be an eigenvector corresponding to Λ and let u = [1, 1, . . . , 1]T be the vector of all 1s. Then Λ=

ΛvT u vT Au Σci di = = ≤ Δ. T T Σci v u v u  

The next theorem (see [70]) involves minimum, rather than maximum, eigenvalues, a topic that is especially significant for line graphs, as we shall see in the next two sections. Theorem 4.3 If G is a nontrivial connected graph with minimum eigenvalue λ, then the following hold: (a) λ ≤ −1 with equality if and only√ if G is complete. (b) If G is not complete, then λ ≤ − 2 with equality if and only if G = K1,2 . (c) If G is not a complete graph and is not either K1,2 or K2 · K3 , then λ(G) < −1.5.

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4 Spectral Properties of Line Graphs

4.3 The Spectrum of a Line Graph When one has a mathematical concept (in this case a matrix) and one is interested in its line graph features, it seems natural to look at the edge aspects, and this is indeed the case with matrices. Of course, given a graph G, there is the adjacency matrix of the line graph A(L(G)), but in addition to this, there is a second matrix that contains the information. If G is a graph with the n vertices v1 , v2 , . . . , vn and the m edges e1 , e2 , . . . , em , its incidence matrix B = B(G) = (bij ) is the n × m matrix in which bij = 1 if vertex vi and edge ej are incident, and bij = 0 if not. Figure 4.2 shows a graph G and its adjacency and incidence matrices A and B. Obviously the transpose of a matrix carries precisely the same information as the original matrix itself; in addition, their two products are also closely related. One way in which this appears is this result from matrix theory: If M is an n × m matrix with m ≥ n, then every eigenvalue of MT M is nonnegative and φ(MT M) = x m−n φ(MMT ). Another connection is through the adjacency matrices of both a graph and its line graph. Here, D = D(G) is the degree matrix of G, defined to be the n × n diagonal matrix with dii equal to the degree of vertex vi . Theorem 4.4 For any graph G, (a) A(G) = BBT − D; (b) A(L(G)) = BT B − 2I. Proof (a) Assume first that vi and vj are adjacent vertices in G (so i = j ) and that ek is the edge vi vj . Then the (i, j ) entry of BBT is 1, while if vi and vj are not adjacent (and i = j ) the entry is 0. On the other hand, the i-th diagonal entry of BBT is clearly the degree of the vertex vi , so (a) follows. (b) Similarly, if ei and ej are adjacent edges of G (so that i = j ), then (BT B)ij = 1. Clearly it is also the case that each diagonal entry of BT B is 2, which establishes (b).   Our next result involves line graphs of regular graphs.

v1 e4 G:

e1 e5

v4 e3

v2 e2

(G) =

0 1 1 1

1 0 1 0

1 1 0 1

v3 Fig. 4.2 The adjacency and incidence matrices of a graph

1 0 1 0

(G) =

1 1 0 0

0 1 1 0

0 0 1 1

1 0 0 1

1 0 1 0

4.4 An Extension of Line Graphs

55

Theorem 4.5 If G is an r-regular graph with n vertices and m edges, then φ(L(G); x) = (x + 2)m−n φ(G; x − r + 2). Proof Let AL denote the adjacency matrix of L(G). From Theorem 4.4(b) we have φ(L(G); x) = det(xI − AL ) = det((x + 2)I − (2I + AL ) = det((x + 2)I − BT B). From the result on reversing the product of matrices cited earlier, it follows that det((x + 2)I − BT B) = φ(BT B; x + 2) = (x + 2)m−n φ(BBT ; x + 2). Finally, from Theorem 4.4(a), we have φ(BBT ; x + 2) = φ(A + rI; x + 2) = det((x + 2)I − A − rI) = φ(A; x − r + 2), which completes the proof.

 

We conclude this section with bounds on the eigenvalues of line graphs. The lower bound was established by Sachs [154], along with other results. As will be seen in what follows in the next section, this has important ramifications. Theorem 4.6 If λ is an eigenvalue of the line graph L(G) and ΔL is the maximum line-degree in G, then −2 ≤ λ ≤ ΔL . Proof The upper bound follows from Theorem 4.2. We now establish the lower bound. As noted earlier, the eigenvalues of the matrix BT B are all real. It is also the case that its characteristic polynomial satisfies the equation φ(BT B; x + 2) = φ(BT B − 2I; x). Hence the eigenvalues of BT B − 2I are all at least −2, proving the result.  

4.4 An Extension of Line Graphs It is an interesting fact that the line graph of a tetrahedron is the graph of an octahedron, K2,2,2 (also denoted K3(2)), the complete graph K6 with a perfect matching removed. The extension of this to four dimensions as the graph K4(2) , however, is not a line graph—the removal of the vertices of a triangle leaves the forbidden graph L3 (that is, K5 − e). Nonetheless, the spectrum of this graph [6, 0, 0, 0, 0, −2, −2, −2], so the minimum eigenvalue is again −2. To formalize this and extend it further, we define the d-dimensional octahedron to be the graph Kd(2) . We note that this is also known as the cocktail party graph CP (d). (The name arises from the situation of a cocktail party of d married couples where each person talks to all of the other persons present except their own spouse, and thus the graph is the complete graph K2d with a perfect matching removed. We

56

4 Spectral Properties of Line Graphs

Fig. 4.3 A petal and a flower

also note that the same name has been given to the complete bipartite graph Kd,d with a perfect matching removed, for the situation in which each person talks only to all of the other persons of the opposite sex except their own spouse.) Thus, CP (1) consists of two isolated vertices, CP (2) is a 4-cycle, and, as noted, CP (3) is an octahedron. With this as background, it would be desirable to have these graphs as the result of some type of extension of the line graph operation. In this, we follow the lead of Cvetkovi´c [64], but introduce some of our own terminology with edges as the basic unit (as with line graphs). Thus, we would like a set of an even number of edges with each related to all but one of the others, in other words, the only unrelated edges are in pairs. One way to visualize this is as the multigraph obtained from the star K1,d ∗ in Fig. 4.3. Our goal is by doubling each edge, as shown by the graph G = K1,4 ∗ achieved if we take the graph L (G) whose vertices are these eight edges and two are adjacent if and only if the corresponding edges are in different pairs, which is the desired CP (4) (that is, Kd(2)). Another way of viewing such a multigraph is by putting d double edges at a single vertex and then taking these edges as vertices and joining two with an edge if and only if they have exactly one common vertex. We extend this to a more general situation by adding such double edges at vertices starting with other graphs. Formally, we define a petal to consist of a pair of vertices joined by two edges and a flower to consist of a set of petals with one common vertex, as indicated in Fig. 4.3. Let G be a graph with ordered vertex set {v1 , v2 , . . . , vn }, and let A = (a1 , a2 , . . . , an ) be an n-tuple of non-negative integers that are not all 0. The floral graph G(A) is the multigraph obtained by adding a flower with ai petals at vertex vi for i = 1, 2, . . . n. Note that this structure always has at least one double edge and that there are no edges of multiplicity greater than 2. Furthermore, every petal has one vertex of degree 2. The line graph operation is now extended to the floral line graph L(G(A)) as the graph whose vertices are the edges of G(A)) and two are adjacent if the corresponding edges have exactly one vertex in common. An example is shown in Fig. 4.4, where G is the labeled graph K4 − e (that is, K1,1,2 ) and A is the sequence (0, 2, 1, 0), and the three structures are the graph G, the floral graph G(A), and the floral line graph L(G(A)). We remark that Cvetkovi´c, Doob, and Simi´c [65], and Cvetkovi´c, Rowlinson, and Simi´c [66] used similar terminology (petal and blossom) for the concepts we described above. The term generalized line graph is also used extensively in

4.5 Graphs Whose Spectrum Is Bounded Below by −2

57

v1 f a

d G: v4

v2

e c

a

d G(0, 2, 1, 0):

b

g

e l

h

c

b k

v3

m

d h

a f L(G(0.2.1, 0)): e g b

k

c m l

Fig. 4.4 A floral graph and its floral line graph

literature for what we have called floral line graph. We prefer the latter term since in this book we discuss several generalizations of line graphs. For a more detailed and excellent discussion of such graphs, please see the papers cited above. We now turn to the spectra of floral graphs, and we begin by defining the incidence matrix B of a floral graph G(A). The rows of B are indexed with the n vertices of G as well as the p ends of the petals (note that p equals the sum of the elements of A). The columns of B are indexed with the edges of G(A). Each row of B that corresponds to a vertex of G has 1s in each column that corresponds to an edge of G(A) and 0 in each of the other columns. If w is a petal end, and if e and f are the edges in the petal at w, then the row corresponding to w has entries −1 and 1 in the columns for e and f , and 0 elsewhere. It is easy to check that the adjacency matrix of L(G(A)) is BT B − 2I. We thus have the following result. Theorem 4.7 If λ is an eigenvalue of a floral line graph, then λ ≥ −2. Now that we know that the smallest eigenvalue of line graphs and floral line graphs is at least −2, we ask a converse question. Are there other graphs with this property? We discuss this question in the next section.

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4 Spectral Properties of Line Graphs

4.5 Graphs Whose Spectrum Is Bounded Below by −2 The objective of this section is to determine all graphs whose minimum eigenvalue is at least −2. The answer to this question was given by Cameron et al. [48] in a very impressive paper. In fact, the review of this paper in Mathematical Reviews begins “Occasionally a paper appears which, by virtue of its superior view-point, not only clarifies previously known results but also gives the knowledge of the mathematical area under discussion a quantum increase in understanding and new results. This is such a paper.” (Doob [71]) Consequently, the material in this section is based largely on this paper and its review and also a chapter by Doob [70]. Let G be a graph with adjacency matrix A and minimum eigenvalue λ(G) ≥ −2. It follows that A + 2I is positive semidefinite (that is, all of its eigenvalues are nonnegative). From a result in linear algebra, A + 2I = XXT , where X has n rows and r = rank(A + 2I) columns. If the rows of X are x1 , x2 , . . . , xn , it follows that xi , xi  = 2, while, for i = j , xi , xj  is 1√when vi and vj are adjacent and 0 if not. Hence, the xi are vectors in Rr of length 2 that meet at angles of 60◦ or 90◦ . Now if x and y are two such vectors that meet at 60◦, then we can add four other vectors, −x, −y, x − y and y − x, to form a star of six vectors. See Fig. 4.5. For n vectors x1 , x2 , . . . , xn as above, we add more vectors to form stars and repeat this process until no new vectors can be added. It is easily seen that this results in a finite set S, which we call star-closed. It can be verified that S has the following properties: (a) For any x and y in S, 2 x,y x,x is an integer. (b) If x and cx are both in S, then c = ±1. (c) For any x and y in S, y − 2 x,y x,x x is in S. A finite set of vectors in Rr that satisfies the above properties is called a root system. Our set S has the additional property that all of its vectors have the same length. All such root systems are known (see [48] or [70] for more details). The following are the three root systems that will be needed here. For k ≥ 1, let e1 , e2 , . . . , ek be an orthonormal basis of Rk . Fig. 4.5 A star

4.5 Graphs Whose Spectrum Is Bounded Below by −2

59

(1) The root system An is the set of n(n + 1) elements {ei − ej |1 ≤ i, j ≤ n + 1, i = j }. For instance, Fig. 4.5 represents A2 , with (say) x = (1, 0, −1) and y = (0, 1, −1). (2) The root system Dn is the set {±ei ± ej |1 ≤ i < j ≤ n}.   (3) The root system E8 is defined as the set D8 ∪ { 12 8i=1 i ei | i = ±1, 8i=1 i = 1}. We are now ready to discuss the connection between root systems and graphs. We say that a root system represents a graph G if the adjacency matrix A of G satisfies A(G) = XXT −2I, where the rows of X are taken from a root system. The following theorems characterize line graphs of bipartite graphs in term of root systems. Theorem 4.8 A graph G is represented by the root system An if and only if G is the line graph of a bipartite graph. Proof First assume that G is a bipartite graph with partite sets {x1, x2 , . . . , xr } and {yr+1 , yr+2 , . . . , yr+s }. For an edge xi yj , consider the vector ei − ej in Rr+s . Let X be the matrix with these vectors as rows. Then it follows that XXT = 2I + A(L(G)). The rows of X are vectors in An where n = r + s − 1. Conversely, assume that a graph G is represented by An . Then A(G)+2I = XXT where X is a matrix with rows that are elements of An . Hence the inner products of the rows of X are all nonnegative. It follows that in each column of X, all the 1s have the same sign. With r the number of positive columns and s the number of negative columns, define an r-by-s bipartite graph H with partite sets X and Y corresponding to the positive and negative columns respectively and, for xi ∈ X and yj ∈ Y, xi is adjacent to yj if and only if there is a row of X that is ei − ej . It then follows that G = L(H ).   We next turn to graphs that are represented by Dn . From Theorem 4.1(b), we have that if B is the incidence matrix of G, then A(L(G)) + 2I = XXT where X = B T . Hence each row of X is an element of Dn , where n is the order of G. Thus line graphs are represented by Dn . It follows from a similar discussion in Sect. 4.4 that floral line graphs are also represented by Dn . It turns out that the converse is also true, which gives us our next result (see [70] for a proof). Theorem 4.9 A graph G is represented by a root system Dn if and only if it is a line graph or a floral line graph. We conclude this chapter with the characterization of graphs with minimum eigenvalue at least −2. In fact, the only graph not already covered is for the root system E8 . For more details and a proof, see [48] or [70]. Theorem 4.10 If λ(G) ≥ −2, then G is an ordinary line graph, a floral line graph, or is represented by E8 .

Chapter 5

Planarity of Line Graphs

5.1 Introduction One of the most interesting questions that is asked about a graph is whether it can be drawn without any of its connections crossing. In more formal terms, this problem asks whether a graph is planar or not. Among mathematicians, it is well known that a primary criterion for planarity is whether or not a graph contains one of a pair of configurations, graphs that can be reduced to K5 , the complete graph with five vertices, or K3,3 , the complete bipartite graph with two sets of three vertices each. Our primary interest here lies of course in determining whether or not a given line graph is planar. As it happens, there are two elegant sets of criteria for graphs with this property, one involving forbidden subgraphs, the other involving degrees and connectivity properties of individual vertices. Following a brief section on the background of planarity, these criteria for planarity of line graphs are developed.

5.2 Graphs Having Planar Line Graphs We begin with some background terminology and results on graphs in the plane. Two operations related to planarity (and similar graph concepts) are ‘homeomorphic to’ and ‘contractible to’. For our purposes, it will be convenient to define them in terms of elementary operations. An elementary subdivision of a graph G is the replacement of an edge e = vw with a path vuw, where u is a new vertex, and a subdivision of G is the result of a finite number of elementary subdivisions. Further, two graphs G and H are homeomorphic if both are subdivisions of a common graph. An elementary contraction of G is the replacement of an edge e = vw and the vertices v and w with a new vertex u adjacent to each of the vertices that is a neighbor of v or w, and a contraction of G is the result of a finite number of elementary contractions. © Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_5

61

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5 Planarity of Line Graphs a a

g

e j

j

b

e

h g

i b

d

h

f i

f c

c

(a)

d (b)

Fig. 5.1 The Petersen graph as a subgraph of L(K5 )

A graph is planar if it can be drawn in the plane in such a way that none of its edges cross. Kuratowski’s classic theorem [129] on planar graphs states that there are basically just two barriers to a graph being planar, the complete graph K5 and the complete bipartite graph K3,3 , barriers being in terms of subdivisions. A follow-up result was observed by Wagner [171] a few years later with the same two graphs being barriers in terms of contractions. While other characterizations of planar graphs have been established, these are the ones that we will use here, and we formalize them in the following characterization theorem. Theorem 5.1 The following statements are equivalent for a graph G: (1) G is planar. (2) No subgraph of G is homeomorphic to K5 or K3,3 . (3) No subgraph of G is contractible to K5 or K3,3 . We begin our discussion of planarity and line graphs by showing that the line graphs of the two Kuratowski graphs are both nonplanar. One way to show that L(K5 ) is not planar is to show that it contains the Petersen graph. To see this, label the edges of K5 as in Fig. 5.1a. Then in its line graph, abcdea and fghijf are cycles. Furthermore, a is adjacent to j , b to h, c to f , d to i, and e to g. As shown in Fig. 5.1b, these vertices and edges form a copy of the Petersen graph in L(K5 ), and that graph is well known to be nonplanar. A similar argument can be used to show that L(K3,3 ) contains a subgraph homeomorphic to K3,3 . Alternatively, as shown in Chap. 1, L(K3,3 ) ∼ = C3 × C3 , which is known to be nonplanar (in fact, to have crossing number 3). Thus, the line graphs of both Kuratowski graphs are nonplanar. The following lemma provides some information on the line graph of a subdivision of a graph. Lemma 5.1 Let H be a subdivision of graph G. (a) The line graph L(H ) is contractible to L(G). (b) If L(H ) is planar, then so is L(G).

5.2 Graphs Having Planar Line Graphs

63

w

w g

e

G:

u

H: f

v

L(G):

v

e

L(H):

f

g

Fig. 5.2 An elementary subdivision of an edge

Proof Assume that H is a subdivision of graph G. By definition, H can be obtained from G by a sequence of elementary subdivisions, and so we may assume that H is the result of inserting one new vertex u into an edge e = vw of G. Now let f and g be the edges vu and uw in H . Clearly, the contraction of the edge fg in L(H ) is isomorphic to L(G), (see Fig. 5.2), which proves (a). Now suppose that (b) is false, and let G be a minimal counterexample. It follows that G is then either L(K5 ) or L(K3,3 ). Furthermore, by the result of Harary and Tutte, since by (a) L(H ) is contractible to L(G), L(H ) must be nonplanar, and so L(G) must in fact be planar.   The following theorem is a consequence of the lemma. Theorem 5.2 The line graph of a nonplanar graph is also nonplanar. This theorem was first proved by Sedláˇcek [158] using a somewhat different method of proof. He also proved the equivalence of the first two statements in the next theorem; the equivalence of the third is due to Greenwell and Hemminger [85]. Theorem 5.3 The following statements are equivalent for a graph G: (1) L(G) is planar. (2) G is planar, has maximum degree at most 4, and every vertex of degree 4 is a cut-vertex. (3) G does not contain a subgraph homeomorphic to K3,3 , K1,5 , K1,1,3 , or K1 ∗ P4 (see Fig. 5.3).

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K1,5 :

K3,3 :

K1,1,3 :

K1 ∗ P4 :

Fig. 5.3 Forbidden subgraphs for the planarity of line graphs a

K1,1,3 :

a

e

b

f

c

g d

f

b L(K1,1,3 ) :

e g

c

d

Fig. 5.4 The non-planarity of L(K1,1,3 )

Proof We show that (1) ⇒ (3), (3) ⇒ (2), and (2) ⇒ (1). (1) ⇒ (3): We first show that the line graph of each of the four given graphs is nonplanar. (a) L(K3,3 ): Since K3,3 is itself nonplanar, this follows from the lemma. (b) L(K1,5 ): This line graph is nonplanar since it is K5 itself. (c) L(K1,1,3 ): Consider the edge-labeling of K1,1,3 in Fig. 5.4. This figure also shows its line graph with its vertices labeled and three of its edges (df, dg, and fg) colored blue. It is easy to verify that the subgraph with only the black edges is homeomorphic to K5 . Hence the line graph of K1,1,3 is nonplanar. (d) L(K1 ∗ P4 ): Similarly, consider the edge-labeling of K1 ∗ P4 in Fig. 5.5 along with its line graph. Once again the black edges constitute a homeomorph of K5 , so L(K1 ∗ P4 ) is nonplanar.

5.2 Graphs Having Planar Line Graphs a

f

e

65

g

e f

b K1 ∗ P4 :

a

c

b

L(K1 ∗ P4 ) :

d

g c

d

Fig. 5.5 The non-planarity of L(K1 ∗ P4 )

x

u=y

z

x

y

z

u w

w

v

v

(a)

(b)

Fig. 5.6 Case 2 of (3) ⇒ (2)

(3) ⇒ (2): Let G be a graph that does not contain a subgraph homeomorphic to any of the four graphs in Fig. 5.3. Suppose that G is nonplanar. By hypothesis it does not have a subgraph homeomorphic to K3,3 , so it must have a subgraph homeomorphic to K5 . But then it would contain a homeomorph of K1,1,3 (and one of K1 ∗ P4 ), which is forbidden. Therefore, G must be planar. Also, G cannot have a vertex of degree 5 or more since it does not contain K1,5 . Therefore what remains is to show that every vertex of degree 4 in G is a cut-vertex. We consider three cases: Case 1 Some path in G − v contains all four neighbors of v. Then clearly there is a subgraph of G that is homeomorphic to K1 ∗ P4 , so this case cannot occur. Case 2 Some path P in G − v contains three neighbors of v but no path contains all four neighbors. Assume that P goes from x to y to z. Since G − v is connected, there is a path from the fourth neighbor w to P . Let Q be a shortest such path, and assume that it goes from z to u. It is easy to see that if u = y, then G has a subgraph homeomorphic to K1,1,3 (see Fig. 5.6a), while if u = y, there is a subgraph homeomorphic to K1 ∗ P4 (see Fig. 5.6b). Hence, this case is also impossible. Case 3 No path in G−v contains more than two of the four neighbors of v. Let P be a w-x path and let Q be a y-z path. If they have a common vertex, it must be internal

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5 Planarity of Line Graphs z

w

y

x

z

w y

x

v

v

(a)

(b)

Fig. 5.7 Case 3 of (3) ⇒ (2)

to both, so G will contain a homeomorph of K1,1,3 (see Fig. 5.7a). Otherwise there is a shortest path joining P and Q, and so G has a subgraph homeomorphic to K1 ∗ P4 (see Fig. 5.7b), also an impossibility. (2) ⇒ (1): We actually prove a stronger statement than this: If G is a planar graph with maximum degree 3 or 4, then L(G) has a plane embedding in which the three edges that arise from each vertex of degree 3 in G bound a face. (One example of this is given by the octahedron as the line graph of K4 .) This statement is clearly true for k = 0 by our earlier observation regarding planar graphs in which all degrees are 3 or less. Hence, we assume that k > 0 and that the statement holds for all graphs with fewer than k vertices of degree 4. Let G be a graph with k vertices of degree 4 that satisfies the given conditions, and let v be a vertex of degree 4. We consider two cases: Case 1 One of the edges at v is a cut-edge e. Let H be the subgraph of G induced by v and the set S of those vertices w for which e is on a v-w path, and let F = G − S. Then both H and F have fewer than k vertices of degree 4, and so have the desired property. Therefore, there is a plane embedding of L(F ) in which the vertices corresponding to the three edges at v other than e lie on a face T of the embedding, and we can assume that this is the unbounded face. Similarly, there must be a plane embedding of L(H ) with the vertex corresponding to e lying on the unbounded region. Thus e can be joined to the vertices of T so that the result is an embedding of L(G) with the desired properties. Case 2 None of the edges at v is a cut-edge. Since v is a cut-vertex, this means that two of the edges at v, say e and e , are in one block, and the other two, f and f , are in another. Similar to what was done in Case 1, let H be the subgraph induced by the vertex v and the set S of those vertices w for which e or e is on a v-w path, and let F = G − S. Then by the induction hypothesis, both L(F ) and L(H ) have plane embeddings with the desired properties. These can be arranged so that the edges ee and ff are both on the unbounded face, and hence e can be joined to f and e

joined to f in such a way that the result is again an embedding of L(G) with the desired properties.

5.3 Regular Planar Line Graphs

67

By the Principle of Mathematical Induction, the desired result follows, and this completes the proof of the theorem.  

5.3 Regular Planar Line Graphs In this section we look at regular planar line graphs from two perspectives, first from that of their root graphs, then from the graphs themselves. This material is based on the work of Deogun [68]. We again restrict our attention to connected graphs. A moment’s reflection reveals that if a connected graph G has a regular line graph and is not itself regular, then there must be two positive integers s and t such that every edge has one vertex of degree s and the other of degree t. Such a graph is called (s, t)-biregular (or just biregular); and, as can be easily seen, every biregular graph is necessarily bipartite. Thus, if G has an r-regular line graph, G is either q-regular with r = 2q − 2, or (s, t)-biregular with r = s + t − 2. Because every planar graph has a vertex of degree less than 6, it must be that r ≤ 5; in addition we assume r ≥ 2 since the only connected graph with a 0-regular line graph is K2 and the only one with a 1-regular line graph is K1,2 . These restrictions mean that we need only consider those graphs G that are q-regular for q = 2, 3, or 4, or those that are stars, or those that are (s, t)-biregular with 2 ≤ s < t ≤ 4. These observations will be useful in proving the characterization theorem to follow, but first we restrict what needs to be considered even further through this lemma. Lemma 5.2 If a graph is either (2, 4)- or (3, 4)-biregular, then its line graph is nonplanar. Proof Let G be either (2,4)- or (3,4)-biregular, and let B be an end-block of G (or all of G if it is nonseparable). Then B must have a cycle of length at least 4, and hence at least two vertices of degree 4. Since B has at most one cut-vertex of G, it follows that G has a vertex of degree 4 that is not a cut-vertex. From Theorem 5.3 we see that L(G) is nonplanar.   The following theorem of Deogun [68] tells us which graphs have regular planar line graphs. Theorem 5.4 Let G be a connected graph. (a) L(G) is a planar 2-regular graph if and only if G is a cycle or is K1,3 . (b) L(G) is a planar 3-regular graph if and only if G is a planar (2, 3)-biregular graph or is K1,4 . (c) L(G) is a planar 4-regular graph if and only if G is a planar 3-regular graph. (d) For r ≥ 5, there are no graphs with a planar r-regular line graph. Proof (a) This is obvious. (b) Let G be a graph with a planar 3-regular line graph. Then by Theorem 5.2, G must be planar, and by our earlier observations, it must be biregular. It is

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easy to see that it then must be either (1, 4)- or (2, 3)-biregular, so the necessity of the statement follows. The sufficiency follows from Sedláˇcek’s criterion in Theorem 5.3. (c) Assume that G has a planar 4-regular line graph. Then, much as in (b), it follows that G must be planar and either 3-regular or (1, 4)- or (2, 3)-biregular. By the lemma, the last of these is impossible, and so the necessity follows. The sufficiency is proved in the same way as that in (b). (d) As noted earlier, there are no r-regular planar graphs for r = 6, so we assume that G is a graph with a planar 5-regular line graph. Since G cannot have any vertices of degree greater than 4, the only possibility is for it to be (3,4)biregular. However, the lemma precludes this, so there can be no such graph.   Having answered the question of which graphs have regular planar line graphs, we turn to the question of which graphs are regular planar line graphs. Because of the non-existence of such graphs that are r-regular for r ≥ 5 and the simplicity of the graphs with 0-, 1-, and 2-regularity, we are left with only the cases r = 3 and r = 4 to consider. It turns out that for 3-regular planar line graphs there is a family of forbidden induced subgraphs with just three graphs from the set of nine for line graphs in general (see Fig. 3.2) Theorem 5.5 A 3-regular planar graph is a line graph if and only if it does not contain any of the three graphs L1 , L4 , and L8 (shown in Fig. 5.8) as an induced subgraph. Proof Let G be a 3-regular planar graph. By the line graph characterization theorem (Theorem 3.1), if G has any of the three graphs in Fig. 5.8 as an induced subgraph, it cannot be a line graph, and this proves the necessity. For the converse, assume that G is nonplanar and that it does not contain L1 , L4 , or L8 as an induced subgraph. Since five of the other six of the nine forbidden subgraphs for a line graph all have vertices of degree greater than 3, it follows that G must contain the only remaining forbidden subgraph, namely L2 in our list, K4 with one edge subdivided (see Fig. 5.9). Let v be the vertex of degree 2 in L2 and let w be the third neighbor of v in G. The other two neighbors of v already have degree 3 in L2 so neither of them can be adjacent to w in G. Since they are not adjacent

L1 :

L4 :

Fig. 5.8 Forbidden subgraphs for planar 3-regular line graphs

L8 :

5.3 Regular Planar Line Graphs

69

L2 :

Fig. 5.9 The forbidden graph L2

L1 :

L4 :

L8 :

: Fig. 5.10 Graphs showing the necessity of L1 , L4 , and L8 for 3-regular planar line graphs

to one another, it follows that G contains K1,3 , that is, L1 , as an induced subgraph. This contradiction completes the proof.   The set of the three graphs in the theorem is minimal in being obstacles to line graphs of planar 3-regular graphs, as is shown by the three graphs in Fig. 5.10. Graph L 1 , the graph of a cube, is a 3-regular planar graph that contains only L1 from the family of three line-forbidden graphs. Similarly L 4 contains only L4 , and L 8 only L8 . The 4-regular case is considerably more complicated, and so we lead up to the result with some lemmas. Each of the lemmas concerns planar claw-free (that is, K1,3 is not induced) 4-regular graphs. If such a graph G is not a line graph, then, by the odd-triangle condition of van Rooij and Wilf, it must contain two odd triangles, which we assume have vertices u, v, w and v, w, x (see Fig. 5.11a). We also assume that y is a vertex that makes triangle uvw odd and that z makes vwx odd (note that y and z could be the same vertex). Now since G is 4-regular, v must

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Fig. 5.11 Labeling a van Rooij-Wilf subgraph

u

u

v

w

x (a) Fig. 5.12 Arguing the impossibility of L3

v

w

x

t (b)

u

v

w

t

x

have a fourth neighbor t, and since G is claw-free, t must be adjacent to at least one of u and x. Hence, we assume without loss of generality that t ∼ x, and so adopt this convention in all the lemmas of this labeling (see Fig. 5.11b). One further observation that we use several times is that 4-regular graphs, being Eulerian, cannot have any cut-edges. Lemma 5.3 Let G be a planar claw-free 4-regular graph. Then G does not contain the line-forbidden graph L3 as an induced subgraph, and if it contains L2 , then it contains L4 . Proof We begin with L3 , which is K5 with one edge removed, and there is thus only one way for it to be embedded in the plane (see Fig. 5.12). With this labeling, we note that if it is an induced subgraph of G, then the fourth edge at u must be a cut-edge since the other three neighbors of u already have degree 4. Hence, L3 cannot be an induced subgraph of G. Now assume that L2 is an induced subgraph of G. Since L2 is homeomorphic to K4 , we may assume that is embedded and labeled as in Fig. 5.13a. From our convention that the fourth neighbor t of v is adjacent to x, it follows that t cannot be adjacent to both u and w since that would imply that L3 is an induced subgraph, which is impossible. Suppose that t is adjacent to neither u nor w. Then the fourth neighbor s of w must be a new vertex. It must also be adjacent to u in order to avoid there being a claw (note the vertices that already have degree 4). Furthermore, the absence of claws means that both s and y must be adjacent to y, so there must be a subgraph like that in Fig. 5.13b. However, arguing as in Case 1, we see that there must be a cut-edge, and that is impossible. Hence, t must be adjacent to w and not to u, which means that L5 is an induced subgraph.  

5.3 Regular Planar Line Graphs

71

u

u

v

y

w

x

t

v

w

s

y

x

t

(a)

(b)

Fig. 5.13 Arguing the impossibility of L2 y

u

v

w

y

u

v

w z

x (a)

z

x (b)

Fig. 5.14 L4 forcing L5 and L5 forcing L4

Lemma 5.4 Let G be a planar claw-free 4-regular graph that does not have L8 as induced subgraph. Then G contains L4 as an induced subgraph if and only if it contains L5 as an induced subgraph. Proof First assume that G contains L4 as an induced subgraph, labeled as in Fig. 5.14a. As before, we take t to be adjacent to v and x. If t is not adjacent to w, then the fourth neighbor s of w must be also be adjacent to u. But as this yields L8 as an induced subgraph, it follows that t must be adjacent to w. It cannot be adjacent to u since that implies that L3 is an induced subgraph. Hence, L5 must be induced. Now assume that G contains L5 , labeled as in Fig. 5.14b. Then the fourth neighbor s of w cannot be adjacent to any of the other vertices, so it completes L4 as an induced subgraph.   Lemma 5.5 Let G be a planar claw-free 4-regular graph that does not have L4 as induced subgraph. Then G contains L7 as an induced subgraph if and only if it contains L8 as an induced subgraph. Proof Assume that G contains L7 as an induced subgraph, labeled as in Fig. 5.15a. Since it is homeomorphic to K4 , we may assume this embedding. Hence, if t (adjacent to v and x as before) is adjacent to w, it must lie inside the face bounded by v, w, and x. But then the fourth edge at t would be a cut-edge. Therefore t is

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5 Planarity of Line Graphs u

u

y

v

w

y

v

z

x

w

t

z

x (b)

(a) Fig. 5.15 L7 forcing L8

u

u

z

v

w

z

v

w x

x y

y

r (b)

(a) s u

z

v

w x

y

r (c)

Fig. 5.16 L8 forcing L7

not adjacent to w. But then the fourth neighbor s of w must be a different vertex that must also be adjacent to u. But then the subgraph s, t, u, v, w, x is L8 (see Fig. 5.15b). Now assume that G contains L8 , labeled as in Fig. 5.16a. Let r be the fourth neighbor of x. Then, for there not to be a claw at x, r must be adjacent to y. Now if r is adjacent to u, it must also be adjacent to z. But then (see Fig. 5.16b) the fourth edge at z must be a cut-edge, so r cannot be adjacent to u. Now the fourth neighbor s of u must also be adjacent to z, as in Fig. 5.16c. But then, depending on whether or not r and s are adjacent, the subgraph r, s, u, v, w, x  is isomorphic to either L7 or L4 . Since L4 is prohibited by hypothesis, G must contain L7 .  

5.3 Regular Planar Line Graphs

73

We are now in a position to give the minimal forbidden sets of graphs for a 4regular planar graph to be a line graph. Interestingly, there are four such sets of three graphs from the set of nine graphs forbidden as induced subgraphs for any line graphs. Theorem 5.6 Let G be a planar 4-regular graph. Then G is a line graph if and only if it does not contain as an induced subgraph L1 , one of L4 and L5 , or one of L7 and L8 . Proof Clearly if G contains any of the graphs in any of the five graphs L1 , L4 , L5 , L7 , or L8 as an induced subgraph, it is not a line graph. For the converse, we assume that G is not a line graph, and suppose that it is clawfree. Then it must contain one of the other eight forbidden subgraphs in the family L as an induced subgraph. This cannot be either L6 or L9 since they both contain vertices of degree 5, and by Lemma 5.3, it cannot be L3 . Also, by that lemma, if G contains L2 , then it contains L4 as an induced subgraph. Consequently, G must contain one of L4 , L5 , L7 , or L8 . However, by Lemmas 5.4 and 5.5, if it contains one of the first pair of these, then it contains the other, and likewise for the second pair. Therefore, if G is claw-free and doesn’t contain one of the graphs from each of these pairs, it must be a line graph, which proves the theorem.   The three graphs, H1 , H2 , H3 , in Fig. 5.17 show that each of the sets of forbidden graphs in this theorem are minimal. First, H1 is a 4-regular planar graph that contains only the claw L1 ∼ = K1,3 from the set L of forbidden graphs. Next, H2

H1 :

H2 :

H3 :

J:

Fig. 5.17 Graphs showing the necessity of L1 , L4 , and L7 for 4-regular planar line graphs

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is a 4-regular planar graph that contains only H7 and H8 . Finally, H3 , obtained by appending a copy of J at each of the starred vertices of degree 2, contains only H4 and H5 . Note that the graph H2 is the square of an 8-cycle. In fact, the square of any longer even cycle would also suffice, and somewhat surprisingly, only such graphs. The following corollary puts these facts (without proof) in other terms. Corollary 5.1 Let G be a planar 4-regular claw-free graph that is not the square of an even cycle of length 8 or more. Then G is a line graph if and only if it does not contain L4 or L5 as an induced subgraph.

5.4 Outerplanar Graphs In this brief section, we look at an interesting subfamily of planar graphs. A graph G is called outerplanar if it can be embedded in the plane in such a way that all of its vertices lie on the unbounded face (the embedding itself called outerplane). For these graphs, one analogue to Kuratowksi’s theorem (due to Chartrand and Harary [55]) is that a graph is outerplanar if and only if it does not contain a subgraph homeomorphic from K4 or K2,3 (see Fig. 5.18). In this characterization, the word “from” is important since the outerplanar graph K1,1,2 (K4 with an edge deleted) is itself homeomorphic to K2,3 . We prefer a variation on this characterization. A theta graph consists of a pair of vertices of degree 3 joined by three internally disjoint paths, each of length at least 2 (see Fig. 5.19 for an example). Since theta graphs are those graphs that are homeomorphic from K2,3 and every graph that is homeomorphic to K4 other than K4 itself contains a theta graph, we have the following result: A graph is outerplanar if and only if it does not contain K4 or a theta graph as a subgraph. The family of 2-connected graphs is particularly interesting in this context; for, it is a fact that every nonseparable graph of order at least 4 is either a cycle or contains a homeomorph of K1,1,2 . A related family of Fig. 5.18 Forbidden graphs for outerplanarity

Fig. 5.19 A theta graph

5.4 Outerplanar Graphs

75

graphs consists of those with no nonseparable graphs of the latter type. Formally, a cactus is a connected graph in which every block is either a single edge or a cycle. These will be of interest in what follows. Our first theorem says which outerplanar graphs are line graphs. Theorem 5.7 An outerplanar graph is a line graph if and only if it does not contain as an induced subgraph L1 , L4 , or L8 from the set of forbidden graphs for line graphs. Proof This result follows from the fact that none of the other six forbidden graphs is outerplanar.   We now turn to the more interesting question of which graphs have outerplanar line graphs. Theorem 5.8 The following statements are equivalent for a connected graph G: (1) (2) (3) (4)

L(G) is outerplanar. G does not contain K1,4 nor a subgraph homeomorphic to K1,1,2 . G has maximum degree at most 3, and every vertex of degree 3 is a cut-vertex. G is a cactus with maximum degree Δ(G) ≤ 3.

Proof We show that (1) ⇒ (2), (2) ⇒ (3), (3) ⇒ (4), and (4) ⇒ (1). (1) ⇒ (2): It is straightforward to verify that if G contains K1,4 , then L(G) contains K4 . Further, if G contains a homeomorph of K1,1,2, then L(G) has a subgraph homeomorphic either to the wheel K1 ∗C4 or the triangular prism K2 ×K3 . Since each of these three graphs is non-outerplanar and one of them must be in G, the implication follows. (2) ⇒ (3): Assume (2) holds. Then G cannot have a vertex of degree 4 or greater. Suppose that it has a vertex v of degree 3 that is not a cut-vertex. Then there must be a nonseparable subgraph of order at least 4 that is not a cycle. Hence, it must contain a graph homeomorphic to K1,1,2 , which contradicts (2). (3) ⇒ (4): Assume (3) holds. Then no block of G on its own can have a vertex of degree 3 so each block must be either a single edge or a cycle. Hence (4) holds. (4) ⇒ (1): This is clearly true if G is a cycle or a tree of maximum degree 3 or less, so we consider cacti with maximum degree 3. Note that then at least one edge at each vertex of degree 3 is a cut-edge. We prove this implication by proving a stronger result by induction on the number m of edges in such graphs. To this end, we define a special edge of the line graph of a cactus of maximum degree 3 to be an edge ef with e and f being adjacent edges of a cycle in G with their common vertex of degree 3 (see Fig. 5.20). We now prove the following: Claim If G is a cactus with m edges and maximum degree 3, then L(G) has an outerplane embedding in which all edges except the special edges are on the unbounded face. Clearly, this is true when m = 1. Let k ≥ 1, assume that the statement is true when m = k, and let G be a cactus with k + 1 edges and maximum degree 3. If G has an end-vertex v, then by the induction hypothesis L(G − v) has an outerplane

76

5 Planarity of Line Graphs e e L(G):

G: f

f Fig. 5.20 A special edge

embedding in which all edges except the special edges are on the unbounded face. If e is the edge at v in G, then e can be added as a new vertex to L(G − v) in a way that preserves the desired property. Now assume that G has no end-vertices, in which case G has more than one block. Consider an end block B. All of its vertices but one have degree 2, so it has an edge e having both vertices of degree 2. Again by the induction hypothesis, L(G − e) has an outerplane embedding in which all edges except the special edges are on the unbounded face, and so e can be added as a new vertex to L(G − v) in a way that preserves the desired property. Thus, in either case, L(G) has an outerplane embedding with the desired property, and so, by induction, the implication is proved, and this completes the proof of the theorem.  

5.5 Crossing Numbers The crossing number of a graph G is the minimum number of crossings in any drawing of G in the plane. In an analysis of nonplanar line graphs, it is natural to begin with those line graphs having crossing number 1. We find it convenient to follow history in this topic, and consider separately planar and nonplanar graphs because of the nature of the results and their proofs. The planar case was established by Kulli et al. [128] (although the hypothesis of their theorem omitted stating the hypothesis of planarity explicitly). Their theorem specifies two families of graphs, both analogous to Sedláˇcek’s characterization of planar line graphs (Theorem 5.3), that is, that the line graph of a planar graph G is planar if and only if its maximum degree is 4 and every vertex of degree 4 is a cut vertex. Theorem 5.9 The line graph of a planar graph has crossing number 1 if and only if either (a) the maximum degree Δ(G) = 4 and G has exactly one vertex of degree 4 that is not a cut-vertex, or (b) the maximum degree Δ(G) = 5, every vertex of degree 4 is a cut-vertex, there is exactly one vertex of degree 5, and at most three of its edges are in the same block.

5.5 Crossing Numbers

77

Fig. 5.21 Only one crossing when Δ(G) = 4

a

c

a

c e

b

d

b

d

(a)

(b)

a

c

a

d

b

c

e b (c)

d (d)

Proof First assume that G is a planar graph that satisfies (a) or (b). Then by the planar line graph characterization theorem, L(G) has at least one crossing. We show that there is only one in the two cases separately. Case 1 Δ(G) = 4. Within G, let v be the vertex of degree 4 that is not a cut-vertex and that its edges a, b, c, and d are as indicated in Fig. 5.21a. Form a new graph G by splitting v into two vertices joined by a new edge e as in (b) in the figure. By Theorem 5.3, the line graph of G is planar, and hence a plane embedding must contain the configuration shown in (c). If this portion of the plane embedding of L(G ) is replaced by (d), the result is a plane embedding of L(G) with just one crossing. Case 2 Δ(G) = 5. We now assume that in G, the vertex v has degree 5 and that the edges at v can be split into one set of three and one set of two so that no edge in one set is in the same block as an edge in the other set. Proceeding as in Case 1, we split the vertex v into two vertices, each incident with the edges in one of the sets as indicated in Fig. 5.22a,b, forming the new graph G . Again, by Theorem 5.3, the line graph L(G ) is planar, and a plane embedding will contain configuration (c) in the figure. Replacing that portion by (d) results in a plane drawing of L(G) with a single crossing. Consequently the crossing number of G is 1. For the converse, assume that G is a planar graph such that L(G) has crossing number 1. Clearly, G cannot have a vertex of degree 6 or more, and must have at least one vertex of degree greater than 3. We consider the two cases of the maximum degree being 4 and 5 separately. Case 1 Δ(G) = 4. By the planar line graph characterization theorem, G has at least one vertex of degree 4 that is not a cut-vertex. Suppose that it has two, v and w. Consider a drawing of L(G) in the plane with just one crossing. It follows from

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5 Planarity of Line Graphs

b

c

c

d

d f

b a

e

e

a

(a)

(b)

c

d

c d b

f

b

e a

a

e

(c)

(d)

Fig. 5.22 Only one crossing when Δ(G) = 5 e

e

f (a)

f (b)

Fig. 5.23 Two crossings when Δ(G) = 4 for contradiction

the fact that the line graph of any nonseparable graph with a vertex of degree 4 is nonplanar that v and w must line in the same block of G. There are just two possible types of proper drawings of K4 in the plane, either with no edges crossing, which we call Type 0 here, or with one pair of edges crossing, which we call Type 1. Let A be the drawing in L(G) of the edges at v and B the drawing of those at w. Clearly not both A and B can be of Type 1, so we first suppose that both are of Type 0 with e and f vertices as in Fig. 5.23a. Since these are in the same 2-connected subgraph of L(G), they must lie on a common cycle C. However, a cycle in G that contains the edge e can have only one other edge at v and so there must be a path from e in L(G) that crosses one of the edges of the triangle around e. Since the same argument must apply to f , the drawing must have at least two crossings, which contradicts our supposition. The other possibility is that L(G) contains a drawing of each type, say A is of Type 0 and B is of Type 1, as in Fig. 5.23b. But then as before there must be a path from e in L(G) that crosses one of the edges of the triangle around e, and hence the drawing must have more than one crossing.

5.5 Crossing Numbers Fig. 5.24 Two crossings when Δ(G) = 5 for contradiction

79

b

c e

a

d

Case 2 Δ(G) = 5. We let v be a vertex of degree 5. Again consider a drawing of L(G) in the plane with just one crossing. If four edges at v, say a, b, c, and d lie in the same block, then, as we have seen, L(G) must have a configuration like that in Fig. 5.24. Now there must be a cycle containing a and c but not b or d, and so L(G) must have a second crossing, which contradicts our assumption. Therefore at most three edges at v lie in the same block. All that remains is to show that all vertices (if any) of degree 4 in G are cut vertices. Suppose that this is not the case for some vertex w. Given an optimal drawing of L(G) in the plane, the five edges at v generate a crossing that uses four of the edges, and so at least one of the four is not in the block containing w. Therefore, its deletion from G results in the elimination of one crossing in the drawing. However, by Theorem 5.3, the line graph of the resulting graph is still nonplanar, so the crossing number of L(G) is again at least 2. This completes the proof.   As noted earlier, Greenwell and Hemminger [85] found the four graphs that prevent a graph from having a planar line graph (see Theorem 5.3). Akka and Panshetty [4] proved a similar result, characterizing those planar graphs whose line graph has crossing number 0 or 1. Their set consists of 26 graphs. Returning to Theorem 5.2, we note that another way of stating it is this: If G is a graph with crossing number 1, then its line graph L(G) has crossing number greater than or equal to 1. This raises the natural question of what happens if 1 is replaced by 2. It comes as something of a surprise that the corresponding statement is false. This was shown by Jendro´l and Klešˇc [112] with K3,3 , one of the two basic nonplanar graphs, as an example. In Chap. 1, while considering the line graphs of the complete bipartite graphs, we observed that the line graph of K3,3 is the product of two 3-cycles, C3 × C3 , shown in Fig. 1.4 with three crossings. As noted earlier, it was verified by Harary et al. [96] that the crossing number of this graph is indeed 3. Since it is nonplanar, any graph homeomorphic to K3,3 is also nonplanar, and one might expect that such a graph would also have crossing number 3. However, Jendro´l and Klešˇc showed that this is not the case. In fact, the line graph of the graph obtained from K3,3 by subdividing a pair of independent edges with one new vertex each actually has crossing number 1. We analyze this example in some detail because it contains the essence of a related theorem on nonplanar graphs. Let G be the 8-vertex graph homeomorphic to K3,3 with its edges labeled as shown in Fig. 5.25. This of course means that L(G) has seven vertices of degree 4

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G:

L(G):

Fig. 5.25 Reduction of the crossing number of a line graph by subdivision

and four of degree 3. It can be seen by checking the labels that the drawing of L(G) in the figure has crossing number 1. It is worth noting that each of the crossing edges has a vertex of degree 2, a feature that is significant. In fact, K3,3 provides an excellent illustration of the crossing numbers of its homeomorphs (as is established in [112]). As already noted, the crossing number of K3,3 itself is 3, and it follows from our example that any homeomorph in which there are two independent edges that have been subdivided has crossing number 1. An interesting fact is that all other homeomorphs have crossing number 2. It is convenient to have a name for the type of crossing in the drawing of the line graph in Fig. 5.25. We define a crossing of a pair of edges in a drawing of a graph in the plane to be a fundamental crossing if both edges have a vertex of degree 2. Thus, since the line graph of a nonplanar graph is nonplanar, if a graph G is nonplanar and its line graph has a drawing with one fundamental crossing and no other crossings, then the crossing number of L(G) is 1. It turns out that if both a graph and its line graph have crossing number 1, then the crossing in any drawing that realizes this must be fundamental. There is another stipulation that the graph G must satisfy for the conclusion to hold, and it is part of the planar line graph theorem, namely, that G have maximum degree at most 4 and that all vertices of degree 4 be cut-vertices. We state this as a theorem. Theorem 5.10 If G is a nonplanar graph whose line graph has crossing number 1, then the maximum degree of G is at most 4 and every vertex of degree 4 is a cut-vertex. Therefore we build this into the hypothesis of the characterization theorem due to Jendro´l and Klešˇc [112]. The proof is complicated and so it is not included here. Note that the conclusion requires that the graph G have at least two vertices of degree 2. Theorem 5.11 Let G be a nonplanar graph with maximum degree at most 4 and every vertex of degree 4 a cut-vertex. Then L(G) has crossing number 1 if and only if every optimal drawing of L(G) has a fundamental crossing.

5.6 Planarity of Iterated Line Graphs

81

A most remarkable generalization of this theorem (which we state here without proof) is due to Huang et al. [109]; it is an extension of this result to all positive integers. Theorem 5.12 Let G be a nonplanar graph with crossing number k, maximum degree at most 4, and with every vertex of degree 4 a cut-vertex. Then L(G) has crossing number k if and only if every crossing in an optimal drawing of L(G) is fundamental.

5.6 Planarity of Iterated Line Graphs In Sect. 1.5, the iterated line graphs Lk (G) of a graph G were introduced. The first to ask about their planarity appears to have been Oystein Ore in his book [145] on the four-color problem. In answering his questions, Kulli and Sampathkumar [127] gave descriptive solutions and Kulli and Akka [125] provided excluded-subgraph criteria. We combine (and slightly strengthen) their results in the following theorem for the case k = 2. As elsewhere, we consider only connected graphs here—the general case then follows easily. We also note that since the line graph of a nonplanar graph is nonplanar, so is its second-order line graph, so there is no real restriction in stating the theorem only for planar graphs. Recall that the line-degree of an edge in a graph G is the sum of the degrees of its vertices less 2. For our theorem, we let G1 , G2 , and G3 be the three graphs in Fig. 5.26. We also define, for i = 3, 4, 5, . . ., the graph Hi to be the cycle Ci augmented by two end vertices added at one vertex of the cycle, and similarly let Ji be the cycle Ci augmented by one end vertex added at each of two adjacent vertices of the cycle. These are shown in Fig. 5.27. Theorem 5.13 The following statements are equivalent for a planar connected graph G of order at least 3: (1) The second-order line graph L2 (G) is planar. (2) In G, the degree of every vertex is at most 4, the line-degree of every edge is at most 4, and every edge of line-degree 4 is a cut-edge. (3) G does not contain a subgraph homeomorphic to G1 , G2 , G3 , any Hi (i ≥ 3), or any Ji (i ≥ 3) as a subgraph (see Figs. 5.26 and 5.27).

G1 :

G2 :

G3 :

Fig. 5.26 Three graphs that cannot appear in a planar second-order line graph

82 Fig. 5.27 Two families of graphs that cannot appear in a planar second-order line graph

5 Planarity of Line Graphs

Hi :

Ji : Ci

Ci

Proof We show that (1) ⇒ (3), (3) ⇒ (2), and (2) ⇒ (1). (1) ⇒ (3): Assume that L2 (G) is planar. Then G cannot contain G1 because if it did, L(G) would contain K5 and L2 (G) would be non-planar. Similarly, if G contained G2 , then L(G) would have a vertex of degree 5 and its line graph would be non-planar, and also if G contained G3 , then L(G) would either have a vertex of degree 4 that was not a cut-vertex or a vertex of degree 5, and so L2 (G) would again be non-planar. This last argument also applies to every Hi and every Ki , with the vertex in question corresponding to any of the edges with line-degree 4 in the graph. (3) ⇒ (2): Assume (3). If some vertex of G has degree 5 or more, then G contains G1 , a contradiction. Hence the maximum degree of G is at most 4. Suppose that some edge e = vw has line-degree 5 (or more). The only way to avoid a vertex of degree 5 in G is for either v or w to have degree 3 and the other vertex degree 2 (or 3). But then G must contain either G2 or, if v and w have a common neighbor, H3 . Now suppose that some edge vw has line-degree 4 but is not a cut-edge of G. This implies that either (i) one of its vertices has degree 2 and the other 4 or (ii) both of its endpoints have degree 3, as in Fig. 5.27. Consider first the labeling in (i). Then in G − e there must be a path from v to w, which implies that G contains some Hi . Now consider the labeling in (ii). If both neighbors of v are also neighbors of w, then G3 results, so we assume that this is not the case. However, then a shortest v − w path in G − e forms a cycle when e is included. Together with the other edges at v and w, this gives some Ji as a subgraph. Therefore, G satisfies (2). (2) ⇒ (1): Now assume that G satisfies (2), and let H = L(G). We first show that H is planar. By hypothesis, G is planar and Δ(G) ≤ 4. Suppose G has a vertex v of degree 4 that is not a cut-vertex, and let w be one of its neighbors. Then w cannot be an end-vertex of G since then v would be a cut-vertex. Thus w has degree (at least) 2, so the edge vw has line-degree 4, and hence is a cut-edge. But this means that v is a cut-vertex. Therefore by Theorem 5.3, H is planar. Furthermore, since Δ(H ) = Δ (G), Δ(H ) ≤ 4. In addition, since in a line graph a vertex is a cut-vertex if it corresponds to an edge that is a cut-edge but not an end-edge in the original graph, it follows that a vertex of degree 4 in H is a cut-vertex. Therefore, by Theorem 5.3 again, L(H ) is planar. This completes the proof.   We now turn to the third-order line graph, for which we can say precisely which graphs possess planarity, a result due in essence to Ghebleh and Khatirinejad [81]. As we shall see, these are the graphs Ai and Bi defined as follows: For i ≥ 1, form

5.6 Planarity of Iterated Line Graphs

83

Ai : i

1 vertices

Bi : i

1 vertices

Fig. 5.28 Graphs for which L3 (G) is planar

F1 :

F3 :

F2 :

F4 :

F5 :

Fig. 5.29 Forbidden graphs for third-order line graph planarity

Ai by adding two new vertices of degree 1 adjacent to one end of the path Pi+2 of length i + 1, and form Bi by adding two such vertices at each end of Pi+2 . The graphs A3 and B3 are shown in Fig. 5.28. As before, in stating our theorem, we consider only planar connected graphs, and furthermore, because the line graphs of paths and cycles are so simple, we assume that the maximum vertex degree is at least 3. Theorem 5.14 The following statements are equivalent for a planar connected graph G with maximum degree at least 3: (1) The third-order line graph L3 (G) is planar. (2) The maximum degree of a vertex in G is 3, and if v is a vertex of degree 3, then the sum of the degrees of its neighbors is at most 4. (3) None of the five graphs in Fig. 5.29 is a subgraph of G. (4) G is either K1,3 or Ai or Bi for some i ≥ 1 (see Fig. 5.28).

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5 Planarity of Line Graphs

Proof We prove the theorem through this sequence of implications: (1) ⇒ (3), (3) ⇒ (2), (2) ⇒ (4), and (4) ⇒ (1). (1) ⇒ (3): It is straightforward to see that the line graph L(Fi ) of each graph Fi in Fig. 5.29 contains one of the graphs in Fig. 5.26 or Fig. 5.27, all of which are, by Theorem 5.13, forbidden subgraphs for a graph to have a planar second-order line graph. Hence, if G contains Fi , then its third-order line graph L3 (G) cannot be planar. (3) ⇒ (2): Assume that G does not contain any of the graphs in Fi in Fig. 5.29 as a subgraph. Since it does not contain F1 , the maximum degree of a vertex is 3. Let v be a vertex of degree 3, and suppose that the sum of the degrees of its neighbors is 5 or more. If two of these vertices have degree 1, then the third has degree at least 3, and then F2 is a subgraph. Hence, at least two of the neighbors of v have degree greater than 1. If these vertices are adjacent, then G contains F5 ; if they have a common neighbor in addition to v, F4 results; and if neither of these possibilities occurs, F3 is a subgraph. Hence the sum of the degrees of the neighbors of v is either 3 or 4. (2) ⇒ (4): Assume that (2) holds, and let v be a vertex of degree 3. If the sum of the degrees of its neighbors is 3, then the graph is simply K1,3 , so we assume that the sum is 4. If v is the only vertex of degree 3, then it is easy to see that G must equal Ai for some i ≥ 1. If there is another vertex w of degree 3, then v and w must be joined by a path, and from this it follows that G must be Bi for some i ≥ 1. Obviously, there cannot be more than two vertices of degree 3 in a graph satisfying (2). Hence, G is either some Ai or some Bi . (4) ⇒ (1): The line graphs of Ai and Bi are shown in Fig. 5.28. Clearly, these satisfy condition (2) of Theorem 5.13, so their second-order line graphs are planar, which establishes this implication.   Moving on, we note that by Theorem 5.14 the third-order line graphs of the graphs in Fig. 5.28 are not themselves planar. From this observation, we deduce the following corollary. Corollary 5.2 If G is a connected graph with planar fourth-order (or higher) line graph, then it is either a path, a cycle, or K1,3 . A slightly different perspective on some of the results in this section was taken by Ghebleh and Khatirinejad [81]. We define the line graph planarity index ξ(G) of a graph G to be the smallest value of k for which Lk (G) is nonplanar if there is such a value (we follow customary practice that L0 (G) = G). One can easily see that this parameter is defined for all connected graphs except paths, cycles, and K1,3 . (For the path Pn , the kth-order line graph does not exist for k ≥ n, but of course is planar when it does exist, while for cycles and K1,3 , it is always a cycle when k ≥ 1.) The following theorem follows from the preceding three theorems. Theorem 5.15 Let G be a connected graph other than a path, a cycle, or K1,3 . (a) ξ(G) = 0 if and only if G is nonplanar. (b) ξ(G) = 1 if and only if G is planar and Δ(G) ≥ 5 or G has a vertex of degree 4 that is not a cut-vertex.

5.6 Planarity of Iterated Line Graphs

K1,4 :

85

H:

(a)

(b)

Ck+ :

Ck

(c)

Fig. 5.30 Forbidden graphs for a second-order line graph to be outerplanar

(c) ξ(G) = 2 if and only if G is planar, the maximum degree Δ(G) ≤ 5, every vertex of degree 4 is a cut-vertex, and the maximum line-degree Δ (G) ≥ 5 or some edge of line-degree 4 is not a cut-edge. (d) ξ(G) = 3 if and only if G is planar, Δ(G) ≤ 4, Δ (G) ≤ 4, every edge of line degree 4 is a cut-edge, and G is not one of the graphs Ai or Bi in Fig. 5.29 for any i ≥ 1. (e) ξ(G) = 4 if and only if G is one of the graphs Ai or Bi in Fig. 5.29. (f) There are no graphs with ξ(G) ≥ 5. We conclude this chapter by considering those iterated line graphs that are outerplanar. The following theorem is due (in a slightly different form) to Kulli and Akka [126]. For this result, we consider the graphs in Fig. 5.30, K1,4 , the tree + K1,3 obtained from K1,3 by subdividing one edge, and a graph Ck+ , the cycle Ck (k ≥ 3) with one end edge attached. Theorem 5.16 The following statements are equivalent for a connected graph G that is neither a path nor a cycle: (1) L2 (G) is outerplanar. (2) G has both maximum vertex degree and maximum edge degree at most 3. + (3) G does not contain K1,4 , K1,3 , or Ck+ (the graphs in Fig. 5.30). Proof We show that (1) ⇒ (3), (3) ⇒ (2), and (2) ⇒ (1). (1) ⇒ (3): If G contains K1,4 or Ck+ for some k ≥ 3, then L(G) is not a cactus, while if G contains H , then L(G) has a vertex of degree at least 4. In any case, by Theorem 5.8, the line graph of L(G) is not outerplanar, which establishes the implication.

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5 Planarity of Line Graphs

(3) ⇒ (2): Assume (3). Since G does not contain K1,4 , Δ(G) = 3 (since by hypothesis G is connected and neither a path nor a cycle). But then, since G does not contain any Ck+ and is connected, G must be a tree. Furthermore, since G does not contain H , no two vertices of degree 3 can be adjacent. (2) ⇒ (1): If G satisfies (2), then L(G) must be a cactus in which every cycle is a triangle, no two of which have a common vertex. Consequently, it satisfies Theorem 5.13(2), and so L2 (G) is outerplanar.   Finally, we note that, similar to the analogous result for planar graphs, any graph G for which Lk (G) is outerplanar for any k ≥ 3 is a path, a cycle, or K1,3 .

Chapter 6

Connectivity of Line Graphs

6.1 Introduction Connectivity properties of graphs are among the most important measures of vulnerability and reliability in the field of communication networks. In general terms, such measures provide information on a network’s resistance to disruption when vertex or edge failures occur. In this chapter we study these properties for line graphs. The two most familiar graph parameters in this field are the vertex-connectivity and the edge-connectivity. After we review their definitions, we develop theorems on these concepts as they involve graphs and line graphs. Many of them involve equalities or inequalities from among the parameters of connectivity, edge-connectivity, and minimum degree for both graphs in general and line graphs in particular. The concluding section is devoted to the behavior of different connectivities in iterated line graphs.

6.2 Background For the sake of completeness, we provide some basic definitions here. More details and examples may be found in standard graph theory textbooks. The connectivity κ(G) of a graph G is the smallest number of vertices whose removal from G results in a disconnected graph or the trivial graph K1 . For G = K1 , the edge-connectivity λ(G) is the smallest number of edges whose removal from G results is a disconnected graph, with λ(K1 ) defined to be 0. For k ≥ 1, a graph G is said to be k-connected (or k-vertex-connected) if κ(G) ≥ k. Similarly, G is l-edge-connected if λ(G) ≥ l. The examples of complete graphs and complete bipartite graphs illustrate these concepts and will be useful later. For the complete graph Kn , it is easy to see that, κ(Kn ) = λ(Kn ) = n − 1, and for the complete bipartite graph Kr,s with r ≤ s, © Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_6

87

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6 Connectivity of Line Graphs

κ(Kr,s ) = λ(Kr,s ) = r. Thus, in these cases both types of connectivity equal the minimum degree. The following result ties the three parameters together. Theorem 6.1 If G is a graph with minimum vertex degree δ(G), then κ(G) ≤ λ(G) ≤ δ(G). For edge-connectedness, we have the following result that is in essence equivalent to the definition. Theorem 6.2 A graph G is k-edge-connected if and only if for any nonempty proper subset S of V , there are at least k edges joining S and V − S. This suggests the concept of separation, for sets of vertices as well as sets of edges. If v and w are nonadjacent vertices in a graph G, then a v–w separating set S is a subset of V − {v, w} for which v and w lie in different components of G − S. Separating sets of edges are defined similarly. The concept of connectivity is closely related to the existence of paths in a graph. This is shown by well-known classic results of Menger and Whitney. Menger’s theorems date back to 1927 and Whitney’s to 1932, but they are in fact equivalent, both having vertex and edge versions. Menger’s theorems can be thought of as global results and Whitney’s as local. First we give the vertex versions. Theorem 6.3 If v and w are nonadjacent vertices in graph G, then the minimum number of vertices in a v–w separating set equals the maximum number of internally disjoint v–w paths in G. Theorem 6.4 A nontrivial graph G is k-connected if and only if for each pair of vertices v and w there are at least k internally disjoint v–w paths in G. We next give the corresponding edge versions. Theorem 6.5 If v and w are vertices in graph G, the minimum cardinality of a v–w separating set of edges equals the maximum number of pairwise edge-disjoint v–w paths in G. Theorem 6.6 A nontrivial graph G is k-edge-connected if and only if for each pair of vertices v and w there are k pairwise edge-disjoint v–w paths in G. We note in passing that similar concepts can be defined for directed graphs, with analogous results.

6.3 Vertex- and Edge-Connectivity Since the line graph of K1 is empty and since κ(L(K2 )) = λ(L(K2 )) = 0, we assume that all graphs in this section are connected and have at least three vertices. From the definition of a line graph, it seems natural that results about the edge-connectivity of a graph should lead to corresponding results about the vertexconnectivity of the line graph. Chartrand and Stewart [58] and Zamfirescu [177]

6.3 Vertex- and Edge-Connectivity

89

were the first to investigate connectivity properties of line graphs, and we begin with their results. Theorem 6.7 For any graph G and positive integer k, the following statements hold: (a) If G is k-edge connected, then L(G) is k-connected. (b) If λ(G) ≥ 2, then κ(G) ≤ λ(G) ≤ κ(L(G)) ≤ δ(L(G)). (c) If G is k-connected, then L(G) is k-connected. Proof From Theorem 6.6, we observe that edge-disjoint paths in G yield vertexdisjoint paths in L(G), (a) follows from this. It is easy to see that (b) and (c) follow from (a) and Theorem 6.1.   The next result, due to Zamfirescu [177], describes relationships between λ(G) and λ(L(G)). Theorem 6.8 If G is agraphG for which L(G) has no vertices of degree λ(L(G)), then λ(L(G)) ≥ λ(G) λ(G) . 2 Proof The proof is so with the given condition, we suppose that  by contradiction,  λ(G) λ(L(G)) < λ(G) 2 . We also assume that λ(G) > 1 since the result is trivial

otherwise. Let X be a nonempty proper subset of vertices in L(G), and let X denote the corresponding set of edges in G. For a vertex v in G, let μ(v) and μ(v) denote the number of edges in X and E(G) − X that are incident with v. Let Y = {v ∈ V (G) : μ(v)μ(v) > 0}. We observe that Y is nonempty since λ(G) > 1. We consider two cases. Case 1 No edges in G are induced by Y . In this case, since the degree  of each vertex λ(G) in G is at least λ(G), at least one of μ(v) and μ(v) is at least . It follows that 2

μ(v)μ(v) ≥

v∈Y

  λ(G) ∗ λ(G) μ (v) ≥ λ(G) λ(G) ≥ λ(L(G)) + 1, 2 2 v∈Y

where μ∗ is μ or μ. Case 2 There is an edge uw in the graph induced by Y . By the hypothesis, the vertex in L(G) corresponding to the edge uw has degree at least λ(L(G)) + 1 so that

μ(v)μ(v) ≥ μ(u)μ(u)+μ(w)μ(w) ≥ μ(u)+μ(u)−1+μ(w)+μ(w)−1 ≥ λ(L(G))+1.

v∈Y

The inequalities in the two cases show that L(G) is (λ(L(G)) + 1)-edge connected, which is a contradiction.   The next result [58] follows from the previous theorem.

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Theorem 6.9 For any connected graph G, the following statements hold: (a) λ(L(G)) ≥ 2λ(G) − 2. (b) If λ(G) > 2, then λ(L(G)) = 2λ(G) − 2 if and only if G has a pair of adjacent vertices of degree λ(G). Proof If the inequality in (a) does not hold, then

λ(L(G)) < 2λ(G) − 2 ≤ λ(G)

 λ(G) . 2

It follows from Theorem 6.8 that there is a vertex in L(G) of degree λ(L(G)). If vw is the edge of G corresponding to this vertex, we have deg v + deg w − 2 = λ(L(G)) < 2λ(G) − 2. It follows that the degree of at least one of v and w is at most λ(G) − 1, which is a contradiction.   . Hence, if For (b), assume that λ(G) = 2. Then 2λ(G) − 2 < λ(G) λ(G) 2 λ(L(G)) = 2λ(G) − 2, it follows from Theorem 6.8 that there is an edge vw in G with deg v +deg w = λ(L(G))+2 = 2λ(G). It follows that deg v = deg w = λ(G). Conversely, if vw is an edge and deg v = deg w = λ(G), then the degree of the vertex in L(G) corresponding to this edge is, 2λ(G) − 2, which implies that λ(L(G) ≤ 2λ(G) − 2. The equality follows from part (a) of the theorem.   Figure 6.1 shows that the condition above does not extend to λ(G) = 2 since the graph G doesn’t have any adjacent vertices of degree 2. Zamfirescu [177] noted that the above result can be extended. For example, one can use the above proof technique to show that if λ(G) ≥ 3 then λ(L(G)) = 2λ(G) − 1 if and only if there are two adjacent vertices in G, one having degree λ(G) and the other degree λ(G) + 1. The next theorem is a more general result of Theorem 6.9.

G:

L(G):

Fig. 6.1 A graph G with λ(L(G)) = 2λ(G) − 2 = 2

6.3 Vertex- and Edge-Connectivity

91

Theorem 6.10 For a graph G,   , then λ(L(G)) = δ(L(G)). (a) If δ(L(G)) ≤ λ(G)  2    λ(G) (b) If δ(L(G)) > λ(G) , then < λ(L(G)) ≤ δ(L(G)). 2 2 Proof The result follows readily from Theorem 6.8.

 

Bauer and Tindell [21] improved Theorem 6.9. Our next theorem gives their result, and we refer the reader to their paper for a proof. Theorem 6.11 For a graph G, if κ(G) ≥ 2 or λ(G) ≥ 4, then λ(L(G)) ≥ 2δ(G) − 2. In Theorem 6.7 we saw that if G is k-connected, then so is L(G). It turns out that the difference between κ(L(G)) and κ(G) can be arbitrarily large. For instance, we observe that G = K1,n has connectivity 1, while L(G) = Kn is (n − 1)-connected. Bauer and Tindell [20] showed that, subject to the conditions in Theorem 6.7, there exist graphs with arbitrary connectivity and line graph connectivity. They also obtained a corresponding result for edge-connectivity. The proof of the first result uses a basic construction of two complete graphs joined by a selection of edges. Theorem 6.12 For all r and s with 1 < r < s, there is a graph G with κ(G) = r and κ(L(G)) = s. Proof We prove the result in two cases. Case 1 s ≥ 2r − 2. Let G = Kr,s−r+2. Then κ(G) = r since r ≤ s − r + 2. Furthermore, since κ(L(Kr,s )) = κ(Kr × Ks ) = r + s − 2, κ(L(G)) = κ(L(Kr,s−r+2)) = s, so this case is settled. Case 2 r < s < 2r − 2. We note that when r = 2 or 3, Case 1 applies, so we assume that r ≥ 4. We start with a 2r-by-2r bipartite graph with partite sets X = {x1 , x2 , . . . , x2r } and Y = {y1 , y2 , . . . , y2r } and the set S of r + s edges xi yi for i = 1, 2, . . . , r, and xj yj +1 for j = 1, 2, . . . , s − r. To this we add the edges of complete graphs on X and Y to form the graph G. Figure 6.2 shows the graph when r = 4 and s = 5. We see that κ(G) ≤ r since the removal of {x1, x2 , . . . , xr } leaves a disconnected graph. A simple application of Menger’s theorem shows that κ(G) = r. Since the removal from G of the edges in S leaves a disconnected graph, we see that λ(G) ≤ s and therefore that κ(L(G)) ≤ s. Again, it is straightforward to verify that, due to the size of the complete subgraphs, there cannot be a smaller set of edges that disconnects G, and this completes the proof.   The corresponding edge-connectivity theorem has a simpler proof. Theorem 6.13 For all r and s with s ≥ 2r − 2, there is a graph G with λ(G) = r and λ(L(G)) = s.

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x8

x1 y1

x7

x2

y8

y7 y2

G: x6

x3 x5

y6

y3

x4 y4

y5

Fig. 6.2 A graph G with κ(G) = 4 and κ(L(G)) = 5

Proof Let G = Kr,s−r+2. Then λ(G) = δ(G) = r. From the previous proof we know that κ(L(G)) = s and that κ(L(G)) = λ(L(G)) = δ(L(G)) for such graphs, and so λ(L(G)) = s.  

6.4 Connectivity of Iterated Line Graphs Iterated line graphs are defined in Chap. 1, and analysis of their behavior appears in other chapters. Some of those results from Chap. 1 will be used in what follows here. Since L(Cn ) = Cn , L(Pn ) = Pn−1 , and L(K1,3 ) = C3 , we assume that the graphs in this sections are connected and are not cycles, paths, or claws. We start with one of the earlier results on the growth of the minimum degree in iterated line graphs, repeating Theorem 1.8 for completeness here. Theorem 6.14 If G is a connected graph that is not a cycle, a path, or K1,3 , then lim δ(Lk (G)) = ∞.

k→∞

Chartrand and Stewart [58] were also the first to obtain connectivity properties of iterated line graphs. The following theorem contains their results for both vertices and edges. Theorem 6.15 Let r ≥ 1. (a) If κ(G) ≥ k then, for r ≥ 1, κ(Lr (G)) ≥ 2r−1 (k − 2) + 2 and λ(Lr (G)) ≥ 2r (k − 2) + 2. (b) If λ(G) = l, then for r ≥ 1, λ(Lr (G)) ≥ 2r (l − 2) + 2. (c) If λ(G) ≥ l > 2 and λ(Lr (G)) = 2r (l −2)+2 for some positive integer r = r0 , then the formula holds for all r < r0 .

6.4 Connectivity of Iterated Line Graphs

93

Proof (a) The proof is by induction. For r = 1, since G is k-connected, L(G) is kconnected by Theorem 6.7. Since G is k-edge-connected, L(G) is (2k − 2)-edge-connected by Theorem 6.9. The general case follows by a similar argument. (b) For r = 1, the result follows from Theorem 6.9. The general case follows by a simple induction argument. (c) Assume that the result holds for r = r0 . It then suffices to show that it holds for r = r0 − 1. Suppose that this is not the case. From (b), we know that the given expression is a lower bound, so we assume that λ(Lr0 −1 (G)) > 2r0 −1 (l −2)+2. But then by applying Theorem 6.9 to λ(Lr0 −1 (G)), we have a contradiction to the value of λ(Lr0 (G)).   Knor and Niepel [118] proved several results about the connectivity of L2 (G) and L3 (G). Before describing these, we discuss some general properties of L2 (G). It is easily seen that the vertices of L2 (G) are in one-to-one correspondence with 2-paths in G. For a vertex w of L2 (G), we denote the corresponding 2-path in G by W , where (say) W = w1 w2 w3 , and w1 w2 and w2 w3 are adjacent edges in G. Knor and Niepel [118] call W the history of w. We also observe that two vertices u and w in L2 (G) are adjacent if and only if their corresponding histories U and W in G have an edge in common. For a path P = v0 v1 . . . , vl in G, we say that a path w0 w1 . . . , wl in L2 (G) is a P -based path if, for 0 ≤ i ≤ l, the corresponding history Wi contains an edge of P . We next state a technical lemma from [118]. Lemma 6.1 Let G be a graph G with minimum degree δ. If G has a path P : v0 v1 . . . vk of length at least 2, then there are δ − 1 disjoint P -based paths P1 , P2 . . . Pδ−1 in L2 (G), with Pi = wi,0 , wi,1 , . . . , wi,ki , such that, for i = 1, 2, . . . , δ − 1, Wi,0 contains the edge v0 v1 and Wi,ki contains the edge vk−1 vk . Proof We first construct the path P1 . For j = 0, 1, . . . , k − 2, let W1,j be the 2path vj , vj +1 , vj +2 in G. We let P1 be the path wi,0 wi,1 . . . , wi,ki . Clearly, P1 is P -based. For j = 1, 2, . . . , k − 1, we denote the δ − 2 neighbors in G other than vi−1 and vi+1 by x2,j , x3,j , . . . , xδ−1,j . Now for i = 2, 3, . . . , δ − 1 and j = 0, 1, . . . , 2(k − 2) + 1, for even values, j = 2h, let Wi,j = vh vh+1 xi,h+1 , and for odd values, j = 2h + 1, Wi,j = xi,h+1 vh+1 vh+2 . It can be easily verified that the paths Pi satisfy the required conditions.   Theorem 6.16 For a connected graph G with δ(G) ≥ 3, κ(L2 (G)) ≥ δ(G) − 1. Proof Since δ(G) ≥ 3, L2 (G) has nonadjacent vertices. Let u and v be any nonadjacent vertices in L2 (G) with their corresponding histories U = u0 u1 u2 and V = v0 v1 v2 in G. From our remarks above, U and V have no edges in common. Let P be a path from a vertex of U to a vertex of V such that P is a shortest such path. It is easy to see that P can be extended to a path P such that the end edges of this path are edges from U and V , and so P has length at least 2. From Lemma 6.1

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6 Connectivity of Line Graphs

we have δ(G) − 1 vertex-disjoint P -based paths in L2 (G) that satisfy the conditions of the lemma. It follows that there are δ(G) − 1 disjoint paths in L2 (G) between u and v.   Knor and Niepel [118] also show the sharpness of the result in Theorem 6.16. To see this, consider a connected graph G with a cut-edge vw with deg(v) = deg(w) = δ(G). If Gv and Gw denote the components of G − vw, then it easily follows that there are at most δ(G) − 1 edge-disjoint paths in L(G) joining edges of L(Gv ) and L(Gw ), which gives κ(L2 (G)) ≤ δ(G) − 1. It is shown in Chap. 1 that for a graph G with δ(G) ≥ 3, δ(Li (G)) ≥ 2i (δ(G) − 2) + 2. From Theorem 6.7, we have that κ(Li (G)) ≤ κ(Li+1 (G)). Hence κ(Li (G)) grows exponentially as a function of i. Furthermore, since δ(L2 (G)) ≥ 4δ(G) − 6, Theorem 6.7 shows that κ(Li (G)) is approximately at most one-fourth of δ(Li (G)). If one makes additional assumptions about the connectivity of the base graph, then a better bound can be obtained, as shown in our next theorem below. The proof of this theorem follows from Lemma 6.2. For the sake of completeness, we just include the statement of the lemma. Its proof can be found in [118]. Lemma 6.2 Let G be a graph with κ(G) ≥ 4. For vertices u and w in L2 (G) with histories U = u0 u1 u2 and W = w0 w1 w2 , suppose one of the following conditions is satisfied: (a) The distance in G between u1 and w1 is at least 2, (b) The distance in G between u1 and w1 is 0 or 1 and u and w are non-adjacent in L2 (G). Then there are 4δ(G) − 6 internally-vertex-disjoint uw paths in L2 (G). Theorem 6.17 For a graph G with κ(G) ≥ 4, κ(L2 (G)) ≥ 4δ(G) − 6. A conjecture of Niepel et al. [144] says that if G is not a path, cycle, or claw, then for sufficiently large i (depending on G) δ(Li (G)) ≥ 2δLi (G)−2). If the conjecture is true, then it follows from Theorems 6.16 and 6.17 that κ(Li (G)) = δ(Li (G)) for sufficiently large i (depending on G). We now return to the result in Theorem 6.15 which, together with the results above in this section, implies that κ(Li (G)) ≥ 2i (δ(G) − 2) + 2 for κ(G) ≥ 4 and i ≥ 2. On the other hand, for any graph G with δ(G) ≥ 3, we get δ(L2 (G)) ≥ 5 and hence κ(L4 (G)) ≥ 4. The following theorem of [118] improves this. We first need another lemma from the same paper which we state without proof. Lemma 6.3 Let G be a graph with λ(G) ≥ 2 and δ(G) ≥ 3. Then κ(L2 (G)) ≥ 4. Theorem 6.18 For a connected graph G with δ(G) ≥ 3, (a) If λ(G) ≥ 2 or δ(G) ≥ 5, then κ(L2 (G)) ≥ 4. (b) If λ(G) = 1 and 3 ≤ δ(G) ≤ 4, then κ(L3 (G)) ≥ 4.

6.4 Connectivity of Iterated Line Graphs

95

Proof If δ(G) ≥ 5, then the result follows from Theorem 6.16. Similarly, if λ(G) ≥ 2, then the result follows from Lemma 6.3. This proves (a). Suppose that λ(G) = 1, and 3 ≤ δ(G) ≤ 4. It follows that δ(L(G)) ≥ 4 and λ(L(G)) ≥ 2. The result again follows from Lemma 6.3, proving (b).   The sharpness of Theorem 6.18 follows from the example described after the proof of Theorem 6.16.

Chapter 7

Traversability in Line Graphs

7.1 Introduction In this chapter, the word traversability refers primarily to the Eulerian and Hamiltonian properties in line graphs. As we shall see, the connections between Eulerian and Hamiltonian graphs are considerably stronger in line graphs than in graphs in general. We begin with Eulerian graphs, where (as is the case with the family of all graphs) the results are quite simple. Characterizations of both Eulerian line graphs and the line graphs of Eulerian graphs are interesting but not complicated. Most of this chapter is devoted to Hamiltonian graphs, including the key property that a graph must hold for some walk if its line graph is to be Hamiltonian. One interesting fact is that just as the line graph of an Eulerian graph is Eulerian, so the line graph of a Hamiltonian graph is Hamiltonian. Section 7.4 of this chapter is devoted to the topic of links between the connectivity of a graph and its line graph being Hamiltonian, including what is arguably the most famous conjecture on line graphs: that every 4-connected line graph is Hamiltonian. A lot of research has been done on this conjecture, with some interesting results. These include the facts that at one extreme every 4-connected line graph of a planar graph is Hamiltonian and at the other extreme that every 7connected line graph is Hamiltonian. Of course, this leaves a large gap in which there are some special cases that are known to be Hamiltonian. There are some interesting Hamiltonian facts known about iterated line graphs, including that, with the exception of paths, every connected graph has a line-graph iteration that is Hamiltonian and then all successive iterations are also Hamiltonian. The least iteration that is Hamiltonian is called the Hamiltonian index, and this is the subject of the last section. In general, determining the value of this parameter is hard, and so we look at some interesting partial results.

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_7

97

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7 Traversability in Line Graphs

7.2 Eulerian Graphs Which graphs have an Eulerian line graph is an easy question to answer. Theorem 7.1 The line graph of a connected graph G is Eulerian if and only if the degrees of the vertices of G all have the same parity. Proof First assume that L(G) is Eulerian. By the Eulerian graph characterization theorem, the degree of every vertex e in L(G) must be even, and since, if e = vw, the degree of e equals deg v + deg w − 2, v and w must both be even or both odd. Since G is assumed to be connected, it follows that all of the vertices of G must be even or all must be odd. The converse is easily seen to hold.   Note that it follows that the line graph of an Eulerian graph is always Eulerian. The next question we answer is which line graphs are such that their root graph is Eulerian. Theorem 7.2 A connected graph G is the line graph of an Eulerian graph if and only if there is a partition of the edges of G into complete graphs of even order with each vertex in exactly two of the subgraphs. Proof Assume first that G is such that its edge set can be partitioned as described. Then by Krausz’s characterization, Theorem 3.1, G is a line graph, and by the root graph construction, the vertices of the resulting root graph F correspond to the complete graphs of the partition and the degree of a vertex is the order of that subgraph, which by hypothesis is even. Hence, F is Eulerian. Now assume that G is the line graph of an Eulerian graph F , and let e = vw be an edge of F . Then in G (= L(F )), the edges at e form two complete subgraphs, one of order deg v and the other of order deg w. But since F is Eulerian, these are both even.   We illustrate this result in Fig. 7.1, where F is the root graph of G. Fig. 7.1 The root graph of an Eulerian line graph G = L(F) :

F:

7.3 Hamiltonian Graphs

99

7.3 Hamiltonian Graphs In analyzing Hamiltonian cycles in a line graph, it is useful to begin by looking at paths. If ef is an edge in L(G), then by definition, there are three vertices u, v, and w in G with e = uv and f = vw (and u = w) so that v is the common vertex of e and f . We find it convenient to use the notation of (uv)(vw) for the edge ef in L(G), with the order within the parentheses immaterial, that is, in effect, (vw) = {v, w}. Given a cycle C in L(G), we say that a vertex v of G is basic to C if it is in at least two consecutive occurrences in the expanded representation of C. For example, consider the cycle C = abcda in the line graph L(G) in Fig. 7.2. Its expanded form is (vw)(wx)(wy)(vy)(vw), so w, y, and v are basic vertices but x is not. Note that a vertex can satisfy the condition for being basic in more than one appearance in a cycle, just on different edges. These ideas will be beneficial in the proof of the following theorem of Harary and Nash-Williams [92] that gives a nice necessary and sufficient condition for a graph to have a Hamiltonian line graph. Before presenting the theorem, we give one more definition. A walk W in a graph is said to cover those edges that have at least one vertex on W and to be an edgecovering walk if it covers all of the edges of G. We also recall that a circuit is a closed trail, that is, a closed nontrivial walk in which no edge is repeated. Theorem 7.3 The line graph L(G) of a connected graph G is Hamiltonian if and only if G has a circuit that covers every edge. Proof First, assume that G has a circuit A = v1 v2 . . . vr v1 . Then, if ei = vi vi+1 (with vr+1 = v1 ), e1 e2 . . . er e1 is a cycle C in L(G). Now consider v1 v2 . If G has any edges at v2 that are not on A, insert them into C immediately after e1 . The result is a longer cycle in L(G). If there are no such edges, or when that has been done, move on to the other vertices of A in succession, repeating the process, adding any edges at a vertex that are not already in the cycle at that stage. Since every edge of G has at least one vertex on the circuit A, the end result must be a Hamiltonian cycle in L(G). For the converse, let C be a Hamiltonian cycle in L(G), and let w1 , w2 , . . . , wt , w1 be the basic vertices of G encountered in traversing C. By definition, consecutive vertices in this sequence are adjacent in G since e = wi wi+1 must be an vertex of C. Since C is a cycle, it follows that the walk W = w1 w2 . . . wt w1 Fig. 7.2 Basic vertices in a cycle in a line graph

x

b b

w

G:

L(G): a

a v

c

c y d

d

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7 Traversability in Line Graphs

Fig. 7.3 An edge-covering circuit

w c

b G:

v

u

y

x

a

g

f e

d z

c

b

L(G):

g

a f

e

d

must be a circuit in G. Furthermore, since every edge of G is a vertex of C, this circuit must cover all of the edges of G.   The proof contains the elements of algorithms both for constructing a Hamiltonian cycle in the line graph of a graph with a closed edge-covering trail and for finding such a trail in the root graph given a Hamiltonian cycle in a line graph. For example, consider the graph G in Fig. 7.3, with a = uv, b = vw, c = wx, d = xz, e = vz, f = vx, and g = xy. Then the cycle vwxv covers the edges, and bcf b is a cycle in L(G). Now, following the procedure indicated in the proof of the theorem, in the sequence b, c, f, b, insert d and g after c and then insert e and a after f . This results in the sequence b, c, d, g, f, e, a, b, and in G, each of its edges shares a vertex with the next edge, and so bcdgf eab is a Hamiltonian cycle in L(G), shown in red in the figure. Now consider the Hamiltonian cycle C = bcfgdeab in L(G), where G is as in the figure. Hence C = (vw)(wx)(vx)(xy)(xz)(yz)(uv)(vw), so v, w, x and z are the basic vertices of C, and the circuit (in this case a cycle) vwxzv covers the edges of G, illustrating the condition in the theorem. The following result is a consequence of Theorems 7.1 and 7.3: the Eulerian conclusion was noted earlier, and the Hamiltonian conclusions follow from the fact that an Eulerian circuit or a Hamiltonian cycle in a graph is clearly an edge-covering circuit.

7.3 Hamiltonian Graphs

101

Theorem 7.4 (a) The line graph of an Eulerian graph is both Eulerian and Hamiltonian. (b) The line graph of a Hamiltonian graph is Hamiltonian. We conclude this section with two examples of interesting families of Hamiltonian line graphs; the first is due to Nebeský [141] and the second was discovered independently by Greenwell [84] and Nebeský [142]. Theorem 7.5 If G is a connected graph with at least three edges, then both L(G2 ), the line graph of its square, and (L(G))2 , the square of its line graph, are Hamiltonian. Theorem 7.6 If G is a connected graph with a cycle, then its line graph L(G) has a spanning subgraph homeomorphic to G. We conclude this section with a discussion of graphs such that the complement of their line graph is Hamiltonian. In Section 3.5 several connections between line graphs and graph complements were discussed. Chartrand et al. [56] were the first to investigate this question, as one section of a paper on the complements of line graphs (which they called jump graphs). Using Dirac’s classic theorem on Hamiltonicity and degrees in graphs, they observed that if G is a graph that has n ≥ 7 vertices and m ≥ 5 edges, then L(G) is Hamiltonian provided that the maximum degree in G is at most m+2 4 . In an impressive result, Wu and Meng [175] provided a characterization of all graphs for which the complement of their line graph is Hamiltonian. As a reminder, we note that G + H denotes the disjoint union of graphs G and H and K2 · K3 the



‘musical triangle’, and we let K3 be the graph obtained by adding one end edge at each vertex of K3 . Theorem 7.7 A non-null graph G with m edges and maximum degree Δ has the complement of its line graph Hamiltonian if and only if the following conditions are satisfied: (a) G is not K5 , K3 + P3 , K3 + 2K2 , or C4 + K2 .



(b) G does not contain the following subgraphs: K2 · K3 if m = 6; K1,1,2 or K3 if m = 7; nor K4 if m = 8. (c) Δ ≤ m2 and, if equality holds, then no two adjacent vertices have degree Δ. Liu [135] compiled the graphs that fall under the description of (b) in the theorem. Liu’s conclusion is that there are twenty-two graphs for which (c) is satisfied but the complement of the line graph is not Hamiltonian: three have 5 edges, four 6, ten 7, four 8, and one 10. Since none of the forbidden subgraphs has more than ten edges, we have the following corollaries, the second of which was a conjecture in [56]. Corollary 7.1 Let G be a graph with m ≥ 11 edges, and maximum degree Δ. Then L(G) is Hamiltonian if and only if Δ ≤ m2 and, if equality holds, then no two adjacent vertices have degree Δ.

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Corollary 7.2 If G is a Hamiltonian graph of order n ≥ 7 and at least 2n − 2 edges, then L(G) is Hamiltonian. Liu extended Hamiltonian results of Wu and Meng [175] to pancyclic results of complements of line graphs. It turns out that most of those that are Hamiltonian are also pancyclic.

7.4 Connectivity and Hamiltonian Cycles Arguably the most interesting standing conjecture on Hamiltonian line graphs is that of Thomassen [167], made in the early 1980s. Conjecture 7.1 Every 4-connected line graph is Hamiltonian. At about the same time as this, Mathews and Sumner [137] conjectured the same conclusion but for a larger family of graphs. Conjecture 7.2 Every 4-connected claw-free graph is Hamiltonian. In fact, the two conjectures are equivalent, with a fundamental part of the proof using a closure operation on a graph introduced by Ryjáˇcek [152]. Although this is not the same as the closure of Bondy and Chvátal [42], the two operations have a lot in common, including the fact that both can be defined as a recursive process of adding edges with the end result being unique, and, important for us, that if the closure of a graph is Hamiltonian, so is the original graph. There is a major difference between the two operations however in that the Ryjáˇcek closure is only defined for certain graphs. Some terminology is useful in defining the operation (we note that our terminology varies somewhat from that of Ryjáˇcek). A vertex v in a graph G is called neighborhood-connected (also called in the literature locally connected) if the induced subgraph N(v) of G is connected. If v is neighborhood-connected but N(v) is not a complete graph, the addition of all edges joining pairs of non-adjacent neighbors of v is said to close the neighborhood of v. For a simple example we consider the graph G shown in Fig. 7.4, where the blue edges show the

G:

G:

v Fig. 7.4 Closing the neighborhood of v

v

7.4 Connectivity and Hamiltonian Cycles

103

neighborhood of v, which is connected but not complete, and in G the red edges are added to close the neighborhood. The Ryjáˇcek closure R(G) is defined only for claw-free graphs and is defined as follows: Let G be a claw-free graph, and successively close vertices until there are none to close. It turns out that the result does not depend on the order in which the closing of vertices takes place, and so R(G) is well-defined. The following is an important result of Ryjáˇcek [152], providing an alternate definition equivalent to the original. Theorem 7.8 The Ryjáˇcek closure R(G) of a claw-free graph G is the minimal spanning diamond-free supergraph of G. The next theorem, also due to Ryjáˇcek [152], gives the intimate connection between closure and line graphs. Theorem 7.9 The Ryjáˇcek closure of a claw-free graph is the line graph of a triangle-free graph. Proof Let H = R(G) be the closure of the claw-free graph G. It is easy to see that H too must be claw-free. Furthermore, as earlier observed, H must also be diamond-free. Therefore, by Theorem 3.4 (characterizing line graphs of trianglefree graphs), H is the line graph of such a graph.   We now show the equivalence not only of Conjectures 7.1 and 7.2, but yet another with an even weaker hypothesis: Theorem 7.10 The following statements are equivalent: (1) Every 4-connected line graph of a triangle-free graph is Hamiltonian. (2) Every 4-connected line graph is Hamiltonian. (3) Every 4-connected claw-free graph is Hamiltonian. Proof Clearly, (3) ⇒ (2) and (2) ⇒ (1). Now assume that (1) holds but that but that (3) does not, that is, there is a non-Hamiltonian 4-connected claw-free graph G. As Ryjáˇcek showed, its closure R(G) is also non-Hamiltonian. Since by the preceding theorem, R(G) is a line graph, this is a contradiction. Therefore, (1) ⇒ (3), and the proof of the equivalence of all three statements is complete.   As a consequence of this theorem, we now have a third equivalent conjecture, namely the following: Conjecture 7.3 Every 4-connected line graph of a triangle-free graph is Hamiltonian. Adding to the list, Plummer [147] observed (using some other results) that the next conjecture is also equivalent to the other three. Conjecture 7.4 Every 4-connected 4-regular claw-free graph is Hamiltonian. In fact, there are many other equivalent conjectures; a survey article by Broersma et al. [46] contains more than 30! Some are similar to these four, others quite

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different. As their paper points out, some of the conjectures appear to be stronger, others weaker, and thus it seems that there is “evidence” in both directions, that the conjectures are true (with hypotheses being less general than in the original conjectures) and that they are false (because of the more general consequences if true). We now turn to some related positive results. The earliest result in this direction was proved independently by Bill Jackson (unpublished) in 1989, and by Zhan [180] published in 1991. Theorem 7.11 Every 7-connected line graph is Hamiltonian. Returning to the problem with 4-connected line graphs, there is the following interesting 1994-result of Lai [131]. Theorem 7.12 Every 4-connected line graph of a planar graph is Hamiltonian. Further results of this type can be found elsewhere, including in papers by Šol´tes and Lai [163] and Krylov [124]. A stronger property than being Hamiltonian that a graph can have is this: A graph G of order n is pancyclic if it has a cycle of each length 3, 4, . . . , n. Nebeský [140], [143] discovered some interesting results on the pancyclicity of line graphs and their complements, one of which is the following: Theorem 7.13 If G is a connected graph with at least six vertices, then the line graph of either G or its complement G is pancyclic. There are also some other stronger conclusions than just the property of a graph being Hamiltonian. A graph G is called Hamilton-connected (“Hamiltonian-pathconnected” might be a more descriptive name) if every pair of vertices are joined by a Hamiltonian path. It is a fact that every Hamilton-connected graph is Hamiltonian. Although some of these results can be extended to all claw-free graphs, we state them here for line graphs. The first of these results is an extension of a theorem of Zhan [180]: Theorem 7.14 Every 7-connected line graph is Hamilton-connected. A stronger result of this type (in at least one sense) is due to Kaiser and Vrána [115]: Theorem 7.15 Every 5-connected line graph with minimum degree at least 6 is Hamilton-connected. The bow-tie graph (also called the hourglass graph), shown in Fig. 7.5, consists of two triangles with one common vertex. Kriesell [123] proved the following Fig. 7.5 Bow-tie graph

7.5 The Hamiltonian Index

105

theorem, which implies that Thomassen’s conjecture above holds for graphs that do not contain an induced bow-tie graph. Theorem 7.16 Every 4-connected bow-tie-free line graph is Hamilton-connected.

7.5 The Hamiltonian Index An interesting consequence of the criterion for Hamiltonicity of line graphs gives a simple property for a graph that guarantees that even if its line graph is not Hamiltonian, the line graph of its line graph will be. The following theorem is an extension of a result of Chartrand and Wall [59]. Theorem 7.17 If G is a connected graph with at least three vertices and no vertices of degree 2, then its second-order line graph L2 (G) is Hamiltonian. Proof It follows from the definition of a line graph that the edges at each vertex v of G yield a complete subgraph of L(G). Unless v has degree 2, the subgraph is either trivial (when deg v = 1) or it has a spanning cycle (when deg v ≥ 3). Consider a subgraph F of L(G) consisting of one spanning cycle from each of these nontrivial subgraphs. Each vertex in L(G) is in either one or two of these cycles, and so it follows that F is a spanning subgraph of L(G) in which each vertex has degree 2 or 4. Hence, F has an Eulerian circuit. Furthermore, this circuit covers the edges of L(G), and therefore, by Theorem 7.3, the line graph of L(G) is Hamiltonian.   In his doctoral dissertation, Chartrand [53] (see also [54]) proved a theorem that has important ramifications, namely, that, except for paths, every connected graph has an iterated line graph that is Hamiltonian. Theorem 7.18 If G is a connected graph other than a path, Lr (G) is Hamiltonian for some r. Proof Let G be a connected graph that is not a path. Since the line graph of a cycle or the claw K1,3 is a cycle, we may assume that G is prolific. It follows from Theorem 1.8 that G has an iterated line graph Lr (G) with no vertices of degree 2. But then it follows from Theorem 7.17 that Lr+2 (G) is Hamiltonian.   Because the line graph of a Hamiltonian graph is Hamiltonian (Theorem 7.4), it follows that for any connected graph G other than a path, there is a least number r for which Lr (G) is Hamiltonian, and so is Lk (G) for all k > r. This number, denoted h(G), is called the Hamiltonian index of G and was first introduced by Chartrand [53] in his doctoral dissertation. There he showed that the maximum possible Hamiltonian index of a connected graph of order n is n − 3 and that this is achieved only by the tree obtained from the path Pn−1 by doubling one of its end vertices (that is, the chain Q (n − 3) of type 1 introduced in Chap. 1). (See Q (4) in Fig. 1.11). This result was generalized by Saražin [155] as stated in the next theorem.

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Fig. 7.6 A comet

Theorem 7.19 Among all connected graphs of order n ≥ 4 and maximum degree Δ ≥ 3, the maximum Hamiltonian index is n − Δ. The only extreme graph in any case is a generalization of Chartrand’s graph, what is sometimes called a comet (a star with a tail): a star K1,Δ with the path Pn−Δ tacked on to one of the Δ end-vertices of the star (see Fig. 7.6). One of the most important results on the Hamiltonian index was proved by Bertossi [40] in 1981: Theorem 7.20 The problem of determining the Hamiltonian index of a graph is NP-complete. Because of this fact, there is no general formula for the Hamiltonian index of a graph, and therefore some special families are of interest. Here we consider just one family of these, trees, for which there is an elegant formula. As noted in Sect. 1.5, vertices of degree 2 are of particular importance, so we begin with trees that don’t have any, in fact, trees in which all of the degrees are 1 or 3 (but they could as well be all odd). If T is such a tree of order at least 4, then each of its edges has degree either 2 or 4, and hence its line graph L(T ) is Eulerian. Therefore, by Theorem 7.4 its second line graph is Hamiltonian. Not surprisingly, the more general result also holds: The Hamiltonian index of every tree without any vertices of degree 2 and that is not a star has Hamiltonian index 2. We next consider the trees that are not paths but have vertices of degree 2, and these are the chains Q (l) of type 1 and Q

(l) of type 2 introduced in Chap. 1, with l denoting the length of the path P of vertices of degree 2. The following result follows from the definitions. For convenience we include the cases in which l = 1 even though these have no vertices of degree 2. Lemma 7.1 If the path P has length l, then h(Q (l)) = l and h(Q

(l)) = l + 1. Theorem 7.21 Let T be a tree that is not a path or a star, and let l1 be the maximum value of l for which there is a Q (l) in T and let l2 be the maximum value of l for which there is a Q

(l) in T . Then the Hamiltonian index of T is h(T ) =

l2 + 1 if l2 ≥ l1 , otherwise. l1

This result is illustrated by the two trees in Fig. 7.7. In the first, l1 = 4 and l2 = 2, and the Hamiltonian index, as can be readily verified, is 4, while in the second, l1 = 2 and l2 = 4, but the Hamiltonian index is 5.

7.5 The Hamiltonian Index

107

Fig. 7.7 Computing the Hamiltonian index of trees T1

T2

The argument used for the Hamiltonian index of trees actually applies also to cacti, a cactus being tree-like as a connected graph in which every block is either one edge or a cycle. Clearly, a chain of 2s that lies on a cycle does not have the same effect on non-Hamiltonicity as the chains in a path for example. Therefore we call a chain of 2s in a connected graph G breakable if it contains a cut-edge of G. Thus, the Hamiltonian index of a cactus has the same formula as that for trees except that the maximum values l1 and l2 are taken only over the breakable chains. The strongest theorem on the Hamiltonian index of graphs in general is due to Saražin [155] and has the result on cacti as an obvious corollary. Theorem 7.22 Let G be a graph that is not a path, and let l1 be the maximum value of l for which G has a breakable chain Q (l) of type 1 and let l2 be the maximum value of l for which G has a breakable chain Q

(l) of type 2. Then the Hamiltonian index of G is h(T ) =

l2 + 1 if l2 ≥ l1 , otherwise. l1

Xiong et al. [176] raised the question of how various operations affect the Hamiltonian index of a graph. Here, we discuss the simple but interesting question of whether the addition of an edge can result in an increase in the index. Rather surprisingly, it can, but only under very limited circumstances. Theorem 7.23 Let G be connected graph that is not complete, and let G be the result of joining two non-adjacent vertices of G with the sum of their degrees at least 3. If the Hamiltonian index of G is greater than that of G, then h(G) = 1 and h(G ) = 2. Otherwise, h(G ) ≤ h(G). We give in Fig. 7.8 an example (from [176]) of a graph G in which the addition of an edge results in an increase in the index. It is easily seen that G does not have a

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Fig. 7.8 Example of non-monotonicity of the Hamiltonian index

e v

w

Hamiltonian cycle since such a cycle C would have to contain all of the edges with a vertex of degree 2, and that cycle does not include v and w. However, it is easy to verify that L(G) is Hamiltonian since the aforementioned cycle is edge-covering, and therefore h(G) = 1. Now consider the graph G obtained by adding the edge e = vw to G. Like G, G is non-Hamiltonian, and in fact, the same reasoning shows that any circuit that covers the edges with both vertices of degree 2 must contain those edges and hence must be the cycle C. However, this circuit does not cover e, and so L(G ) too is non-hamiltonian. It is straightforward to show that L2 (G ) is however Hamiltonian; that is, h(G ) = 2. As a corollary to the theorem, we see that if the line graph of a graph G is nonHamiltonian and if H is a spanning supergraph of G, then h(H ) ≤ h(G).

Chapter 8

Colorability in Line Graphs

8.1 Introduction Graph colorings constitute one of the most important areas of graph theory, and in this chapter we explore the aspects of this topic related to line graphs. Recall that a coloring of a graph G is an assignment of colors to the vertices of G in such a way that no two adjacent vertices get the same color, and the chromatic number χ(G) of a graph G is the minimum number of colors needed to properly color G. Similarly, an edge-coloring of a graph G (with at least one edge) is an assignment of colors to the edges of G so that no two adjacent edges get the same color, and the chromatic index χ (G) of G is the minimum number of colors needed to color the edges of G so that no adjacent edges have the same color. The words ‘edge’ and ‘adjacent’ here suggest a possible connection between line graphs and coloring, and indeed this is the case: χ(L(G)) = χ (G). We begin with the classic Four Color Theorem in an equivalent form as a theorem on line graphs. We then move on to the chromatic number of the line graphs of cubic graphs in the guise of graphs that are regular of degree 4. Vizing proved that the chromatic number is then either 3 or 4, but determining which is in general a difficult problem, and several aspects of this are considered in the third section. The next section looks at arguably the most significant edge-coloring theorems (other than the edge coloring version of the four color theorem). In line graph terms, they are König’s theorem on the exact value of the chromatic number of the line graph of bipartite graphs and Vizing’s theorem on the two possible values of the chromatic number of the line graph of graphs in general. We conclude the chapter with an excursion into some variations and generalizations of graph coloring. Graph theorists have studied many different types of colorings, but we look at only two here, one that relaxes the requirement that adjacent vertices get different colors and another that puts additional restrictions on allowable colorings. The chapter concludes with a problem in coloring the line graphs of multigraphs. © Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_8

109

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8.2 Line Graphs of Planar Cubic Graphs For the sake of completeness, we begin this section with formal definitions regarding coloring vertices and edges of graphs and line graphs. • A coloring of a graph G is an assignment of colors to the vertices of G so that adjacent vertices always have different colors. (It is understood that if there are exceptions to this definition, that will be made clear.) • The chromatic number χ(G) of a graph G is the minimum number of colors needed for a coloring of G. • An edge-coloring of a graph G is an assignment of colors to the edges of G so that adjacent edges always have different colors. • The chromatic index χ (G) (also known as the edge-chromatic number) of a graph G is the minimum number of colors needed for an edge-coloring of G. Even though the connection between these definitions made by the words “adjacent” and “edge” is elementary, we state it as a theorem. Theorem 8.1 The chromatic number of the line graph L(G) of a graph G is the chromatic index of G; that is, χ(L(G)) = χ (G). It follows that for every theorem on the chromatic index of a graph there is a corresponding theorem on the chromatic number of a line graph. The study of the chromatic index goes back at least as far as P. G. Tait and the four color conjecture. Tait’s creative idea [165] was to consider a map in which only three countries meet at a corner as a cubic planar graph, and then to consider coloring the edges. He proved that every cubic planar graph is 3-edge-colorable if and only if the four color conjecture holds. Because of its interest and its impact on coloring line graphs we include Tait’s result and its proof in the following theorem. We assume that all maps here have no cut-edges since it is clear that including them would not affect the validity of the basic argument. Theorem 8.2 The following statements are equivalent: (1) Every planar map is 4-colorable. (2) Every cubic planar graph is 3-edge-colorable. (3) The line graph of every planar cubic graph has chromatic number 3. Proof We show first that (1) and (2) are equivalent and then that (2) and (3) are equivalent. (1) ⇒ (2): Assume that every planar map is 4-colorable, and let M be a connected map in the plane in which each corner (vertex) of a region is on three regions. Assume that the regions are colored with four colors, say, red, blue, green, and purple, and let G be the graph of the borders of the regions in the map. Then the regions at each vertex have three of the four colors, and so each edge borders on

8.2 Line Graphs of Planar Cubic Graphs

111

a pair of colors, with the three edges bordering on one pair from each of these three pairs: (a) red-blue and green-purple, (b) red-green and blue-purple, and (c) red-purple and blue-green. Treating a, b, and c as colors, by coloring each edge according to the letters of its bordering regions, we have a proper 3-coloring of the edges at each vertex of G. (2) ⇒ (1): Now assume that every cubic planar graph is 3-edge-colorable, and let M be a planar map. By adding edges to the regions of M if necessary, we may assume that it is triangulated. Let G be a cubic plane graph corresponding to the planar map M. Assume also that G has its edges properly 3-colored, say, with the colors red, blue, and green. We observe that the subgraph of G having the edges of two of the colors consists of a collection of disjoint cycles, and furthermore, each region of the map M lies either in the interior of one of the cycles or lies outside all of them. If we now take two different pairs of the three colors of the edges, say the pair red and blue and the pair blue and green, then there are four possibilities for a region R of M: either (α) it lies inside a cycle of both types, or (β) it lies inside a red and blue cycle but not inside a blue and green one, or (γ ) it lies inside a blue and green cycle but not inside a red and blue one, or (δ) it doesn’t lie inside a cycle of either of these types. Now treating α, β, γ , and δ as colors for the regions of M, we have a proper 4-coloring of the map M, and hence (2) implies (1). (2) ⇐⇒ (3): As we observed at the start of this section, the chromatic number of the line graph of a graph is equal to the chromatic index of the graph itself, so both of these implications follow.   Of course, as a consequence of the four color theorem, we have the following theorem. We note that it is interesting that so many results, including this one, are equivalent to that famous theorem. The fact that the line graph of a cubic planar graph has chromatic number 3 is illustrated in Fig. 8.1, which shows both an edge 3-coloring of the cube Q3 and the corresponding vertex 3-coloring of its line graph.

Q3 :

L(Q3 ):

Fig. 8.1 3-edge-coloring Q3 and 3-coloring its line graph

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8 Colorability in Line Graphs

8.3 Quartic Line Graphs Following up on Tait’s theorem, we now look at cubic graphs in general. As a special case of his theorem, Vizing showed that the chromatic number of the line graph of such a graph is always either 3 or 4. Clearly it must be at least 3, and so the proof involves showing that it is at most 4. We prove the result in the chromatic index version rather than in terms of line graphs. The strategy involves showing that if G has maximum degree 3 and has an edge e for which G − e is 4-edge-colorable, then, perhaps by recoloring, one can find such an edge for which one of the four colors is missing from both vertices of that edge. Theorem 8.3 The chromatic number of the line graph of a cubic graph is 3 or 4. Proof Since all of the edges at a vertex must get different colors in an edge-coloring of a graph, the chromatic index of a cubic graph must be at least 3. To show that every cubic graph is 4-edge-colorable, we prove a slightly stronger result, that this statement is true for all graphs with maximum degree 3. We use induction on the number of edges in a graph. Thus, we may assume that G is a graph with maximum degree 3, and that for any edge e, G − e has a 4-edge-coloring C1 with colors red, blue, green, and purple. Clearly, at each vertex at least one of the four colors is absent. Our goal is to show that, possibly after re-coloring some of the edges, we can eventually find an edge f for which G − f has a 4-edge-coloring with some color missing from both vertices of f . In such a circumstance, G is clearly 4-edgecolorable, so we suppose that this is not the case. This means that, for any edge e, all four colors are present at the end points of G − e, so that G is cubic. Let us assume that e is the edge vw1 and that purple is missing from v and red is missing from w1 . There must then be an edge at w1 colored purple and an edge at v, say vw2 , colored red. (See Fig. 8.2a, which shows a small portion of the graph G and where a dashed line denotes an uncolored edge and a solid line indicates a colored edge but the color is undetermined. Furthermore, a color in brackets at a vertex denotes the presence and a color in parentheses at a vertex denotes the absence of that color there.) We now switch the color red from the edge vw2 to the edge vw1 , leaving vw2 uncolored, and with color red absent from the vertex w2 . This gives coloring C2 (as in Fig. 8.2b). We may assume that the color purple is present at w2 since otherwise vw2 can be colored purple and the graph is 4-edge-colored. Hence, we are in a similar situation to where we were, with purple still missing at v, but now with (say) blue missing at w2 and present at v. Hence, the edge vw3 at v must be blue and green must be missing at v (see Fig. 8.2c). We now switch the color blue from the edge vw3 to vw2 , leaving vw3 uncolored and the color green absent from w3 , giving us coloring C3 (as in Fig. 8.2d). We now consider two cases, depending on whether or not red is present at w3 . We assume first that it is. Then the situation is as in Fig. 8.2e, and we can color the edge vw3 green so that G has a proper edge-coloring. Therefore we assume that red is absent from w3 . Then the situation is as in Fig. 8.2f. Now consider the subgraph

8.3 Quartic Line Graphs w1 [

113 ]

w1 [

]

( ) w2

v

( ) w3

v )

]

w2 [

]

w3 [

w2 [

]

)

(

v (

)

(c)

( )

w1 [

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w2 [

]

v (

( )

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)

w3

(b)

( )

(d)

(

w3

(a)

(

v ( )

( )

w1 [

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c1

w2

v

w1 [

w3 [

]

(

)

)

w1 [

]

w2 [

]

( )

(f)

w3 [

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(

)

Fig. 8.2 Coloring the edges of a cubic graph

H of G generated by the edges colored red and purple. It is not difficult to see that each component is either a path or a cycle of even length. It follows that v and w1 are in the same component, and that is a path with v as an end-vertex. Similarly, w2 is an end-vertex of a path, as is w3 . Hence (at least) one of w2 and w3 is not in the same component as v and w1 . If it is w2 , then we swap the colors red and purple in its component, and color vw2 purple and vw3 blue. If, on the other hand, it is w3 , then we swap red and purple in that component and color the edge vw3 purple. Therefore, G is 4-edge-colorable, which completes the proof of the theorem.   The proof of this theorem contains the essence of a simple algorithm for finding a 4-edge-coloring of an arbitrary cubic graph. However, it was shown by Holyer [108] that determining whether the chromatic number of the line graph of an arbitrary cubic graph is 3 or 4 is NP-complete. We state Holyer’s theorem in general line graph terms. Theorem 8.4 Determination of the chromatic number of a line graph is NPcomplete. In an early paper on the topic, Beineke and Wilson [36] observed that there is a simple condition that guarantees that a graph of odd order has chromatic index greater than its maximum degree. The following result is a special case of this; a more general version will be given in the next section.

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8 Colorability in Line Graphs

Theorem 8.5 If G is a graph with 2r+1 vertices and 3r+1 edges and has maximum degree 3, then it has chromatic index 4. Proof We note that in a proper edge-coloring of a graph of order n, there can be at most  n2  edges of any given color, and the result follows at once from this fact.   The smallest examples of such graphs are the cycle C5 with two independent diagonals and the cycle C7 with three such edges, shown in Fig. 8.3. It is obviously the case that if a cubic graph G has a subgraph that satisfies the bound in the theorem, then χ (G) = 4. However, not all cubic graphs with chromatic index 4 have such a subgraph, and the nicest example of this is the Petersen graph P ∗ , shown in Fig. 8.4. That it is not 3-edge-colorable is not difficult to see. Readers may wish to prove this on their own, but we do give here an outline of one proof. Suppose that P ∗ is 3-edge-colorable. Note that up to symmetry and Fig. 8.3 The smallest graphs of maximum degree 3 and edge-chromatic number 4

Fig. 8.4 Petersen graph

8.3 Quartic Line Graphs

115

interchange of colors, there is really only one way to color a 5-cycle with three colors: 1 − 2 − 1 − 2 − 3 − 1. If the outer 5-cycle in the figure is colored in this way, then the colors on the spokes are forced, after which the colors on the edges of the inside 5-cycle are also forced, and then there is a vertex that will have two edges the same color. We now consider some variations of the Petersen graph first introduced by Watkins [172]. Note that the Petersen graph is based on 5-cycles and can be defined as follows: Take two 5-cycles, C5 = v1 v2 v3 v4 v5 v1 and C5 = w1 w2 w3 w4 w5 w1 , and then join v1 to w1 , v2 to w3 , v3 to w5 , v4 to w2 , v5 to w4 (successively going two steps along the second cycle, with subscripts naturally taken congruent modulo 5). This construction has been generalized to other cycles in the following way: Let P (r, k) be the graph obtained by taking two r-cycles Cr = v1 v2 . . . vr v1 and Cr = w1 w2 . . . wr w1 , and then add “spokes” between them, joining v1 to w1 , v2 to wk+1 , v3 to w2k+1 , and so forth (successively going k steps along the second cycle, with subscripts naturally taken modulo r). Thus, the Petersen graph is P (5, 2); and Fig. 8.5 shows P (7, 2). Readers are invited to determine the chromatic index of this graph. Watkins conjectured that the only graph in the family of graphs P (r, k) that has chromatic index 4 is the Petersen graph itself, and in support of his conjecture, he showed that the graphs in some subfamilies are indeed 3-edge-colorable. His conjecture was eventually proved by Castagna and Prins [51]. Theorem 8.6 The line graph L(Pr,k ) of every generalized Petersen graph has chromatic number 3 except for that of the Petersen graph itself, for which it is 4. Looking further at cubic graphs with chromatic index 3, we recall that Tait’s theorem says that all planar cubic graphs have this property. Focusing on polyhedra (that is, 3-connected planar graphs), we recall that the faces comprise a set of cycles in which each edge occurs exactly twice. This geometric concept led Szekeres [164] Fig. 8.5 The generalized Petersen graph P (7, 2)

v1 v7

v2 w1 w6

w3

w4

w5

v6

v3 w2

v5

w7

v4

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8 Colorability in Line Graphs

Fig. 8.6 An illustration of Szekeres’s theorem

d

g

b a

i

e c

h

f

to the following more general result that applies to graphs in general, not just those that are planar. Theorem 8.7 The chromatic number of the line graph of a cubic graph G is 3 if and only if its edges can be partitioned into cycles of even length. We note that since the line graph of a cubic graph is 4-regular, each of its vertices must be on exactly two of the cycles in such a decomposition. We illustrate the result in Fig. 8.6 for the line graph of the triangular prism. The cycles in one set that satisfies Szekeres’s result are abegida, acehifa, and bdghfcb. One family of cubic graphs with chromatic index 4 has drawn special interest, in particular, because for many years so few graphs in the family were known. In fact, for their presumed rareness, they were given the name ’snark’ (from Lewis Carroll’s “The Hunting of the Snark”). The definition is rather technical (so the family has some desirable properties): A snark is a connected cubic graph for which (a) the shortest cycle has length at least 5, (b) at least four edges must be removed in order to have two components each containing a cycle, and (c) the chromatic index is 4. The smallest snark (and the first one known) was the 10-vertex Petersen graph (1898). Between 1946 and 1973, only four more were found: two of order eighteen in 1946 by Blanuša [41], then one with 210 vertices by Tutte [69] 2 years later, and then a quarter of a century later one with 50 vertices by Szekeres. A breakthrough occurred 2 years after this when Isaacs [110] exhibited two infinite families of snarks. Early on in the history of snarks, when only four were known, Tutte [69] conjectured that all snarks were closely related to the Petersen graph, a result that was stated to be true in 2001 by N. Robertson, D. Sanders, P. Seymour, and R. Thomas (as yet apparently not published in full). Theorem 8.8 Every snark contains a subdivision of the Petersen graph. Our discussion of snarks could also have included their line graphs as 4-regular graphs of chromatic number 4. We conclude this section with a result on 4-regular line graphs in general.

8.4 König’s and Vizing’s Theorems

117

Theorem 8.9 Let L(G) be a connected 4-regular line graph other than the complete graph K5 . Then either (a) G is bipartite with vertices of degrees 2 and 4, and χ(L(G)) = 4, or (b) G is cubic and χ(L(G)) = 3 or 4. Proof Since the degree in L(G) of an edge e = vw in G is deg v + deg w − 2, it follows that deg v + deg w = 6, so, assuming that deg v ≤ deg w, either (1) deg v = 1 and deg w = 5, (2) deg v = 2 and deg w = 4, or (3) deg v = deg w = 3. Furthermore, because L(G) is connected, all of its edges must be of the same type. It follows that in case (1) G can only be K1,5 and so L(G) is the complete graph K5 , which we are not considering. In case (3), G is cubic, and these we have already considered, which leaves only case (2). Here, every edge in G has one vertex of degree 2 and one of degree 4, and so G is bipartite. Drawing on König’s theorem on bipartite graphs from the next section (or the reader’s showing this), the chromatic index of every such graph (and hence the chromatic number of its line graph) is 4, which completes the proof.  

8.4 König’s and Vizing’s Theorems The next major achievement on edge-colorings following the work of Tait in the late 1880s took place just over a 100 years ago when Dénes König proved in 1916 [121] that the chromatic index of every bipartite graph G equals the largest degree among its vertices. Theorem 8.10 The chromatic number of the line graph of a bipartite graph G is the maximum degree Δ(G) of a vertex in G; that is, χ(L(G)) = Δ(G). Proof We prove this result in the equivalent version of the chromatic index χ (G), and let Δ = Δ(G). Since all edges at a vertex must have different colors, clearly χ (G) ≥ Δ. We prove the reverse inequality by contradiction. To that end, we assume that G is a bipartite graph with the fewest edges for which the theorem does not hold, and let e = vw be an edge of G. Then the subgraph H = G − e has an edge-coloring using at most Δ colors. Since v and w have fewer than Δ colors, there is at least one color missing from each of them. If some color is missing from both, then that color is available for e, and we have a proper edge-coloring of G with Δ colors. Hence, we assume that (say) red is missing from v and blue is missing from w. It follows that if v and w are in the same component, there is a blue-red alternating path from v to w of even length, but this contradicts G being bipartite. By switching the colors red and blue on the edges in the component containing w, we have an edge-coloring of G − e in which red is absent from both v and w, and so G can indeed be properly edge-colored with Δ colors.   The next major breakthrough on the chromatic index of graphs was made by Vizing [169] in 1964, and it might well be called the fundamental theorem on edge

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8 Colorability in Line Graphs

colorings. (It was independently discovered by R. P. Gupta in his doctoral thesis at the Tata Institute of Fundamental Research.) Vizing proved that the chromatic index of a graph is either the maximum vertex degree or 1 more. In the previous section we proved the result for the special case of cubic graphs, and the general proof follows the same lines (with more notation needed), so we omit the details (they can be found, for example, in Fiorini and Wilson [79]). We again state the result in line graph terms. Theorem 8.11 The chromatic number of the line graph of a graph G having maximum degree Δ is either Δ or Δ + 1. When there are just two possible values that a parameter can take, it is natural to ask what can be said about each type. In an early paper on the subject (Beineke and Wilson [36]), the term Class 1 was introduced for those graphs whose chromatic index is the maximum degree and the term Class 2 for those for which it is 1 greater. Because there is no easy answer to this question—after all, as pointed out earlier, Holyer [108] proved that this is an NP-complete problem—there has been a large body of research on the problem. Although not explicitly in line graph terms, we nonetheless present some of the results. In fact, in the previous section, the case where Δ(G) = 3 was discussed. Here we generalize some of those ideas. Since our tale of edge-colorings began with Tait’s theorem on planar graphs, we turn to a variation of that topic due to Vizing [170]. The question that he investigated was this: For which Δ are all connected planar graphs with maximum degree Δ of Class 1? It is straightforward to show that they are not for 2 ≤ Δ ≤ 5 by example: for Δ = 2, the cycle C5 ; for Δ = 3, 4, 5, insert a vertex into one edge of an appropriate polyhedron, those being a tetrahedron, an octahedron, and an icosahedron, respectively. In contrast, Vizing showed that the answer is yes for Δ ≥ 8. In his doctoral dissertation at Universität Bielefeld in 2000, Stefan Grunewald (and others at about the same time) showed that 7 can be added to the list. Thus, the case Δ = 6 is the only one that remains open. Theorem 8.12 There exist Class 2 planar graphs of maximum degree 2, 3, 4, and 5, but every connected planar graph of maximum degree 7 or greater is of Class 1. The Classification Problem As reported by Fiorini and Wilson [79], of the 143 connected graphs with up to six vertices, only eight are of Class 2 (two each with maximum degree Δ = 2 and 3, and four with Δ = 4). It thus appears that most graphs are Class 1, and indeed, as Erd˝os and Wilson [72] proved, in the long run this is so. Theorem 8.13 If P (n) is the probability that a random graph of order n is Class 1, then limn→∞ P (n) = 1. A graph G with n vertices, m edges, and maximum degree Δ is said to be overfull if m > Δ n2 . Because a graph of order n cannot have more than  n2  edges of any one color, if it is of Class 1, it cannot be overfull. Thus we have the following observation in Beineke and Wilson [36] (also implicit in Vizing [169]). (The case Δ = 3 was discussed in the previous section.)

8.4 König’s and Vizing’s Theorems

119

Theorem 8.14 Every overfull graph is of Class 2. From this we have the following consequence. Corollary 8.1 If a graph G has an overfull subgraph with the same maximum degree as G, then G is of Class 2. Given that the general problem of classification is seemingly very difficult, partial results and conjectures are of substantial interest. One of those is the following conjecture of Hilton (see Chetwynd and Hilton [61] for example), a partial converse to this theorem. Conjecture 8.1 If G is a graph of Class 2 having n vertices and maximum degree Δ with Δ > n3 , then it has an overfull subgraph. As described by Jensen and Toft [113], this conjecture has been proven for some special cases, including for the large values of Δ = n − 2 and n − 3. On the other hand, for small Δ such as 3, where the graph G obtained from K3,3 by putting a new vertex on each of three independent edges (see Fig. 8.7) has n = 9, m = 12, and Δ = 3 has no overfull subgraphs of maximum degree 3 yet is of Class 2. (We note that this graph can also be obtained from the Petersen graph by removing one of its vertices.) Edge-Critical Graphs We now consider further graphs of a given chromatic index that are minimal with respect to being of Class 2. Formally, a graph G of Class 2 is called edge-critical if χ (G) = Δ(G) + 1 but χ (G − e) = Δ(G) for each edge e of G. In much of the literature, such graphs are called simply critical, but since we are working also with line graphs, including the prefix edge will help keep things clear. Removing an edge from a graph is in essence the same as removing a vertex from its line graph. Thus, we define graph G to be vertex-critical if χ (G) = Δ(G)+1 but χ (G−v) ≤ Δ(G) for each vertex v of G. Thus, formally, a graph G of Class 2 is edge-critical if and only if L(G) is vertex-critical. Fig. 8.7 A graph of Class 2 that is not overfull

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We start with vertex-critical graphs in general, with two conjectures of Hilton and Johnson [105]. Although the line-graph counterparts are not as simple when removing a vertex from a graph as when removing an edge, we include them here for their interest and connection with the concepts involved. Note that the removal of a vertex from a graph G amounts to removing the vertices of a complete subgraph from L(G). Conjecture 8.2 If G is a graph of even order n and maximum degree Δ with Δ > n 3 , then it is not vertex-critical. Conjecture 8.3 If G is a graph of even order n and maximum degree Δ with Δ ≥ n 2 , then it is vertex-critical if and only if it is overfull. Because there are many open questions on edge-colorings of graphs, the literature on the subject is extensive, and in particular that on critical graphs. There are therefore corresponding results and questions on vertex-critical line graphs. Usually, however, the former are simpler to state and to work with than the latter, and so we do not attempt to be comprehensive here. Two excellent references are Chapter 8 in the book Graph Coloring Problems [113] and Chapter 5 by McDonald [138] in Topics in Chromatic Graph Theory. For purposes of illustration in addition to historical interest, we focus our attention primarily on graphs of maximum degree 3 with few vertices, the results due for the most part to Jakobsen [111]. Based on his work, Beineke and Fiorini [34] expanded his results on small graphs. Assume that G is an edge-critical graph of order n, maximum degree 3, and chromatic index 4, but for which the result of deleting any edge has chromatic index 3. For convenience, we call such graphs 3critical. Jakobsen showed that such a graph must be nonseparable and that every vertex must have at least two neighbors of degree 3. From these facts, the following bounds on the number of edges were deduced. Lemma 8.1 Let G be a 3-critical graph with n vertices and m edges. Then

4n 3



≤m≤

 3n − 1 . 2

Considering only small values of n, it is known that there are no 3-critical graphs of even order, only one of order 5, four of order 7, and eighteen of order 9. All those of orders 5 and 7 can be seen to have chromatic index 4 by the overfull criterion, and are obtained by adding either two independent chords to a 5-cycle or three to a 7-cycle in all possible ways, as shown in Fig. 8.3. As for order 9, all but one of the critical graphs consists of a 9-cycle with four independent chords, the odd one having only three such chords, equally spaced around the cycle, as shown in Fig. 8.7 (the result of deleting one vertex from the Petersen graph). Beyond this, it has been shown that there are no 3-critical graphs of order 12, 14, or 16, which supported a conjecture that there are no k-critical graphs of any even order, but this was disproved by Goldberg [83].

8.4 König’s and Vizing’s Theorems

121

From Jakobsen’s lemma, we can deduce some facts about the number of edges in 4-edge-critical graphs with small numbers of vertices; in particular, the values in the following table. We know that all of these values of m are realizable except possibly 15 and 18, and we believe those also to be so.

n m

5 7

7 10

9 12 or 13

11 15 or 16

13 18 or 19

Returning explicitly to line graphs, we recall that each edge coloring of a graph G corresponds to a vertex coloring of the line graph L(G). Thus, we consider the chromatic number χ(L(G)) and a vertex-critical graph F as one for which χ(F − v) < χ(F ) for every vertex v in F . Hence we have the following result on 4-critical line graphs of small order. Theorem 8.15 For n ≤ 17, there is a 4-vertex-critical line graph of order n if and only if n = 10, 12, 13, or 16, and possibly n = 15. Vizing’s Theorem Extended To conclude this section, we take another look at Vizing’s theorem. Because the chromatic number of the line graph of a graph with maximum degree less than 3 is easily determined, we assume that the graphs under consideration have maximum degree Δ ≥ 3. The version of Vizing’s theorem that we gave was the following in terms of line graphs, utilizing Theorem 8.1. Since line graphs can be characterized in terms of forbidden subgraphs, so Vizing’s theorem can also be stated in those terms. As before, we let L denote the set of nine forbidden graphs in the characterization. Also, instead of the maximum degree of the root graph of a line graph, we can use the corresponding parameter in the line graph itself. Recall that a clique in a graph G is a maximal complete subgraph of G, and the order of a largest clique is called the clique number of G, denoted ω(G). (Note: some graph theorists use the term clique for any complete subgraph; here that makes no essential difference.) Since the only way in which a clique of order 4 or more can appear in a line graph L(G) is from the set of edges at some vertex in G, it follows that (still with Δ ≥ 3), Δ(G) = ω(L(G)). Therefore, we can restate Vizing’s theorem in yet another way: Theorem 8.16 If the graph G has none of the nine graphs in the family L as an induced subgraph and has clique number ω, then its chromatic number is either ω or ω + 1. A delightful extension of this version was given by Choudum [62]—he proved that the same conclusion holds when three of the nine graphs are no longer forbidden. Theorem 8.17 If G is a graph with clique number ω that does not have any of the six graphs shown in Fig. 8.8 as an induced subgraph, then χ(G) = ω or ω + 1.

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L1 :

L4 :

L3 :

L2 :

L5 :

L6 :

Fig. 8.8 Six basic forbidden subgraphs

L7 :

L8 :

L9 :

Fig. 8.9 The other three basic non-line graphs

We note that the three basic forbidden graphs that are not cited in this theorem (shown in Fig. 8.9) are not line graphs but they do satisfy the conclusion of the theorem. More precisely, ω(L7 ) = 3 and χ(L7 ) = 4, ω(L8 ) = χ(L8 ) = 3, and ω(L9 ) = χ(L9 ) = 4. We note that the six graphs in Fig. 8.8 do themselves also satisfy the conclusion of the theorem. In fact, in all six cases, the chromatic number equals the clique number. Regarding the theorem, Choudum showed that the first two of the six graphs cannot be omitted. The graph H1 in Fig. 8.10 has L1 = K1,3 as an induced subgraph but not any of L2 , L3 , L4 , L5 , or L6 , and with ω(H ) = 2 and χ(G) = 4, does not satisfy the theorem’s clique range. A similar statement holds for the graph H2 (that is, C5 ∗ C5 ). It has L2 = K1,1,1,2 as an induced subgraph but not any of L1 , L3 , L4 , L5 , or L6 , and, with ω(H2 ) = 4 and χ(H2 ) = 6, also does not satisfy the theorem’s clique range. We note that that just as the problem of determining which connected graphs G satisfy χ(G) = Δ(G) is hard, so too is the problem for χ(G) = ω(G).

8.5 Extensions and Variations

H1 :

123

H2 :

Fig. 8.10 The necessity of two non-line graphs

8.5 Extensions and Variations We conclude this chapter with some variations on coloring line graphs. We first look at two different requirements on what colorings are allowed, one weaker, the other stronger, than the standard definition. We then turn to coloring multigraphs. Weak and Strong Colorings For our purposes here, some of the notation and terminology will be specific to this context. In particular, we will use the terminology of ‘weak’ and ‘strong’ colorings because we are talking about just two of the many variations in the literature. Our notation will then be χw (G) and χs (G) for the corresponding chromatic number, and of course χw (G) and χs (G) for the chromatic index. We first consider the weak case. A coloring of the vertices of a graph G is a weak coloring if every clique with at least three vertices has more than one color. (Recall that a clique in a graph is a maximal complete subgraph.) It is perhaps not surprising, given what we know about the standard chromatic number, that this is a hard problem in general. Bacsó et al. [6] showed that while some weak coloring problems are NP -complete, there are classes of graphs for which efficient algorithms exist for line graphs. We mention the following result of theirs as an illustration. Theorem 8.18 Let G be a connected graph other than an odd cycle, and let k ≥ 2 be an integer. Given a k-edge-coloring of G in with no monochromatic triangles, a weak k-coloring of L(G) can be constructed from the given coloring in polynomial time. The next theorem [6] gives some results about weak 2-colorings. Theorem 8.19 Let G be a nontrivial connected graph. (a) If G is not a 5-cycle and has independence number 2, then χw (L(G)) = 2. (b) If G is not an odd cycle of length 5 or more and if χ(G) ≤ 4, then χw (L(G)) = 2.

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8 Colorability in Line Graphs

We now turn to what we are calling the strong case (sometimes called ‘acyclic’ colorings). A coloring of the vertices of a graph G is a strong coloring if every cycle has at least three colors. The chief problem on strong colorings of line graphs was first posed as a conjecture by Fiamˇcík [74]: Conjecture 8.4 If G has maximum degree Δ, then χs (L(G)) ≤ Δ + 2. Among the known results are the following two theorems; the first is due to Esperet and Parreau [73], the second to Bernshteyn [39]. Theorem 8.20 If G has maximum degree Δ, then χs (L(G)) ≤ 4(Δ − 1). Theorem 8.21 If G is a graph with maximum degree Δ and G does not contain a given bipartite graph F as a subgraph, then χs (L(G)) ≤ 3Δ + o(Δ). We note that the implicit function in this result does depend on the graph F . Coloring Line Graphs of Multigraphs We first need to be clear on what the line graph of a multigraph is in this context, with parallel edges permitted between vertices but loops not allowed. Therefore of course we must have an appropriate definition for the simplest new situation, the line graph of a pair of parallel edges: Is it two edges, or just one, or perhaps none? The most natural would seem to many to be 2; that is, the line graph of a 2-cycle would be a 2-cycle, so 2-cycles behave just as cycles of other lengths do. However, the choice made here is 1 (and in Chap. 4 on spectral properties the choice was 0). An example is shown in Fig. 8.11 (corresponding to the illustration in Fig. 1.1) Formally, the definition of the type-1 line graph L1 (M) of a multigraph M is the graph G whose vertices are the edges of M and two are adjacent if they have at least one vertex in common. An example is shown in Fig. 8.12. Note that under this definition, the line graph of a multigraph does not have multiple edges between vertices, and so it is just a graph in our terminology. The following result is due to King et al. [116]. Theorem 8.22 Let M be a multigraph for which its type-1 line graph L1 (M) has maximum degree Δ and clique number ω. Then its chromatic number satisfies these bounds: 

ω+Δ+1 . ω ≤ χ(L1 (M)) ≤ 2

e

e L(M)

f

Fig. 8.11 The line graph of two parallel edges

f

8.5 Extensions and Variations

125

M:

L(M):

Fig. 8.12 The line graph of a multigraph

Among the facts known about this parameter is the following nice result of Alon et al. [5] for graphs. Theorem 8.23 For almost all Δ-regular graphs G, χ(L(G)) ≤ Δ + 2. Among the various conjectures on this subject that have been made, we draw attention to one by Reed [149]. He conjectured that the upper bound in Theorem 8.22 holds for all graphs, not just for the line graphs discussed here. Conjecture  8.5 If G is a graph with maximum degree Δ and clique number ω, then χ(G) ≤ ω+Δ+1 . 2 An open problem such as this, with much known for line graphs but not in general, seems to be an appropriate place for us to leave the topic of coloring for now.

Chapter 9

Distance and Transitivity in Line Graphs

9.1 Introduction This chapter concludes Part I of our book, the material on line graphs in their own right. The remainder of the book is on line digraphs and other variations of line graphs. Here, the focus is on local relationships between the elements of a line graph. Beginning with the distance between pairs of vertices, we investigate for each vertex the greatest distance that it is from any of the other vertices; thus, this is in the spirit of the maximum of minimum path lengths. Following geometric concepts and terminology, we then turn to the minimum and maximum over the vertices of these path lengths, the radius and diameter of the graph. This then leads to the concept of the center of a line graph. Results are found on the range of values that the radius and diameter of a line graph can have, and how they are related to those of the original graph. Some interesting facts about these parameters for iterated line graphs are also found. Still following geometric terminology, the center of a graph is also defined in a natural way, and some intriguing results on the center of a line graph are found. The other topic in this chapter involves the edges of a line graph more specifically. The key question is this: For which graphs can a given line graph be arranged as a partially ordered set? In graph theory terms, the question becomes this: In which line graphs can the edges be oriented so that the result is a transitive ordering? The answer turns out to be describable in terms of forbidden subgraphs.

9.2 Distance For vertices v and w in a graph G, the distance from v to w, denoted dist(v, w), is the length of a shortest v–w path. It is straightforward to show that in a connected graph this is a metric; that is, for all vertices, u, v, and w, (a) dist(v, w) ≥ 0 with © Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_9

127

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9 Distance and Transitivity in Line Graphs

equality if and only if v = w, (b) dist(v, w) = dist(w, v), and (c) dist(u, w) ≤ dist(u, v) + dist(v, w) (the triangle inequality). Naturally, the distance from one given vertex to another given vertex is of great interest, but so is the farthest that any vertex is from a given one. For convenience, we adopt the usual convention that for two sets of vertices S and T in a graph G, dist(S, T ) is the minimum distance between a vertex in S and a vertex in T . In particular, if edge e = vw, then dist(u, e) denotes the distance from a vertex from u to e and equals dist({u}, {v, w}). The following observations connect distances in a line graph with distances in its root graph. Theorem 9.1 (a) If u and v are vertices and e = vw is an edge in a connected graph G, and if dist(u, v) = d, then dist(u, e) = d or d − 1. (b) If e = vw and f = xy are two edges in a connected graph G, then their distance in L(G) is dist(e, f ) = 1 + dist({v, w}, {x, y}). One of the interesting aspects about this concept is the range of the distances from a given vertex that the other vertices are. Formally, we make the following definitions for a connected graph G: The eccentricity ecc(v) of a vertex v in G is the greatest distance than any of the vertices is from v. The radius of G, rad(G), is the minimum eccentricity among the vertices of G, and the diameter of G, diam(G), is the maximum such eccentricity. It is straightforward to see that for every connected graph G, rad(G) ≤ diam(G) ≤ 2 · rad(G). Eccentricity We now look at how the eccentricity of a vertex in a graph G may be related to that of an incident edge. This question was first explored by Knor, Niepel, and Šoltés [119]. Clearly, by the triangle inequality, the eccentricities of two adjacent vertices in a graph can differ by at most 1. It turns out that similar statements hold for ecc(v) and ecc(vw) in the corresponding graphs. Figure 9.1 shows five examples, all of order six. In the first three, the two vertices of an edge have equal eccentricity, while in the last two, they differ by 1. More specifically, in graphs G1 and L(G1 ), all three eccentricities, ecc(v), ecc(w), and ecc(vw), are 3, while for G2 and G3 , the first two are equal while the third is one less and one more, respectively. Furthermore, for G4 and G5 , the eccentricity of the edge vw equals that of one of the vertices but not of the other. The following theorem summarizes the possibilities of the relationships between the eccentricities of two adjacent vertices in a graph and that of the edge joining them in the line graph.

9.2 Distance

129 vw v

w L(G1 ) :

G1 :

v L(G2 ) :

G2 :

vw

w

v

vw

w

L(G3 ) :

G3 :

v

G4 :

w

vw

L(G4 ) : vw

v w

G5 :

L(G5 ) :

Fig. 9.1 Eccentricity in line graphs

Theorem 9.2 Let G be a connected graph of order at least 3, let e = vw be an edge of G, and assume that ecc(v) ≤ ecc(w). The possibilities for the three eccentricities ecc(v), ecc(w), and ecc(e) are the following: (a) ecc(v) = ecc(w) = ecc(e); (b) ecc(v) = ecc(w) = ecc(e) + 1;

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9 Distance and Transitivity in Line Graphs

(c) ecc(v) = ecc(w) = ecc(e) − 1; (d) ecc(e) = ecc(v) = ecc(w) − 1; (e) ecc(e) = ecc(w) = ecc(v) + 1. Proof Let e = vw be an edge of G. It follows from Theorem 9.1(a) that the distances of a vertex u to two of the three entities v, w, and e can differ by at most 1. Consequently, the eccentricities of the three can differ by at most 1. This means that at least two of three eccentricities must be equal, and the third must be within 1 of these two. By symmetry in the argument, we may assume that ecc(v) ≤ ecc(w), and from this it follows that one of the five possibilities must hold. The five pairs of graphs and line graphs in Fig. 9.1 show that all five cases are possible.   Radius and Diameter We now turn to results on the radius and the diameter of line graphs. We observe that the radius of graph G1 in Fig. 9.2 is 3 while that of its line graph L(G1 ) is 2. On the other hand, the radii of G2 and its line graph L(G2 ) are reversed, that is, 2 and 3 respectively. It was shown by Knor, Niepel, and Šoltés [120] that these differences are extreme, and in fact, a similar result holds for the diameter. Perhaps the most interesting result on the radius of a line graph is that it cannot differ by more than 1 from that of its graph, and the same holds for the diameter. Theorem 9.3 Let G be a non-null connected graph. (a) rad(G) − 1 ≤ rad(L(G)) ≤ rad(G) + 1. (b) diam(G) − 1 ≤ diam(L(G)) ≤ diam(G) + 1. Furthermore, all of these bounds are sharp. Fig. 9.2 Bounds on the radius and diameter of a line graph

G1 :

L(G1 ):

G2 :

L(G2 ):

9.2 Distance

131

As noted above, the diameter of a connected graph lies between the value of the radius and twice that value. Niepel et al. [144] showed that these bounds can be extended to tighter bounds on for iterated line graphs. Recall that the term prolific applies to those connected graphs for which the iterated line graphs increase in size, that is, those that are other than a path, a cycle, or the claw. Theorem 9.4 If G is a prolific graph, then there exist positive constants c1 and c2 for which, for all k, rad(Lk (G)) +

  2 log2 k + c1 ≤ diam(Lk (G)) ≤ rad(Lk (G)) + 2 log2 k + c2 .

Returning to the diameter on its own, Niepel et al. [144] established interesting results on the diameter of iterated line graphs. For instance, the d-dimensional cube Qd with d ≥ 3 has diameter d, and for all k ≥ 2, diam(Lk (Qd )) = diam(L(Qd )) + k − 2. Another example is that for n ≥ 6, diam(Lk (Kn )) = diam(L(Kn )) + k. As it happens, these two examples are the extreme possibilities for the kth iterated line graph, as the following theorem states. Theorem 9.5 If G is a graph with minimum degree at least 3 and if G is neither K4 nor K5 , then for all k ≥ 2, diam(L(G)) + k − 2 ≤ diam(Lk (G)) ≤ diam(L(G)) + k. Recall that the diameter of a line graph can be less than that of the graph itself, even for prolific graphs. It is therefore interesting that, for k sufficiently large, the diameter of Lk+1 (G) is always exactly 1 more than that of Lk (G), a result proved in [144]. Theorem 9.6 If G is a prolific graph, then there exists k0 such that for all k ≥ k0 , diam(Lk+1 (G)) = diam(Lk (G)) + 1. Centers A vertex v in a graph G is called central if its eccentricity is equal to the radius of G, and the center Ctr(G) of G is the subgraph induced by the set of its central vertices. Every graph is the center of some graph, a fact that readily be shown by the following construction. Given a graph G, form H by adding two new edges, each with one vertex joined to all of the vertices of G and the other having only the one adjacency. An example is shown in Fig. 9.3. Then in H every vertex of G has eccentricity 2, while each of the four newly added vertices has eccentricity either 3

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9 Distance and Transitivity in Line Graphs

H:

G:

Fig. 9.3 Construction of a graph with a given center y3

y2 x3

x2

v1

w3

w2 v3

v2 H:

w1 x1 y1

Fig. 9.4 Illustration of proof of Theorem 9.7

or 4. Our next theorem, also due to Niepel et al. [144], gives a similar result for line graphs. Theorem 9.7 If G has at least three vertices and at least one edge, then there is a connected graph H for which L(G) is the center of L(H ). Proof Given G, let V (G) = {v1 , v2 , . . . , vn }. Now, for i = 1, 2, . . . , n, form H by adding n paths wi xi yi to G and joining wi to all vj except for j = i. Let e = vj vk be any edge of G and let f = xi yi for some i. By the construction, wi is adjacent to either vj or vk as well as to xi and so dist(e, f ) = 3. On the other hand, if e = vi wj (so i = j ) and f = xi yi , then dist(e, f ) = 4, and hence the eccentricity of any edge not in G is at least 4. Consequently, Ctr(L(H )) = G.   A connected graph G in which all of the vertices have the same eccentricity is called self-centered . Such graphs have an important role in the next theorem. It addresses the question of when the two operations of taking the line graph of a graph and forming its center commute. A key part of the following theorem (whose proof is omitted here), involves constructions similar to those in Theorem 9.7 (see Fig. 9.4).

9.3 Transitive Orientations of Line Graphs

133

Theorem 9.8 Let G be a graph without isolated vertices. (a) If rad(L(G)) = rad(G) + 1, then Ctr(L(G)) = L(Ctr(G)) if and only if G is self-centered. (b) If rad(L(G)) = rad(G) or rad(G) − 1, then there is a graph H having G as its center and for which Ctr(L(H )) = L(Ctr(H )). Determining the minimum order of a graph H satisfying part (b) of the theorem remains an open problem, although it is shown in [144] that if G is bipartite, then H requires at most six more vertices. We conclude this section with some results on centers in iterated line graphs. A graph G is called k-centerable if there exists a connected graph H for which Lk (G) is the center of Lk (H ). As noted earlier, every graph is 0-centerable and in Theorem 9.7 almost all graphs are 1-centerable. Knor, Niepel, and Šoltés [119] show that this can be extended to k = 2 and partially to k = 3. Theorem 9.9 (a) Every graph G for which L2 (G) has at least one edge is 2-centerable. (b) Every triangle-free graph G for which L3 (G) has at least one edge is 3centerable.

9.3 Transitive Orientations of Line Graphs An orientation of a graph G is an assignment of a direction to each edge of G, and an orientation is called transitive if for all vertices u, v, and w, if arcs uv and vw are present, so is the arc uw. Clearly every complete graph has a transitive orientation, simply as a complete order. The name comparability graph is given to any graph that has a transitive orientation. The reason for this is because such an oriented graph connects pairs of elements that are comparable to each other in a partial order. We note that every bipartite graph is a comparability graph—simply orient all of the edges from one partite set to the other. In particular, every path and every even cycle is a comparability graph, while it is easy to see that no odd cycle of length 5 or more is. The goal of this section is to determine precisely which graphs have line graphs that are comparability graphs, the solution to which was determined by Petkovšek [146] and which will be given here. One example is the complete graph K4 , whose line graph is the octahedron K2,2,2 . Figure 9.5 shows a transitive orientation of this graph. A simpler example, but one that can be generalized, and in that form will be a significant part of the solution, is the line graph of K2 · K3 , the graph obtained by identifying one vertex of K2 with one of K3 (this is sometimes called a musical triangle. Its line graph is the diamond (two triangles with a common edge). In this vein, we note that the diamond K1,1,2 (that is, K4 with one edge removed) has three

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Fig. 9.5 A transitive orientation of the octahedron

Fig. 9.6 The transitive orientations of the diamond

transitive orientations, as shown in Fig. 9.6. (Note that the last two are converses of one another, while the first is self-converse.) In contrast, Fig. 9.7 shows six graphs, none of which is a comparability graph. This can be seen using ad hoc methods, say by starting with a transitive orientation of a triangle and then deducing the impossibility of extending this to the entire graph. Such observations can be useful, both in finding transitive orientations and in proving that none exist. The six graphs in Fig. 9.8 are the root graphs of the six line graphs in Fig. 9.7. Since every induced subgraph of a comparability graph must also have that property, and since the line graph of a subgraph of a graph G is an induced subgraph of L(G), it follows that no graph containing any of the graphs in Fig. 9.8 can have a comparability line graph. As we shall show, these six constitute the basic set of such graphs. In addition to the diamond, there are some other graphs that are important to the solution to our problem, one being the bow-tie (two triangles with a common vertex). For k ≥ 0, we define two graphs, the -path graph Pk , obtained from the path with k vertices Pk by attaching a triangle at one end, and the double--path graph Pk , obtained from Pk by attaching a triangle at each end. Examples are shown in Fig. 9.9. Note that P1 is just K3 and P1  is the bow-tie.

9.3 Transitive Orientations of Line Graphs

G1 :

G3 :

G5 :

135

G2 :

G4 :

G6 :

G6 : Fig. 9.7 Six line graphs that are not comparability graphs

To trim a graph means to delete all of its end-vertices (if any), which here will be called bristles. A caterpillar is a graph that when trimmed is a path, and we call a graph a bristly cycle if trimming it results in a cycle. Theorem 9.10 Let G be a connected graph whose line graph is a comparability graph. If F is the result of trimming G, then F must be one of the following: (a) (b) (c) (d) (e) (f)

a path, an even cycle, K4 , a diamond, a -path, a double--path.

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F1 :

F2 :

F3 :

F4 :

F5 :

F6 :

Fig. 9.8 The root graphs of the six line graphs in Fig. 9.7

Δ P3 :

Δ P2 Δ : Fig. 9.9 A -path graph and a double--path graph

9.3 Transitive Orientations of Line Graphs

137

Proof Assume that G is a connected graph whose line graph L(G) is a comparability graph, and let F be the result of trimming G. It follows from an earlier observation that L(F ) is also a comparability graph. After checking graphs with three or fewer vertices, we may assume that F has order at least 4. We proceed by considering several cases that depend on cycles in F . Case 1 F has no cycles. It follows that F , and therefore G, must be a tree. Since a tree that is not a caterpillar must contain the excluded graph F1 (see Fig. 9.8), it follows that G is a caterpillar, and hence that F is a path. Case 2 The only cycles in F are 3-cycles (and F has a cycle). It is straightforward to verify that if F has three (or more) 3-cycles, it must contain either F1 or F6 , both of which are prohibited. Consequently, F has either one triangle or two. Let C be a triangle in F . Since F4 is excluded as a subgraph, at least one vertex of C must have degree 2. If only one has (and since F has only 3-cycles), then it follows from a simple argument that G must contain F6 . Hence, two of the vertices of C must be of degree 2 (in F ). Therefore, F must be either a -path graph or a double--path graph. Case 3 The longest cycle in F is a 4-cycle. Let C be such a cycle. Since F2 is excluded from G, F cannot have any vertices other than those on the cycle C. Hence, F is either C4 , K4 , or the diamond. Case 4 F has a cycle C of length greater than 4. Since cycles of odd length greater than 3 are prohibited, C must have even length. Since F1 is prohibited from G, no vertex of G can be at distance 2 from C. Thus, if F has a vertex v not on C, it must be adjacent to two vertices, say u and w, on C (and only to vertices on C). This implies that F contains either F2 (if u and w are at distance 2 apart) or F1 (if at distance 4 or more), so there cannot be such a vertex v. Similarly, if there is a chord of C in F , then there is a copy of either F1 or F2 . Hence, F can only be a cycle in this case, which completes the proof of the theorem.   Our next result involves adding bristles to graphs that have comparability line graphs. Theorem 9.11 Let G be the root graph of a comparability line graph. If a line graph L(G) is a comparability graph, and if G has a bristle at a vertex v, and if H is obtained from G by adding more bristles at v, then L(H ) is also a comparability graph. Proof Let edge e0 be a bristle at v in G, and suppose that H has r more bristles at v than G has. Then the line graph L(H ) consists of L(G) together with r more vertices e1 , e2 , . . . , er , each with the same neighbors as e0 , along with all adjacencies between e0 , e1 , . . . , er . Now let L(H ) be oriented as follows, beginning with a transitive orientation on the subgraph L(G) and a transitive orientation on the complete subgraph on e0 , e1 , . . . , er . For each ei with 1 ≤ i ≤ r, orient its other edges just as those at e0 are. Now suppose that the resulting orientation is

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not transitive, that is, L(H ) has arcs ab and bc for which the arc ac is not present. It follows that we may assume that (up to converses) for some i and j , a = ei , b = ej , and c is an edge of G different from e0 . However, by our construction, arc ei c is present if and only if ej c is. Hence, the prescribed orientation of L(H ) is transitive.   We are now in a position to state and prove the characterization theorem of Petkovšek [146]. Theorem 9.12 The line graph of a connected graph G is a comparability graph if and only if G is one of the following: (a) (b) (c) (d)

a caterpillar, an even cycle, possibly with bristles at its vertices, K4 , a diamond, possibly with bristles at the vertices of degree 2 or at the vertices of degree 3, but not both, (e) a graph consisting of a triangle with possibly a caterpillar at one vertex and bristles at another (and nothing at the third), (f) a graph consisting of two triangles joined by a caterpillar (possibly degenerate) and possibly with bristles at one other vertex of each triangle.

Proof We first show that every graph G with a transitively orientable line graph is of one of these six types. In each case, we begin with the trimmed graph G , assuming that G is connected and has order at least 3, and consider the six cases of the theorem. If G is a path, then G is a caterpillar by definition. If G is an even cycle, then G is an even cycle, perhaps with bristles. If G is K4 , then so is G since F6 is excluded as a subgraph. If G is a diamond, then G must be a diamond, perhaps with bristles. However, since F4 is excluded, there cannot be bristles at both a degree-2 and a degree-3 vertex. (e) and (f) If G is a -path graph or a double--path graph, then G is such a graph with bristles. However, again because F4 is prohibited, one of the vertices on each triangle must have degree 2.

(a) (b) (c) (d)

We now show that the line graph of each of these graphs has a transitive orientation. By virtue of Theorem 9.11, we do not need to consider more than one bristle at any vertex. On the other hand, since the property of a graph having a comparability graph as its line graph is a property of all of its subgraphs, it suffices to show that in each family, a graph with one bristle at each eligible vertex has a line graph of the desired type. We start with an eligible trimmed graph of each type.

9.3 Transitive Orientations of Line Graphs

139

Fig. 9.10 A caterpillar with its line graph oriented transitively

Fig. 9.11 A bristled even cycle with its line graph oriented transitively

(a) Paths. In Fig. 9.10, we show this for the case where the trimmed graph is the path P4 and there is one bristle at each of its four vertices. From this it is easy to see the general case. (b) Even cycles. Similarly, Fig. 9.11 shows the 4-cycle with one bristle at each vertex and a transitive orientation of its line graph. Again, the general case of an even cycle C2r can be deduced from a cycle of 2r triangles. (c) The complete graph K4 . As noted earlier, its line graph is the octahedron, and Fig. 9.5 shows a transitive orientation. (d) The diamond K1,1,2 . In this case there are two graphs to be considered, the diamond with one bristle at each vertex of degree 2, and the diamond with one bristle at each vertex of degree 3. These graphs and transitive orientations of their line graphs are shown in Fig. 9.12. (e) -path graphs. Figure 9.13 shows a small example, from which the result for both larger and smaller graphs can easily be deduced. (f) Double--path graphs. Similarly, Fig. 9.14 shows two examples, the first beginning with the bow-tie, and the second the next case up. The result for any longer path between the triangles follows in a straightforward way. (Note the common subgraphs in the second case here and the example in Fig. 9.13.)  

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Fig. 9.12 Two bristled diamonds with their line graphs oriented transitively

Fig. 9.13 A bristly -path graph with its line graph oriented transitively

9.3 Transitive Orientations of Line Graphs

Fig. 9.14 Two bristled double--path graphs with their line graphs oriented transitively

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Part II

Line Digraphs

Chapter 10

Fundamentals of Line Digraphs

10.1 Introduction One of the delights of mathematics is the extension of a concept to new and different objects. Among these is the creation of a version of the line graph to directed graphs (called digraphs). The most natural and productive way of doing this is to consider the line graph operation as shortening paths: a path of length 2 in a graph yields a path of length 1 in the line graph. Consequently, we take the line digraph operation on a digraph to follow the same process: a path of length 2 in a digraph yields a path of length 1 in the line digraph. This turns out to be a rich concept, with many interesting similarities and differences. The two concepts differ in some very interesting ways. For example, unlike in the undirected case where cycles are the only connected graphs that are isomorphic to their line graph, there are many other connected digraphs isomorphic to their line digraph. Furthermore, there are pairs of digraphs with each digraph isomorphic to the other’s line digraph. As will be discussed later, the theory of line digraphs originated in 1960 by Frank Harary and Robert Norman. Many ideas and results have developed since then. In the next section, the basic notation and terminology of digraphs will be outlined in preparation for some of the elementary concepts of line digraphs. These concepts include directed versions of connectedness, degrees, paths and cycles, and distance. In Sect. 10.3, a variety of fundamental properties are established. In the subsequent sections, two isomorphism questions will be answered: When do different digraphs have isomorphic line digraphs? When is a digraph isomorphic to its line digraph? The chapter concludes with some results on the spectra of line digraphs.

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_10

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10.2 Basic Definitions Before making a formal definition of line digraph, we provide some of the terminology and notation that will be used here. Whereas a graph G consists of a set V of vertices and a set E of edges, a digraph D consists of a set V of vertices and a set A of arcs, the difference being that an arc is an ordered pair of vertices. Because there will be little chance for confusion, we use the same notation for arcs as for edges: if the arc a is the ordered pair of vertices v and w, we write a = vw, and (using the terminology of arrows for direction) say that w is the head of a and v is its tail. Unlike the situation with graphs, we allow digraphs to have loops (arcs with both vertices the same). Thus, in this case, in set-theory terms, a digraph is the graph of a finite relation on a set. Occasionally however, it will be convenient to allow more than one arc to have the same ordered pair of vertices (including loops), in which case these arcs are called parallel. Continuing with terminology and notation, the out-degree of a vertex v is the number of arcs of which it is the tail and is denoted d + (v), while the in-degree is the number of arcs of which it is the head and is denoted d − (v). Based on whether the in- and out-degrees are zero or positive, we classify vertices as being of one of four types. An isolated vertex (or isolate) is a vertex with no incident arcs; a source has positive out-degree but no in-coming arcs; a sink is the opposite, positive in-degree but no out-going arcs; and a juncture has both in-coming and out-going arcs. In addition, an arc of which vertex v is the tail is called an out-arc of v and the set of such arcs is denoted A+ (v). The corresponding concept for v being the head of the arcs are in-arc and A− (v). Also, if vw is an arc, then w is called an outneighbor of v and v an in-neighbor of w. The corresponding sets of vertices are denoted N + (v) and N − (w). A digraph D is connected if its underlying undirected graph is connected, and is strongly connected or strong if for any two vertices v and w, there are paths from v to w and from w to v. A subdigraph of D is a (connected) component if it is a maximal connected subdigraph and a strong component if it is a maximal strong subdigraph. Reversal of the arcs in a digraph D results in the converse D defined formally as having the same set of vertices as D but with an arc vw if and only if wv is in A. Distance is a concept that is particularly useful in strong digraphs because of its universality there (appropriate allowances can be made for other digraphs also, but we will not have need for them). In a strong digraph D, the distance from vertex v to vertex w, denoted d(v, w) is the length of a shortest v–w path. The diameter diam(D) of D is the maximum distance from any vertex to any other. For the corresponding least values, there are two parameters of interest. The out-radius rad+ (D) is the minimum distance required for some vertex to reach all of the others, while the in-radius rad− (D) is the minimum distance required for some vertex to be reached by all of the others.

10.2 Basic Definitions a

147 b

L

a

b

Fig. 10.1 The line digraph operation

Fig. 10.2 Small connected digraphs

D = L(D):

Fig. 10.3 An example of a line digraph isomorphic to itself

With that background material, we are now prepared for our key definition. Let D be a digraph with at least one arc. The line digraph of D, denoted L(D), has the arcs of D as its vertices, with an arc in L(D) from vertex a to vertex b if the head of a is the tail of b; that is, if a = vw and b = xy, then ab is an arc in L(D) if and only if w = x. This operation is illustrated in Fig. 10.1 for the basic case in which D is the path of length 2. In contrast to the undirected case in which by definition there is only one connected graph with two edges, there are six connected digraphs with two arcs. In addition to the one in Fig. 10.1, there are the five in the next figure. Applying the line digraph operation to these, we discover that the result for the first two is just a pair of isolated vertices, while for the remaining three, each is isomorphic to the digraph itself. Thus, those examples illustrate interesting facts themselves, contrasting with the undirected line graph operation: (a) the line digraph of a connected digraph may be disconnected; (b) it is not only cycles for which a digraph and its line digraph may be isomorphic (Fig. 10.2). Another example of a line digraph is shown in Fig. 10.3, which also has the interesting feature of being isomorphic to the original digraph itself.

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D1 :

L(D1 ):

D2 :

D3 :

L(D2 ):

L(D3 ):

Fig. 10.4 Line digraphs of properties of relations

Having noted earlier that digraphs are the same as a relation on a finite set, we conclude this section with the line digraphs of three types of relations in Fig. 10.4, the first is reflexive and symmetric, the second symmetric and irreflexive relation, and the third transitive and without cycles.

10.3 Fundamental Properties We begin this section with some simple observations about line digraphs, including the number of arcs at each vertex and thus the total number of arcs.

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Theorem 10.1 Let D be a digraph with n vertices and m arcs (m, n ≥ 1). (a) If a = vw is an arc of D, then the out-degree of a in L(D) is d + (w) and the in-degree is d − (v).  (b) L(D) has m vertices and v d + (v)d − (v) arcs. Proof (a) Clearly, given arc a in D, each arc ba in L(D) comes from an arc b = uv in D, and hence the in-degree of a in L(D) must equal the in-degree of v in D. The analogous fact of course holds for the out-degree of v, and this completes the proof of (a). (b) By definition, L(D) has m vertices and each arc comes uniquely from one arc into a vertex v and one vertex out of the same vertex v. Hence, the vertex v   contributes precisely d + (v)d − (v) arcs to L(D), and (b) follows. As the definition of a line digraph suggests, and as is supported by the examples given, paths and cycles feature prominently in the study of line digraphs. Another family of subgraphs that is extremely important (as the proof of the previous theorem suggests), consists of orientations of complete bipartite graphs. We define a digraph to be bicomplete if it is the result of orienting all of the edges of a complete bipartite graph from one of the partite sets to the other. The bicomplete digraph −−→ having all of its arcs going from the set of r vertices to the set of s is denoted Kr,s . −→ Now consider a digraph, which we denote Sr,s , consisting of a central vertex v along with r in-arcs and s out-arcs. The significance of bicomplete digraphs, analogous to complete graphs for line graphs, is that the line digraph of the arcs at a vertex is a −−→ −→ bicomplete digraph. Figure 10.5 shows S2,3 and K2,3 as its line digraph. The next theorem summarizes these basic line digraphs. Theorem 10.2 For digraphs without isolated vertices, the following hold: ∼ Pn+1 . ∼ Pn if and only if D = (a) L(D) = ∼ Cn if and only if D ∼ (b) L(D) = = Cn . −−→ −→ ∼ (c) L(D) = Kr,s if and only if D ∼ = Sr,s . As noted earlier, when we speak of paths, cycles, and other types of walks when discussing digraphs, unless indicated otherwise, we mean directed walks. We also use terminology that is common for the various types of walk: a path has no vertices

S2,3 :

Fig. 10.5 The line digraph of an oriented star

K2,3 :

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10 Fundamentals of Line Digraphs

repeated; a cycle has at least one arc and the only repetition of a vertex are the first and last; a trail may have repeated vertices but has no repeated arcs, and it is closed or open depending on whether the first and last vertices are the same or different; and a closed trail with at least one arc is called a circuit. The following theorem describes when a line digraph has a path or a cycle of a given length. Theorem 10.3 Let D be a digraph with at least one arc. (a) The arcs of an open trail of length k in D are the vertices of a path of length k − 1 in L(D) and conversely. (b) The arcs of a circuit of length k in D are the vertices of a cycle of length k in L(D) and conversely. Proof It follows that since a circuit D has no repeated arcs, the corresponding vertices in L(D) form a cycle. Furthermore, the arcs in D that correspond to the vertices of a cycle in L(D) must form a closed walk with none of its arcs repeated, that is, those arcs must form a circuit. This proves (b), and the same argument can be modified to prove (a).   We conclude this section with a few results on strongly connected digraphs. We assume that all digraphs here are nontrivial. Theorem 10.4 If D is a strong digraph, then so is L(D). Proof Let D be a strong digraph and let a = vw and b = xy be arcs in D. Since D is strong, there is a shortest w–x path P in D. It follows that the arc a followed by the path P and the arc b is a trail, and so by Theorem 10.3, taking the line digraph yields an a–b path in L(D). Since a and b were arbitrary, the result follows.   Aigner [2] proved the following nice result on the diameter of a strong line digraph. Theorem 10.5 If D is a strong digraph other than a cycle, then diam(L(D)) = diam(D) + 1. Proof Let D be a strong digraph that is more than a cycle, and let k be its diameter and let l be that of L(D). We first show that l ≤ k + 1. Let Q = a1 a2 . . . al+1 be a shortest path from a1 to al+1 in L(D). If, for i = 1, 2, . . . , l, ai = vi−1 vi , then dist(v1 , vl ) = l − 1. Hence, k ≥ l − 1; that is, l ≤ k + 1 as desired. To show the reverse inequality, we begin with vertices v0 and vk that satisfy the diameter of D with the path P = v0 v1 . . . vk . Since D is strong, there must exist arcs uv0 and vk w. If u = w, then with a0 = uv0 , ai = vi−1 vi for i = 1, 2, . . . , k, and ak+1 = vk w, a0 a1 . . . ak+1 is a shortest path from a0 to ak+1 in L(D), and hence l ≥ k + 1 in this case. On the other hand, if u = w, then this vertex together with the path P form a cycle C = v0 v1 . . . vk wv0 of length k + 1. Since D is strong and not itself a cycle, there must be another arc into a vertex of the cycle. Starting with that arc in the line digraph L(D) and traversing the cycle results in a shortest path of length k + 1

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from one vertex to another, and so again l ≥ k + 1. Therefore, a combination of our results yields the theorem.   Aigner also established a corresponding result for the radii of strong digraphs, for which the proof is similar. We note that two values are possible in general here. Theorem 10.6 If D is a strong digraph, then (a) rad+ (D) ≤ rad+ (L(D)) ≤ rad+ (D) + 1; (b) rad− (D) ≤ rad− (L(D)) ≤ rad− (D) + 1. The next question that we consider is whether a line digraph is Hamiltonian or not. It turns out that it is quite easily answered. Theorem 10.7 The following statements are equivalent for a connected digraph D. (1) L(D) is Hamiltonian. (2) D is Eulerian. (3) Every vertex of D has equal in- and out-degree. Proof Let D be a connected digraph with m arcs. By the previous theorem, L(D) has a cycle of length m if and only if D has a circuit of length m. In other words, L(D) is Hamiltonian if and only if D is Eulerian. This shows that (1) and (2) are equivalent. That (2) and (3) are equivalent is just the basic characterization theorem for Eulerian digraphs.   This raises a related question: Which digraphs have Eulerian line digraphs? Not surprisingly, it too has an easy answer in terms of degrees. Theorem 10.8 The line digraph L(D) of a connected digraph D is Eulerian if and only if for every arc vw in D, d + (w) = d − (v).

10.4 Digraphs with Isomorphic Line Digraphs The two digraphs D1 and D2 in Fig. 10.6 are quite difference in appearance, but their line digraphs are isomorphic, as the figure indicates. Another example of two nonisomorphic digraphs with the same line digraph is given in Fig. 10.7. The key idea to this is perhaps clearer in this figure, where the main differences are in the vertices with either in-degree or out-degree 0. As the next theorem states, basically, it is only in such vertices that digraphs with the same line digraph can differ. We use the notation of A+ (v) for the set of arcs going out of vertex v and A− (v) for the set of those arcs going into v. Also, we point out that the green 5-cycles in the two digraphs in Fig. 10.7 is a foreshadowing of a key concept that we describe next. If D is a digraph with at least one juncture, then the subgraph induced by all  Our next result appeared in of its junctures is called its core and is denoted D. Harary and Norman’s original paper [93]; the proof we give follows the lines of that

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D1 :

D2 :

L(D1 ) = L(D2 ):

Fig. 10.6 Digraphs with isomorphic line digraphs

F1 :

F2 :

L(F1 ) = L(F2 ):

Fig. 10.7 Another pair of digraphs with isomorphic line digraphs

of Hemminger [100]. A simple example is provided by the digraphs F1 and F2 in Fig. 10.7, where the core of each is just the cycle. Theorem 10.9 If two digraphs have isomorphic line digraphs, then their cores are either both empty or isomorphic. Proof Let D and F be digraphs with nonempty cores, and assume that φ is an isomorphism from L(D) onto L(F ). We consider φ to be also a one-to-one mapping in the natural way from the arc-set of D onto the arc-set of F . Let v be a vertex in the core of D, and let a be an in-arc of v and b an out-arc. Then a → b in L(D), and so φ(a) → φ(b) in L(F ). Consequently, in F the head of φ(a) is the tail of . If c is any other arc out of v, then a → c φ(b). This vertex, call it v , is thus in F

10.4 Digraphs with Isomorphic Line Digraphs

153

in L(D), and hence φ(a) → φ(c). Consequently, φ is a one-to-one mapping that takes the arcs out of v to the arcs out of v , that is, φ(A+ (v)) ⊆ A+ (v ). By the same argument, φ −1 is a one-to-one mapping from A+ (v ) to A+ (v), and therefore, φ(A+ (v) = A+ (v ). Since, for different vertices v and w, A+ (v) = A+ (w), φ yields a one-to-one  to V (F ), where φ ∗ (v) = v . Furthermore, by symmetry mapping φ ∗ from V (D) ), there is a vertex u in V (D)  for which in the argument, for each vertex x in V (F −1 + + + + φ ((A (x)) = A (u), that is, for which φ(A (u) = A (x). Therefore φ ∗ is onto as well as one-to-one. Finally, we show that φ ∗ preserves both adjacency and non-adjacency. Let v and  If D contains the arc a = vw and if a = φ(a), then, since w be vertices in D. + a is in both A (v) and A− (w), a is in both A+ (φ ∗ (v)) and A− (φ ∗ (w)), that is, φ ∗ (v) → φ ∗ (w). This argument also works in the opposite direction: if x and y are  and x → y, then (φ ∗ )−1 (x) → (φ ∗ )−1 (y) in D.  Thus, D ∼ , which vertices in F =F completes the proof.   Even though a line digraph can have multiple root digraphs, one of them can be chosen as canonical. We define a digraph to be fundamental if every source has outdegree 1, every sink has in-degree 1, and there are no isolated vertices. We let F be the set of all fundamental digraphs. Theorem 10.10 For every non-null digraph D, there is exactly one fundamental digraph F with L(F ) ∼ = L(D). Proof Let D be a non-null digraph. We first consider the case in which its core is empty. Then each of its arcs goes from a source to a sink, and so L(D) is a null digraph. If L(D) has order m, let F be the digraph consisting of m disjoint arcs. Then clearly L(D) ∼ = L(F ). Furthermore, any digraph other than F with m arcs and no isolated vertices either has a vertex v with both d + (v) and d − (v) positive or with one of them 0 and the other greater than 1. Hence, F is the only digraph in F with L(F ) ∼ = L(D). Now we consider the case in which the core of D is non-empty. We form a digraph F from D in two stages: First, we replace each source v and its arcs vw1 , vw2 , . . . , vwk with k new vertices v1 , v2 , . . . , vk and arcs v1 w1 , v2 w2 , . . . , vk wk . Following this, we make the analogous replacements of the sinks. Then clearly L(F ) ∼ = L(D). Furthermore, by Theorem 10.9, any other  with arcs corresponding to digraph H for which L(H ) ∼ = L(D) must contain D those incident with sources and sinks. As in the first case, there is only one such digraph in F with the required number of source- and sink-arcs, so F is unique.   If we were to work in the more general setting of multidigraphs, then we could take the canonical form of the root digraph to have just a single source and a single sink (or none), which is the approach that Hemminger [100] took. The theorem has some corollaries that are worth stating. Corollary 10.1 If every vertex in digraphs D1 and D2 is a juncture, then L(D1 ) ∼ = L(D2 ) if and only if D1 ∼ = D2 .

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Corollary 10.2 For any strongly connected digraphs D1 and D2 , L(D1 ) ∼ = L(D2 ) if and only if D1 ∼ = D2 . We note that the first of these corollaries can be extended to include digraphs with at most one source and at most one sink (and of course no isolated vertices). The following result is a special case of this observation. Corollary 10.3 For any tournaments T1 and T2 , L(T1 ) ∼ = L(T2 ) if and only if T1 ∼ = T2 .

10.5 Connected Digraphs Isomorphic to Their Line Digraphs Figure 10.8 shows two connected digraphs, each of which is isomorphic to its line digraph. As was shown by Harary and Norman [93], there are many such digraphs, but all that have this property resemble these two. In contrast with this variety, every connected graph that is isomorphic to its line graph is a cycle. While describing all disconnected digraphs that are isomorphic to their own line digraph gets complicated, there are some basic ones worth noting, an example of which appears in Fig. 10.9. This is the representation of a function on a set of 16 elements, and what makes it particularly interesting is its being isomorphic to its line digraph. The same can of course be said of each of the connected components, and that is the essence of the next theorem. Figure 10.8 gives two digraphs with the linedigraph preservation property. As noted earlier, a digraph and its converse behave in much the same way, including that the line digraph L(D ) of the converse D

of a digraph D is the same as the converse (L(D)) of the line digraph of D. (In some of the graph theory literature, the terminology uses the prefix of ‘counter’, contrafunction.) Thus, all contrafunction digraphs as well as all function digraphs (and combinations on disjoint sets) have the property of being preserved under the line digraph operation. The following theorem formalizes these facts. ∼ D if and only if the outTheorem 10.11 A connected digraph D satisfies L(D) = degree of every vertex is 1 or the in-degree of every vertex is 1.

Fig. 10.8 Connected digraphs isomorphic to their line digraphs

D1 :

D2 :

10.5 Connected Digraphs Isomorphic to Their Line Digraphs

155

Fig. 10.9 The digraph of a function

Fig. 10.10 A disconnected digraph isomorphic to its line digraph

Proof Assume first that D is a digraph in which every vertex has out-degree 1. Let f : V (D) −→ V (L(D)) be the mapping taking vertex v to the arc a = vw in D. Then clearly v → w in D if and only if f (v) → f (w) in L(D), so L(D) ∼ = D. Clearly the same result holds if every vertex has out-degree 1. For the converse, assume now that D is a connected digraph with  n vertices and m arcs that is isomorphic to its line digraph. Then clearly n = v d + (v)d − (v). It follows that if D has no sinks, that is, if d + (v) ≥ 1 for every vertex v, then all vertices must have out-degree 0. In other words, D would then be the digraph of a function. By the analogous argument, D would be the converse of a function if it had no source vertices. Now suppose that D has both source and sink vertices. It must have a cycle since otherwise the length of a longest path in L(D) would be 1 less than that in D. Also, its underlying graph cannot have more than one undirected cycle since there are no more edges than vertices. Let C = v1 v2 . . . vk v1 be the cycle in D. Then there must be an in-arc and an out-arc at some vertices. If a vertex vi on C has one of each, say there are arcs uv1 and v1 w, then L(D) contains the arc uw (and hence D has a corresponding arc), which violates the number of arcs present. Hence, there cannot be both types of arc at one vertex. Thus (relabelling the vertices if necessary), we may assume that the closest out-arc not on C following an in-arc not on C are uv1 and vi w. However, the corresponding arcs in L(D) are 1 unit closer on the cycle (see Fig. 10.11). Hence, D cannot have both a source and a sink, and this completes the proof.   If the requirement of connectedness for the digraphs is removed, as noted earlier, the situation gets more complicated. This is illustrated by the digraph in Fig. 10.10.

156 Fig. 10.11 Two digraphs that are isomorphic to the other’s line digraph

10 Fundamentals of Line Digraphs

D1 :

D2 :

This topic is pursued further in Chap. 13, where we repeat the procedure of taking the line digraph and see cases in which one encounters a digraph that one had earlier in the process—in effect, one circles back to where one was before. Clearly then, the digraph consisting of the union of the digraphs in such a cycle is its own line digraph. This is illustrated in Fig. 10.11, where each of the digraphs is the line digraph of the other, and so their union has the preservation property.

10.6 Spectra of Line Digraphs In Sects. 4.2 and 4.3 we discussed basic properties of eigenvalues of graphs and, in particular, proved Theorem 4.5 which gives a relationship between the characteristic polynomials of a regular graph and its line graph. In this section, we discuss a similar and somewhat nicer result about digraphs. We first describe some notation and basic properties. In the following, we mostly follow notation and results from Brualdi [47]. Additional related results may be found in Balbuena et al. [17] and Fiol and Mitjana [78]. In this section we assume that digraphs have no loops or parallel arcs. Let D be a digraph with n vertices v1 , v2 , . . . , vn and m arcs a1 , a2 , · · · , am . The adjacency matrix A = A(D) = (aij ) of D is the n × n matrix in which aij is 1 if there is an arc with tail vi and head vj , and is 0 otherwise. Clearly, the ith row and the ith column sums equal the out-degree and the in-degree of the vertex vi . We define two additional associated matrices. The in-incidence matrix Bin = Bin (D) = (bij ) of D is the n × m matrix in which bij = 1 if vertex vi is the head of the arc aj , and bij = 0 otherwise. Similarly, the out-incidence matrix Bout = Bout (D) = (bij ) of D is the n × m matrix in which bij = 1 if vertex vi is the tail of the arc aj , and bij = 0 otherwise. It can be easily checked that Bout (D)BTin (D) = A(D). Figure 10.12 shows two digraphs and some of their associated matrices. As noted in Sect. 10.5, D1 and D2 are line digraphs of each other. Theorem 10.12 For a digraph D, BTin (D)Bout (D) = A(L(D)). Proof If D has n vertices v1 , v2 , . . . , vn and m arcs a1 , a2 , · · · , am , then BTin (D)Bout (D) is of order m × m. The (i, j )th element of this matrix is 1 if the head of the arc ai is the same vertex as the tail of the arc aj . The result follows.   It can be easily verified that in Fig. 10.12, BTin (D1 )Bout (D1 ) = A(D2 ).

10.6 Spectra of Line Digraphs

157

Fig. 10.12 Two digraphs and their associated matrices

Just as for graphs, the matrix terminology carries over to digraphs through the adjacency matrix. Thus, if D is a graph of order n and A is its adjacency matrix, then the characteristic polynomial of D is the characteristic polynomial φ(A; x) = det (xI − A) of the matrix A. Similarly, the eigenvalues and the spectrum of D are those of the adjacency matrix A of D. The next result shows that the characteristic polynomials of a digraph and its line digraph are nicely related. Theorem 10.13 If D is a digraph with n vertices and m arcs, then x n φ(L(D); x) = x m φ(D; x). Proof We denote by In the identity matrix of order n and by O a matrix of all zeros. Let     xIn −Bout In Bout P= and Q = . O Im BTin xIm

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Then     xIn − A(D) O O xIn PQ = and QP = Im BTin xBTin xIm − A(L(D)) It is well-known that det(PQ) =det(QP). Hence, we have x m det(xI − A(D)) = x n det(xI − A(L(D))), and the proof is complete.   As with other chapters, there is more, much more, that could be said. In this case, we mention one more extension, and refer the reader to relevant sources. Fiol and Lladó [77] and Fiol and Mitjana [78] have defined a generalization of the line digraph, the partial line digraph. An extension of Theorem 10.13 for partial line digraphs is obtained in [78]. A survey of the material in the three chapters of Part II may be found in [8].

Chapter 11

Characterizations of Line Digraphs

11.1 Introduction In this chapter, we turn to the characterization question: Which digraphs are line digraphs? Several answers are known, all published in the 1960s, and they vary considerably in nature: The first appeared in the first paper on the subject of line digraphs, published in 1960 by F. Harary and R. Z. Norman (as noted earlier), and involves partitions of the sets of arcs of the digraph. The second, due to C. Heuchenne, appeared 4 years later and involves subdigraphs whose existence necessitates the existence of additional arcs in the digraph. The basis of the third characterization, published by P. I. Richards in 1967, is the adjacency matrix of the digraph under consideration. For reasons of simplicity, we begin our discussion by allowing digraphs to have parallel arcs as well as loops. However, as we shall see, the line digraphs of these ‘multidigraphs’ do not even then have parallel arcs. In the next section, the corresponding characterizations of the line digraphs of special families are given, namely, relations and oriented graphs (in which no pairs of vertices have an arc in each direction). We conclude the chapter with forbidden digraphs in various families of line digraphs. These include relations (in which loops are allowed, but not parallel arcs), digraphs (in the usual sense of neither loops nor parallel arcs), and oriented graphs (digraphs with at most one arc between any pair of vertices).

11.2 Background As noted in the Introduction, two of the conditions equivalent to a digraph being a line digraph involve partitions and matrices, and so we set the stage for the theorem by developing those concepts here.

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_11

159

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11 Characterizations of Line Digraphs

Connection Digraphs The concept that we develop here involves an analogue of the intersection graph of a collection of subsets of a set. Since an arc in a digraph D is oriented, that is, it is an ordered pair of vertices, a vertex in the line digraph L(D) can be associated with two different sets of vertices in D, one consisting of the heads of the arcs at a given vertex, and the other consisting of the tails at a given vertex. Of course, some vertices may have only heads of arcs and some only tails. These ideas suggest the following general concept of pairs of subsets of any set X where one of the two sets in a pair (but not both) may be empty. A double partition of a finite set X consists of a collection of pairs of subsets {(S1 , T1 ), (S2 , T2 ), . . . , (Sr , Tr )} with each element of X in exactly one of the first sets of the pairs and in exactly one of the second sets. An example in which X = {1, 2, . . . , 7} is the four pairs of sets ({1, 2}, {1, 3}), ({4}, {2, 5}), ({3, 6, 7}, ∅), ({5}, {4, 6, 7}). Given a double partition, one can construct a digraph in a natural way, analogous to intersection graphs in the undirected case. The connection digraph of a double partition P = {(Sk , Tk ))rk=1 } has the pairs of P as its vertices vk , k = 1, 2, . . . , r with an arc from vi to vj for each element that is in both the second set of vi and the first set of vj . Thus, the connection digraph of the double partition in our example is the multidigraph in Fig. 11.1. Note that it has seven arcs, including a loop, a pair of opposite arcs, and a pair of parallel arcs. Adjacency Matrices As noted in the introduction, one of the conditions that is equivalent to a digraph being a line digraph involves the adjacency matrix, so we set the stage for this criterion with that topic. A binary matrix (that is, a matrix of 0s and 1s) has the roworthogonality property if every pair of rows are either identical or orthogonal; the column-orthogonality property is of course defined similarly. The following lemma characterizes matrices with these properties. Lemma 11.1 The following statements are equivalent for a binary matrix M: (1) M has the row-orthogonality property. (2) M has the column-orthogonality property. (3) No 2 × 2 submatrix of M has exactly one 0. Fig. 11.1 A connection digraph

v1

v4

v2

v3

v1 = (S1 , T1 ) = ({1, 2}, {1, 3}) v2 = (S2 , T2 ) = ({4}, {2, 5}) v3 = (S3 , T3 ) = ({3, 6, 7}, φ ) v4 = (S4 , T4 ) = ({5}, {4, 6, 7})

11.3 Characterization of Line Digraphs of Multidigraphs

161

Fig. 11.2 An illustration of Lemma 11.1

Proof If a binary matrix has a 2 × 2 submatrix with three 1s and one 0, then clearly the corresponding rows are neither identical nor orthogonal, and the same holds for the corresponding columns. Hence, both (1) and (2) imply (3). Figure 11.2 shows an example of such a matrix. On the other hand, if M is not row-orthogonal, then there are two rows with a 1 in some position in both (since they are not orthogonal) and with different entries in another position (since they are not identical). These four entries show that (3) does not hold. The same argument holds if M is not column-orthogonal, which serves to complete the proof.   We observe (and it is straightforward to show) that if S is a full set of rows that are identical and non-zero, and if T is the set of columns for which there is a 1 in a row of S, then the submatrix corresponding to S and T consists entirely of 1s. Furthermore, there are no other 1s in the columns of T . In our example, the choices for S are the sets of rows {1, 3}, {2, 4}, and {8}, and the corresponding sets T are {1, 2, 8}, {3}, and {6, 7}. We formalize these ideas in the following lemma. Lemma 11.2 Every row-orthogonal matrix M has nonempty disjoint subsets of the rows S1 , S2 , . . . , Sr , and corresponding subsets of the columns T1 , T2 , . . . , Tr such that for each i = 1, 2, . . . , r, the Si × Ti submatrix consists entirely of 1s and every 1 is in such a submatrix.

11.3 Characterization of Line Digraphs of Multidigraphs We are now in a position to state the main result of this chapter, the characterization of the line digraphs of multidigraphs. We have already observed that the line digraph of a multidigraph has no parallel arcs, but now we want to consider pairs of opposite arcs further. Clearly, two vertices v and w in a line digraph L(D) have a pair of

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opposite arcs ab and ba if and only if a and b are themselves opposite arcs in D. Now assume we have two pairs of opposite arcs, ab and ba together with bc and cb, in a line digraph L(D). Then by the definition of line digraphs, in D the tail of b must be the head of a and vice versa, and also the tail of b must be the head of c and vice versa. Hence, a and c must be the same arc. Therefore, a line digraph cannot have more than one pair of opposite arcs at a vertex. The following theorem is a compilation of the characterizations of line digraphs of multidigraphs. Harary and Norman [93] developed condition (2) in the first paper on line digraphs, while (3) was first given by Heuchenne [104], and (4) by Richards [150]. Theorem 11.1 The following statements are equivalent for a digraph D with adjacency matrix M: (1) D is the line digraph of a multidigraph. (2) The arcs of D can be partitioned into bicomplete digraphs in such a way that each vertex belongs to at most one first set and at most one second set. (3) If vx, wx, and wy are arcs in D, then so is vy. (4) Any two rows of M are either identical or orthogonal. Proof The following sequence of implications will be established: (1) ⇒ (2), (2) ⇒ (3), (3) ⇒ (4), (4) ⇒ (1). (1) ⇒ (2): Let D = L(E). For each vertex v in E, let A− (v) denote the set of entering arcs and A+ (v) the set of exiting arcs. Then the set of bicomplete digraphs K(A− (v), A+ (v)) of the junction vertices of E form a partition of the arc-set of D. Furthermore, since an arc has only one tail and one head, each vertex of D is in the first set of at most one of these subdigraphs, and at most one second set. (2) ⇒ (3): Assume that D satisfies (2), and let K(S1 , T1 ), K(S2 , T2 ), . . . , K(Sr , Tr ) be the given partition of the arcs of D. Assume also that vx, wx, and wy are arcs in D, and that wx lies in K(Si , Ti ). Since w can be in only one first set and x can be in only one second set, it follows that y ∈ Ti and v ∈ Si , and hence vy is in K(Si , Ti ) and therefore in D. (3) ⇒ (4): Assume that D satisfies (3), with its vertices labeled v1 , v2 , . . . , vn . Also, let its adjacency matrix be M = (mij ), with row i denoted μi . Suppose that (4) does not hold, say that rows μi and μj are neither identical nor orthogonal. Then there exist h and k such that mih = 1, mj h = 0, mik = 1, and mj k = 1. It follows that vi vh , vi vk , and vj vk are arcs in D, but that vj vh is not, contradicting (3). (4) ⇒ (1): Assume that M has the row-orthogonality property, and let (S1 , T1 ), (S2 , T2 ), . . . , (Sr , Tr ) be the pairs of sets of the partition given in Lemma 11.2. In addition, let S0 be the set of rows of all 0, and T0 the set of columns of all 0. Let P be the set of given pairs together with (S0 , ∅) and (∅, T0 ) as needed, and let E be the corresponding connection digraph. We now verify that D∼ = L(E). By construction, each vertex of D is in exactly one first set and exactly one second set of the pairs in P, and hence is in exactly one intersection of a second set and a first set. Thus there is a natural one-to-one mapping φ from V (D) onto V (L(E)). We now show that φ preserves adjacency and non-adjacency. If vw is an

11.3 Characterization of Line Digraphs of Multidigraphs

163

arc of D, then there exist i, j , and k such that v ∈ Ti ∩ Sj and w ∈ Tj ∩ Sk . Thus, φ(v) is an arc of E from (Si , Ti ) to (Sj , Tj ) and φ(w) is an arc of E from (Sj , Tj ) to (Sk , Tk ). Hence, there is an arc from φ(v) to φ(w) in L(E). Since every arc in L(E) arises from two such pairs of sets, there must be an arc in D that gives rise to it. Hence, D ∼   = L(E). We illustrate the last implication in the proof of the theorem with the matrix M introduced earlier in Fig. 11.2. From the sets of identical rows and the sets of identical columns, the 1s give us the pairs ({1, 3}, {1, 2, 8}), ({2, 4}, {3}), and ({8}, {6, 7}). Because of the rows and columns that are all 0, we add the pairs ({5, 6, 7}, ∅) and (∅, {4, 5}). Labelling these five pairs v1 , v2 , . . . , v5 , we get the connection digraph E, shown in Fig. 11.3. Its line digraph L(E) is also shown, and the reader can check that with the vertices labeled 1, 2, . . . , 8 as in the figure, the adjacency matrix of L(E) is the given matrix M.

v1 = ({1, 3}, {1, 2, 8})

v1

v2 = ({2, 4}, {3}) v3 = ({8}, {6, 7})

E:

v3

v2

v4 = ({5, 6, 7}, ) v5 = ( , {4, 5})

v4

v5 (a)

1

8

6

2

3

7

4

5

L(E):

(b) Fig. 11.3 A root digraph of a connection digraph

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11 Characterizations of Line Digraphs

A1 :

A3 :

A2 :

Fig. 11.4 Fundamental parallel arcs

L(A1 ):

L(A2 ):

L(A3 ):

Fig. 11.5 Line graphs of fundamental parallel arcs

Perhaps the most useful of the three criteria is the third if one only wants to check whether a digraph is a line digraph, and as we shall see in the next section, it leads to a forbidden subgraph criterion. However, if one wants to construct a root graph of a line digraph, then one of the other two criteria is probably more useful, using either the algorithm given in Chap. 10 or the essentials of the proof that (4) implies (1) above. We now look at those line digraphs of multidigraphs with the property that none of its roots is a proper digraph. It was observed in Chap. 10 that many different digraphs can have isomorphic line digraphs but they have a canonical representative which we called fundamental. The key property of these digraphs is that if an arc has in-degree 0, then its out-degree is 1 and vice versa. Obviously, this concept can be extended to multidigraphs, with the condition that if a fundamental multidigraph has parallel arcs from vertex v to vertex w then both v and w are junctures, that is, v has an in-coming arc and w has an out-going arc. The most natural such feature is that shown as A1 in Fig. 11.4 (the first and last vertices could be the same). Another possible feature is that shown as A2 , where the in-coming arc is the same as the out-going one. The only other possibility is that the parallel arcs are loops, as shown in A3 in the figure. It is not difficult to verify that these are the only possibilities. We are of course interested in what each of these structures will transport over to the line digraph of a multidigraph that contains it. Of course, that will be its line digraph which is shown in Fig. 11.5.

11.4 Families of Line Digraphs In this section, we present the restricted characterizations of digraphs themselves, that is, without loops or parallel edges. It is naturally based on Theorem 11.1. We note again that a digraph has a loop if and only if its line digraph does, so we choose to state this theorem (and some later ones) for loop-free digraphs. Obviously, it can easily be adapted to permit loops.

11.4 Families of Line Digraphs

165

Theorem 11.2 The following statements are equivalent for a loop-free digraph D having adjacency matrix M: (1) D is the line digraph of a digraph. (2) The arcs of D can be partitioned into bicomplete digraphs in such a way that (a) each vertex belongs to at most one first set and one second set and (b) a first set and a second set have at most one vertex in common. (3) If vx, wx, and wy are arcs in D, then so is vy, and there do not exist two paths vxw and vyw or two cycles vxv and vyv. (4) Any two rows of M are either identical or orthogonal, and the product of any row and column is at most 1. Proof We again prove the implications sequentially: (1) ⇒ (2), (2) ⇒ (3), (3) ⇒ (4), and (4) ⇒ (1) (in each case using the contrapositive). (1) ⇒ (2): Assume that D is the line digraph of a simple graph E. By the preceding theorem, the arcs of D can be partitioned into bicomplete digraphs in which no vertex is in no more than one first set and no more than one second set. By an earlier observation, we may assume that a pair with an empty first or second set has at most one vertex in its other set. Now suppose that every such partition has pairs, say (A1 , B1 ) and (A2 , B2 ), (which may be the same) having two vertices b and c in both B1 and A2 . It follows from our choice of partition that both A1 and B2 are non-empty, so say a ∈ A1 and d ∈ B2 . It follows that in E the arcs b and c both go from one vertex v to another vertex w, and that the in-degree of v and the out-degree of w are both positive. Hence, E must have a pair of parallel arcs. (2) ⇒ (3): Assume that D satisfies (2), and suppose that there are two paths vxw and vyw. By (2), one bicomplete digraph of the given partition must contain both of the arcs vx and vy, and another must contain xw and yw, and so x and y violate (2). Since the same argument applies if D contains two 2-cycles vxv and vyv, the result follows. (3) ⇒ (4): Assume that (3) holds but that (4) does not. Then by the previous theorem, there must be a row and column whose product is at least 2. Suppose these are row i and column j . Then for some h and k, mi,h = mh,j = 1 and mi,k = mk,j = 1. It follows that D has two paths that violate (3). (4) ⇒ (1): Assume that (4) holds. From the previous theorem it follows that D is a line digraph. Suppose that each of its root digraphs has multiple arcs; that is, there exist arcs b and c with the same first vertex v and the same second vertex w. It follows from earlier observations that v has an in-coming arc a and w has an outgoing arc d. But then in M the row corresponding to a and the column corresponding to d have 1s in the positions corresponding to b and c, contradicting (4).   Directed graphs constitute a useful perspective in the area of binary relations in the study of set theory and functions. It is obvious that the digraph of a reflexive relation has a loop at every vertex and that of an irreflexive relation has no loops. Similarly, the existence (or not) of opposite arcs corresponds to symmetric or antisymmetric relations. What is not so obvious is one way in which transitive triples

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11 Characterizations of Line Digraphs

play a role. This is in Heuchenne’s condition, which says that if a line digraph has arcs vx, wx, and wy, then it must also have the arc vy. We call this fourth arc the Heuchenne arc. Now assume that v = y, in which case w, v, and x form a transitive triple, and hence what was an ordinary arc must be a loop at v. One way of viewing this is that in a loop-free digraph D there cannot be any transitive triples in its line digraph. Heuchenne’s condition can make its presence felt in other digraphs with three arcs in addition to the natural four-vertex version and the aforementioned transitive triple. The other possibilities arise from the creation of loops by the merging of the vertices of arcs. Theorem 11.3 There are six digraphs with three arcs satisfying the hypothesis of the Heuchenne condition. These are shown in black in Fig. 11.6, with the Heuchenne arc shown in color. There are several families of digraphs that have proven to be of special interest in this field. In this book, we took digraphs to be relations without restrictions, but often they are assumed not to have loops. Under varying circumstances, the following are considered to be the most useful, and so they will be analyzed further here and in the next section. • multidigraphs (loops and parallel arcs permitted) • digraphs (no parallel arcs permitted) • loop-free digraphs (irreflexive relations; that is, neither loops nor parallel arcs allowed) • oriented digraphs (irreflexive antisymmetic relations; that is, loops, parallel arcs, and opposite arcs not allowed).

Fig. 11.6 Digraphs of the Heuchenne condition

11.5 Excluded Induced Subdigraphs

167

The following theorem put these families in subgraph perspective using the observations made following Theorem 11.1. Theorem 11.4 Let D be the line digraph of a multigraph. (a) D is the line digraph of a digraph if and only if it does not contain any of the digraphs L(A1 ), L(A2 ), or L(A3 ) in Fig. 11.5. (b) Let D be loop-free. Then D is the line digraph of a loop-free digraph if and only if it does not contain either of the digraphs L(A1 ) or L(A2 ) in Fig. 11.5. (c) Let D be loop-free and without opposite arcs. Then D is the line digraph of an oriented digraph if and only if it does not contain the digraph L(A1 ) in Fig. 11.5.

11.5 Excluded Induced Subdigraphs In contrast to line graphs, which have an elegant characterization as the family of graphs with a particular set of nine excluded subgraphs, the situation is not quite so nice for line digraphs. That there exists such a set follows at once from Heuchenne’s condition, and our quest in this section is for the set of excluded induced subdigraphs for two of the families that we have been investigating. Because a loop and only a loop can result in a loop in a line digraph, we again focus on loop-free digraphs here. Clearly, the same argument applies to pairs of opposite arcs; that is, a digraph has a pair of opposite arcs if and only if its line digraph has. Therefore, we begin this section with oriented digraphs. Theorem 11.5 An oriented graph is the line digraph of an oriented graph if and only if it does not contain any of the five digraphs in Fig. 11.7 as an induced subdigraph. Proof First assume that digraph D is the line digraph of an oriented graph. As observed earlier, any root digraph of either E1 or E2 in the figure will have a pair of parallel arcs, so neither of these digraphs can occur as an induced subdigraph of D. Furthermore, each of the digraphs E3 , E4 , and E5 is easily seen to violate Heuchenne’s condition, so none of them is a line digraph and therefore cannot be an induced subdigraph of one. For the converse, we assume that D is an oriented graph that does not contain any of the five digraphs in Fig. 11.7 as an induced subdigraph, and suppose that it is not the line digraph of an oriented graph. It follows from Theorem 11.4(c) that if D satisfies Heuchenne’s condition, then every root digraph must contain parallel arcs, and so D must contain E1 as a subdigraph. Consequently, either E1 or E2 must be an induced subdigraph, and this has already been forbidden. Therefore, D must violate Heuchenne’s condition; that is, there must be three arcs vx, wx, and wy, for which the arc vy is not present. If not all four of the vertices v, w, x, and y are different, since loops and parallel arcs have been forbidden, it must be that

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11 Characterizations of Line Digraphs

E1 :

E3 :

E2 :

E4 :

E5 :

Fig. 11.7 The forbidden family for oriented line digraphs

v = y, and this is proscribed since the transitive triple E3 is forbidden as an induced subgraph. Now let B = v, w, x, y. Since E4 is forbidden, there must be at least one more arc. It cannot join either v and w or x and y since in either case there would be a transitive triple. Because there is no arc from v to y by hypothesis, it must be the arc yv. Hence, since E5 is forbidden, this is impossible. Therefore, D must satisfy Heuchenne’s condition, and thus must be the line digraph of an oriented graph.   We now turn to the line digraphs of loop-free digraphs. Theorem 11.6 A digraph is the line digraph of a loop-free digraph if and only if it does not contain any of the 19 digraphs in Fig. 11.8 as an induced subdigraph. Proof For the necessity, we assume that D is the line digraph of a digraph H . In order for there to be no parallel arcs in H , D cannot contain any of the digraphs F1 , F2 , F6 , F10 , or F11 . Furthermore, D cannot contain a transitive triple, and so F3 , F6 , F7 , and F8 are all forbidden. Finally, it is straightforward to check that each of the order-4 digraphs F12 , F13 , . . . , F19 can have its vertices labeled v, w, x, and y in such a way that the arcs vx, wx, and wy are present, but vy is not. Hence, all of these digraphs must be forbidden. The proof of the sufficiency proceeds much like that in the preceding theorem. Suppose that none of the 19 digraphs in Fig. 11.8 is an induced subdigraph of digraph D, but that D is not the line digraph of a digraph. If D were the line digraph of a multidigraph with parallel arcs, then by earlier results, it would have to contain F1 or F6 as a subgraph. Since D cannot have any transitive triples, the digraph induced by the vertices in a copy of F1 must be F1 or F2 and one induced by the vertices in a copy of F6 cannot have any more arcs. It follows that D cannot be a line digraph and must therefore violate Heuchenne’s condition, and thus we need only consider induced subdigraphs of orders 3 and 4. As observed earlier, every such digraph of order 3 must contain a transitive triple, and thus must

11.5 Excluded Induced Subdigraphs

F1 :

169

F3 :

F2 :

F5 :

F6 :

F9 :

F4 :

F8 :

F7 :

F10 :

F11 :

F12 :

F13 :

F14 :

F15 :

F16 :

F17 :

F18 :

F19 :

Fig. 11.8 The excluded family for digraphs

be one of the digraphs F7 , F8 , . . . , F11 . Consequently, we turn to a digraph B of order 4 having its vertices labeled v, w, x, and y so that the arcs vx, wx, and wy are present, but vy is not. If the arc yw is also not present, then B must be the result of adding one or more of the reverse arcs xv, xw, and yw, and the only ways this can be done without getting F6 result in F12 , F13 , F14 , or F15 . Similarly, if the arc yw is present, the only possibilities are for B to be F16 , F17 , F18 , or F19 . Since these are all impossible, the result follows.   There is of course a similar theorem for the line digraphs of binary relations; that is, with loops allowed. Its proof is basically just a matter of determining which of the digraphs obtained from members of the family {F1 , F2 , . . . , F19 } by adding

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11 Characterizations of Line Digraphs

loops at some of the vertices are not the line digraph of a relation. The list gets to be quite extensive and its compilation is left to the reader. Therefore, rather than pursue this further, we turn instead to other consequences of Theorems 11.5 and 11.6. We see at once from the fact that the existence of a transitive triple in a line digraph implies the existence of a loop (by Heuchenne’s condition), and therefore no tournament of order 4 or more can be a line digraph; hence, there are only three tournaments that are line digraphs: those of order 1 and 2 and the cyclic triple. Continuing along this line, we consider next bipartite tournaments, orientations of complete bipartite graphs. Let T be an r × s bipartite tournament with 2 ≤ r ≤ s (it is easy to see that every orientation of K1,s is a line digraph). If all arcs go from one partite set to the other, then we have the bicomplete − → digraph K r,s , which is a line digraph. Therefore, we assume that T is a line digraph that is not bicomplete. Since F1 and F5 are forbidden subgraphs, it follows that T has a 4-cycle, say x1 y1 x2 y2 x1 . Now assume there is a fifth vertex, y3 . If it has arcs to both x1 and x2 , or from both, then F5 is induced, while if it has an arc to one but not the other, then F1 is induced. Consequently, the only 2 × 3 bipartite tournament → − that is a line digraph is the bicomplete digraph K 2,3 . The following result is thus a consequence of Theorem 11.5. Corollary 11.1 (a) Every orientation of K1,s is a line digraph. (b) The only orientations of K2,2 that are line digraphs are the 4-cycle C4 and → − K 2,2 . (c) For r ≥ 2 and s ≥ 3, the only orientation of Kr,s that is a line digraph is the → − bicomplete orientation K r,s . As the observation on tournaments suggests, when a digraph is sparse in terms of the number of arcs, it is quite likely to be a line digraph, but not if it is dense. Because induced subdigraphs of orders 3 and 4 play such an important role in the theory of line digraphs, we look at those orders of loop-free digraphs here. Theorem 11.7 A digraph of order 3 is a line digraph if and only if it has at most three arcs and is not a transitive triple. Theorem 11.8 Let D be a digraph of order 4. (a) If D has three or fewer arcs, then it is a line digraph if and only if it is not one of the two digraphs in Fig. 11.9. (b) If D has four or more arcs, then it is a line digraph if and only if it is one of the 15 digraphs in Fig. 11.10.

11.5 Excluded Induced Subdigraphs

171

G1 :

G2 :

Fig. 11.9 The digraphs with four vertices and three arcs that are not line digraphs

H1 :

H2 :

H3 :

H4 :

H5 :

H6 :

H7 :

H8 :

H9 :

H12 :

H10 :

H13 :

H11 :

H14 :

Fig. 11.10 The line digraphs with four vertices and at least four arcs

H15 :

Chapter 12

Iterated Line Digraphs

12.1 Introduction The subject of this chapter is defined precisely as one would expect: analogous to the definition of iterated line graphs in Chap. 1. That is, the second-order line digraph L2 (D) of a digraph D is the line digraph of L(D), and so on. However, as was seen in the preceding two chapters, this is pretty much where the analogy ends; iterating the respective operations on directed versus undirected graphs results in very different outcomes. That makes the material here all the more interesting, as was indicated by the results in the previous two chapters. In the next section, we show two properties for which there are major differences in results given differences in elementary properties. One is on the number of vertices in the iterated line digraphs of a digraph, and the other on the necessity of a particular type of fourth walk given three other walks, a generalization of the Heuchenne condition in Chap. 11. It is interesting that there are digraphs D whose second-order line digraph L2 (D) is isomorphic to D itself but not to L(D), and in fact more generally there are digraphs whose iterated line digraphs are eventually periodic of any given period. Section 12.4 is devoted to a characterization of secondorder line digraphs. This turns out to be quite a complicated result, so we do not proceed to higher orders even though that result is known.

12.2 Basic Properties As stated above, iterated line digraphs are defined recursively just as one would expect: Given a non-null digraph D, L1 (D) = L(D), and if Lk (D) is non-null, then Lk+1 (D) = L(Lk (D)). (If Lk (D) is null, then Lk+1 (D) is undefined.)

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_12

173

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12 Iterated Line Digraphs

D1 :

L2 (D1 ):

L(D1 ):

D2 :

L(D2 ):

D3 :

L(D3 ):

L3 (D1 ):

L2 (D2 ):

L2 (D3 ):

Fig. 12.1 Examples of iterated line digraphs

Some examples are shown in Fig. 12.1. Note that the first example D1 is acyclic, the second D2 consists of two cycles joined by a path, while the third D3 consists of two cycles that share a path. Our first theorem on this topic tells what happens to the order of iterated line digraphs in two extreme cases. (Recall that paths and cycles are considered to be directed unless otherwise specified.) In preparation for its proof, we look further at cycles in digraphs in general. Consider a digraph D in which the vertices in one cycle C can reach the vertices in another cycle C . If the two cycles have no arcs in common, then it follows that there is a path P (possibly of length 0) from (say) C to C , as in D2 in the figure. Now assume that the two cycles have an arc in common. We claim that then there is a an orientation of a theta graph (three internally disjoint paths between a pair of vertices, with two of the paths oriented in the same direction

12.2 Basic Properties

175

and the third in the opposite direction (see D3 in the figure). Let C be a shortest cycle that has an arc uv in common with another cycle C , and without loss of generality we may assume that the next vertex w after v on C is not the next vertex on C. Now let x be the first vertex after v on C that is also on C. It follows that, together with C, the v to x path on C forms the desired oriented theta graph. We now state our theorem. Theorem 12.1 Let D be a digraph. (a) For some k, Lk (D) is a null graph if and only if D has no directed cycles. Furthermore, the value of this k equals the length of a longest directed path in D. (b) Assume that D has a pair of cycles for which there is a path from a vertex of one to a vertex of the other, and let nk be the order of Lk (D). Then lim nk = ∞. k→∞

Proof (a) Clearly a line digraph L(D) has a cycle if and only if D has. We now show that if D has no cycles and if l is the length of a longest path in D, then the length of a longest path in L(D) is l − 1. This follows from two facts: (1) the line digraph of a path is a path of length 1 less, and (2) for there to be a path of length l in L(D), there must be a path of length l + 1 in D. Statement (a) follows from these observations (see D1 in Fig. 12.1). (b) Assume that D is a digraph that has a cycle whose vertices can reach those of another cycle. It follows from the preceding discussion that then D has either a subgraph that consists of two cycles with a path (possibly of length 0) from one to the other (like D2 in Fig. 12.1) or a strongly oriented theta graph (like D3 in Fig. 12.1). In the first of these two cases, the line digraph has the same structure, with two cycles of the same lengths but joined by a path of length 1 greater. In the second case, assume that the path common to the two cycles has length l ≥ 1. Then (as in the figure) L(D) has two cycles of the same lengths, but the path in which they overlap has length 1 less. Eventually the overlap is just a single vertex, and so we are in the previous situation. In any case the linedigraph iterates of these digraphs always increase by 1, and since D contains at least one of these, its line-digraph iterates get arbitrarily large, proving (b).   It turns out that the converse of (b) in the theorem also holds, but its proof is quite complicated and its validity will follow from material in the next section. The Heuchenne Conditions Recall from the line digraph characterization theorem (Theorem 11.1) that if vx, wx, and wy are arcs in a line digraph L(D), then so is vy. The following theorem shows that a generalized version (which will be useful in the next section) must hold for iterated line digraphs. Theorem 12.2 If the k-th iterated line digraph Lk (D) has internally disjoint v-x, w-x, and w-y paths, then it also has a v-y path internally disjoint from the others.

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12 Iterated Line Digraphs

Proof We give an outline of the proof. The theorem can be proved by induction, with the case k = 1 already established in Theorem 11.1. However, since the notation in the general induction step gets rather nasty, we show only the case k = 2 with accompanying figures, as it contains the main ideas of the induction step. Given three paths of length 2 as in the hypothesis for L2 (D), these can be taken as having the following arcs of L(D) as successive vertices: first path, st, tu, and uv; second path, wx, xu, and uv; and third path, wx, xy, and yz, as in Fig. 12.2a. It then follows that L(D) must contain the digraph shown in (b) in the figure. Since this is a line digraph and contains the arcs tu, xu, and xy, it must also contain the arc ty. This arc must be a vertex in L2 (D), and by the definition of line digraph, there must be Fig. 12.2 Illustration of the proof of Theorem 12.2

st

uv

tu

xu

xy

wx

yz

(a) In L2 (D) s

t

u

v

w

x

y

z

(b) In L(D) st

tu

uv

xy

yz

ty xu

wx

(c) In L2 (D)

12.3 Periodic Line Digraphs

177

the two arcs of the path st, ty, and yz (as shown in (c) in the figure), which is the required path.   We call the condition given in the theorem the kth Heuchenne condition.

12.3 Periodic Line Digraphs As we saw in Chap. 2, the only graphs that are isomorphic to their line graphs are those in which each component is a cycle. It is thus natural to ask which connected digraphs D are such that L(D) ∼ = D. It turns out that the situation is much more complicated than for graphs. For historical reasons, we note that the first result of this type was the connected case, first proved by Harary and Norman [93]: A connected digraph D is isomorphic to its line digraph if and only if every vertex has out-degree 1 or every vertex has in-degree 1. It has since become a corollary to the general result, most of the credit for which goes to Hemminger [101]. It is useful to have some additional terminology and notation, beginning with in-tree and out-tree. First, we define in-trees recursively: A single vertex is an intree, and given an in-tree Tk− with k vertices, the addition of a new vertex with an arc to one vertex of Tk− results in an in-tree of order k + 1. Less formally, we can consider in-trees to be orientations of trees with a root vertex having all paths oriented towards the root. Out-trees are of course defined similarly, and are the converse digraphs of in-trees. (An out-tree is sometimes called an arborescence, and by analogy an in-tree is a counter-arborescence.) An example of each is shown in Fig. 12.3. A related concept is the digraph of a function on a finite set, which we call a function digraph; that is, a digraph in which each vertex has out-degree 1. They are characterized by each component comprising a cycle and (possibly) in-trees at some of the vertices. An example is given in Fig. 12.4. It is straightforward to show that

Fig. 12.3 An in-tree and an out-tree

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12 Iterated Line Digraphs

Fig. 12.4 The digraph of a function

every such digraph is isomorphic to its line digraph, as is that of its converse. The result of Harary and Norman says that these are the only connected digraphs with this property. However, there are other digraphs with this property. Not only are there disconnected digraphs D with a single cycle for which L(D) ∼ = D, but for any p > 1, there are digraphs D with Lp (D) ∼ = D (and Lk (D)  D for k < p), and more. In fact, there are digraphs whose iterated line digraphs eventually repeat even though they do not at the start. We make the following definition. A line digraph D is periodic if there are positive integers k0 and p for which, for all k ≥ k0 , Lk+p (D) ∼ = Lk (D). When this occurs, the minimum value of p is called the period of D. For example, consider the digraph D and the iterated line digraphs shown in Fig. 12.5. Although all five of the digraphs are different, we can see that L5 (D) ∼ = L3 (D) (since 4 2 L (D) is isomorphic to L (D) except for its isolated vertex), and hence, for k ≥ 3, Lk+2 (D) ∼ = Lk (D) (and Lk+1 (D)  Lk (D)), so D is periodic with period 2. Unicyclic Digraphs The general purpose of this section is to investigate digraphs whose line digraphs have a given period. As it happens, by far the nicest result is for unicyclic digraphs, those with just one cycle, and so for now we restrict our attention to these structures. It should be clear from our discussion that within a digraph D having a single cycle, the paths going to that cycle get maintained under the line digraph operation, and therefore the unions of those paths are of interest. Of course, a similar statement holds for the paths going from the cycle. We therefore define the foundation F (D) of a unicycle digraph to be that subdigraph formed by the union of its cycle C and all paths to and from the vertices of C. A unicyclic digraph and its foundation are shown in Fig. 12.6.

12.3 Periodic Line Digraphs

179

L2 (D) :

L(D) :

D:

L3 (D) :

L4 (D) :

Fig. 12.5 A digraph with period 2

D:

Fig. 12.6 The foundation of a unicyclic digraph

F(D) :

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12 Iterated Line Digraphs

We now consider a special type of unicyclic digraph: an eddy digraph consists of a cycle C = v0 v1 . . . vk−1 v0 together with an in-tree Ai and an out-tree Bi (possibly trivial) at each vertex vi of C. Theorem 12.3 If D is a unicyclic digraph, then for k sufficiently large the foundation of Lk (D) is an eddy digraph. Proof If the foundation of a unicyclic digraph D is not an eddy digraph, then there is a vertex for which there are two different paths from it to the cycle C or two different paths to it from C. Without loss of generality, we assume the former and consider such a vertex v that is closest to C. We now show that in L(D), any such vertex is at a greater distance from the cycle in L(D). This is done in two cases. First, we assume that v has positive in-degree, and let a = uv be an in-coming arc. Then a has the same property in L(D) that v had in D, except that its distance to the cycle is 1 greater. On the other hand, if v has in-degree 0, then the paths from v to the cycle split. It follows that eventually every vertex in the foundation digraph that has a nontrivial path to the cycle C will have exactly one path to C. In other words, these vertices induce a collection of in-trees. By duality, eventually the vertices on paths from the cycle will induce a collection of out-trees. Consequently, the foundation digraph F (Lk (D)) will eventually be an eddy digraph.   Figure 12.7 illustrates the two cases for paths to C in the proof, going from the first case to the second. Since the line digraph of a path is another path and a path in a line digraph can only come from a trail, it follows that the foundation F (L(D)) of a line digraph

u

a

a v

D:

L(D):

Fig. 12.7 Illustration of eddy formation

L2 (D):

12.3 Periodic Line Digraphs

F(D):

181

F(L2 (D)):

F(L(D)):

Fig. 12.8 The foundations of iterated line digraphs

is isomorphic to the line digraph L(F (D)) of the foundation of D. Therefore, in illustrating this theorem and its proof, we show in Fig. 12.8 only the foundations of the iterated line digraphs. We now let D be an eddy digraph with cycle C = v0 v1 . . . vr−1 v0 , and let Ai and Bi be the in-tree and out-tree respectively rooted at vi . (Note that some of these may be trivial.) Furthermore, let {A0 , A1 , . . . , Ar−1 } be the cyclic (modulo k) sequence of in-trees and define the in-tree index of D to be the least positive integer l for which Ai+l ∼ = Ai for all i. The sequence of out-trees {B0 , B1 , . . . , Br−1 } and the out-tree index are defined similarly. Lemma 12.1 The foundation of the line digraph of an eddy digraph is isomorphic to the digraph obtained by shifting the in-tree sequence one step along the cycle relative to the out-tree sequence. The proof of the lemma is straightforward, and the result is illustrated in Fig. 12.9.

D:

L(D):

L2 (D):

Fig. 12.9 The foundation of the line digraph of an eddy digraph

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12 Iterated Line Digraphs

Theorem 12.4 If D is a unicyclic digraph, then the period of D is the greatest common divisor of its out-tree and in-tree indices. Proof We first observe that we may assume that the foundation of D itself is an eddy digraph (and hence so is that of each of its iterated line digraphs). Therefore, it is sufficient to find the minimum positive integer p for which Lp (D) ∼ = D. Let a be the in-tree index of D, and let b be its out-tree index. If the in-trees are shifted any multiple of a steps along the cycle of D (forward for a positive multiple, backward for a negative multiple), the resulting foundation is isomorphic to D; and a similar result holds if the out-trees are shifted through a multiple of b units (in the reverse directions). Hence, if c is any positive linear combination of a and b, then the foundation of Lc (D) is isomorphic to D. Since gcd(a, b) is the least such c, it follows that the period of D is at most gcd(a, b). Suppose now that the period of D is less than this; that is, the foundation of Ld (D) is congruent to D with d < gcd(a, b). But then so are the foundations of L2d (D), L3d (D), . . .. Taking the iterations modulo the length r of the cycle in D, either d = 1 or there is a smaller number than d with the same property. Hence, the period of D is gcd(a, b).   The period-1 case is of course of special interest: Corollary 12.1 A unicyclic digraph has period 1 if and only if its out-tree and intree indices are relatively prime.  An example is shown in Fig. 12.10, with an eddy digraph D having in-tree index 3, out-tree index 2, and a 6-cycle, and the foundation of L(D). The theorem of Harary and Norman mentioned earlier also follows as a corollary since iterated line digraphs of eddy digraphs having both a nontrivial in-tree and a nontrivial out-tree are eventually disconnected. The proof is illustrated in Fig. 12.11. Multicycled Periodic Digraphs From Theorem 12.1(b), we know that if a digraph D contains a pair of cycles having a path from one to the other (including having a path in common), then

D:

F(L(D)):

Fig. 12.10 The line digraph of a digraph with a 6-cycle and relatively prime in-tree and out-tree indices

12.3 Periodic Line Digraphs

183

D:

L(D):

L2 (D):

L3 (D):

Fig. 12.11 Illustration of a result of Harary and Norman

its iterated line digraphs get arbitrarily large. Hence we now conclude our analysis by considering digraphs without two such cycles, which we call cycle-separated. Theorem 12.5 If D is a cycle-separated digraph, then for k sufficiently large, no component of Lk (D) contains more than one cycle. Proof Let D be a connected cycle-separated digraph with two cycles in the same component. We first consider the case where some vertex has paths to both cycles or has paths from both. Without loss of generality, we assume the former and let d be the shortest distance from such a vertex to a cycle. Using the same argument as that in Theorem 12.3, we see that in L(D) the shortest distance from such a vertex to a cycle is d + 1. Since the lengths of paths in a cycle-separated digraph is bounded, eventually there are no such vertices. Therefore we may assume that the foundation of the digraphs generated two cycles C and C must be disjoint. Since we have assumed that C and C are in the

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12 Iterated Line Digraphs

same component of D, the underlying graph is connected, and so there are other arcs in D. The extreme case is where there is a path P from a vertex on a path to C in F (C) to a vertex on a path from C in F (C ). The image of P in Lk (D) gets further from the cycles and eventually there is no way to connect C and C in the underlying graph of Lk (D). It follows that if there is no path such as P , then the separation of the cycles occurs by the time k is 1 more than the length of a longest path whose arcs are not in any F (C).   Theorem 2.2 also follows as a corollary since iterated line digraphs of eddy digraphs having both a nontrivial in-tree and a nontrivial out-tree eventually are disconnected. We now consider the general question of which digraphs have period 1. By Theorem 12.1(b), such a digraph must be cycle-separated, and since eventually its iterated line digraphs are such that no component has more than one cycle, we need only consider such digraphs. Let A be a digraph of period p with Lk+p (A) ∼ = Lk (A). Then the digraph D = Lk+1 (A) + Lk+2 (A) + . . . + Lk+p (A) (where + denotes disjoint union) clearly has period 1. Of course, one can combine more such digraphs and get other digraphs isomorphic to their line digraph. Likewise, by taking various combinations, one can get digraphs with a desired period p. For example, if D1 has period p1 and D2 has period p2 , then D1 + D2 has period the least common multiple of p1 and p2 .

12.4 Characterization of Second Order Iterated Line Digraphs The focus of this section is a characterization of second-order line digraphs; that is, those digraphs D for which there is a digraph F with D ∼ = L2 (F ). This result is due to Beineke and Zamfirescu [38], and our proof follows the one in their paper closely. Before going into the primary fundamental concept for our characterization of second-order line digraphs, we observe that just as a line digraph cannot have two arcs from one vertex to another, a second-order line digraph cannot have a pair of paths of length 2 from one vertex to another. Therefore, in what follows, we assume that neither of these configurations can be present in the digraphs we consider as candidates for being second-order line digraphs. One of the characterizations of a line digraph in Theorem 11.1 states that if vx, wx, and wy are arcs in a line digraph D, then the arc vy must also be present. We call this the first Heuchenne condition and extend it to the second Heuchenne condition: For any vertices v, w, x, and y (not necessarily different from each other), if there are walks of length 2 from v to x, from w to x, and from w to y, then there must also be one from v to y. The case in which not only are the vertices v, w, x, and y different from each other but the three given walks of length 2 are internally disjoint

12.4 Characterization of Second Order Iterated Line Digraphs

185

Fig. 12.12 The second Heuchenne condition

Fig. 12.13 The necessity of isolated vertices

D:

E:

D+ :

E+:

F:

is shown in Fig. 12.12, where the existence of the three paths with black arcs implies the existence of the one with red arcs. As we will show in our characterization theorem, a second-order line digraph cannot have two walks of length 2 from one vertex to another (the effect would be a pair of parallel arcs). Even though it is necessary and sufficient for a digraph D to be a line digraph is only that it satisfy the first Heuchenne condition, for D to be a second-order line digraph the corresponding statement does not hold even with the aforementioned restriction on walks of length 2. That is, given that it is necessary for a digraph D to be a second-order line digraph that it not have multiple 2-paths from any vertex to another, and that it satisfy both the first and second Heuchenne conditions, those are not sufficient. We illustrate this with two examples. First consider digraph D in Fig. 12.13. It is clearly the line digraph of E in the figure, but E is not itself a line digraph, as it violates the first Heuchenne condition. However, if an arc is added to E, the result E + is the line digraph F . The existence of that arc, however, necessitates that D be augmented by an isolated vertex. It can be verified that the resulting digraph F in the figure satisfies D + ∼ = L2 (F ). This possible necessity of isolated vertices (and their number) will be spelled out in condition (e) of our theorem. Now consider digraph D in Fig. 12.14. As in the previous example, it is the line digraph of digraph E in the figure, but E is not itself a line digraph. In order for this

186 Fig. 12.14 The necessity of sinks and sources

12 Iterated Line Digraphs

D:

E:

D+ :

E+:

F:

to be the case, E must have an additional arc as indicated in E + . This entails there being another arc in D, one going to a sink. (So that the procedure is systematic, we assume that the sink is a new vertex.) The result is that E + is the line digraph of the digraph F in the figure and hence D + ∼ = L2 (F ). Naturally, the same argument applies to the converse situation when an arc to a sink vertex is added. The necessity of these additional arcs is covered in condition (f) of our theorem. To facilitate the statement of the theorem, we say that a vertex is an end-source if it has out-degree 1 and in-degree 0, and an end-sink if those values are switched. In stating our theorem, we also find it useful to introduce a particular type of subgraph of a digraph D. We say that two 2-walks P : v0 v1 v2 and Q : w0 w1 w2 are hooked if vi = wi for at least one value of i = 0, 1, 2. In addition, we say that P and Q are hook-related, denoted P ∼ Q, if there is a sequence of 2-walks P = P0 , P1 , . . . , Pr = Q with the property that for each j = 1, 2, . . . , r, Pi−1 and Pi are hooked. Clearly ∼ is an equivalence relation on the set of 2-walks in D. The union of the 2-walks in an equivalence class is called a tri-level subdigraph of D, with the first vertex in a 2-path called first-level, the middle vertex secondlevel, and the last vertex third-level. The digraph D in Fig. 12.15 is an example of a tri-level digraph, with the diagram T showing all of the 2-walks and how they are hooked. (The digraph D can be obtained from this by identifying the vertices that are labeled with the same letter.) Note that the digraph D does not itself satisfy the two Heuchenne conditions, but as the digraph D + and the corresponding diagram T + show, it can be completed so that it does. We now have the background that we need to state our characterization theorem. Theorem 12.6 A line digraph D is a second-order line digraph if and only if it satisfies the following four conditions: (a) For any two vertices v and w, there is at most one v-w 2-walk.

12.4 Characterization of Second Order Iterated Line Digraphs

187

g

a

c

b

c d b

D:

T:

b

a

c

d

f

a e

f

c

g

a

b

e d

f

g

a

c

b

c d b

D+ :

T +:

b

a

f

a f g

d

c

e c

b

a

f

e d

Fig. 12.15 A tri-level digraph

(b) (Second Heuchenne condition) For all vertices v, w, x, y, if there is a v-x 2walk, a w-x 2-walk, and a w-y 2-walk, then there is also a v-y 2-walk. (c) For each tri-level subgraph, D has rs isolated vertices, where r is the number of sinks at its first-level vertices and s is the number of sources at its third-level vertices. (d) Within each tri-level subgraph, there are the same number of sinks at all firstlevel vertices and the same number of sources at all third-level vertices. Before proving the theorem, we show in Fig. 12.16 examples of each of the four properties. Proof We first show the necessity of the conditions, so we assume that D is a second-order line digraph, say L2 (F ) = D, and let E = L(F ). To show that (a) holds, we suppose that D has two 2-walks v → x → w and v → y → w. Then for some i and j , v ∈ Si and w ∈ Tj , and so v and w are both in Ti and in Sj . This means that there are two arcs from vi to vj in E, which violates our definition of a digraph. To establish (b), we assume that D contains three 2-walks v → r → x, w → s → x, and w → t → y. Let v ∈ Sh , w ∈ Si , x ∈ Tj , and y ∈ Sk . From the above observation, it follows that E contains the arcs vh vj , vi vj , and vi vk . Again since E is a line digraph, by the first Heuchenne condition, it must have the arc vh vk ,

188

12 Iterated Line Digraphs

(a) a forbidden configuration

(b) a necessary 2-path

p

p (c) a necessary isolated vertex

(d) equinumerous sink neighbors

Fig. 12.16 Examples of four conditions for a second-order line digraph

and therefore there must be a 2-walk from v to y in D. Consequently, the second Heuchenne condition must hold. It turns out that for our proof it is convenient to prove (d) before (c). For (d), it is sufficient (by duality and the transitivity of isomorphism) to show that if v → r → x and w → s → y are 2-walks in D with v = w then v and w must have the same number of sink neighbors. Consequently, any two isomorphism-related 2walks must have that property, and, in addition, the dual statement holds by duality. We first note that if r = s, then v and w must have the same set of sink neighbors (and thus the same number) by virtue of the first Heuchenne condition. Therefore we assume that r = s, and thus that x = y. We let z be a sink neighbor of w and assume that v ∈ Sh , w ∈ Si , x ∈ Tj , and y ∈ Sk . It follows that Tk = ∅. We now show that there is exactly one sink neighbor of v in Sk . Since z ∈ Ti ∩ Sk , E contains the arc vi vk , and by our earlier observation, E contains the arcs vh vj and vi vj . Hence, by the first Heuchenne condition, vh vk is an arc. Therefore, D must have a vertex u in both Th and Sk . Since all of the sink neighbors of v must be in Th , it follows that Sk cannot contain a second such vertex (since if it did E would have parallel arcs). For the same reason, Sk contains exactly one sink neighbor of w. Similarly, every such Sk contains the same number of sink neighbors of v and w, which completes the proof that (d) is necessary. We now establish (c). We first show that if v → r → w is a 2-walk in D, and if s is a sink neighbor of v and t is a source neighbor of w, then there is an isolated vertex in both the set Sk that contains s and the set Th that contains t. To this end,

12.4 Characterization of Second Order Iterated Line Digraphs

189

assume that r ∈ Si and u ∈ Tj . We then have r, s ∈ Ti , r, t ∈ Sj , and Tk = Sh = ∅. As before, if follows that E must contain the arc vh vk , and hence D must have a vertex y in Th ∩ Sk . Furthermore, since both Sh and Tk are empty, y must be an isolated vertex. Since E cannot have any parallel arcs, it follows (as in the latter part of the proof of (d)) that every such Sk contains exactly one sink neighbor of v and also that every such Th contains exactly one source neighbor of w. Hence, if v has p sink neighbors and w has q source neighbors, then there must be at least pq isolated vertices associated with the 2-path v → r → w. We next show that an isolated vertex x for the tri-level subgraph of the 2-walk v → r → w cannot serve as an isolated vertex for any other tri-level subgraph. To this end, we assume that there is a subdigraph with these arcs: v → s , v → r → w , and t → w with s a source in Sk (which contains s and x) and t a sink in Th (which contains t and x). Let i be such that v is in Si and r and s are in Ti , and let j be such that t and w are in Sj and v is in Tj . It follows as before that there must be arcs vi → vj , vh → vj , and vi → vk . Because of the v → r → w tri-level subgraph we must also have vi → vj , vh → vj , and vi → vk in D, so from the first Heuchenne condition it follows that E must have the arcs vi → vj , and vi → vj . Hence D must contain vertices y and z with y ∈ Ti ∩ Sj and z ∈ Ti ∩ Sj . It follows from the observation made at the beginning of the proof that then we have v → r → w and v → z → w. It follows that we have these loop-related 2-walks: (v → r → w) ∼ (v → y → w ) ∼ (v → r → w ). Therefore, each isolated vertex can serve only one tri-level subgraph, a fact that completes the proof of the necessity of the four given conditions. To prove the sufficiency, we assume that D is a line digraph that satisfies (a)–(d), and then proceed to construct from D a connection digraph E in such a way that it too is a line digraph. Since D is a line digraph, it has double partitions, and we specify one for which the connection digraph satisfies the first Heuchenne condition. For a given double partition {(Si , Ti )}ki=1 of D, it is readily seen that if a vertex v ∈ Si has positive out-degree, then Si is determined (as the set of vertices with arcs to precisely the same vertices as v), and similarly that if v ∈ Ti has positive in-degree, then Ti is determined. Therefore in constructing our double partition, the only flexibility that we have is in partitioning the sinks into sets Si (with the corresponding Ti = ∅) and the sources into sets Ti (with the corresponding Si = ∅). We begin with the sinks. First consider those sinks that have arcs to them from a first-level vertex of a given tri-level subgraph. We observe that each of these sinks is connected in this way with only one tri-level subgraph. For, since D satisfies the first Heuchenne condition, all neighbors of such a sink must be first-level vertices in the same tri-level subgraph. Hence, we consider just one tri-level subgraph and the sink neighbors of its first-level vertices. The sets of sink neighbors of two first-level vertices must be either disjoint or identical, again because of the first Heuchenne condition. Therefore, if p is the number of sink neighbors of the first-level vertices of a this at each sink neighbor of this tri-level subgraph can be assigned a number between 1 and p.

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12 Iterated Line Digraphs

If there are two sets Si and Tj that arise from some tri-level subgraph with Si containing a sink and Tj a source, we add an isolated vertex to each. Note that by (f), there are enough isolated vertices to do this. To each remaining sink x we assign the pair ({x}, ∅) and to each remaining source y the pair (∅, {y}). Having thus defined a double partition {(Sk , Tk )}rk=1 (which is also a partition of the arcs of D), we must show that its connection digraph E is itself a line digraph. We first show that E doesn’t have any multiple arcs. Our construction of the partitions means that there cannot be two sink neighbors of any vertex in the same set Sk , two source neighbors of any vertex in the same Tk , or two isolated vertices in both an Sk and a Tk for any k. Therefore, the only way that E could have multiple arcs would be for D to have multiple 2-walks, and this is prohibited by (c). Hence, E can have no multiple arcs. Now suppose that E has vertices v1 , v2 , v3 , and v4 and arcs v1 v3 , v2 v3 , and v2 v4 , and we show that v1 v4 must also be present. Note that although the four vertices need not all be different, we must have v1 = v2 and v3 = v4 since otherwise E would have multiple arcs. As before, we assume that the vertex vk in E corresponds to the pair {(Sk , Tk )} in the double partition. It follows that there exist vertices u, v, and w in D with u ∈ S3 ∩ T1 , v ∈ S3 ∩ T3 , and w ∈ S4 ∩ T3 , and we must show that there exists a vertex x ∈ S4 ∩ T1 , thereby establishing the existence of the arc v1 v4 in E. Since S3 and T2 are not singleton sets and v ∈ S3 ∩ T2 , u, v, and w must all be associated with the same tri-level subgraph. Whether they are sinks, sources, or mid-level vertices depends on which, if any, of the four sets S1 , S2 , T3 , and T4 are empty. Because of symmetry in the argument when all arcs are reversed, only the following cases need to be considered: (0) None of the four sets is empty. (1) Exactly one of the sets is empty: (a) T4 , (b) T3 . (2) Exactly two of the sets are empty: (a) T3 and T4 , (b) S2 and T4 , (c) S1 and T4 , (d) S2 and T3 . (3) Exactly three of the sets are empty: (a) all except S1 , (b) all except S2 . (4) All four sets are empty. We adopt the convention that if one of the four sets being considered here is nonempty, then one of its vertices will be denoted as follows: s1 ∈ S1 , s2 ∈ S2 , t3 ∈ T3 , and t4 ∈ T4 . We observe that if some of the vertices vi are the same, then the proofs of the cases given below also apply when the following substitutions are made: • • • •

if v1 if v1 if v2 if v2

= v4 , let s1 = v3 , let s1 = v3 , let s2 = v4 , let s2

= z and t4 = u; = t3 = u; = t3 = v; = t4 = w.

The diagrams in Fig. 12.17 may be useful in the proof of the ten cases; the notation v(i, j ) means that v is a vertex in both Si and Tj .

12.4 Characterization of Second Order Iterated Line Digraphs

191

x(4, 1) u

s1

t3

s1

s1

u

x t4

w

s2

x(4, 1) u(3, 1)

s2

x(4, 1)

s1

u(3, 1)

w(4, 2)

Case (2b)

Case (2a)

u(3, 1)

s1

t3

x(4, 1)

v(3, 2)

s2

t4

w(4, 2)

w(4, 2)

x(4, 1)

v(3, 2)

Case (2c)

Case (2d) u(3, 1) y( , 1)

x(4, 1) u(3, 1)

t3

v(3, 2)

v(3, 2) w(4, 2)

u(3, 1)

t4

v Case (1b)

c

s2

v(3, 2) w(4, 2) x(4, 1)

c y( , 2) v(3, 2) w(4, 2)

Case (3a)

w

Case (1a)

s1

s1

x s2

w

Case (0)

c

t3

v

v s2

u(3, 1)

u(3, 1) v(3, 2) w(4, 2) a(3, ) d(4, )

c

b( , 1) c( , 2) x(4, 1)

Case (3b)

Fig. 12.17 Cases in the the proof of sufficiency

Case (4)

192

12 Iterated Line Digraphs

Case (0) In this case, D must have 2-walks from s1 to t3 , s1 to t3 , and s1 to t3 , and hence by (d) in the hypothesis, there must be a 2-walk s1 → x → t4 with x ∈ S4 ∩ T1 . Case (1a) T4 = ∅. In this case, w is not a sink neighbor of s2 . Furthermore, since v1 = v2 , it is not a sink neighbor of s1 either. Hence, by our construction, there must be a sink neighbor x of s1 in the same Sk as w; that is, x ∈ S4 ∩ T1 . Case (1b) T3 = ∅. In this case, u and v are both sinks in S3 , so by our construction, there must be arcs to them from first-level vertices s1 and s2 . Thus (see the figure) there are 2-paths from them to some vertex p, and so, by (d) there must exist a 2-walk from s1 to t4 , and its middle-level vertex must be in S4 ∩ T1 . Case (2a) T3 = T4 = ∅. We have a vertex p as in the preceding case, and by condition (e) and our construction, a sink neighbor of s1 must be paired with w. Case (2b) T3 = T4 = ∅. In this case, a vertex w in S4 ∩ T2 is an isolated vertex. Since s1 is a first-level vertex associated with the same tri-level subgraph and w ∈ S4 , our construction guarantees the existence of a sink neighbor of s1 , that is, a vertex in T1 that is also in S4 . Case (2c) S2 = T4 = ∅. In this case, u is a source and w is a sink and both are attached to the same tri-level subgraph. It follows from condition (f) that there is a corresponding isolated vertex x and it must be in S4 ∩ T1 . Case (2d) S2 = T3 = ∅. In this case, u is a sink, v is an isolated vertex, and w is a source. Furthermore, they must all be associated with the same tri-level subgraph in which there is again a 2-walk from s1 to t4 whose middle-level vertex is in S4 ∩ T1 . Case (3a) S2 = T3 = T4 = ∅. Here u is a sink and v and w are isolated vertices. There must be a source neighbor of a last-level vertex z with y ∈ T2 to account for v ∈ S3 along with u. But then s1 must have a sink neighbor x to account for w and y being in the same Ti . Hence x ∈ S4 ∩ T1 . Case (3b) S1 = T3 = T4 = ∅. In this case, u is an isolated vertex and v and w are sink neighbors of s2 . It follows as in the preceding case that there is a source y in conjunction with u and v, and so there must be another isolated vertex x for y and w. Hence, once again x is in the required two sets. Case (4) All four sets are empty. In this case, all three of the vertices u, v, and w are isolated. Since u and v are both in S3 , there must also be a sink in S3 and two corresponding sources b and c in T1 and T2 , all attached to the same tri-level subgraph. Similarly, associated with the same subgraph, since v and w are both in T2 , there must be a second sink neighbor at each of the first-level vertices. It follows that there is a fourth isolated vertex x, and it must be in S4 ∩ T1 . This completes the proof that E is a line digraph and therefore that D is a secondorder line digraph.  

12.5 Characterizations of Families of Second-Order Line Digraphs

193

It is natural to try to extend the characterization of iterated line digraphs beyond the second order. In a remarkable tour de force, Hemminger [102] was able to achieve this for all values. As one might expect, his characterization is similar to Theorem 12.6, but substantially more complicated, and his proof has some 20 lemmas. Consequently, the general characterization is well beyond the scope of this book.

12.5 Characterizations of Families of Second-Order Line Digraphs In this section we focus on subgraph characterizations of iterated line digraphs of some restricted families of digraph, starting with those having no loops. We begin with the root digraphs of several small digraphs relevant to this topic. These can be obtained by means of the algorithm presented in Chap. 11 using connection digraphs, but here we just show the results, which can of course be confirmed by checking that L(Hi ) ∼ = Fi for each of the digraphs in Fig. 12.18. Fig. 12.18 Some root digraphs of line digraphs F1 :

H1 :

F2 :

H2 :

F3 :

F4 :

H3 :

H4 :

194

12 Iterated Line Digraphs

A1 :

A3 :

A2 :

A5 :

A4 :

A6 :

A7 :

Fig. 12.19 Digraphs for condition (a )

Theorem 12.7 A loop-free line digraph D is a second-order line digraph if and only if it satisfies the following four conditions: (a ) D does not contain any of the seven digraphs in Fig. 12.19. (b ) (Second Heuchenne condition) If a subdigraph F of D contains any of the seven digraphs with six solid arcs in Fig. 12.20, then it must contain the additional vertex and pair of arcs shown. (c ) For each configuration C1 or C2 in Fig. 12.21, if v has p sink neighbors and w has q source neighbors, then D has pq associated isolated vertices, and for each configuration C3 or C4 , if v has r sink neighbors and s source neighbors, then D has rs associated isolated vertices. In addition, the sets of isolated vertices for different configurations are disjoint. (d ) For each configuration D1 or D2 in Fig. 12.22, the vertices v and w have the same number of sink neighbors in D, and for each configuration D3 or D4 , the vertices x and y have the same number of source neighbors. Proof Assume that digraph D is the second-order line digraph of some loop-free digraph F . Considering the digraphs in Fig. 12.19, we know that D being itself a loop-free line digraph, cannot contain A1 . Furthermore, D cannot contain A2 or A6 by the prohibition on there being two different paths of length 2 from one vertex to another. Now if A3 is a subdigraph of D, then as shown in Fig. 12.18, its root digraph must contain a transitive triple, and so A3 , and hence D itself cannot be a second-

12.5 Characterizations of Families of Second-Order Line Digraphs

B1 :

B2 :

B3 :

B4 :

B5 :

B6 :

195

B7 :

Fig. 12.20 Digraphs for condition (b )

v C1 :

C2 : w

v

Fig. 12.21 Digraphs for condition (c )

order line digraph. Likewise, for A4 and A5 , their root digraphs must contain A2 , and so they cannot be in a second-order line digraph. A similar argument applies to A7 since it is the line digraph of A6 , which itself cannot be in a line digraph. Therefore, condition (a ) must hold. The seven digraphs in Fig. 12.20 are the ways in which the second Heuchenne condition can occur in a loop-free digraph without any of the digraphs in Fig. 12.19, so D must satisfy condition (b ). Similarly, D must satisfy conditions (c ) and (d ) as they are the digraphical interpretation for (c) and (d) in the general characterization of Theorem 12.6. Therefore the four conditions are all necessary for D to possess.

196

12 Iterated Line Digraphs

Fig. 12.22 Digraphs for condition (d )

v D1 :

D2 : v

w

w

x D3 :

D4 : y

x

y

For the sufficiency, we now assume that D is a loop-free digraph satisfying conditions (a )–(d ) and show that D meets the four conditions of Theorem 12.6 (that is, (a)–(d)) and so therefore must be a second-order line digraph. That (a) is satisfied follows from the fact that the only way that a loop-free digraph can have two paths of length 2 from one vertex to another is as in digraph A2 or A6 in Fig. 12.19. We now assume that D contains three 2-walks vax, wbx, and wcy, and determine which configurations, in addition to the one in which all seven vertices are different as shown in the digraph B1 in Fig. 12.20, are necessary in order to meet the second Heuchenne condition. We look at the vertices in two groups, the three center vertices, a, b, c, and the four end vertices v, w, x, y, of the three walks. There are thus three types of identifications of two vertices to consider: Case (i) One vertex from each of the two sets. If v = c or a = y, then the forbidden digraph A3 in Fig. 12.19 would be present, while in all of the other cases, there is either a loop or a transitive triple, so there cannot be any configurations from this group. Case (ii) Two central vertices. If a = c, then there are two paths of length 2 from w to x, and this is forbidden. If a = b, then, as shown in B1 in Fig. 12.20, since D is a line digraph, the first Heuchenne condition dictates the presence of the arc vc and so the second Heuchenne condition is already satisfied in this case by the path vcy. Since the same argument applies to the case b = c, there are no further considerations here. Case (iii) Two end vertices. Clearly, v = w and x = y since for either of these the digraph A2 would be present. Each of the other four single identifications require one of the configurations in Fig. 12.20 as follows: v = x → B5 ; v = y → B7 ; w = x → B6 ; w = y → B4 ; as do the other two pairs of identifications: v = x, w = y → B2 ; v = y, w = x → B3 .

12.5 Characterizations of Families of Second-Order Line Digraphs

197

Thus, the second Heuchenne condition (that is, Condition (b) from Theorem 12.6) now follows from Conditions (a ) and (b ). Turning to the conditions involving tri-level digraphs, since by hypothesis D has no loops, every walk of length 2 is either a path or a cycle. It follows that Condition (c ) implies (c) of Theorem 12.6, and likewise Condition (d ) implies (d). It follows that D is a second-order line digraph, which completes the proof.   One of the most interesting families of digraphs are oriented graphs discussed in Chap. 11, which are loop-free and asymmetric, that is, they have at most one arc between any pair of vertices. In other words, if vw is an arc in an oriented graph, then wv is precluded from being present (technically, this also prohibits the presence of any loops). The following result is therefore a corollary to the preceding theorem. Clearly, the line digraph of an oriented graph is also an oriented graph, and vice versa. Even though the digraphs in this theorem appear in other figures, we present them again so that the results are self-contained. Theorem 12.8 An oriented graph D is a second-order line digraph if and only if it satisfies the following four conditions: (a

) D does not contain any of the six digraphs in Fig. 12.23. (b

) (Second Heuchenne condition) If a subdigraph F of D contains a subdigraph with the six solid arcs in Fig. 12.24, then it must contain the additional vertex and arcs shown. (c

) For each appearance of the digraph in Fig. 12.25, if v has p sink neighbors and w has q source neighbors, then D has pq associated isolated vertices. (d

) For each configuration D1 in Fig. 12.26, the vertices v and w have the same number of sink neighbors in D, and for each configuration D3 , the vertices x and y have the same number of source neighbors. Proof Let D be the second-order line digraph of an oriented graph. Then directly from the theorem, it cannot contain any of the first five digraphs in Fig. 12.19. The sixth digraph cannot appear in D since by the second Heuchenne condition a pair of opposite arcs would be present. The other three conditions are simply the result of eliminating from consideration those subdigraphs with pairs of opposite arcs. Thus, the stated conditions are necessary. Since the other conditions for sufficiency for a digraph to be the second-order line digraph of a loop-free digraph in the theorem are vacuously satisfied, these four conditions are sufficient for a digraph to be the second-order line digraph of an oriented graph.   To conclude this chapter, we note that because the complexity of characterizations of iterated line digraphs (beyond the first) involves sources, sinks, and isolated vertices, junctional digraphs, those in which every vertex has both an in-arc and an out-arc, have simple characterizations of their iterated line digraphs. We state it for the second-order case, but with the corresponding statement holding if both (a) and (b) in this theorem hold for walks of lengths i = 1, 2, . . . , k.

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12 Iterated Line Digraphs

A1 :

A2 :

A3 :

A4 :

B7 :

A5 :

Fig. 12.23 Digraphs for Condition (a

)

B1 :

Fig. 12.24 Digraph for Condition (b

)

12.5 Characterizations of Families of Second-Order Line Digraphs

199

v C1 : w

Fig. 12.25 Digraph for Condition (c

) v D1 : w

x D3 : y

Fig. 12.26 Digraphs for Condition (d

)

Theorem 12.9 A junctional digraph D is a second-order line digraph if and only if it satisfies the following conditions: (a

) For any two vertices in D, there does not exist more than one walk of length 2 from the first to the second. (b

) D satisfies both the first and second Heuchenne conditions.

Part III

Generalizations

Chapter 13

Total Graphs and Total Digraphs

13.1 Introduction The study of line graphs has yielded rich results on graphs of a certain type as well as deeper insight into properties of their root graphs. It is not surprising that the concept of line graphs has been generalized in many ways. In this third part of the book, we consider a variety of generalizations of line graphs. We begin here with the observation that an edge in a graph can be viewed in several ways, such as an element of the graph, as a subset of two vertices that are adjacent, as a path of length 1, or as a subgraph of edge-size 1. The concept of adjacency of edges can then be viewed accordingly. In this chapter, we consider the first generalization, in which both the vertices and edges of a graph G are the basic objects of a new graph called the total graph of G. Adjacency of these objects are then defined in a natural composite way. A convenient alternative description is given in terms of subdividing the edges of G and then joining vertices at distance 2. This assists in obtaining information about total graphs, including the number of edges and triangles and the degrees of the vertices. The next section gives information about the total graph of graphs in some elementary families, after which some questions about isomorphisms are answered. This is followed by sections on planar total graphs and on Eulerian and Hamiltonian total graphs. The last section of the chapter is devoted to the directed analogue of the subject, that is, total digraphs, whose vertices are the vertices and arcs of a given digraphs, with an appropriate definition of directed adjacency. Because of the difference in the definition of adjacency in total digraphs as opposed to total graphs, some of theorems are quite different in the two cases.

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_13

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13.2 Basics of Total Graphs In his 1965 doctoral dissertation, Mehdi Behzad [22] generalized the concept of the chromatic number of a graph to the total chromatic number and defined the total graph of a graph. The total graph T (G) of a graph G is the graph that has as its vertex set V (G) ∪ E(G) with a pair of these elements adjacent if in the graph G they are either (i) adjacent vertices, (ii) adjacent edges, or (iii) an incident vertex and edge. This is illustrated schematically in Fig. 13.1. It follows from the definition that the total graph of graph G is spanned by the sum (disjoint union) G + L(G) of G and its line graph L(G), as is illustrated in Fig. 13.2. The edges in T (G) can be partitioned into three sets: those in G (red), those in L(G) (green), and those that join vertices in G with vertices in L(G) (blue). We shall use this partition in several illustrations in this chapter. We observe that the set of edges of the total graph T (G) that are not in G or L(G) consists of two edges from each vertex of L(G) to vertices of G. As we shall see, there is a second way in which to view total graphs, one that involves the subdivision graph S(G) of a graph G, defined as the result of replacing each edge by a path of length 2 by inserting one new vertex. The following characterization of subdivision graphs in terms of distances will be useful later. Lemma 13.1 A connected graph G is a subdivision graph if and only if it is bipartite and the distance between any two vertices whose degrees are not 2 is even. Proof It easily follows from the definition of the subdivision graph that the stated conditions are necessary. For convenience in the proof, we call a vertex whose degree does not equal 2 a ‘non-2’ vertex. Clearly the converse holds for cycles and graphs with just one non-2 vertex. Let G be a graph with at least two non-2 vertices. Every edge of G must either be on a cycle with only one non-2 vertex or on a path joining two non-2 vertices with all other vertices of degree 2. Since each of these subgraphs is a subdivision graph, it follows that G is.  

v

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Fig. 13.1 The total graph operation

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13.2 Basics of Total Graphs

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L(G) :

u c

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w c

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G:

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Fig. 13.3 The square of the subdivision of a graph

The total graph T (G) is then constructed by taking the square (S(G))2 of the subdivision graph S(G), which is obtained by adding edges joining the pairs of vertices at distance 2 from one another. An illustration of this for the graph G in our example is shown in Fig. 13.3. As observed earlier, G and L(G) are disjoint induced subgraphs of T (G). Thus, we have the following result.

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Theorem 13.1 The total graph of any graph G is the square of its subdivision graph; that is, T (G) ∼ = (S(G))2 . Several observations can be made from the view of total graphs that this result provides. For instance, the total graph of the trivial graph K1 is K1 itself, while that of K2 is K3 . Furthermore, C3 is the only cycle that is a total graph and K1 and K3 are the only complete graphs that are. Additionally, it follows from the squaring property that the total graph of every non-trivial connected graph is 2-connected and hence is also 2-edge-connected. The following result gives a variety of additional properties of total graphs, involving the numbers of edges and triangles and the degrees of the vertices. Theorem 13.2 Let G be a graph with n vertices, m edges, and t triangles. Let the vertices be v1 , v2 , . . . , vn with deg vi = di for each i, and let Δ = max deg G. Then the following hold: (a) (b) (c) (d) (e) (f) (g) (h)

T (G) has n + m vertices. For any vertex v of G, degT (G) (v) = 2 deg v. For any edge e = vw of G, degT (G) (e) = deg v + deg w. For any edge e = vw of G, if degT (G) (e) is maximum, then deg v = deg w = Δ. max deg T (G) = 2Δ. G is r-regular if and only if T (G) is 2r-regular. T (G) has 2m + 12 ni=1 di2 edges.       T (G) has m + 2t + ni=1 d2i + ni=1 d3i triangles.

Proof The first six statements (a) through (f) follow at once from the definition of the total graph. For (g), we take the sum of the degrees of vertices in T (G) in two parts n corresponding to vertices and edges of G. From (b), the first part yields 2 i=1 di =  4m. From (c), the second part gives the sum (di + dj ), taken over the edges of G.  Since a vertex ui of G appears in di edges, the second sum yields ni=1 di2 , and this establishes (g). Finally, for (h), we observe that a triangle in T (G) can be one of the four types: (i) The vertices of this triangle are vertices in G. There are t such triangles. (ii) The vertices are edges in G. Three adjacent edges in G are a triangle or a K1,3    in G. There are t + ni=1 d3i of these triangles. (iii) The triangle is formed from two vertices and one edge of G. In this case the two vertices must be the endpoints of the edge, and there are m such triangles. (iv) The triangle is formed  by one vertex of G and two edges incident with that vertex. There are ni=1 d2i triangles of this type. This completes the proof.

 

In the next two sections, the total graphs of some common families are determined, and we prove an analogue for total graphs of one of Whitney’s theorems about graphs that have isomorphic line graphs.

13.3 Special Families

207

13.3 Special Families We begin with the structure of the total graphs of paths, cycles, and complete graphs. Theorem 13.3 A connected graph G is the total graph of a path Pn (for n ≥ 3) if and only if G has order 2n − 1 and has two disjoint paths v1 v2 . . . vn and w1 w2 . . . wn−1 having the edges of the path v1 w1 v2 w2 . . . vn−1 wn−1 vn . Proof If G = T (Pn ), then the conditions are clearly satisfied by definition of the total graph. The converse follows by observing that the line graph of the path v1 v2 . . . vn is the path w1 w2 . . . wn−1 and that the edges between these two paths yield the path of length 2(n − 1).   The total graph T (P6 ) is shown in Fig. 13.4. Obviously, the total graph of a cycle is similar; indeed, it is simpler. Theorem 13.4 A connected graph G is the total graph of a cycle Cn with n ≥ 3 vertices if and only if G is 4-regular of order 2n, has two vertex disjoint cycles v1 v2 . . . vn v1 and w1 w2 . . . wn w1 for which v1 w1 v2 w2 . . . wn v1 is a cycle. Proof This follows directly from the definition of a total graph. Figure 13.5 shows   T (C9 ). Our next result shows that the line graph and the total graph of complete graphs are directly related [28]. Theorem 13.5 For all n, T (Kn ) ∼ = L(Kn+1 ). Proof Let F be a complete graph with n vertices v1 , v2 , . . . , vn and let H be a complete graph with n + 1 vertices w1 , w2 , . . . , wn+1 . Define a mapping from V (T (F )) to V (L(H )) as follows: vi → wi wn+1 for 1 ≤ i ≤ n, and vj vk → wj wk for 1 ≤ j < k ≤ n. It is straightforward to verify that this mapping is an adjacencypreserving one-to-one correspondence.   The above result also implies that T (Kn ) is the edge-disjoint union of n + 1 copies of Kn with each vertex in precisely two of the copies. This is illustrated for P6 :

T (P6 ):

Fig. 13.4 The total graph of a path

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Fig. 13.5 The total graph of C9

T (K4 ):

Fig. 13.6 K4 s in T (K4 )

n = 4 in Fig. 13.6. Properties of the line graphs of complete graphs were discussed in Chap. 1; specifically see Fig. 1.3 and Theorem 1.2.

13.4 Ordinary Graphs Having covered the total graphs of paths, cycles, and complete graphs, we now turn to the other graphs, which for convenience we call ordinary. We denote the neighborhood of a vertex v in a subgraph H of a graph G by NH (v), and the graph induced by vertices v1 , v2 , . . . , vk of G by v1 , v2 , . . . , vk G . The following result is due to Behzad and Radjavi [26, 27].

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Lemma 13.2 If G is an ordinary connected graph, and if v is a vertex of maximum degree 2d in the total graph T (G), then the subgraph of T (G) induced by the neighborhood NT (G) (v) contains exactly one complete graph Kd if and only if v is a vertex of G. Proof The given conditions imply that d ≥ 3. If x = vw is an edge of G, then from Theorem 13.2, we have deg v = deg w = d, so that the set of edges (in G) incident to v and the set incident to w each forms a copy of Kd in H . On the other hand, if x is a vertex of G, then the d edges incident to x form Kd in H . If the d vertex neighbors of x in G induce Kd in H , then x and its neighbors in G induce Kd+1 . Since G is not complete, at least one of the vertices of this complete subgraph Kd+1 has degree greater than d in G, a contradiction. Finally, if H has a copy of Kd that includes at least one vertex (say v) and at least one edge (say e) of G, it follows that e = xv. But then e and v have no common neighbor in H , a contradiction. This completes the proof.   Theorem 13.6 If G is an ordinary connected graph and H is a subgraph of T (G) for which T (H ) = T (G), then H ∼ = G. Proof From Theorem 13.2, the maximum degree of T (G) is even, say 2d, with d ≥ 3. Let v0 be a vertex in G of maximum degree d. From Lemma 13.2 it follows that the graph of T (G) = T (H ) induced by NT (G) (v0 ) has exactly one copy of Kd , which implies that v0 ∈ V (H ). Let NT (G) (u0 ) = {u1 , u2 , · · · , ud , e1 , e2 , · · · , ed } where u1 , u2 , · · · , ud are vertices in G adjacent to u0 and ei is the edge u0 ui (1 ≤ i ≤ d) in G. We denote {u0 , u1 , · · · , ud } by Ad . In H , if at least d − 1 of u1 , u2 , · · · , ud are edges at u0 , we get another Kd in T (H ), a contradiction. Hence at least two of the edges at u0 are from the set {e1 , e2 , · · · , ed }. Without loss of generality, suppose e1 and e2 are edges at u0 in H . Since e1 , e2 , · · · , ed induce Kd in T (H ), it follows that NH (u0 ) = {u1 , u2 , · · · , ud }. We observe that the set Ad satisfies that properties that (1) Ad T (G) is a subgraph of G as well as H , and (2) the set of edges in Ad G are also edges in < Ad >H . If < Ad >G = G, it follows that G = H . The rest of the proof shows that we can iteratively add vertices to Ad one at a time, and the sets so obtained satisfy properties (1) and (2). Accordingly, suppose there is a vertex ud+1 in V (G) − Ad . Let Ad+1 = {u0 , u1 , · · · , ud+1 }. Since G is connected, there is an edge uj ud+1 , for some j . Let e = uj uk be an edge in G as well as in H . Then e is not adjacent to ud+1 in T (G). If ud+1 is not a vertex in H , then in T (G) it corresponds to an edge of H , which is adjacent to the vertex uj . Hence, ud+1 is adjacent to the edge e, a contradiction. It follows that ud+1 is also a vertex in H . Thus, Ad+1 satisfies (1). By a similar argument it can be shown that Ad+1 satisfies (2). This completes the proof.   We can now state the analogue for total graphs of an isomorphism theorem of Whitney for line graphs, Theorem 2.2. Note that, unlike for line graphs, the theorem for total graphs has no exceptions.

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Theorem 13.7 For connected graphs G and H , T (G) ∼ = T (H ) if and only if G ∼ = H. ∼ H is sufficient for T (G) and T (H ) to be Proof Clearly the condition G = isomorphic. We now prove the necessity. Assume T (G) ∼ = T (H ). If G is an ordinary graph, the result follows from Theorem 13.6. If G is a path, the result follows from Theorem 13.3. Finally, if G is a cycle or a complete graph, the result follows from Theorem 13.2. This completes the proof.   Having characterized the total graphs of paths, cycles, and complete graphs, we next consider total graphs of ordinary regular graphs. If G is an r-regular graph of order n with 3 ≤ r ≤ n − 2, then from Theorem 13.2 we know that T (G) is 2r-regular of order n(r+2) 2 . Also, from Lemma 13.2, for each vertex x of T (G), the graph induced by NT (G) (x) has exactly one Kr if and only if x ∈ V (G). We thus have the following result [27]. Theorem 13.8 An ordinary connected regular graph G is the total graph of an ordinary regular graph if and only if the following three properties hold: (a) There exist integers r and n with 3 ≤ r ≤ n − 2 for which G is 2r-regular and has order n(r+2) 2 ; (b) G has exactly n vertices v with the property that the subgraph of G induced by the closed neighborhood of v has exactly one Kr ; (c) G is the total graph of the graph generated by these n vertices. We conclude this section by noting that Gavril [80] found a recognition algorithm for total graphs by using a result of Behzad [24].

13.5 Planarity In this section we present some basic results related to the planarity of total graphs. In Chap. 5 we defined planar graphs and discussed the planarity of line graphs. It is useful to have one further definition, namely, that a graph is biplanar if it is the union of two planar graphs (under this definition, planar graphs are also biplanar). Theorem 13.9 For any graph G, if L(G) is planar, then T (G) is biplanar. Proof By Theorem 5.2, the planarity of L(G) implies that of G. In a plane embedding of G, for each edge, alongside each edge e = vw, add a new path vxw to form a new graph that we call G∗ . See Fig. 13.7. Clearly G∗ is planar. Furthermore, the newly added vertices are in one-to-one correspondence with the edges of L(G). Hence, L(G) can be added to G∗ . It is not hard to see that the result is T (G). Since both G∗ and L(G) are planar, it follows that T (G) is biplanar.   We next present a result of Behzad [23] that gives necessary and sufficient conditions for the planarity of total graphs. We use the fact that, for a graph G, T (G) ∼ = (S(G))2 (Sect. 13.2), where S(G) is the subdivision graph of G.

13.5 Planarity

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a

a

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G∗ :

G: c

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L(G): c

b

c

b

d d

d

T (G):

Fig. 13.7 The biplanarity of the total graph

Harary, Karp, and Tutte [95] obtained useful necessary and sufficient conditions for the square of a graph to be planar. Theorem 13.10 A graph G has a planar square if and only if the following three conditions hold: (a) the maximum degree of G is at most 3; (b) every block of G with more than four vertices is an even cycle; (c) G does not have three mutually adjacent cut vertices. This theorem gives one set of necessary and sufficient conditions for a graph to have a planar total graph; another was found by Behzad [23]. Theorem 13.11 The following are equivalent for a connected graph G: (1) T (G) is planar. (2) The maximum degree Δ(G) ≤ 3 and every block of S(G) with more than four vertices is an even cycle. (3) The maximum degree Δ(G) ≤ 3 and every vertex of degree 3 is a cut-vertex. Proof We show that (1) ⇒ (2), (2) ⇒ (3), and (3) ⇒ (1). (1) ⇒ (2): Assume that G has a planar total graph. Since T (G) ∼ = (S(G))2 , (2) holds by Theorem 13.10.

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(2) ⇒ (3): Assume that (2) holds and suppose that there is a vertex v of degree 3 that is not a cut-vertex. Then v and its three neighbors are all in the same block of G, and hence there is a block of S(G) that violates (2). (3) ⇒ (1): We now assume that (3) holds, and show that the subdivision graph S(G) satisfies the three conditions of Theorem 13.10. First, that Δ(S(G)) ≤ 3 follows from the fact that Δ(G) ≤ 3. Next, if B is a block of S(G) with more than four vertices, then B cannot contain a vertex of degree 3 or more. Finally, S(G) cannot have three mutually adjacent cut vertices.  

13.6 Traversability In Chap. 7 we described several results regarding Eulerian and Hamiltonian line graphs. As we will see in this section, there is a parallel set of results for total graphs. Our treatment largely follows that of Behzad [22]. We begin with a result on Eulerian total graphs that is directly related to Theorem 7.1. Theorem 13.12 The following statements are equivalent for a connected graph G: (1) T (G) is Eulerian. (2) L(G) is Eulerian. (3) The degrees of the vertices of G all have the same parity. Proof Theorem 7.1 on line graphs shows the equivalence of (2) and (3). If T (G) is Eulerian, then the degree of every vertex e in T (G) must be even, and since, if e = vw, the degree of e equals deg v + deg w, it must be that v and w are both even or both odd. Since G is connected, it follows that same must be true of all of the vertices of G, which proves (3). That (3) implies (1) follows from the definition of T (G).   Clearly, this result also implies that if a graph G is Eulerian, then so are L(G) and T (G). Theorem 7.2 provides a characterization of those line graphs whose root graph is Eulerian. We leave a similar characterization for total graphs as an open problem. We now turn to Hamiltonian total graphs. The next theorem, analogous to Theorem 7.3 on line graphs, has several interesting special cases. Theorem 13.13 If a graph G contains a spanning Eulerian subgraph, then T (G) is Hamiltonian. Proof Given a spanning Eulerian circuit e0 , e1 , . . . , er−1 , e0 of edges in G, we denote the common vertex of ei and ei+1 by vi , 0 ≤ i ≤ r − 1. We create an ordered sequence of the edges and vertices of G such that consecutive elements are adjacent in T (G). Starting with e0 as the first element in the sequence, we add v0 as the second element and then all the edges (except e1 ) at v0 which are not already

13.7 Total Digraphs

213

in the sequence, followed by e1 . It is not hard to see that a repetition of this process results in a Hamiltonian cycle of T (G).   It can be easily seen that for s ≥ 3 the total graph T (K1,s ) is not Hamiltonian. However, T (T (G)) is always Hamiltonian, as our next result shows. Corollary 13.1 If G is a connected graph, then T (T (G)) is Hamiltonian. Proof Since G is connected, the spanning subgraph H of T (G) obtained by removing all the edges of L(G) is connected and the degree of every vertex in H is even. The result follows from Theorem 13.13.   We close this section with the observation that some connectivity properties of total graphs have also been investigated. The interested reader may refer to Bauer and Tindell [21], Hamada et al. [87], and Simões Pereira [160].

13.7 Total Digraphs In 1965, Chartrand and Stewart [57] extended the concept of total graphs to total digraphs. The total digraph T (D) of a digraph D has the vertices and arcs of D as its vertices; as is indicated in Fig. 13.8, an arc from a vertex v to a vertex w can occur in one of four ways: Both v and w are vertices in D and there is an arc from v to w in D; both v and w are arcs in D and the head of v is the tail of w; v is the tail of arc w in D; or w is the head of arc v in D. Note that the total digraph T (D) of a digraph D is spanned by D and L(D) as disjoint subgraphs and then each vertex in the line digraph portion has one arc to and one arc from the digraph portion. In Fig. 13.9 two total digraphs are shown, one of an opposite pair of arcs and the other of a transitive triple. For simplicity, the digraphs we consider here may have pairs of opposite arcs, but no loops.

v

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Fig. 13.8 The total digraph operation

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Fig. 13.9 The total digraph of a digraph

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Fig. 13.10 Every arc in a transitive triple

We observe that in all five situations that yield the arc in a total graph, there is a third element x that together with v and w generates a transitive triple, as indicated in Fig. 13.10. From this we deduce that every arc in a total digraph is on a transitive triple, a fact that we return to later.

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The following theorem gives basic results on numbers and degrees in total digraphs. Theorem 13.14 The following statements hold for a nontrivial digraph D with n vertices and m arcs.  (a) T (D) has m + n vertices and 3m + v d + (v)d − (v) arcs. (b) If v is a vertex of D, then its out-degree in T (D) is 2d + (v) and its in-degree is 2d − (v). (c) If e = vw is an arc of D, then its out-degree in L(D) is d + (w) + 1 and its in-degree is d − (v) + 1. Observe that each arc in D gives rise to a transitive triple in T (D), each transitive triple also gives rise to a transitive triple, and a directed 3-cycle gives rise to two 3cycles. It is also not hard to see that each 2-cycle in D gives rise to four 3-cycles and four transitive triples in T (D). Theorem 13.15 If D is a digraph with m arcs, a transitive triples, b 2-cycles, and c 3-cycles, then T (D) has a + 4b + m transitive triples and 4b + 2c 3-cycles. Again, similar to the undirected case, total digraphs can be characterized in terms of the subdivision digraph of a digraph. The square D 2 of a digraph D is obtained from D by adding an arc from vertex v to vertex w if the (directed) distance from v to w in D is 2. Also, the subdivision digraph S(D) of D is obtained from D by adding a new vertex on each arc of D; that is, replace each arc vw with a path of length 2 from v to w. It is useful to have a name for vertices like those added in a subdivision: a vertex is a carrier if its in-degree and its out-degree are both 1. Now that we have seen many parallels between total graphs and total digraphs, the following result is not surprising. Theorem 13.16 The total digraph of a digraph D is the square of its subdivision digraph; that is, T (D) ∼ = (S(D))2 . We now turn to connectedness properties of total digraphs. Recall that a digraph is said to be connected if its underlying graph is connected, and strongly connected if there is a directed path from each vertex to each of the others. Assume now that D is strongly connected. If v and w are vertices in T (D) that are also vertices in D, then there is v-w path in both digraphs. If v and w are both arcs in D, we similarly get a v-w path in L(D) that begins with the arc from v to head(v), followed by a path in D from head(v) to tail(w), and ends with the arc from tail(v) to v. The cases when exactly one of v and w is a vertex and the other an arc in D are similar. Consequently, T (D) is also strongly connected. Conversely, it follows directly from the definitions that if T (D) is strongly connected, then so is D. This and similar facts lead us to the next result, due to Chartrand and Stewart [57].

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Theorem 13.17 For a digraph D, (a) T (D) is connected if and only if D is connected. (b) T (D) is strongly connected if and only if D is strongly connected. (c) T (D) is acyclic if and only if D is acyclic. The next result gives a characterization of total digraphs given by Skowro´nska, Sysło, and Zamfirescu [161]. We require some additional terminology and notation. If a transitive triple in a digraph consists of the arcs uv, vw and uw, then uv and uw are called legs of the triple and vw is called a hypotenuse. We observed earlier that each arc of a total digraph is in a transitive triple. Figure 13.10 illustrates that if uv is a hypotenuse in the total digraph T (D), then u and v are either both vertices of D or both arcs of D and hence uv is the hypotenuse of just one transitive triple in T (D). It was shown in [161] that these properties constitute part of a set that characterize total digraphs. For this, it is useful to have some additional notation regarding transitive triples. Given a digraph D, let Dhyp be the spanning sub-digraph for which the arcs are the hypotenuses of D, and let DNhyp be the subdigraph generated by the arcs that are not hypotenuses (that is, each of its vertices is on an arc that is not a hypotenuse). Theorem 13.18 A digraph H is a total digraph of a digraph if and only if the following conditions are satisfied: (a) (b) (c) (d) (e)

Every arc of H is on a transitive triple. Every hypotenuse is the hypotenuse of exactly one transitive triple in H . If arcs uv and vw are arcs of H that are not hypotenuses, then uw is in H . Hhyp is disconnected. If u and v are not carriers in HNhyp and if Hhyp has a path from v to a vertex w, then uw and wu are not arcs in HNhyp .

It is illustrative to verify the above result for the digraphs in Fig. 13.11. In particular, by noting the colors of the arcs, it can be seen that T (D)Nhyp = S(D) and T (D)hyp = D + L(D). Using this characterization, an efficient algorithm was developed (see [161]) for testing whether a given digraph is a total digraph. We conclude this section by noting that Liu and Meng [136] have studied connectivity properties of total digraphs.

13.7 Total Digraphs

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v5

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Fig. 13.11 A digraph and its friends

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f T (D) = S(D)2 :

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Chapter 14

Path Graphs and Path Digraphs

14.1 Introduction We now turn to a variation of line graphs in which the basic element is a path of length 2, just as the basic element of a line graph being an edge is a path of length 1. With this interpretation, an edge appears in a line graph when two of these paths overlap in a vertex (a path of length 0). Implied in this is that their union is a path of length 2. The corresponding behavior for 2-paths is taken to be that they overlap in a path of length 1 and that their union be either a path of length 3 or a cycle of length 3. The concept can of course be extended to paths of longer length: informally, a k-path graph has for vertices paths of length k (thus paths with k + 1 vertices) in a given graph G, with adjacency arising from the two paths overlapping in a path of length k − 1 and their union being a path or cycle of length k + 1. The main focus in this chapter is the case k = 2, for which the richest theory has been found, and therefore we call them just ‘path graphs’ for convenience. After proving some basic results in Sect. 14.2, we turn to characterizing path graphs in Sect. 14.3. The main theorem is an analogue of Krausz’s partition characterization for line graphs themselves, but more complicated. The next section is on some isomorphism results, followed by a section on path digraphs, the directed analogue of path graphs. The chapter concludes with material on clique graphs, which are yet another variation of line graphs. The vertices of the clique graph of a given graph G are (naturally) the cliques of G (that is, the maximal complete subgraphs), with adjacency being their having a nonempty intersection. The main results here are the characterization of clique graphs and some variants.

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_14

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14.2 Definition and Basic Properties As noted in the Introduction, one description of the formation of the line graph L(G) of a graph G is to begin with vertices u, v, and w in graph G with u ∼ v ∼ w, and then take the edges e = uv and f = vw as vertices in L(G) with e ∼ f . This concept has as a natural extension taking vertices u, v, w, and x in G with u ∼ v ∼ w ∼ x (allowing x = u) and then in a new graph Λ(G), taking the paths p = uvw and q = vwx with p ∼ q. Thus, the basic process consists of forming a new graph Λ(G) in which two paths of length 2 go to one of length 1, as is illustrated in Fig. 14.1, unless the two paths form a 3-cycle C3 instead of the path P4 , in which case, repetition of the procedure twice more yields a 3-cycle in Λ(G) as illustrated in the figure. It is sometimes convenient to show different 2-paths in a graph G by different colors and for the colors to serve as the vertices of the path graph as we have done in these figures. One of the advantages of this is that the colors can be used as labels on the vertices of Λ(G) to indicate where an edge comes from in G. Of course this means that edges of G can have multiple colors, as in these two examples. We formalize this concept in a slightly different way in the following definition: The path graph Λ(G) of a graph with at least one pair of adjacent edges has the paths of length 2 as its vertices, with two of these vertices adjacent if their intersection is an edge and their union is either a path of length 3 or a cycle of length 3. Thus, vertices p and q in Λ(G) are adjacent if they are paths of length 2 in G and can be labeled as uvw and vwx where x may or may not equal u. The path graph Λ(Pn ) of a path of n vertices (n ≥ 3) is clearly Pn−2 (analogous to L(Pn ) being Pn−1 ). Another parallel between the two concepts is that both operations preserve cycles: Λ(Cn ) = L(Cn ) = Cn . Before looking at other graphs, we explain some aspects of our definition and terminology. The concept was introduced by Broersma and Hoede [43] in a general form (based on paths of arbitrary fixed length r) under the name Pr -path graphs, with our Λ(G) being denoted P3 (G) and called a P3 -graph. While this general notation is quite natural, it does have the awkwardness of path graphs of paths being Pk (Pn )

P4 :

C3 :

Λ

Λ

Fig. 14.1 The operation of forming the path graph

Λ (P4 ) :

Λ (C3 ) :

14.2 Definition and Basic Properties

221

G:

Λ (G) :

H:

Λ (H) :

Fig. 14.2 Two graphs and their path graphs

in symbols. Hence, because there are not many results known in the general case (or even in the case r = 4), we use the simpler term “path graph” for the name and the single letter Λ that suggests a path of length 2 for notation. We turn now to the question of why the original definition was taken to include having Λ(C3 ) equal C3 rather than adjacency of two 2-paths in Λ(G) only when there is a 3-path in G. In the spirit of line graphs, it seems desirable for the operation to take cycles to cycles, that is, for 3-cycles not to be an exception to the isomorphism property of cycles for path graphs as it is for line graphs. In fact, without this convention, it would be the case that all path graphs would be trianglefree (a fact that we encourage readers to verify for themselves). This is illustrated by stars, the graphs K1,s . Since these graphs have neither paths nor  cycles of length 3, their path graphs have no edges; that is, for s ≥ 2, Λ(K1,s ) = 2s K1 . To give more of the flavor of the Λ-graph operation, we give two examples in Fig. 14.2 (note that H = Λ(G)). Just as it is assumed that the graphs of which one is taking the line graph have at least one edge, so with taking path graphs, it is assumed that there is at least one pair of adjacent edges. Since any pair of adjacent edges e and f at a vertex v in a graph  G gives rise to a vertex ef in the path graph Λ(G), v gives rise to deg2 v vertices in Λ(G). Since each vertex in Λ(G) arises from just one vertex in G, this gives a nice formula for the order of Λ(G). The number of edges in Λ(G) is also easily counted since each edge corresponds to the middle edge of just one path or one edge of a cycle of length 3. Thus, an edge e = vw in G produces (deg v −1)(deg w −1) edges in Λ(G). Therefore the numbers of vertices and edges can both be simply expressed in terms of the degrees of the vertices of the base graph G. Theorem 14.1 If the path graph Λ(G) of graph G = (V , E) has n vertices and m edges, then n =

deg v  v∈V

2

and m =

e=vw∈E

(deg v − 1)(deg w − 1).

222 Fig. 14.3 The path graph of a comet

14 Path Graphs and Path Digraphs

K1,4 :

Λ (K1,4 ) :

An extension of the result that the path graph of a star is totally disconnected is that the path graph of any connected graph G having one vertex adjacent to two vertices of degree 1 will have an isolated vertex and hence will be disconnected as long as it has at least one more vertex. On the other hand, if G is a connected graph without any paths of length 2 joining two vertices of degree 1, then its path graph Λ(G) can easily be seen to be connected since every path of length 2 in G is a subpath of a longer path. Theorem 14.2 The path graph Λ(G) of a connected graph G is connected if and only if no vertex of G has two neighbors of degree 1. Corollary 14.1 The path graph Λ(G) of a connected graph G has at most one nontrivial component, and has one if and only if G is not a star.

, obtained from a star K Another interesting example is a comet K1,s 1,s by adding a tail, in this case adding a single vertex adjacent to one ofthe end vertices of the isolated vertices. star. Its path graph is then the star K1,s−1 together with s−1 2 Figure 14.3 shows the example with s = 4. Even more interesting is the subdivision graph S(K1,s ), the results of inserting a vertex on each of the edges in the star K1,s . Then the path graph Λ(S(K1,s)) has two kinds of vertices, the s 2-paths, each with a vertex of degree 1, and the 2s 2-paths whose middle vertex is the center of the graph. The path graph is bipartite with the vertices of the first kind of degree 2 and those of the second kind of degree s − 1. In fact, it is the subdivision of the complete graph Ks . Thus we have the rather neat result involving subdivision graphs: For s ≥ 3, Λ(S(K1,s )) ∼ = S(Ks ). The cases s = 3 and 4 are shown in Fig. 14.4). We note that Λ(S(K1,3 )) is a 6-cycle, and this suggests that non-isomorphic graphs having isomorphic path graphs may be more complicated than the corresponding concept for line graphs. As we saw in finding a formula for the number of vertices in a path graph Λ(G) in terms of the base graph G, the center vertices of 3-paths play a useful role. This is especially so in the case of bipartite graphs: If paths uvw and vwx are adjacent vertices in a path graph Λ(G), then v and w are in different partite sets of G. In other words, the partite sets of G induce sets for the vertices of Λ(G) according to the center vertices of the corresponding 3-paths. On the other hand, if a graph G has a cycle of odd length, then its path graph has a cycle of the same length, so its path graph Λ(G) is not bipartite either.

Theorem 14.3 The path graph Λ(G) of a graph G is bipartite if and only if G is itself bipartite.

14.3 Characterization of Path Graphs

S(K1,3 ):

223

Λ (S(K1,3 ):

S(K1,4 ):

Λ (S(K1,4 )):

Fig. 14.4 The path graph of an augmented star

Returning to the topic of triangles, recall that there is a one-to-one correspondence between those in a graph G and those in its path graph Λ(G). In fact, the triangles in a path graph are all disjoint. This is demonstrated in Fig. 14.5, which shows the path graphs of the two smallest graphs that have two intersecting triangles.

14.3 Characterization of Path Graphs In his pioneering work on properties of line graphs, one of the key features of the work of Krausz [122] was his observation that the set of edges at a vertex of a graph G gives rise to a complete subgraph of L(G), (and this is the basis of his characterization theorem). A partial counterpart to his result is that for a path graph Λ(G) the set of edges at  a vertex gives rise to independent vertices, and if the vertex has degree d these are d2 in number. However, these individual sets are not enough – it turns out that the building blocks for path graphs are complete bipartite graphs, as we shall now see.

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14 Path Graphs and Path Digraphs

Fig. 14.5 The path graphs of intersecting triangles G:

Λ (G) :

H:

Λ (H) :

Fig. 14.6 Complete bipartite sets G:

Λ (G) :

In fact, there is a property of sets of order n and their 2-element subsets that is extremely useful in characterizing path graphs: Let A = {a1, a2 , . . . , an }, and for i = 1, 2, . . . , n, let Ai be the set of the (unordered) pairs of elements of A that contain ai . Then it is readily seen that any pair {aj , ak } is in exactly two of the sets Ai and each pair Aj and Ak have precisely the pair {aj , ak } in common. It is also convenient to introduce some terminology involving complete bipartite subgraphs of graphs. Returning to earlier observations, the set of vertices of the path  graph Λ(G) is the union of the sets of d2 independent vertices arising from the d edges at those vertices of degree at least 2. Broersma and Hoede called these sets binomial sets, but since there are also other binomial numbers besides these, we choose to shorten the name to bins here. Consider now the contribution to Λ(G) of an edge e = v1 v2 of G as the center vertex of paths uv1 v2 w. Assuming that deg v1 = d1 and deg v2 = d2 with d1 , d2 ≥ 2, we observe that there are d1 −1 possible vertices u and d2 −1 possible vertices w. Hence, this configuration around e yields the complete bipartite graph Kd1 −1,d2 −1 in Λ(G). (See Fig. 14.6.) Furthermore, each of the two partite sets of Kd1 −1,d2 −1 lies in a bin, and so we say that the partite set fits into that bin. We are now prepared for the characterization theorem. Theorem 14.4 A graph G is a path graph if and only if its vertices can be partitioned into bins in such a way that its edges are partitioned  complete    into and s+1 bipartite graphs Kr,s whose partite sets are in bins of orders r+1 2 2 respectively, and the following two properties hold: (a) Each vertex in a bin is in at most two partite sets that fit into that bin.

14.4 Graphs with Isomorphic Path Graphs

225

(b) Two partite sets of vertices that fit into the same bin have at most one vertex in common. Proof (Necessity) Let G be the path graph   of graph H . As observed above, a vertex of degree d in G gives rise to a set of d2 independent vertices in Λ(G), and an edge e whose vertices both have degree at least 2 gives rise to a complete bipartite graph whose partite sets fit into the corresponding sets of vertices. By the properties of sets described above, (a) and (b) must hold. (Sufficiency) Now assume that G is a graph for which the given conditions hold. Construct a graph H as follows: First, let H have vertices that correspond to the bins described. For each of the bipartite graphs in the partition of the edges of G, join the two vertices that correspond to the two bins into which the partite sets fit. If  a bin B has d2 vertices, at most d sets of d − 1 pairs of vertices fit into B and any two of these have exactly one pair in common. Now if there are fewer than d sets, say c, then add d − c vertices adjacent to the vertex that corresponds to that bin B. This completes the construction of H . We now show that the the path graph of H is to G. By construction,  isomorphic  every vertex of H that represents a bin of order d2 has degree d, so there is a natural correspondence between the bins of H and the vertices of G. An end edge in H does not contribute to the edges of Λ(H ), while an edge that joins two vertices of degrees d1 and d2 , both at least 2, contributes     a complete bipartite graph Kd1 −1,d2 −1 to Λ(H ) which fits into bins of orders d21 and d22 . Two subsets that correspond to edges of H with a common vertex thus have one vertex of the associated bin in common, as described in the set property given above. Those vertices in a bin that do not correspond to a pair of edges in this way are either isolated vertices or end vertices. In either case, we have a one-to-one correspondence between the vertices of Λ(H ) and G. Furthermore, adjacency of the vertices is the same by the construction, which completes the proof.   As Li and Lin point out in their paper [132], this characterization leaves open the question of whether there is a good algorithm for determining whether a given graph is or is not a path graph, in contrast to the situation with line graphs.

14.4 Graphs with Isomorphic Path Graphs Recall that the two graphs K3 and K1,3 have line graphs that are isomorphic (and they are the only connected non-isomorphic graphs with this property). It is an interesting curiosity that the subdivision graphs of these two graphs, that is, C6 and S(K1,3 ), have isomorphic path graphs, the 6-cycle. However, the similarity ends there. Broersma and Hoede [43] showed that there are infinitely many pairs of connected non-isomorphic graphs having isomorphic path graphs. One pair of their smallest examples is shown in Fig. 14.7 and another in Fig. 14.8. Note that both graphs G1 and H1 have one cycle, while G1,2 and H1,2 are a bit larger, but have the additional asset of being trees. Broersma and Hoede in

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14 Path Graphs and Path Digraphs

G1 : Λ (G1 ) = Λ (H1 ) : H1 :

Fig. 14.7 A pair of graphs with isomorphic path graphs

H1,2 :

G1,2 :

Λ (G1,2 ) = Λ (H1,2 ) :

Fig. 14.8 Another pair of graphs with isomorphic path graphs

F:

v

w

Fig. 14.9 The foundation of one pair of graphs with isomorphic path graphs

fact extended these examples to infinitely many pairs. Their examples of unicyclic graphs can be described quite easily by appropriately adding starlike graphs at different vertices in the same graph. Starting with the star K1,s , we take both its subdivision S(K1,s ) and its double subdivision S2 (K1,s ) obtained by inserting two new vertices on each edge. Now let F be the graph shown in Fig. 14.9, and form Gs by attaching the center of S(K1,s ) at the vertex labeled v in one copy of F and form Hs by attaching the center of S2 (K1,s ) at the vertex labeled w in another copy of F . The graphs with s = 1 are in fact those shown in Fig. 14.7. Examples of trees are particularly interesting; the examples involve adding two generalized stars to each graph. Let T be the tree shown in Fig. 14.10, and let s < t. Now form Gs,t by attaching the center of S(K1,s ) at v and the center of S2 (K1,s ) at w in one copy of T and form Hs,t attaching the center of S(K1,s ) at w and the

14.5 Path Digraphs

227

T:

v

w

Fig. 14.10 The foundation of a pair of trees with isomorphic path graphs

center of S2 (K1,s ) at v in a second copy of T . The graphs in Fig. 14.8 are for the case s = 1, t = 2.

14.5 Path Digraphs The concept of path graphs was first extended to path digraphs by Broersma and Li in 2002 [44]. We will consider only the case for paths of length 2 here, both for the sake of simplicity and we are aware of few results for paths of other lengths that would merit their inclusion here. When we speak of paths or cycles in this section, they are assumed to be directed unless otherwise specified. We now give the definition of a path digraph analogous to that of a path graph. Let D be a digraph with at least one path of length 2. The path digraph of D, denoted Λ(D), has the paths of length 2 as its vertices and has an arc from vertex p to vertex q if the second arc of P is the first arc of q. Note that it follows that their union is either a path or a cycle of length 3. Figure 14.11 illustrates the operations of forming the path digraph Λ(D). As with the edges of 2-paths in graphs, coloring the arcs of 2-paths in digraphs facilitates making the illustrations of path digraphs clearer. It follows from the definition that the path digraph of a path of length at least 2 is again a path, but of length 2 less. On the other hand, the path digraph of a cycle is again a cycle of the same length. We assume here that the digraphs are loop-free but may have pairs of opposite arcs. A simple example of a digraph and its path digraph are shown (with details left to the reader) in Fig. 14.12.

P4 :

C3 :

Λ

Λ

Fig. 14.11 The operation of forming the path digraph

Λ (P4 ) :

Λ (C3 ) :

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14 Path Graphs and Path Digraphs

Λ (D):

D:

Fig. 14.12 An example of a path digraph

D:

D:

Λ (D):

Fig. 14.13 Another example of a path digraph

We begin with some elementary observations on the number of vertices and edges in a path graph, and for this we use the notation that with the digraph D = (V , A) having n vertices and m arcs, its path graph Λ(D) equals (VΛ , AΛ ) and has nΛ vertices and mΛ arcs. To get things started, we first consider oriented digraphs, those in which there is at most one arc between any two vertices, for which we have the following result. Theorem 14.5 Given an oriented digraph D, the number of vertices and the number of arcs in Λ(D) are  nΛ = v∈V (d − (v)d + (v)), and mΛ = vw∈A (d − (v)d + (w)). Turning now to the general case (still loop-free), we introduce some additional terminology. We call an arc vw and its converse wv opposite arcs, and by extension v and w are opposite vertices. Note that although an arc can have only a single opposite, a vertex can have many. However, it follows from the definition of a path digraph that even if D has opposite arcs, Λ(D) does not. This is illustrated in Fig. 14.13. For clarity here, we show the digraph D itself before showing the different 2-paths with different colors. In a digraph D, we let B denote the set of arcs that have opposites, and let 2b be its cardinality. Our next result is the generalization of the previous theorem.

14.5 Path Digraphs

229

Theorem 14.6 Given a digraph D with b pairs of opposite arcs, the number of vertices and the number of arcs in Λ(D) are  nΛ = v∈V (d − (v)d + (v)) − 2b, and mΛ = vw∈A (d − (v)d + (w)) − vw∈B (d − (v) + d + (w)) + 2b. The next result gives the in- and out-degrees of the vertices of a path digraph. Theorem 14.7 If P = uvw is a 2-path in digraph D, then in the path digraph Λ(D), −



d (P ) =

d − (u) − 1 if u and v are opposite vertices d − (u) otherwise,

and +

d (P ) =



d + (w) − 1 if v and w are opposite vertices otherwise. d + (w)

The next result gives some of the cycle structure in Λ(D) for small cycles; similar results are possible for longer cycles. It is convenient to have notation for the underlying graph of a digraph D; we denote it [D]. Theorem 14.8 Let D be a digraph with at least one path of length 2. Then the following hold: (a) Λ(D) has no 2-cycles. (b) Λ(D) has no transitive triples. (c) Each 4-cycle in [Λ(D)] is chordless and is a (directed) 4-cycle or has alternating arc directions in Λ(D). (d) No cycle of length 5 or more in [Λ(D)] is both chordless and oriented with alternating arc directions. In [44], the authors also investigated isomorphism questions such as when is Λ(D) ∼ = D, and for which digraphs does Λ(D1 ) ∼ = Λ(D2 ) imply D1 ∼ = D2 . We observe that if a digraph D is strongly connected, then D contains a spanning in-tree and a spanning out-tree. We conclude with three theorems involving isomorphisms between a path digraph and another digraph. We assume that all given path digraphs have at least one juncture vertex, that is, a vertex with both in-degee and out-degree positive (in other words, a 2-path). Theorem 14.9 If D is a connected digraph with no sources or sinks that has an in-tree or an out-tree, then Λ(D) ∼ = D if and only if it is a cycle. As we observed earlier, every strongly connected digraph satisfies the hypotheses and so is not its own path digraph unless it is a cycle. The next theorem gives a partial answer to the question of when do two digraphs have isomorphic path digraphs. We define a digraph to be arc-extendible if every arc is the middle arc of a path or cycle of length 3.

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Theorem 14.10 If D and F are connected arc-extendible digraphs, then every isomorphism from Λ(D) to Λ(F ) is induced by an arc-isomorphism from D to F . The third isomorphism theorem combines iterated line digraphs and path digraphs. Theorem 14.11 If D is a digraph with a juncture vertex, then Λ(D) ∼ = L2 (D). For a more detailed discussion we refer the reader to the papers by Broersma and Li [44] and Li, Liu, and Zhao [134].

14.6 Clique Graphs When one first encounters line graphs, one sees that complete graphs play a major role in their development since the edges at each vertex in a graph yield a complete subgraph of the line graph. This suggests taking complete subgraphs rather than just edges as the basis of an analogue to line graphs, and this was the basic reason for the introduction of cliques in Sect. 3.4. We recall that a clique in a graph G is a maximal complete subgraph of G and that the clique graph K(G) is the intersection graph of the cliques of G. An example of a graph and its clique graph is shown in Fig. 3.10. Clique graphs were first introduced in 1968 by Hamelink [88], who, in seeking a characterization of clique graphs, also introduced an important intersection property of subgraphs of a graph. A collection A of complete subgraphs A1 , A2 , . . . , Ar in a graph is said to have the common intersection property if all of the subgraphs in every subcollection of pairwise intersecting graphs in A have at least one vertex in common. As we will see, every clique graph has such a family that covers the entire graph. Note that checking this condition for graphs does not appear to be an easy matter since many subgraphs need to be checked. On the other hand, the graph in Fig. 14.14 shows a graph that does not have the common intersection property; the three outer triangles clearly intersect pairwise, but they have no vertex in common. Working from Hamelink’s ideas, Roberts and Spencer [151] characterized clique graphs (and the proof of the sufficiency is similar to Hamelink’s proof).

Fig. 14.14 A graph that does not have the intersection property

14.6 Clique Graphs

231

Theorem 14.12 A connected graph G is a clique graph if and only if it has a collection of complete subgraphs that cover the edges of G and satisfy the common intersection property. Proof (Necessity) Assume that G is the clique graph of the graph F . Further, let v1 , v2 , . . . , vn be the vertices of G, which represent the cliques A1 , A2 , . . . , An , respectively. Then vi and vj are adjacent if and only if Ai and Aj intersect. Now let the vertices of F be u1 , u2 , . . . , ur , and for each i, let Bi = {vj : ui ∈ Aj }. Each subgraph Bi is complete since if vk and vl are in Bi , then ui ∈ Ak ∩ Al , and so vk and vl are adjacent. Now if vk vl is an edge of G, then Ai ∩ Aj = ∅ and so the edges of G are covered by the Bi . Finally, we show that the common intersection property holds. Assume that Bi1 , Bi2 , . . . , Bis intersect pairwise. It follows that for all j and k there is a vertex vh in both Bij and Bik . Hence, uij and uik are both in Ah and therefore are adjacent. It follows that all of the vertices ui1 , ui2 , . . . , uir are in some clique Al of F , and therefore the corresponding vertex vl is in all of the subgraphs Bit for t = 1, 2, . . . , s. (Sufficiency) Now assume that the connected graph G has a family of complete subgraphs B1 , B2 , . . . , Br that include all of the edges of G and have the common intersection property. Let v1 , v2 , . . . , vn be the vertices of G. Construct a new graph F with n + r vertices u1 , u2 , . . . , un , w1 , w2 , . . . , wr and these adjacencies: no vertices ui and uj are adjacent, ui is adjacent to wj if and only if vi ∈ Bj , and wi and wj are adjacent if and only if Bi ∩ Bj = ∅. We now show that F has G as its clique graph. To this end, associate with each vertex ui the subgraph Fi of F that is induced by the union of ui and those wj for which Bj contains ui . It is not difficult to see that each Fi is a clique of F . We now show that these are the only cliques of F . Let H be any complete subgraph of F . If H contains some vi , then H is contained in Bi , and otherwise H is contained in some Bj by the common intersection property. Hence, G is the clique graph of F .   In checking whether a graph has the common intersection property, it is in fact not necessary to verify it for all sets of pairwise intersecting complete subgraphs in the family under consideration, only those sets below a certain number. Recall that the First defined in Ch. 8 ω(G) of a graph G is the greatest order of any complete subgraph of G. Roberts and Spencer [151] showed that if a graph has clique number k then it is sufficient to verify the property for sets with at most k complete subgraphs. This is especially useful when the graphs of interest are K4 -free. As observed earlier, if a graph is triangle-free, then its clique graph is isomorphic to its line graph, so we assume now that G is a graph whose largest cliques are triangles. Theorem 14.13 Let G be a connected graph with clique number 3. Then the following statements are equivalent: (1) G is a clique graph. (2) G has a collection of complete subgraphs that cover its edges and satisfy the common intersection property for sets of three graphs. (3) The graph in Fig. 14.14 is not a subgraph of G.

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14 Path Graphs and Path Digraphs

Sato [156] discovered some related interesting results involving iterated clique graphs and line graphs. Here are two examples with which we conclude the chapter. Theorem 14.14 A connected triangle-free graph G is isomorphic to its secondorder clique graph K 2 (G) if and only if it has no vertices of degree 1. Theorem 14.15 A connected graph G is isomorphic to the clique graph of its line graph K(L(G)) if and only if it has no vertices of degree 1, every triangle has exactly one vertex of degree 2, and no triangles share an edge.

Chapter 15

Super Line Graphs and Super Line Digraphs

15.1 Introduction The generalization of line graphs discussed in this chapter closely resembles the original concept. Just as the vertices of a line graph L(G) are the individual edges of the graph G, the vertices of a super line graph of G are the sets of edges of some fixed cardinality r (called the index), with adjacency between two vertices being defined in terms of the existence of at least one pair of adjacent edges in the corresponding two sets. This yields some interesting structures, some of which have quite a rich set of properties involving degrees, subgraphs, independence, paths and cycles, and graph partitions. After presenting results on those topics for super line graphs of indexes in general, we turn to the specific case of index 2. The main results in this case are facts about the super line graph of common families of regular graphs, in particular, cycles, complete graphs, complete symmetric bipartite graphs, and d-dimensional cubes. This is followed by a section on paths and cycles, with a focus on various Hamiltonian properties. In contrast to concentrating on a specific index, we next consider the “linecompletion number” of a graph G, the least index r for which the super line graph is complete, which turns out to be a particularly interesting concept. This is especially so in the case of complete bipartite graphs, which turns out to be equivalent to a purely number-theoretic problem which we call the ‘bisection’ problem: Loosely speaking, the problem is to split two given numbers s and t into two parts so that the minimum of the products of the parts is as large as possible. It turns out to be a deviously difficult problem for many cases unless s and t are both even. We conclude the chapter with sections on super line digraphs and super line multigraphs.

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_15

233

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15 Super Line Graphs and Super Line Digraphs

15.2 Definition and Basic Properties Let G be a graph with m edges and no isolated vertices and let r be an integer with 1 ≤ r ≤ m. Then the super line graph Lr (G) of index r has the sets of r edges in G as its vertices, and two of these are joined by an edge if some edge in one set is adjacent to some edge in the other. Clearly, L1 (G) is the  ordinary line graph L(G) and Lm (G) = K1 . In general, the order of Lr (G) is mr . Figure 15.1 shows a graph G with m = 4 edges and its super line graphs Lr (G) for 1 ≤ r ≤ 4. In this figure, we use the simpler notation ab for the vertex {a, b} in L2 (G), abc for the vertex {a, b, c} in L3 (G), and so on. Super line graphs were first defined by Bagga, Beineke, and Varma in [10]. Note that our definition suggests other possibilities such as forming a multigraph by joining two vertices with as many edges as there are adjacencies between the corresponding two sets of edges, and also by allowing loops. These variations were first studied in [13] and we consider this in greater detail in Sect. 15.8. Our first result lists several basic properties of Lr (G).

Fig. 15.1 Super line graphs

15.2 Definition and Basic Properties

235

Theorem 15.1 Let G be graph with at least r edges with r ≥ 2, and let S and T be two vertices in Lr (G). (a) If S and T are not adjacent in Lr (G), then their intersection is either empty or a set of isolated edges in the graph of their union. (b) If neither S nor T consists only of r isolated edges of G, then the distance between S and T in Lr (G) is 1 or 2. (c) If fewer than r components of G are isolated edges, then Lr (G) has diameter 1 or 2. (d) At most one component of Lr (G) is nontrivial. Proof (a) This follows from the observation that if an edge in S ∩ T is adjacent to some edge in S ∪ T , then S and T contain an adjacent pair of edges, and so S would be adjacent to T in Lr (G). (b) From the hypothesis it follows that G contains an edge e adjacent to some edge in S and an edge f adjacent to some edge in T . Now let R be any set of r edges of G containing e and f . Then both S and T are adjacent to R in Lr (G), and so the distance between them is at most 2. (c) This follows at once from (b). (d) This follows from the above and the fact that a set of r isolated edges in G constitutes an isolated vertex in Lr (G).   It turns out that super line graphs have several nice properties involving containment. Before getting to some of those results, we prove a lemma on sets that is not only of interest in its own right but will be useful in our next theorem. Although its statement does not involve graphs, our proof does. Lemma 15.1 Let S be a set of m elements and let r < m/2. Then there exists a one-to-one mapping φ from the set of r-subsets of M into the set of (r + 1)-subsets of M such that for each r-subset X, X ⊆ φ(X). Proof Given a set M with the stated properties, let B be the bipartite graph having as its two partite sets the r-subsets and the (r + 1)-subsets of S and with adjacency being determined by set inclusion. Then each of the r-subsets is adjacent to m − r of the (r + 1)-subsets and each of the (r + 1)-subsets is adjacent to r + 1 of the r-subsets. If S is a subset of the partite set of r-subsets, then there are (m − r)|S| edges of B incident to S and this number should be at most (r + 1)|N(S)|, where N(S) denotes the set of neighbors of S in B. It follows that |S| ≤ |N(S)|. By Hall’s marriage theorem, there is a matching in B that covers the collection of r-subsets, and this gives the desired mapping φ.   Theorem 15.2 (a) If graph G is a subgraph of graph H , then for all r, Lr (G) is an induced subgraph of Lr (H ). (b) If G is a graph with m edges and if r < m2 , then Lr (G) is isomorphic to a subgraph of Lr+1 (G).

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15 Super Line Graphs and Super Line Digraphs

Proof (a) This follows from the observation that each set of r edges in G is also in H , and, two such sets are adjacent in H if and only if they are adjacent in G. (b) By the lemma, there is a one-to-one mapping from V (Lr (G)) into V (Lr+1 (G)) with each r-set of edges of G being mapped to a set containing it. Now if two r-sets have edges that are adjacent in G, then their images (because of containment) must also have adjacent edges. Hence, Lr (G) is a subgraph of Lr+1 (G).   We make two observations about the conclusion in part (b) of the theorem. First, it cannot be guaranteed that the subgraph in the conclusion be induced. This is illustrated by the graph G of Fig. 15.1, where L1 (G) is not an induced subgraph of L2 (G). Secondly, if m is odd, then for r = m−1 2 , Lr (G) is a spanning subgraph of Lr+1 (G). The next result involves spanning subgraphs in a different way, by creating supergraphs. We define the splitting of a vertex v of degree at least 2 in a graph G as the result of replacing v and its edges vw1 , vw2 , . . . , vwd by two new vertices v1 and v2 and replacing each edge vwi by either v1 wi or v2 wi (and not both) so that v1 and v2 each have degree at least 1 in the resulting graph. Theorem 15.3 Let G be a graph with m edges, and let F be the result of splitting a vertex of G. Then for each r ≤ m, Lr (F ) is a spanning subgraph of Lr (G). Proof Since F and G have the same number of edges, Lr (F ) and Lr (G) have the same number of vertices, with the natural bijection between the two sets. Furthermore, if A and B are adjacent sets of r edges in Lr (F ), then the corresponding sets of edges in Lr (G) are also adjacent. Hence, Lr (F ) is a spanning subgraph of Lr (G).   The following result is an elegant consequence of the theorem. Theorem 15.4 If G is a connected graph with m edges, then there is a tree T of order m + 1 for which Lr (T ) is a spanning subgraph of Lr (G). As we shall see later, a useful application of the theorem works in the opposite direction: one can deduce a property of Lr (G) from knowing that a subgraph has the same property.

15.3 Independence Number Recall that the independence number α(G) of a graph G is the maximum number in a set of non-adjacent vertices in G, and the edge-independence number α (G) is the maximum order in a set of non-adjacent edges. Clearly, the independence number of L(G) is the same as the edge-independence number of G; that is, α(L(G)) = α (G). In this section we extend this result to super line graphs. If vertices S and T in

15.4 Degrees of Index-2 Super Line Graphs

237

Lr (G) induce vertex-disjoint subgraphs in G, then it follows from Theorem 15.1 that S and T are nonadjacent in Lr (G). However, as the next result [12] shows, with a few exceptions, a maximum independent set of vertices in Lr (G) arises   from a maximum set of independent edges of G. For a subset X of E(G), we let Xr denote the set of subsets of X having r elements. Theorem 15.5 Let G be a graph with at least r edges.   . (a) α(Lr (G)) = α (G) r (b) If Y is a maximum independent set of vertices in Lr (G), then either   (i) Y = Xr for some maximum independent set X of edges in G, or (ii) Y consists of r + 1 disjoint stars K1,r , or (iii) r = 3 and every vertex in Y is K1,3 or K3 . Furthermore, if Y satisfies (ii) or (iii), then each edge of G not in any element of Y is incident with the center of a star in Y .   is clear from our remarks above. Now Proof The inequality α(Lr (G)) ≥ α (G) r assume that S1 , S2 , . . . , Sk are independent vertices in Lr (G). Also, let a be the number of the Si that are matchings in G, let b = k − a, and  let c be the number of edges in G in the union of the a matchings. Clearly, a ≤ cr . Let e and f be two edges in S1 ∪S2 ∪. . .∪Sk . Since S1 , S2 , . . . , Sk are independent in Lr (G), it follows that if e and f are adjacent, then there is precisely one Si containing both e and f . Hence, if we select one such edge from each of the b sets, we have b + c ≤ α (G).    α (G) Thus, k = a + b ≤ rc + b ≤ b+c ≤ r , and this proves the equality in (a). r Now assume that Y = {S1 , S2 , . . . , Sk } is a maximum independent set in Lr (G).   With the above notation, we have k = a + b = α (G) . If b = 0 or r = 1, then r it is easy to see that Y satisfies (i), so we assume that b > 0 and r > 1. Then    b + cr = b+c implies that c = 0 and b = r + 1. It also follows that a = 0 r and α (G) = r + 1. If some Si is not a star, then it must have two independent edges or it must be a K3 . In the former case, we get r + 2 independent edges, which is a contradiction. In the latter case, r = 3 and each Si is a star or a K3 . The last statement in the theorem now follows since an edge of G not satisfying that property would lead to r + 2 independent edges in G.  

15.4 Degrees of Index-2 Super Line Graphs In this section and the next, we focus on super line graphs of index 2. While the general cases get much more complicated, the index-2 super line graphs are representative of the general case. Following the notation of neighborhoods of vertices, we denote the set of edges that are adjacent to a given edge e in graph G by N(e) (we note that in the line graph L(G), e is a vertex and its neighborhood is precisely the set of vertices corresponding to the edges in N(e)). As we observed in Chap. 1, the degree of e

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15 Super Line Graphs and Super Line Digraphs

in L(G) is deg v + deg w − 2, where e is the edge vw in G. The following theorem gives a formula for degrees in L2 (G). While it should be possible to generalize this to Lr (G), the expression for the degree formula would be much more involved. Theorem 15.6 Let G be a graph with m edges. For a vertex ef in L2 (G), let δ1 = |N(e)∪N(f )|, and let δ2 = 1 if e and f are adjacent in G and let δ2 = 0 otherwise. Then deg ef =

δ1 (2m − δ1 − 1) − δ2 . 2

(15.1)

Proof  A vertex ab in L2 (G) is adjacent to ef if a or b is in N(e) ∪ N(f ). There are δ21 + δ1 (m − δ1 ) such pairs, and these terms can be combined to yield the first term in the theorem. The correction term δ2 is needed since, for e ∼ f , ef is also included in N(e) ∪ N(f ).   We give an illustration of this theorem using the graph G in Fig. 15.1. For the adjacent edges a and b in G, which has 4 edges, the vertex ab in L2 (G) has δ1 = 3 and δ2 = 1. The formula in the theorem then gives 5, which is the degree of the vertex ab in the figure. Similarly, for the non-adjacent vertices a and d, δ1 = 1 and δ2 = 0, so the formula gives 3, which is the degree of ad. Obviously, the number of edges in the super line digraph of index 2 can then be easily computed (in our example this is 12). Using this theorem to find the degrees of all of the vertices in L2 (G) for graphs of any complexity requires either considerable patience and effort or a computer, and neither of those is appropriate for us here. However, we are able to use the theorem to find the degrees of the vertices in some of the common families of graphs, and hence the number of their edges. The graphs that we have chosen here are regular: the complete graphs Kn , the complete bipartite graphs Kt,t , the cycles Cn , and the hypercubes Qn , and so there are only a few types of neighborhood unions in each case, The results for the first three families were computed by Bagga, Beineke, and Varma [11], while the fourth is due to Bagga and Vasquez [9]. As noted, the approach used in the proofs is to find the different types of pairs of edges of the graph G and how many there are, and then use the theorem to find the degrees. Theorem 15.7 (a) The index-2 super line graph L2 (Kn ) of the complete graph Kn has  3 n3 vertices of degree 3(n − 2)3 and   3 n4 vertices of degree 2(n − 2)2 (n − 3), and therefore has 18 n(n − 1)(n − 2)(n4 − 8n3 + 32n2 − 72n + 67) edges. (b) The index-2 super line graph L2 (Kr,r ) of the regular complete bipartite graph Kr,r has r 2 (r − 1) vertices of degree 2r − 3 and  2 2 2r vertices of degree 4(r − 2), and therefore has 12 r 2 (r − 1)(2r 3 − r 2 − 6r + 4) edges.

15.4 Degrees of Index-2 Super Line Graphs

239

(c) For n ≥ 5, the index-2 super line graph L2 (Cn ) of the cycle Cn has n vertices of degree 4n − 11, n vertices of degree 3n − 6, and n(n−5) vertices of degree 4n − 10, 2 and therefore has 18 n(n − 2)2 edges. Proof As noted earlier, the approach being used in this proof is, for each graph G, to find the different types of pairs of edges of G that are possible with respect to their number and degree. The number of edges in L2 (G) then follows at once as half the sum of the degrees. and (a) In the complete graph Kn , two edges are either adjacent or independent   all pairs within each group are alike. We first observe that there are n3 different sets ofthree vertices in Kn , and each has three pairs of adjacent edges, for a total of 3 n3 vertices of this type in L2 (Kn ). If v = {e, f } is a pair of such edges in Kn , then in Theorem 15.6, δ1 = 3(n − 2) and δ2 = 1, and so deg v = 3(n − 2)3 . n Similarly, there are 4 different sets of four  vertices in Kn and each has three pairs of non-adjacent edges for a total of 3 n4 vertices in L2 (Kn ). In this case δ1 = 4n − 10 and δ2 = 0, and so deg v = 2(n − 2)2 (n − 3). The result (a) now follows. (b) As in complete graphs, in the regular complete bipartite graph Kr,r all pairs of adjacent edges are alike, as are all pairs of independent edges. For a pair of edges in Kr,r there are 2r possibilities for the shared vertex and then adjacent r possibilities for the other two vertices, to make a total of r 2 (r − 1). For each 2 of these δ1 = 3r − 2 and δ2 = 1, and so deg v = 2r − 3. By the same argument,  2 there are 2 2r pairs of non-adjacent edges in Kr,r , and each has δ1 = 4(r − 1) and δ2 = 0; consequently, the corresponding vertex in L2 (Kr,r ) has degree 4(r − 2), and the conclusion follows. (c) Let e and f be a pair of edges on the cycle Cn with n ≥ 5 and let d be the distance between them. Our approach is to divide the pairs into three groups, depending on this distance. Then for each group, we first determine the number of vertices that it contains and then the degree of each of those vertices using the formula in the theorem. Type 1. The distance is 0; that is, e and f are adjacent edges on the cycle. Then there are n such pairs and for each, δ1 = |N(e) ∪ N(F )| can be determined to be 4. Since δ2 = 1 in this case we have from Eq. 15.1 in Theorem 15.6 that the degree of each of these vertices in L2 (Cn ) is 4n − 11. Type 2. The distance is 1. Again, there are n pairs of edges and in this case δ1 = 3 and δ2 = 0, so the degrees are all 3n − 6. Type 3. The distance is greater than 1. Here we consider the odd and even values of n separately, beginning with the odd case, n = 2k + 1. Since the greatest distance between a pair of edges on C2k+1 is k, there are k − 2 distances greater than 1, and there are n pairs of edges for each of them, there is a total of n(n−5) vertices of this type in L2 (C2k+1 ). In the even case, with 2

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n = 2k, there are again k − 2 different distances greater than 1, but for the largest distance, k −1, there are only half as many pairs of edges. Hence the n(n−5) n total number of pairs is n−6 2 · n + 1 · 2 , which equals the same total, 2 , as in the odd case. In both cases, δ1 = 4 and δ2 = 0, and so the degrees are all 4n − 10. This establishes the total number of edges in L2 (C2k+1 ) and completes the proof of the theorem.   Theorem 15.8 Let m = n2n−1 , the number of edges of Qn (n ≥ 2). The index-2 super line graph L2 (Qn ) of the hypercube Qn has m(n − 1) (4m2 − 4m(3n − 2) + 3(5n2 − 9n + 3)) 4 edges. Proof For a vertex ef of L2 (Qn ), we consider four cases depending on the the type of subgraph induced by the edges e and f of Qn . Type 1. Type 2. Type 3. Type 4.

e and f e and f e and f e and f

are adjacent so the subgraph induced by e and f is P3 . are nonadjacent and the subgraph induced by e and f is Q2 . are nonadjacent and the subgraph induced by e and f is P4 . are nonadjacent and the subgraph induced by e and f is 2K2 .

Figure 15.2 illustrates the four types, where a dashed line represents no edge. For 1 ≤ i ≤ 4, we denote by Ai the set of vertices ef of L2 (Qn ) of Type i. Clearly, this gives a partition of the set of vertices of L2 (Qn ). Fig. 15.2 Types of edge pairs in Qn

e

e

f f Type 1

Type 2

e

e

f

f

Type 3

Type 4

15.5 Paths and Cycles in Super Line Graphs

241

We first compute deg ef for each type. For a pair e, f in A1 , it is easy to verify that |N(e) ∩ N(f )| = n − 2. Hence, |N(e) ∪ N(f )| = |N(e)| + |N(f )| − |N(e) ∩ N(f )| = 2(n − 1) + 2(n − 1) − (n − 2) = 3n − 2. For a pair e, f in A2 , A3 and A4 the corresponding values of |N(e) ∩ N(f )| are 2, 1, and 0, which gives the three values |N(e) ∪ N(f )| as 4n − 6, 4n − 5 and 4n − 4. From Theorem 15.6, it follows that • • • •

for e, f for e, f for e, f for e, f

∈ A1 , deg ef ∈ A2 , deg ef ∈ A3 , deg ef ∈ A4 , deg ef

= 12 (3n − 2)(2m − 3n + 1) − 1; = (2n − 3)(2m − 4n + 5); = (4n − 5)(m − 2n + 2); = (2n − 2)(2m − 4n + 3).

n We next determine |Ai | for each 1 ≤ i ≤ 4. Since n n there are 2 pairs of adjacent edges at a vertex of Qn , it follows that |A1 | = 2 2 . For A2 , we can easily count the number of Q2 s in Qn . Once two sides of a Q2 are selected   at a vertex, the other two are uniquely determined. Hence there are 14 n2 2n = n2 2n−2 of these since the same Q2 is obtained from each of its four  corners. Because each Q2 yields two elements of A2 , it follows that |A2 | = n2 2n−1 . We observe that the elements in A3 are in one-to-one correspondence with vertex pairs  v, w in Qn with d(v, w) = 3. The number of such pairs is easily seen to 3 n3 2n . Finally, we note that |A4 | = m   2 − (|A1 | + |A2 | + |A3 |). The theorem follows.

15.5 Paths and Cycles in Super Line Graphs In this section we discuss the structure of paths and cycles in super line graphs. In Chap. 7 we described conditions on a graph that make its line graph and its iterated line graphs Eulerian or Hamiltonian. Most known results for such properties are restricted to super line graphs of index 2. We first review some definitions and terminology. A graph G of order n ≥ 3 is called pancyclic if it has cycles of all lengths from 3 to n. It is vertex-pancyclic if every vertex lies on a cycle of length l for l = 3, 4, . . . , n. An edge-pancyclic graph is similarly defined. A graph is called Hamilton-connected if every pair of vertices are connected by a Hamiltonian path. G is said to be panconnected if, between each pair of vertices, there exist paths of all lengths greater than or equal to the distance between the vertices. G is called path-comprehensive if every pair of vertices are joined by paths of all lengths 2, 3, . . . , n (but not necessarily length 1). Clearly complete graphs are path comprehensive. It is easily checked that wheels W1,n = K1 ∗ Cn (n ≥ 1) of order n + 1 are also path-comprehensive, where ∗ denotes the join operation. We note that the path-comprehensive property is the strongest of these properties in that path-comprehensive graphs are panconnected, vertex-pancyclic, and edgepancyclic. Broersma, Ryjáˇcek, and Schiermeyer [45] provide a good survey of this topic.

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15 Super Line Graphs and Super Line Digraphs

The main results of this section are (a) if G has no isolated edges, then L2 (G) is path-comprehensive and (b) if G has at most one isolated edge, then L2 (G) is vertex-pancyclic. To prove the main results, we will first handle a number of special cases. The following result was proved in [133]. Theorem 15.9 If G and H are path-comprehensive, then so is G ∗ H . Proof We assume without loss of generality that G has no more vertices than H . First assume that G = K1 . If H = K1 or K2 , the result is clear. If H has at least three vertices, then it has a spanning cycle so that G ∗ H has a spanning wheel. Suppose now that G has at least two vertices. Let v and w be two vertices in G ∗ H . We consider three cases: Case 1 Both v and w are in G. Then v and w are joined by paths of lengths 2, 3, · · · , |G| − 1 in G. Choose two adjacent vertices x and y in H . Let P be a spanning v-w path in G, and let Q be a spanning x-y path in H . By inserting appropriately into P the vertex x, the edge xy, and subpaths of Q of lengths 2, 3, . . . , |G| − 1 we obtain v-w paths in G ∗ H of lengths |G|, |G| + 1, . . . , |G| + |H | − 1. Case 2 Both v and w are in H . This is similar to Case 1. Case 3 v is in G and w is in H . Choose a vertex x in G adjacent to v and a vertex y in H adjacent to w. Then v and x are joined by paths of lengths 2, 3, . . . , |G| − 1 in G. Let P be a spanning v-x path in G. By appropriately appending paths in H to P , we obtain v-w paths in G ∗ H of lengths |G|, |G| + 1, . . . , |G| + |H | − 1.   Our next two results determine conditions for complete multipartite graphs with at least three parts to be path-comprehensive. Theorem 15.10 (a) For r ≤ s ≤ t, the complete tripartite graph Kr,s,t is path-comprehensive if and only if r + s > t. (b) For r ≥ 1, the complete 4-partite graph G = Kr,r,r,r is path-comprehensive. (c) Let G be the graph obtained from K1,r,r+1 by adding one edge within the partite set of r + 1 vertices. Then G is path-comprehensive. Proof (a) was proved in [12], and (b) can easily be shown to follow from (a). We now prove (c). Denote the edge within the partite set of r + 1 vertices by uv, and let w be the vertex in the singleton partite set. Then G − u ∼ = G−v ∼ = K1,r,r is path-comprehensive by (a). Let x and y be any two vertices in G. If {x, y} = {u, v} and (say) x = u, then, in G − u, w and v are joined by paths of lengths 1, 2, . . . , 2r. Adding the edge uw to these paths gives us the desired xy paths. The case {x, y} = {u, v} follows in a similar way.   Theorem 15.11 A complete multipartite graph with at least three parts is pathcomprehensive if and only if no partite set contains half or more of the vertices.

15.5 Paths and Cycles in Super Line Graphs

243

Proof The case when the graph is tripartite follows from Theorem 15.10. We now give the general proof by induction on the number of partite sets. Assume that the result holds for all such graphs with r partite sets (with r ≥ 3). Suppose  that G = Kn1 ,n2 ,...,nr+1 , (with n1 ≤ n2 ≤ . . . ≤ nr+1 ) and 2nr+1 < n = i=r+1 ni . Let i=1 n∗ = n1 + n2 . If n∗ ≤ nr+1 , then (after a suitable reordering of partite set sizes) the complete r-partite graph Kn∗ ,n3 ,...,nr+1 is path-comprehensive by the induction hypothesis. Since this is a spanning subgraph of Kn1 ,n2 ,...,nr+1 , we are done. If n∗ > nr+1 , we consider the complete r-partite graph Kn3 ,...,nr+1 ,n∗ . Observe that n3 + . . . + nr+1 + n∗ ≥ n3 + nr+1 + n∗ ≥ 2n∗ . Two cases arise. First, suppose that n3 + . . . + nr+1 + n∗ > 2n∗ . Then the result follows again by induction as before. The only other possibility is that n3 + . . . + nr+1 + n∗ = n3 + nr+1 + n∗ = 2n∗ . This leads to r = 4 and G = Kn1 ,n1 ,n1 ,n1 , which is path-comprehensive by Theorem 15.10.   A family of graphs that will turn out to be useful for our main results is L2 (rP4 ), for r ≥ 1. In [12] it was shown that this graph is vertex-pancyclic., and then in [133] this result was strengthened by showing that it is path-comprehensive. Theorem 15.12 For every positive integer r, the index-2 super line graph L2 (rP4 ) is path-comprehensive. Proof The result can be proved by induction on r, but we show only the first two cases. For r = 1, L2 (P4 ) = K3 , and for r = 2, Fig. 15.3 shows two wheels in L2 (2P4 ), where the one in (c) is a spanning wheel. Li, Li, and Zhang give the general proof in [133].   We are now ready to discuss our two main results in this section, as stated earlier. It was shown in [12] that if G has no isolated edges, then L2 (G) is vertex-pancyclic. We describe here a brief outline of the proof. First, Theorems 15.11 and 15.12 and an induction on the number of edges are used to show the result holds for such forests. It is then extended to graphs by induction on the number of cycles. In [133] this was further extended in two ways: that the same hypothesis implies

a1 c 1 a1 b1 a1 a2

b1 a2 a1 a2

b1 b2

c1

c 1 a2

a1 b2

c1 a2

a1 c2 c1 c2

Fig. 15.3 Wheels in L2 (2P4 )

a1 b2 a2 b2 a2 c2

b1 b2

c 1 b2

(a)

a1 a2

b1 a2

b1 b2

c2

b1 c1

b1 c2 (b)

a1 c2 b2 c2

c1 b2 c1 c2

b1 c2 (c)

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15 Super Line Graphs and Super Line Digraphs

the path-comprehensive property, and that a weaker hypothesis suffices for vertexpancyclicity. Theorem 15.13 If G is a graph with no isolated edges, then the index-2 super line graph L2 (G) is path-comprehensive. Theorem 15.14 If G has at most one isolated edge, then L2 (G) is vertexpancyclic. We next show that these results are sharp. First consider the graph G = P4 ∪K2 of Fig. 15.1. Then L2 (P4 +K2 ) (recall that + denotes the disjoint union) is isomorphic to K6 − C3 , and there is no Hamiltonian path between the vertices ab and bc since the subgraph induced by {ac, ad, bd, cd} has no path of length 3. Hence the edge {ab, bc} cannot be on a cycle of length 6. It follows that L2 (G) is vertex-pancyclic but not edge-pancyclic. Now consider L2 (P4 + 2K2 ). It has an isolated vertex and a component of order 9 that is isomorphic to K1,1,2,5 , which is not pancyclic since there is no Hamilton cycle. Theorem 15.13 implies that if G has no isolated edges, then L2 (G) is edgepancyclic and Hamilton-connected. It would be interesting to explore extensions of the results in this section to super line graphs of index r > 2.

15.6 The Line Completion Number In this section we discuss the problem of which super line graphs are complete. It is easily seen that the only connected graphs with complete line graphs are the stars K1,t and the complete graph K3 . If a graph G has three independent edges, then in L2 (G) there are nonadjacent vertices. The same is true if G has 2K1,2 as a subgraph. On the other hand, if S and T are nonadjacent in L2 (G), and S ∩ T has a single edge, then by Theorem 15.1, it follows that G has 3K2 as a subgraph. If S ∩ T is empty, then each of S and T is 2K2 or 2K1,2 , and again G has 3K2 or 2K1,2 as a subgraph. One can look for similar results for super line graphs of higher indices, and indeed, we state a result for index 3 below. However, the description of the exceptions gets more involved. Nevertheless, for any graph there are indices for which the super line graph is complete. This follows from the observation that if Lr (G) is complete, then so is Lr+1 (G). This led Bagga, Beineke, and Varma [11] to define the line completion number lc(G) of a graph G to be the least index r for which Lr (G) is complete. The following result (see [11]) gives those graphs whose line completion number is small or large. (Recall that our notation is that G + H denotes the disjoint union of G and H and G ∗ H denotes the join of G and H .)

15.6 The Line Completion Number

245

Theorem 15.15 Let G be a graph with m edges. (a) lc(G) = 1 if and only if G is K3 or K1,s (s ≥ 1). (b) lc(G) ≤ 2 if and only if G does not have 3K2 or 2P3 as a subgraph. (c) lc(G) ≤ 3 if and only if G does not have any of the following as a subgraph: 4K2 , K2 + K1,2 , 2K3 , K3 + P4 , K3 + K1,3 , 2K1,3 , K1,3 + P4 . (d) lc(G) = m − 1 if and only if G is P3 ∗ (m − 2)K2 or 2P3 ∗ (m − 4)K2 . (e) lc(G) = m if and only if G is mK2 . Our next theorem gives sharp bounds for the line completion number in terms of the edge-independence number α , and the order and number of components. Theorem 15.16 If G is a graph with m edges, c components, and no isolated vertices, then α ≤ lc(G) ≤

m + c . 2

Furthermore, the bounds are sharp for all m and c. Proof We first consider the upper bound. Let G be a graph with m edges (not all independent) and c components for which Lr (G) is not complete. Then Lr (G) has nonadjacent vertices S and T , and it follows from Theorem 15.1 that S ∩ T is a set of independent edges. If they are l in number, then m ≥ (2r − l) + (l + 2 − c), so r ≤ m+c−2 . Therefore, if s > m+c−2 , then Ls (G) is complete, which establishes 2 2 the upper bound. The lower bound is obvious. For sharpness, first suppose c = 1. If m ≤ 2, thenequality clearly holds. Let edges. Form G by joining m ≥ 3, and let H be a connected graph with r = m−1 2 two copies of H with one or two edges according as m is odd or even. Then Lr (G) is not complete, since the sets of edges from the two copies of H are not adjacent. Hence lc(G) = r + 1. For c > 1, first observe that m ≥ c since there are no isolated vertices. If m − c is even, say m − c = 2t, let H be a connected graph of t + 1 edges, let G = (c − 2)K2 + 2H , and let r = t + c − 1. Then G has c − 2 + 2t + 2 = m edges and Lr (G) is not complete. Hence lc(G) = r + 1 = m+c 2 . If m − c is odd, add an extra edge to one of the components of the graph G in the even case. Now we turn to the lower bound. The bound is clear for α = 1. For a graph G with α > 1, any two subsets of α − 1 edges in an independent set of α edges in G are nonadjacent in Lα −1 (G). For sharpness, let G = (c − 1)K2 + K1,m−c+1 .  

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15 Super Line Graphs and Super Line Digraphs

Observe that for G = mK2 , the two bounds of Theorem 15.16 are equal. On the other hand, as we shall see below, the line completion number of a double star formed by adding an edge joining the centers of two stars K1,k has lc(T ) = k + 1 while α (T ) = 2, so the difference can be arbitrarily large. Theorem 15.17 For any graph G, lc(G) ≥ 1 + max{||S||}, where the maximum is over all sets S ⊂ V for which ||S|| ≤ ||V − S||. Proof Clearly, if S and T are two sets of r edges in a graph G that have no vertices in common, then the corresponding vertices in Lr (G) cannot be adjacent. Consequently, the line completion number of G is at least r + 1.   The above bounds are useful in establishing other results. For example, we have the following theorem on trees which includes an intermediate value result on the range of the line completion numbers of trees. Theorem 15.18 If T is a tree of order n ≥ 2, then lc(T ) ≤  n2 . Furthermore, for any integer k satisfying 1 ≤ k ≤  n2 , there exists a tree of order n and line completion number k. Proof The bound follows from Theorem 15.16. For 1 ≤ k ≤  n2 , let T be the double star obtained by joining the central vertices of Kk−1 and Kn−k−1 with an edge. It is easily checked that lc(T ) = k.   In our next result, we give line completion numbers of some common families of graphs [11]. The fan Ft , the wheel Wt , and the windmill Mt are the graphs resulting from adding one vertex adjacent to all of the vertices in the path Pt , the cycle Ct , and the graph tK2 , respectively (see Fig. 15.4). The general idea of the proof is as follows: We first get a bound by using Lemma 15.17, and then we show that the line completion number cannot be better than the bound. For more details, see [11].

Fig. 15.4 Fans, wheels and windmills

15.6 The Line Completion Number

247

Fig. 15.5 The hypercube Q4

Fig. 15.6 The grid P4 × P5

Theorem 15.19 The line completion number of some elementary families of graphs are the following:     Complete graphs: lc(Kn ) = p2 + 1, where p = n2 ;   Paths and cycles: lc(Pn ) = lc(Cn ) = n2 ;  2t  Fans and wheels: lc(Ft ) = lc(Wt ) = 3 ;   Windmills: lc(Mt ) = 3t4 + 1. The line completion number of two other families of graphs have also been determined, the hypercubes Qd by Tapadia and Waphare [166], and the grid graphs Pk × Pl by Kureethara and Sebastian [130]. We state these in the next two theorems and illustrate the results in Figs. 15.5 and 15.6. By Theorem 15.21, lc(P4 ×P5 ) = 14. Figure 15.6 shows two sets of (colored) edges that are nonadjacent vertices in L13 (P4 × P5 ). Figure 15.5 provides a similar illustration for Q4 . Theorem 15.20 The line completion number of the d-dimensional cube Qd is lc(Qd ) = 1 + (d − 1)2d−2.

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15 Super Line Graphs and Super Line Digraphs

Theorem 15.21 The line completion number of the grid graph Ps × Pt with 1 < s ≤ t is ⎧ st ⎪ ⎪ ⎨ st lc(Ps × Pt ) = ⎪ st ⎪ ⎩ st

+ 1 − (s + 2t ) − (s + t +1 2 ) − (s + t −1 2 ) + 1 − (s + 2t )

if s if s if s if s

is even and t is even, is odd and t is odd, is even and t is odd, is odd and t is even.

Notably missing from these results is the family of complete bipartite graphs Ks,t , and there’s a good reason for that: The problem is difficult and only partially solved. It is in fact equivalent to an enticing problem in number theory. Bisection Problem For positive integers s and t, let s1 , s2 , t1 and t2 be nonnegative integers with s1 + s2 = s and t1 + t2 = t. Determine the bisection number β(s, t) = max{min{s1 t1 , s2 t2 }}. Because of the structure of complete bipartite graphs, it turns out that the bound in Theorem 15.17 is important. Since every induced subgraph of a complete bipartite graph is also complete bipartite, it follows that lc(Ks,t ) = 1 + β(s, t). Therefore, we will work with β(s, t) here rather than the line completion number itself. A natural starting point is to split s and t into two numbers as nearly equal as possible (which of course depends on their parity). It is not difficult to show that when s and t are both even, then indeed β(2p, 2q) = pq. When at least one of s and t is odd, then we have these bounds: (a) when both are odd then β(2p +1, 2q +1) ≥ p(q + 1), while if they have different parities, β(2p, 2q + 1) ≥ pq. Gutiérrez and Lladó [86] conjectured that equality holds, but this is not what happens. However, in the mixed parity case, for a given value of s even, inequality holds for only a finite number of odd values of t. (When the pairs are in the opposite order, the situation is more complicated.) Furthermore, when s and t are both odd, the conjecture almost never holds, and in fact can be off by an arbitrarily large amount. Bagga, Beineke, and Varma [16] considered the problem in greater detail, assuming that s ≤ t and exploring the four cases separately, with the bounds being base cases. A few results will be proved here, but only a few because many of the exact results have lengthy computations; see [16] for details. The idea of many proofs is this: If a base case (a, b) (having value ab) does not hold for β(s, t), then for some x ≥ 0 and y ≥ 0, it must be that both (a + x)(b − y) > ab and (a − x)(b + y) > ab. Here are the four types of parity (with s ≤ t): • Type I: s = 2p and t = 2q, with base case (p, q). • Type II: s = 2p + 1 and t = 2q + 1, with base case p(q + 1).

15.6 The Line Completion Number

249

• Type III: s = 2p and t = 2q + 1, with base case (p, q). • Type IV: s = 2p + 1 and t = 2q, with base case (p, q). A pair (a, b) with 0 ≤ a ≤ s and 0 ≤ b ≤ t is called optimal if ab = β(s, t). It is easily seen that within each of the four types the product of the two numbers in the base case gives a lower bound for β(s, t). One question that we consider is when does equality hold; that is, when is the base case optimal. Type I Both s and t are even. The solution for this type is straightforward: simply split each number in half. Theorem 15.22 If s = 2p and t = 2q, then β(s, t) = pq. Proof Suppose that β(s, t) > pq. Then by the observation made above, there must exist x ≥ 0 and y ≥ 0 for which both (r + x)(s − y) > rs and (r − x)(s + y) > rs, and a little algebra shows that this is impossible.   That is really all that is to be said in this case. Type II Let s = 2p + 1 and t = 2q + 1 with s ≤ t). It turns our that the base case holds some of the time, but not always. A computer program verified that if s ≤ 25, then for all t, β(s, t) = p(q + 1); that is, the base case holds. However, it does not hold for the pair (27, 45). That is, the base case has the value 13 · 23 = 299, but β(27, 45) = 300 (= 12 · 25 = 15 · 20). (In terms of graphs, this says that splitting K27,45 into K12,25 and K15,20 is better for our purposes than splitting it into K13,23 and K14,22 , and consequently lc(K27,45) = 301.) The main results that we have are some pairs for which the base case holds and some for which it does not (and some for which it is not known). Note that for fixed m, eventually the base case never holds. Theorem 15.23 Let s = 2p + 1 and t = 2q + 1, with 13 ≤ p ≤ q. (a) If q ≤ 2p + 1, then the base case holds (that is, β(s, t) = p(q + 1)), for s = r, r + 1, . . . , r + 6 and s = 2r − 3, 2r − 2, . . . , 2r + 1. (b) If q ≥ 2p + 2, then the base case never holds. Some sporadic values other than those given in the theorem are known. Type III s is even and t is odd with s < t; thus, we assume that s = 2p and t = 2q + 1. Here, for fixed s, the base case eventually holds. Theorem 15.24 Let s = 2p and t = 2q + 1 with p ≤ q. 2

(a) If p is odd and q ≥ (q−1) 2 , then the base case β(s, t) = pq holds. (b) If p is even and q ≥ (q−1)(q−2) , then the base case β(s, t) = pq holds. 2 Furthermore, the bounds on q are sharp. Some sporadic values other than those given in the theorem are known.

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15 Super Line Graphs and Super Line Digraphs

Type IV s is odd and t is even with s < t, thus, we assume that s = 2p + 1 and t = 2q with q > p. Here, for fixed s, the base case only holds for one value of t, namely when t = s + 1. Because the proof illustrates the methodology well, showing both when the base case holds and when it does not, we include it here. Theorem 15.25 Let s = 2p + 1 and t = 2q with p < q. (a) If q = p + 1, then the base case β(s, t) = pq holds. (b) If q ≥ p + 2, then the base case does not hold. Proof Let s = 2p + 1 and t = 2q with q > p. (a) Assume that t = s + 1, that is, q = p + 1, and suppose that β(s, t) > pq. Then there exist integers x and y for which the products (p − x)(q + y) and (p + x + 1)(q − y) both exceed pq. It follows from the first of these inequalities that y≥

x2 − x + 1 xp + x + 1 =x+ . p−x p−x

Since the second fraction is positive and y must be an integer, it follows that y ≥ x + 1. On the other hand, the second inequality implies that y≤

xp + p + x x2 + x + 1 =x+1− , p+x+1 p+x+1

and from this we deduce, again since the second fraction is positive, that y ≤ x. Therefore, there cannot exist such x and y, and so β(2p+1, 2p+2) = p(p+1), that is, the base case holds. (b) Assume now that t > s + 1, that is, q ≥ r + 2. In order to show that the base case does not hold, all that needs to be done is to find one pair of numbers x and y for which both of the products (p + x)(q − y) and (p − x + 1)(q + y) exceed pq. It turns out that taking both x and y to be 1 suffices. For, with these values, we have (p + x)(q − y) − pq = (p + 1)(q − 1) − pq = q − p − 1 ≥ 1, since q ≥ p + 2. We also have (p + 1 − x)(s + y) − pq = (p + 1)q − pq = p ≥ 1, and this completes the proof that the base case does not hold.

 

We observe that not only does the base case not always hold, but there are values of s and t for which it is off by an arbitrarily large amount. In fact, the following result shows that β(s, t) can exceed the base case by nearly 12 s 2 . We observe that all

15.6 The Line Completion Number

251

that is needed for this is an appropriate lower bound, but as determining equality is straightforward, we include it in the theorem. Theorem 15.26 Let s = 2p + 1 and t = 2q = 2s(s − 2) (so q = 4p2 − 1), then β(s, t) = pq + 2p2 − p. Proof The split of (s, t) (that is, (2p + 1, 2q)) into (p, q + 2p − 1) and (p + 1, q − 2p + 1) gives the two equal products of pq + 2p2 − p. Hence this is a lower bound for β(2p + 1, 2(4p2 − 1)). Suppose this is not optimal. Then there exist non-negative integers x and y for which (p + x)(q + 2p − 1 − y) ≥ pq + 2p2 − p + 1, and (p + 1 − x)(q − 2p + 1 + y) ≥ pq + 2p2 − p + 1. Solving these inequalities for y yields qx + 2px − x − 1 qx + x − 2px + 1 ≤y≤ , p+1−x p+x further simplification leads to 2qx(x −1)+2p +1 ≤ 0, which is impossible. Hence, such an x and y cannot exist, and this completes the proof of the theorem.   The determination of the exact value of the bisection number of all pairs of positive integers appears to be a hard problem, even those pairs for which the bisection number is the base case. What we have found however is what eventually happens when the smaller number is fixed. We summarize this in the accompanying table. s even odd even odd

t even odd odd even

Base case Always holds Never holds for t ≥ 2s + 3 Always holds beyond some critical value t ≥ t0 (s) Holds only for t = s + 1

Thus, for any fixed value of s, the answer to the base case question is known for all but finitely many values of t. In loose terms, we can say that the base case holds about half the time, depending on the parity of the smaller of the two numbers. That is, if the smaller number is even, then the base value is known to hold in all but a finite number of cases, while if the smaller number is odd, then the base value is known to hold in only a finite number of cases. The exact value of β(s, t) is known for a variety of pairs for which it is not the base value, and we refer the reader to [16] for many of these.

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15 Super Line Graphs and Super Line Digraphs

To conclude this section, we return to the problem that led to the bisection number, namely super line graphs and the line completion number. (Recall that lc(Ks,t ) = β(s, t) + 1.) The following theorem is a summary of key results on this topic, focusing on what eventually happens when the size of the smaller partite set is fixed. Theorem 15.27 The line completion number lc(Ks,t ) of the complete s-by-t bipartite graph with s fixed and t ≥ s satisfies the following: (a) For s even, (i) if t is also even, then lc(Ks,t ) = 14 st + 1 for all t; (ii) if t is odd, then lc(Ks,t ) = 14 s(t − 1) + 1 for

t ≥ 14 (s 2 − 6s + 12) if s ≡ 0 (mod 4), t ≥ 14 (s 2 − 4s + 8) if s ≡ 2 (mod 4).

(b) For s odd, (i) if t is even, then lc(Ks,t ) ≥ 14 t (s − 1) + 2 for all t ≥ s + 3; (ii) if t is also odd, then lc(Ks,t ) ≥ 14 (s − 1)(t + 1) + 2 for all t ≥ 2s + 3.

15.7 Super Line Digraphs We now turn to super line digraphs, a concept that is a natural extension of super line graphs. We saw in the previous sections an extensive set of results and literature on super line graphs. By comparison, the generalization to super line digraphs is relatively unexplored. We consider digraphs without loops. Given a digraph D with m ≥ 1 arcs and an integer r with 1 ≤ r ≤ m the super line digraph Lr (D) of index r is the digraph whose vertices are r-subsets of A(D). For two vertices S and T of Lr (D) there is an arc from S to T if and only if there exist s ∈ S and t ∈ T such that s and t form a 2-path in D, that is, head(s) = tail(t). Clearly, L1 (D) = L(D). We observe that our definition does not allow loops. Figure 15.7 illustrates a digraph D and its super line digraphs. We list below some basic properties. We first describe some notation. For an arc a in D, we denote by N + (a) the set of arcs that have the same tail as the head of a. The notation N − (a) has a similar meaning. + − For a vertex ab of L2 (D), we define ∂ab = |N + (a) ∪ N + (b)|, ∂ab = |N − (a) ∪ − N (b)| and μab = 1 if a, b induce a 2-path or a 2-cycle in D, and μab = 0, otherwise. More generally, for r ≥ 2, if S is a vertex of Lr (D), we use the notation ∂S+ and ∂S− .

15.7 Super Line Digraphs

D:

253 x

y

a

b

axy

bxy

abx

aby

y

x a b

ab

ax

xy

ay

by bx

abxy Fig. 15.7 A digraph and its super line digraphs

Theorem 15.28 If D is a digraph with m arcs and if H is a subdigraph of D, then (a) Lr (H ) is an induced subdigraph of Lr (D); (b) if 1 ≤ r < m/2, thenLr (D) is a subdigraph of Lr+1 (D). Theorem 15.29 If ab is a vertex of L2 (D), ∂ +  + + (a) d + (ab) = 2ab + ∂ab (q − ∂ab ) − μab ; −  ∂ab − − − (b) d (ab) = 2 + ∂ab (q − ∂ab ) − μab . Aigner [2] showed that if D is a digraph with at least three vertices (none of which is isolated), then L(D) is strongly connected if and only if D is strongly connected. However, for r > 1, the situation for Lr (D) is quite different, as the next result of Ferrero [75] shows. Theorem 15.30 If r ≥ 2 and if D is a digraph with the property that for any vertex S of Lr (D), ∂S+ > 0 and ∂S− > 0, then Lr (D) is strongly connected and the distance between any two vertices is at most 2.

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15 Super Line Graphs and Super Line Digraphs

We now consider the special case when the distance between any two vertices in Lr (D) is at most 1, that is, Lr (D) is complete. It is easily seen that if Lr (D) is complete, so is Lr+1 (D). If D has m arcs, then Lm (D) = K1 . We define the line-completion number of D as the least r for which Lr (D) is complete. Clearly, the line completion number of D is 1 if and only if D is P2 or C2 . Theorem 15.31 If D is a digraph with at least three arcs, then L2 (D) is complete if and only if D does not have any disjoint 2-cycles and every subdigraph of D generated by three arcs contains C2 , C3 , or P4 .

15.8 Super Line Multigraphs For a graph G with m edges, and for an integer r with 1 ≤ r ≤ m, the super line multigraph Mr (G) of index r has the sets of r edges in G as its vertices, and two such sets S and T are joined by as many edges as pairs of adjacent edges s ∈ S and t ∈ T . Unlike in the definition of Lr (G), we do allow loops in super line multigraphs. Figure 15.8 shows a graph and its super line multigraph of index 2. We observe that M1 (G) = L1 (G) = L(G). Our first result counts the edges in super line multigraphs. Theorem 15.32 Let G be a graph with n vertices and m edges, and degrees d1 , d2 , . . . , dn . The number of edges in Mr (G) is     n   m−1 2 m − 2 di , + r −1 2 r −2 i=1

of which 2

m−2 n i=1

r−2

di  2

are loops.

ac

a

G:

b c

ab

bc

ad

cd

d

bd Fig. 15.8 An example of a super line multigraph of index 2

15.8 Super Line Multigraphs

255

Proof Given a vertex v of degree d in G, consider a pair of edges e, f in G that are incident at v. We count the number of edges and loops of Mr (G) that arise due to this adjacency of e and f at v. Consider the (ordered) pairs (S, T ) of edges in 2    Mr (G) such that e ∈ S and f ∈ T . There are m−1 such pairs. Of these, m−2 r−1 r−2 pairs have S = T , for each of which we get two loops in Mr (G). Each pair (S, T ) where S = T gives an edge in Mr (G) due to the adjacency of e and f in G. Note that pairs (S, T ) with {e, f } ⊆ S ∩ T occur as (S, T ) and as (T , S) giving us two edges in Mr (G), one for the adjacency   of e ∈ S and f ∈ T and another for f ∈ S and e ∈ T . Finally, since there the d2 such pairs of adjacent edges e and f for each vertex v of degree d in G, the result follows.   We next discuss some spectral properties of Mr (G). We use the terminology and notation from Sect. 4.2. For more details, see [157] and [37]. We also proved the following two results in Chap. 4 about spectral properties of line graphs. We restate those here for completeness. Theorem 15.33 If G is an r-regular graph with n vertices and m edges, then φ(L(G); x) = (x + 2)m−n φ(G; x − r + 2). Theorem 15.34 If λ is an eigenvalue of the line graph L(G) and ΔL is the maximum line-degree in G, then −2 ≤ λ ≤ ΔL . Bagga, Ellis, and Ferrero [14, 15] generalized the above results to Mr (G). Theorem 15.35 Let G be a graph with m edges. If λ is an eigenvalue of Mr (G), then   m−2 λ ≥ −2 . r −1 Theorem 15.36 Let G be a d-regular graph with n vertices, m edges and eigenvalues d = β1 ≥ · · · ≥ βn , where β1 corresponds to the eigenvector 1. Then φ(Mr (G); λ) = (λ)p−m

m 

(λi − λ),

i=1

where λ1 = 2bm(d −1)+2(c −b)(d −1), λi = (c −b)(d −2+βi ) for i = 2, . . . , n, and λn+1 = · · · = λm = −2(c − b). The results in this chapter show recent progress in the relatively new area of super line graphs and super line digraphs. Clearly, much more remains to be investigated. Among the problems to be explored are extensions of the results on existence of paths and cycles in Lr (G) for r > 2, and a complete determination of the line completion number of Ks,t .

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15 Super Line Graphs and Super Line Digraphs

We conclude this chapter by observing that many graph operations similar to line graphs and super line graphs have been studied. The book Graph Dynamics by Erich Prisner [148] provides a comprehensive treatment of several such graph operations. A survey of some other generalizations appeared in [7].

Chapter 16

Line Graphs of Signed Graphs

16.1 Introduction Signed graphs are another interesting variation of graphs, usually taken to be graphs in which each edge is either positive or negative. When taking the line graph of such a graph, a rule is needed for the sign on an edge of the line graph. In the literature, there is more than one option for the rule, and two of these will be considered in this chapter. Because of this and for the sake of consistency, it is useful for us to use different numerical values as the signs in the two cases, with correspondingly different arithmetic formulas for the line-graph rule. The first we call the normal labeling, with +1 and −1 (usually just + and −) as the edge labels, and a graph with such a labeling will be denoted G+/− . The label on an edge of the line graph of this graph will be the algebraic product of the labels of the two edges that produce it. This type of line graph is the subject of Sect. 16.2, and the primary property of interest in these line graphs is whether all of their cycles have a positive value. The second we call the Boolean labeling, with 0 and 1 as the edge labels, and here a graph with such a labeling will be denoted G0/1 . The label on an edge of the line graph of such a graph will be the Boolean algebraic sum of the labels of the two edges that it comes from. This is the subject of the last section, and two of the results of particular interest there are the characterization of the line graphs of Boolean graphs and special colorings of Boolean graphs.

16.2 Line Graphs of Signed Graphs Interestingly, the origin of signed graphs appears to lie in social psychology with Harary and Cartwright’s generalization of Heider’s theory of balance [49]. As described by Harary [89], the relationships within a group of people can be © Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_16

257

258

16 Line Graphs of Signed Graphs

Fig. 16.1 A signed graph

G+/− :

represented by a square binary matrix M with an entry mij being 1 if Person i likes Person j and 0 otherwise. It seems reasonable to assume that entries on the diagonal are 0, that our interest is only between two different people. Hence, such a matrix is the adjacency matrix of a directed graph. In a more sophisticated version, the entries of the matrix M are allowed to be −1, 0, and 1, with 1 representing likes, −1 representing dislikes, and 0 neither likes nor dislikes. This is then represented in a digraph with an arc being labeled + or − according as the corresponding entry of the matrix M is +1 or −1. Here we consider only undirected graphs, so that the matrix M is symmetric. We define a signed graph G+/− as having each edge of the graph G being either positive or negative. In our figures, we color the negative edges red and the positive edges blue. An example is shown in Fig. 16.1. Given a signed graph G+/− , the graph G with the same adjacencies as G+/− but with no signs on its edges is called the underlying graph. We begin with a special family, assuming that the underlying graph is a complete +/− graph Kn ; that is, G+/− = Kn . Taking as a model motivated by the saying “friends of my friends are my friends and enemies of my enemies are also my friends”, we see that every triangle that has a negative edge must have one and +/− only one of the other two edges negative. Let v be a vertex of Kn with at least two negative edges and let the set Y consist of the vertices joined to v by a negative edge, +/− and let the other vertices form the set X. It follows that since Kn has no triangles with all edges negative, all edges in the induced subgraph Y  must be positive. +/− Similarly, it follows that since Kn has no triangles with just one edge negative, all edges in the subgraph X must also be positive. Furthermore, by our construction, all edges between X and Y  are negative. In other words, the negative edges of +/− Kn constitute the edges of a complete bipartite graph. In terms of international relations, this models the situation where a group of countries is divided into two blocs (which may or may not lead to a stable situation). +/− The key feature of our signed graph Kn is the basic property of bipartite graphs, namely, that every cycle has an even number of edges; in other words, in this +/− situation of the entire signed graph Kn , this means an even number of negative edges. This is the motivation for the following definition: A signed graph G+/− is balanced if every cycle has an even number of negative edges. It turns out that balanced signed graphs have a property not unlike that of bipartite graphs for the negative edges. An example is given in Fig. 16.2.

16.2 Line Graphs of Signed Graphs

259

Fig. 16.2 A balanced signed graph G+/− :

Obviously, any signed graph that has only positive edges is balanced, so we say that a signed graph is nontrivial if it has at least one negative edge. Another way of viewing a signed cycle is through the product of the signs on its edges, so we say that a cycle is a positive cycle if it has an even number of negative edges and is a negative cycle if an odd number. We extend these definitions to paths, and make the following definition: A signed graph G+/− has the signed-path property if for every pair of vertices v and w all v-w paths have the same sign. We now make one further definition, generalizing the observation made above for complete signed graphs: A signed graph G+/− has the signed-partition property if its vertices can be partitioned into two sets in such a way that every positive edge joins vertices in the same set and every negative edge joins vertices in different sets. The following theorem shows that each of these two properties is equivalent to a signed graph being balanced. Theorem 16.1 The following statements are equivalent for a connected nontrivial signed graph G+/− : (1) G+/− is balanced. (2) G+/− has the signed-path property. (3) G+/− has the signed-partition property. Proof We show the equivalence in this order: (1) ⇒ (2), (2) ⇒ (3), and (3) ⇒ (1). (1) ⇒ (2): Let v and w be two vertices of G+/− . Assume that P and Q are two v-w paths and let F +/− be the subgraph of G+/− generated by the symmetric difference of P and Q. It follows that F +/− consists of a collection of edge-disjoint cycles C and for each cycle, each edge is either in P or in Q and the product of the edges are the same for both. Therefore, both P and Q must have the same sign. (2) ⇒ (3): This is proved by induction on the number of edges in connected graphs with the signed-path property. For convenience, we call the signed-path property A and signed-partition property B. We first show that A implies B for trees. Clearly, a signed tree with just one edge satisfies B. Assume that all signed trees with k edges that satisfy A also satisfy B, and let T +/− be a signed tree with k + 1 edges. Let v be an end vertex of T +/− and let e = vw be the edge at v. Then T +/− − v satisfies B with sets X and Y and with an edge in different sets if and only if it is negative. Assume that w ∈ X. If e is a positive edge, assign v to X and

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if e is negative, assign it to Y . In either case, T +/− then satisfies B. Hence all trees that satisfy A also satisfy B. Now assume that this is true for all signed graphs with k edges, and let G+/− be a graph with at least one cycle and k + 1 edges that satisfies A . Let vw be an edge on a cycle C +/− , and consider the signed graph H +/− = G+/− − e. Clearly, H +/− satisfies A . By the induction hypothesis, H +/− satisfies B with an edge in different sets if and only if it is negative. Assume first that e is negative. Then the v-w signed path that is obtained from C +/− when vw is removed must also be negative, and hence, since this path has an odd number of negative edges, v and w must be in different sets and therefore G+/− must satisfy B. By the same line of reasoning, if e is positive, then v and w must be in the same set. Therefore, by the principle of induction, A implies B. (3) ⇒ (1): Assume that the signed partition property holds. Then every cycle that contains a negative edge must have an even number of negative edges and hence must be positive, and thus the implication follows.   Harary [90] proved that in fact in using the definition of balanced, one doesn’t need to check that all cycles are positive. In particular, he proved the following result. Theorem 16.2 If G+/− is a nonseparable signed graph with the property that for any given vertex v, every cycle through v is positive, then G+/− is balanced. There is more than one way to define a line graph of a signed graph. We begin with the one that seems most natural to us, continuing to use the product of adjacent edges (as in paths and cycles). The line graph L(G+/− ) of the signed graph G+/− has the edges of G+/− as its vertices, with two joined by a positive edge if the two edges have the same sign in G+/− and by a negative edge if their signs are different. This is illustrated for the possible pairs of adjacent edges in Fig. 16.3. In drawing L(G+/− ) we use blue and red nodes to respectively represent the positive and negative edges of G+/− . An example of a signed graph G+/− and its line graph L(G+/− ) is given in Fig. 16.4. We note that in this example G+/− is not balanced. It follows immediately that L(G+/− ) has the signed-partition property. Fig. 16.3 The signed graph line operation

L

L

L

16.3 Boolean Signed Graphs

261

L(G+ ):

L(G− ): L(G+/− )

G+/− Fig. 16.4 The line graph of a signed graph

G+/− :

L(G+/− ):

Fig. 16.5 Positive subgraphs in the line graph of a signed graph

The fact that there are three possible pairs of adjacent edges can be used to advantage in constructing the line graph of a signed graph: both edges are positive, both edges are negative, and there is one edge of each type. In a signed graph G+/− , let G+ be the subgraph generated by the positive edges and let G− be the subgraph generated by the negative edges. Then the line graph L(G) of the underlying graph G contains the line graphs of G+ and G− as induced subgraphs (with all edges positive). Furthermore, this pair of subgraphs constitutes the subgraph of positive edges in L(G+/− ), and the remaining edges of L(G) are the negative edges of L(G+/− ). The examples in Figs. 16.4 and 16.5 show many of the aspects of line graphs of signed graphs. The following theorem of Ferrero [76] therefore is a consequence of Theorem 16.1. Theorem 16.3 The line graph of every signed graph is balanced.

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16.3 Boolean Signed Graphs We noted earlier that the line graph of a signed graph has been defined in more than one way, a second method being the following (see Behzad and Chartrand [25]): Given a signed graph G+/− with underlying graph G, then in this case an edge ef of L(G) is taken to be negative if both of the edges e and f are negative and positive if at least one of e and f is positive. Because this is so different from how signed operations generally behave, and in particular in the signs of paths and cycles in signed graphs, we introduce different terminology and symbols here. The operations here act very much like sums in Boolean algebra, where, with the two numbers 0 and 1, we have 0 + 0 = 0, and 0 + 1 = 1 + 0 = 1 + 1 = 1. We therefore adopt the following notation and terminology. If each edge of a graph G is assigned either 0 or 1 the result is called a Boolean signed graph and is denoted G0/1 . An edge is called a 1-edge if its label is 1 and a 0-edge if the label is 0. In our figures for Boolean signed graphs, 1-edges will be black (as usual), but 0-edges will be dotted. An example is the graph G0/1 in Fig. 16.6. The line graph of a Boolean signed graph will often be referred to as the Boolean line graph. Formally, it is the result of making an edge ef of L(G) a 0-edge if both e and f are 0-edges in G0/1 and making ef a 1-edge otherwise (that is, if e or f is a 1-edge). This is illustrated for the possible pairs of adjacent edges in Fig. 16.7. Figure 16.8 shows the Boolean line graph L(G0/1 ) of the given Boolean signed graph G0/1 . Note that if e is an isolated 0-edge of G0/1 , then the Boolean line graph of G0/1 is the same as if e were switched from being a 0-edge to being a 1-edge. Fig. 16.6 A (0, 1)-graph

Fig. 16.7 The Boolean line operation

L

L

L

16.3 Boolean Signed Graphs

263

Fig. 16.8 A (0, 1)-graph and its Boolean line graph

b

G0/1 :

a

e

g

d

c

f

b

e

L(G0/1 ): a

g

d

c

f

By definition, the underlying graph of a Boolean line graph is a line graph. This observation yields the following informal algorithm for finding the Boolean line graph L(G0/1 ) given L(G). Let J be the subgraph of G generated by the 0-edges. In L(G) make the edges of L(J ) 0-edges and all of the other edges 1-edges. The result is L(G0/1 ). We illustrate this in Fig. 16.9. One of the interesting features of a Boolean line graph G0/1 = L(F 0/1 ) is that no 1-edge has a 0-edge at each end. For suppose there is such a configuration with edges a, b, c in G0/1 , where b is a 1-edge, and a and c are 0-edges at the two ends of b. Then, in F 0/1 , there are edges e, f, g, and h (where h could also be e), such that e and f are 0-edges in F 0/1 as are g and h. By definition, this means that b must also be a 0-edge, a contradiction. A further consequence of this is that the subgraph of G0/1 induced by the vertices on 0-edges cannot contain any 1-edges. Acharya and Sinha [1] proved that each of these properties guarantees that a Boolean signed graph with that property is a Boolean line graph. Theorem 16.4 The following statements are equivalent for a Boolean signed graph for which the underlying graph G is a line graph: (1) G0/1 is the Boolean line graph of a Boolean signed graph F 0/1 . (2) No 1-edge in G0/1 has a 0-edge at each of its ends. (3) There are no 1-edges in the subgraph of G0/1 induced by the vertices on 0edges. Proof We again show that (1) ⇒ (2), (2) ⇒ (3), and (3) ⇒ (1).

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G0/1 :

G:

L(G):

L(J):

J:

L(G0/1 ):

Fig. 16.9 From L(G) to L(G0/1 )

(1) ⇒ (2) and (2) ⇒ (3): These implications follow from the comments made before introducing the theorem. (3) ⇒ (1): Assume that statement (3) holds for a Boolean signed graph G0/1 for which the underlying graph G is a line graph, say G = L(F ). Let G0 be the subgraph of G generated by the 0-edges of G0/1 . Since by (3) no 1-edge joins two of its vertices, G0 is an induced subgraph of G0/1 . Therefore it is the line graph of a subgraph of F all of whose edges are labeled 0. Label all of the other edges of G with 1s. Now each of these 1-edges must have at least one end vertex without any 0-edges. Therefore G0 is the Boolean line graph L(G0/1 ). This completes the proof of the theorem.   What we are calling Boolean signed graphs originated in a paper by Cartwright and Harary [50] in the context of vertex colorings. We now describe these colorings; they have some features of traditional graph colorings, but they also have one major difference. A Boolean coloring of a Boolean signed graph G0/1 is an assignment of colors to the vertices so that two vertices joined by a 1-edge have the same color and two vertices joined by a 0-edge have different colors. An example of a Boolean coloring is shown in Fig. 16.10. Clearly, if a Boolean signed graph G0/1 has a path v-w of 1-edges, and if vw is a 0-edge, then no Boolean coloring of G0/1 is possible. It was shown by Cartwright and Harary [50] that this is a necessary condition as well as a sufficient one.

16.3 Boolean Signed Graphs

265

Fig. 16.10 A Boolean colored (0, 1)-graph

Fig. 16.11 Boolean line graphs with no Boolean colorings

Theorem 16.5 A Boolean signed graph can be Boolean colored if and only if no cycle has just one 0-edge. Proof We assume that the graph G0/1 has 0-edges, since otherwise the result is trivially true. We have seen that the given condition is necessary. Assume now that the graph does not have a cycle with just one 0-edge. Let H 0/1 be the subgraph obtained by removing all of the 0-edges. We claim that H 0/1 must have at least two components. For otherwise, if H 0/1 is connected, and vw is a 0-edge in G0/1 , then there is a v-w path of 1-edges in H 0/1 and we get a cycle in G0/1 with just one 0edge, and this contradicts our assumption. Now color each component of H 0/1 with a different color. Since each 0-edge joins different components, we are done.   Suppose that a Boolean signed graph G0/1 contains a 3-star (that is, a copy of K1,3 ) with one 1-edge and two 0-edges. Then, as is shown in Fig. 16.11, its Boolean line graph L(G0/1 ) has a 3-cycle with just one 0-edge and therefore is not Boolean colorable. The same result holds if G0/1 contains a cycle in which there is only one pair of adjacent 0-edges. Therefore, as is illustrated in Fig. 16.11, L(G0/1 ) has a 3-cycle with just one 0-edge and so is not Boolean colorable. Behzad and Chartrand [25] showed that these are the only obstacles to this type of coloring.

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Theorem 16.6 Let G0/1 be the Boolean line graph of a (0, 1)-graph F 0/1 . The following statements are equivalent for G0/1 : (1) G0/1 is Boolean colorable. (2) There is no cycle in G0/1 with just one 0-edge. (3) F 0/1 does not contain either (a) a vertex incident with one 1-edge and two 0-edges or (b) a cycle in which there is one and only one pair of adjacent 0-edges. Proof From the previous theorem, it follows that (1) ⇐⇒ (2). We show that (2) ⇐⇒ (3) through their contrapositives. (∼ 3) ⇒ (∼ 2): Suppose first that F 0/1 contains three edges at a vertex v with only one of them a 1-edge. Then the Boolean line graph G0/1 has a triangle with only one 0-edge. Similarly, if F 0/1 has a cycle with a pair of adjacent 0-edges and no other 0-edges on the cycle are adjacent, then again G0/1 has a cycle with only one 0-edge. Hence, (2) does not hold. (∼ 2) ⇒ (∼ 3): Now suppose that G0/1 has a cycle with just one 0-edge, and assume that C is a shortest such cycle. We consider two cases. Case 1 C is a 3-cycle. Since it has just one 0-edge, its source in F 0/1 must be either a 3-cycle or a 3-star with two 0-edges and one 1-edge, both of which contradict (3) (see Fig. 16.11). Case 2 C has length at least 4. Then C must contain a sequence of three edges, say e, f, and g, with f a 0-edge and e and g 1-edges, as in Fig. 16.12. Recall that if a line graph has a path of length 3, then its root graph has either a path of length 4, a 3-star K1,3 with one edge subdivided, or a 4-star K1,4 . This means that in the root graph F 0/1 , e, f, and g come from four edges a, b, c, and d in that order with a and d 1-edges and b and c 0-edges and (at least) these adjacencies: a ∼ b, b ∼ c, c ∼ d, as illustrated in Fig. 16.12. In the first case, since C is a

Fig. 16.12 Root graphs of a subgraph of a Boolean line graph

16.3 Boolean Signed Graphs

267

shortest cycle in G0/1 which has only one 0-edge, it must come from a cycle in F 0/1 in which the only adjacent 0-edges can be b and c, and this violates condition (b) in (3). On the other hand, the other two possibilities imply the existence of the forbidden 3-star. Therefore (∼ 2) ⇒ (∼ 3), and the proof is complete.   For more results on signed graphs and line graphs of signed graphs, see [178] and the comprehensive bibliography by Zaslavsky [179].

Chapter 17

The Krausz Dimension of a Graph

17.1 Introduction As we noted in Chap. 1, the first paper on line graphs per se was by J. Krausz in 1943. As its title indicates, as does the review by Erd˝os (MR18403) in Math Reviews, its focus was on Whitney’s theorem that K3 and K1,3 are the only nonisomorphic connected graphs that are edge-isomorphic. Since that was not a new result, the real impact of Krausz’s paper was in its characterization of line graphs: a graph is a line graph if and only if its edge set can be partitioned into complete subgraphs in such a way that each vertex is in just one or two of the subgraphs. In this chapter, we extend this concept and consider lifting the constraint on the number of the subgraphs that a vertex is in when the edges of a graph are partitioned into complete subgraphs. One question that can be asked about a specific graph is among all of the partitions of its edge sets into complete subgraphs, what is the greatest number of these that any of the vertices is in? It is natural to have as a goal for a given graph getting the greatest number that any vertex is in to be as small as possible, which we will call the ‘Krausz dimension’ of the graph. As the problem is known to be hard in general, we concentrate on bounds for some families of graphs. We conclude with an investigation into graphs that are critical with respect to their Krausz dimension.

17.2 Some Properties of the Krausz Dimension We begin with some definitions and examples. A Krausz partition of a graph is a partition of its edges into complete subgraphs. Given such a partition, the complete subgraphs are called clusters. Note that every graph has a Krausz partition: simply take each edge as a cluster on its own. Given a Krausz partition P of a graph G, the cluster number of a vertex v in G is denoted c(v) and is defined as the number © Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4_17

269

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17 The Krausz Dimension of a Graph

Fig. 17.1 The Krausz dimension of a wheel

w1

w5

w2 v

w4

w3

of clusters in P containing it. The dimension dim P of a Krausz partition P is the maximum cluster number among the vertices of G. The Krausz dimension of a graph G, denoted K-dim G, is defined to be the least dimension of any of the Krausz partitions of G. Clearly, a connected graph has Krausz dimension 1 if and only if it is complete, and it has Krausz dimension 2 if and only if it is a line graph that is not complete. An example of such a graph is the wheel graph W5 shown in Fig. 17.1. Let P be the partition {v, w1 , w2 , v, w3 , w4 , v, w5 , w2 , w3 , w4 , w5 , w5 , w1 }, where each color in the figure represents a cluster. The cluster numbers of v and w5 are 3, while the other cluster numbers are 2. Consequently, dim P = 3, and since W5 is not a line graph, K-dim W5 = 3. Clearly, the set E(G) of the edges of any graph G constitutes a Krausz partition of that graph, so the concept is well-defined. It also has an elementary interpretation. A newly formed club has a large number of members and a set S of prescribed pairs need to get acquainted. The officers decide to have some social gatherings with two requirements: each gathering will have only unacquainted members and no pair will be together in more than one gathering. Because the members don’t want to attend many events, the officers want to have the most events that any one person must attend kept to a minimum. Of course, a Krausz dimension of the graph of unacquainted members gives a solution satisfying the given requirements. The following theorem gives some elementary facts about the Krausz dimension. Theorem 17.1 Let G be a graph with maximum degree Δ. (a) (b) (c) (d)

If G is triangle-free, then its only Krausz partition is the set of its edges. K-dim G ≤ Δ(G), with equality if G is triangle-free. If H is an induced subgraph of G, then K-dim H ≤ K-dim G. If the star K1,s is an induced subgraph of G, then K-dim G ≥ s.

17.3 Some Interesting Families

271

Proof (a) This is obvious since the only complete subgraphs of G are its edges and vertices. (b) Clearly, the cluster number of any vertex is no greater than its degree, so the Krausz dimension of G is no greater than Δ. Furthermore, if G has no triangles, then by (a) equality holds. (c) This follows from the fact that the restriction to H of a Krausz partition of G is a Krausz partition of H . (d) This is an immediate consequence of (c) and (a).   The computational complexity of the Krausz dimension was investigated by Hlinˇený and Kratochvíl [106], with some interesting results. The following were among the problems that they considered: • K-dim: Determine the Krausz dimension of an arbitrary graph. • K-dim(r): Determine the Krausz dimension of a graph given that its Krausz dimension is at most r. • K-dim(r; s): Determine the Krausz dimension of a graph given that its Krausz dimension is at most r and its maximum degree is at most s. And here are some of their results: • K-dim is NP-complete; in fact, K-dim (3) is NP-complete. • K-dim(3; 4) is polynomial (O(n4 )). • K-dim(r; r + 2) is NP-complete, even for planar graphs. Further results were found by Glebova, Metelsky, and Skums [82]. The complexity of the general problem certainly adds to its interest for particular families, which we turn to next.

17.3 Some Interesting Families In this section we explore the Krausz dimension of some familiar families of graphs. Complete Multipartite Graphs We begin with complete bipartite and complete tripartite graphs. Theorem 17.2 K-dim Kr,s = max{r, s} and K-dim Kr,s,t = max{r, s, t}. Proof The bipartite result is an immediate consequence of Theorem 17.1 (b). To establish the tripartite result, we first show that the regular complete tripartite graph Kr,r,r is the union of r 2 edge-disjoint triangles. Let A, B, and C be the three partite sets of vertices and let F1 , F2 , . . . , Fr be a 1-factorization of the edges joining B and C. Now form r triangles by joining the ith vertex of A to both ends of the edges in Fi . Together these form a decomposition of the 3r 2 edges of Kr,r,r into r 2 triangles, with each vertex on r triangles. Consequently, K-dim Kr,r,r ≤ r. Now

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17 The Krausz Dimension of a Graph

if r = max{r, s, t}, then K-dim Kr,s,t ≤ r. On the other hand, by Theorem 17.1 (c), K-dim Kr,s,t ≥ max{r, s} = r, so equality holds.   The situation gets more complicated when a complete multipartite graph has more than three partite sets. For instance, it is easy to see that the Krausz dimension of K1,1,1,2 (also known as K5 − e) is 3. However, it is also the case that K-dim K1,1,1,3 is 3: Let the partite sets be {u}, {v}, {w}, and {x1 , x2 , x3 }, with the Krausz partition {u, v, x1 , v, w, x2 , , u, w, x3 , u, x2 , v, x3 , w, x1 }. In fact, it can be shown that for all r > 2, K-dim K1,1,1,r = r. Thus, for some complete 4-partite graphs (in addition to the obvious case of K1,1,1,1) the Krausz dimension equals the order of a largest partite set, just as it does for all complete bipartite and complete tripartite graphs, but for others it doesn’t. In this context we also briefly mention the general octahedral graph Kr(2) (the complete r-partite graph with two vertices in each partite set), and note, without giving details of the proof, that K-dim K4(2) = 3 and K-dim K5(2) = 4. Outerplanar Graphs We begin with those that are maximal, that is, triangulations of polygons. They can also be defined inductively, starting with a triangle in the plane and then in the unbounded region repeatedly adding a new vertex adjacent to both vertices of an edge on the boundary. See Fig. 17.2 for an illustration. Thus, each of the bounded faces is a triangle, and a property that will be useful and is easily proved by induction is that those faces can be properly 2-colored. The next theorem contains results on the Krausz dimension of these graphs, the value depends very much on the maximum degree of the graph. Figure 17.3 shows three outerplanar graphs all with Krausz dimension 3, but with maximum degrees respectively being 4, 5, and 6. Those examples may suggest that there can be considerable variation in the maximum degree of a maximal outerplanar graph with specified Krausz dimension, but in fact, as the following theorem indicates, there is not. Theorem 17.3 If G is a maximal outerplanar graph with maximum degree Δ, then Δ + 1 2 Fig. 17.2 A maximal outerplanar graph

≤ K-dim G ≤

Δ + 1 2

.

17.3 Some Interesting Families

F:

G:

H:

273

Δ =4 K-dim = 3

Δ =5 K-dim = 3

Δ =6 K-dim = 3

Fig. 17.3 Examples of the Krausz dimension of outerplanar graphs

Proof We first establish the upper bound. Consider an embedding of a maximal outerplanar graph G in the plane. Clearly, the triangular faces of G have a 2coloring. Let P be the Krausz partition consisting of the triangles of one of the color classes together with the edges that are not on any of these triangles. In G, any vertex v and its neighbors induce a fan graph (in which the neighbors of v form a path). Hence, if v has odd degree 2r + 1, it must be on r of the triangles and on one edge in P, while if it has even degree 2r, it is either on r − 1 triangles and two edges, or on just r triangles. It follows that K-dim G ≤ Δ2 + 1. The lower bound follows from the fact that triangles are the largest complete subgraphs in a maximal outerplanar graph, and hence K-dim G ≥ Δ2 . The form of the bounds in the theorem follows from elementary integer considerations.   The following result is an immediate consequence of the theorem. Corollary 17.1 Let G be a maximal outerplanar graph with maximum degree Δ and Krausz dimension ζ . (a) If Δ = 2r + 1, then ζ = r + 1. (b) If Δ = 2r, then ζ = r or r + 1. Figure 17.3 illustrates the possibilities. First, the graph G has Δ = 5 and ζ = 3, and in fact the corollary tells us that all maximal outerplanar graphs with a given odd maximum degree have the same Krausz dimension. On the other hand, F and H have different even maximum degrees, but the same Krausz dimension. However, there are only the two possibilities. We can restate the corollary in this way: If G is a maximal outerplanar graph with Krausz dimension ζ , then Δ(G) = 2ζ − 2, 2ζ − 1, or 2ζ . The three examples can all be extended in a natural way to higher dimensions.

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17 The Krausz Dimension of a Graph

Fig. 17.4 A 2-tree

2-Trees By removing the restriction of planarity, the definition of maximal outerplanar graphs results in what are known as 2-trees , defined as follows: Starting with a single edge and then successively adding new vertices of degree 2 adjacent to the two vertices of an existing edge, the resulting graph is a 2-tree. An example is shown in Fig. 17.4. (For further properties and characterizations of 2-trees, see [35].) Theorem 17.4 If G is a 2-tree with maximum degree Δ, then Δ 2

≤ dim G ≤ Δ − 1.

Furthermore, both bounds are sharp. Proof The lower bound again follows from the fact that a triangle is the largest cluster that one can have in a Krausz partition of a 2-tree. For the upper bound, we first show that every 2-tree has a set of edge-disjoint triangles meeting every vertex of degree greater than 2 (and perhaps some vertices of degree 2): This is clearly true when the 2-tree has order 2. Assume it is true for every 2-tree of order n and let G be a 2-tree of order n + 1. Then for some vertex v, G − v is a 2-tree of order n; let u and w be the neighbors of v, and let (by the induction hypothesis) S be such a set of triangles for G − v. If the edge uw is on a triangle in S , then S also works for G; while if not, then the triangle u, v, w can be added to S to obtain a set that works for G. The upper bound follows immediately from the existence of this set of triangles. The sharpness of the lower bound has already been shown in Fig. 17.2, while the 2-tree consisting of Δ−1 triangles with a common edge proves that the upper bound is sharp.   This result can easily be extended to all graphs that do not contain four mutually adjacent vertices. Corollary 17.2 If G is a K4 -free graph with maximum degree Δ, then  Δ2  ≤ Kdim G ≤ Δ, and both bounds are sharp.

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275

Complete Graphs Missing One Edge Even though complete graphs have Krausz dimension 1, it is somewhat surprising that if just one edge is removed, the problem is frequently difficult, and affine planes sometimes provide a solution. We denote the graph resulting from the removal of one edge from the complete graph Kn by Kn− (in the literature this is sometimes denoted Kn − e). It follows from the induced subgraph property of Krausz partitions − that K-dim Kn− ≤ K-dim Kn+1 . Another key idea that we use is the following lemma—its proof is straightforward. Lemma 17.1 If C is a cluster in a Krausz partition P of Kn− and v is a vertex not in C, then every edge from v to C must be its own cluster in P. We begin by considering some small graphs. It is easy to see that K-dim K3− = K-dim K4− = 2, and we observed earlier that K-dim K5− = 3, it being the graph K1,1,1,2. We observe further that K-dim K6− is also 3: Let its vertices of degree 4 be u and v and let those of degree 5 be w, x, y and z; then a suitable Krausz partition has these clusters: {u, w, x, u, y, z, u, w, y, v, x, z, w, z, x, y}. We now show that K-dim K7− > 3: Suppose that it equals 3, and let P be an optimal partition. Then since every vertex has degree at least 5, by the lemma no cluster in P can have more than three vertices. Consequently, each of the five vertices of degree 6 must be on three triangles in P and the other two vertices (which cannot share a cluster) must lie on two triangles and one edge. This is clearly impossible, so K-dim K7− ≥ 4. We leave it to the reader to show that equality holds. The preceding argument can be extended to prove the following result. Detailed proofs of other results on this topic may be found in Beineke and Broere [33]. − Lemma 17.2 If p ≥ 4, then K-dim Kp(p−1) ≥ p + 1.

The following lower bound is a consequence of this lemma.  √  Theorem 17.5 If n ≥ 7, then K-dim Kn− ≥ 3+ 24n+1 . We turn now to the promised application of finite geometries. As background, we note that an affine plane of order r consists of r 2 points and r(r + 1) lines with the property that any given pair of points are on exactly one line and there is a partition of the lines into r + 1 sets of r parallel (that is, pairwise disjoint) lines. One useful fact here is that there exist affine planes of all orders that are powers of primes. The following theorem puts this into the terminology of Krausz partitions. Theorem 17.6 If an affine plane of order r exists, then Kr 2 has a Krausz (r + 1)partition consisting of r(r + 1) clusters, each of order r, such that (a) each vertex is on exactly r + 1 clusters, and (b) there is a partition of the clusters into r + 1 families of r disjoint clusters each. One consequence of this is the following upper bound for some graphs that are joins of certain complete graphs.

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Theorem 17.7 If an affine plane of order r exists, and if G is a graph of dimension d and order at most r + 1, then K-dim (Kr 2 ∗ G) ≤ r + d. The graph G of interest to us here is 2K1 , the null graph with two vertices, which together with earlier results gives the following theorem. Theorem 17.8 If there exists an affine plane of order r, then K-dim Kn− = r + 1 for r 2 − r ≤ n ≤ r 2 + 2 except that K-dim K6− = 3. Since there exists an affine plane of order r whenever r is a prime power, this theorem gives the Krausz dimension of infinitely many of these nearly complete √ graphs and says that in general dim Kn− ≈ n. The accompanying table summarizes what is known about dim Kn− for n ≤ 83. Note that in all of these cases, the value is known exactly or to be one of two consecutive numbers. n 3, 4 5, 6 7–11 12–18 19–27 28, 29 30–41 42–51 52–55 56–66 67–71 72–83

K-dim Kn− 2 3 4 5 6 6 or 7 7 or 8 8 8 or 9 9 9 or 10 10

Except for a few cases, these values follow from Lemma 17.2 and Theorem 17.8. For n < 6, the values were noted earlier, while the upper bound of 7 when n = 29 follows from Theorem 17.8 by taking r = 5 and G to be K4− . The value for n = 19 was computed separately.

17.4 The Krausz Dimension and Graph Operations One of the important questions often asked about a graphical property (such as the chromatic number) is whether any of its proper subgraphs has the same value. An investigation of this topic for the Krausz dimension was begun by Beineke and Broere [32]. In this section, we consider how various graph operations affect the Krausz dimension. We begin with a simple result on the disjoint union of graphs.

17.4 The Krausz Dimension and Graph Operations

277

Theorem 17.9 If the components of graph G are G1 , G2 , . . . , Gk , then K-dim G = maxi {K-dim Gi }. Next we look at the product of two graphs G and H . Note that in G × H , every complete subgraph of order greater than 2 must lie entirely within a copy of G or a copy of H . Theorem 17.10 If G and H are non-null graphs, then max{K-dim G, K-dim H }+ 1 ≤ K-dim (G × H ) ≤ K-dim G + K-dim H . Furthermore, both bounds are sharp. Proof We first observe that dim(G × H ) > dim G, since at each vertex of a copy of G there is an edge that is not on a triangle with any edge of G, so the inequality holds. Since the same holds for H , the lower bound is established. On the other hand, if we take an optimal Krausz partition of each copy of G and another for each copy of H , we have a Krausz partition of G × H and so the upper bound holds. Furthermore, both bounds are sharp since if either graph is complete, then the two bounds are equal.   The theorem has the following interesting corollaries, with the second leaving some open questions on the dimension of a product of a graph with itself. Corollary 17.3 For any graph G, if r ≥ 2, then K-dim (G × Kr ) = K-dim G + 1. Corollary 17.4 For any graph G, K-dim G + 1 ≤ K-dim (G × G) ≤ 2K-dim G. Clearly the addition of an edge to a graph cannot increase the Krausz dimension by more than one, but as we have seen even in the case of complete graphs, deleting an edge can increase the dimension by an arbitrarily large amount (but can only decrease it by 1). The next theorem involves the deletion of a vertex. Theorem 17.11 Let v be a vertex of graph G and let F = G − v. Then K-dim F ≤ K-dim G ≤ max{deg v, K-dim F + 1}. Furthermore, these bounds are sharp. Proof The lower bound follows from the fact that F is an induced subgraph of G. For the upper bound, we take an optimal Krausz partion of F and augment it with the individual edges at v. The result is a Krausz partition of G, and its cluster number is either deg v or K-dim F , whichever is larger. Examples for which K-dim G equals K-dim F or K-dim (G − v) + 1 are easily found. As for an example G with K-dim G the maximum vertex degree Δ(G) and greater than K-dim F + 1, simply take a tree with one vertex of maximum degree and all other vertices of sufficiently small degree.  

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17 The Krausz Dimension of a Graph

17.5 Krausz Critical Graphs A graph G is dimension-critical if K-dim (G − v) < K-dim G for every vertex v in G; it may be called d-critical for short if its dimension is d. Since the only connected graphs of Krausz dimension 1 are the nontrivial complete graphs, the only graph that is 1-critical is clearly K2 . It also follows that the result of deleting any vertex from a 2-critical graph G must be a complete graph, and consequently G must be K1,2 . Restricted to bipartite graphs, it easily follows from Theorem 17.1(b) that this observation can be extended as follows. Theorem 17.12 The only d-critical bipartite graph is the star K1,d . Since Krausz’s characterization of line graphs as being those graphs of dimension 2 or less, it follows that each of the nine graphs in the forbidden subgraph characterization (see Theorem 3.1 and Fig. 3.2) is 3-critical. Our next result gives us a way to create new critical graphs from old ones by adding edges. Theorem 17.13 Let v and w be vertices at distance at least 3 in graph G, and let H be the graph obtained from G by joining v and w. (a) If, in G, deg v < K-dim G and deg w < K-dim G, then K-dim H = K-dim G. (b) If, in G, deg v < d − 1 and deg w < d − 1, then H is d-critical if and only if G is. The graphs in Fig. 17.5 show that this theorem is best possible in several ways. In each case, Hi is the result of joining two vertices in Gi that are at distance at least 3 from each other. • K-dim G1 = 4 and K-dim H1 = 3, which shows that the condition vertices at distance least 3 is necessary. • K-dim G2 = 2 and K-dim H2 = 3, which shows that the degree conditions in (a) are sharp. • G3 is 4-critical but H3 is not (since it contains a copy of K1,4 as an induced proper subset), which shows that the degree conditions in (b) are sharp. Now we consider merging two vertices. Theorem 17.14 Let v and w be vertices at distance at least 4 in graph G, and let H be the graph obtained from G by merging v and w. (a) If, in G, deg v + deg w ≤ dim G, then dim H = dim G. (b) If, in G, deg v + deg w < d and G is d-critical, then H is. We show with the graphs in Fig. 17.6 that this theorem too is best possible in several ways. In each case Hi is obtained from Gi by merging the vertices v and w to form u.

17.5 Krausz Critical Graphs

279

G1 :

H1 :

G2 :

H2 :

G3 :

H3 :

Fig. 17.5 The effect of adding an edge

• K-dim G1 = 4 and K-dim H1 = 3, which shows that the condition vertices at distance at least 4 is necessary. • K-dim G2 = 2 and K-dim H2 = 3, which shows that the degree conditions in (a) are sharp. • G3 is not 3-critical but H3 is, which shows that the converse of (b) does not always hold.

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17 The Krausz Dimension of a Graph

Fig. 17.6 The effect of merging vertices v

u H1 :

G1 : w

w

H2 :

G2 :

u

v

v

G3 :

u

w

H3 :

Bases We now consider the set of graphs of a given dimension that are minimal in a sense beyond being critical. To this end, we define two operations for the joining or merging of vertices, as described in the previous two theorems. A type A d-operation consists of joining two vertices by an edge if the distance between them is at least 3 and their degrees are at most d − 2. A type M d-operation consists of merging two vertices into one if the distance between them is at least 4 and the sum of their degree sums is at most d − 1. A graph H is said to be d-dependent on graph G if it can be obtained from G by a sequence of d-operations of type A or M. The next result is a consequence of part (b) in Theorems 17.13 and 17.14.

17.5 Krausz Critical Graphs

281

Fig. 17.7 Creating critical graphs

G:

H:

Theorem 17.15 If G is a d-critical graph and if H is a graph that is d-dependent on G, then H is also d-critical. For example, the graph H in Fig. 17.7 is 4-critical since it can be obtained from the 4-critical graph G by a combination of operations of types A and M. We now define a d-critical graph to be d-basic if it is not d-dependent on any other d-critical graph, and the set of all d-basic graphs is called the Krausz d-basis. For example, since the graph G in Fig. 17.7 is 4-critical, so is the graph H . The first interesting case is the 3-basis since there is only one 1-critical graph and only one 2-critical graph. Clearly all of the nine graphs L1 − L9 in Fig. 3.2 have Krausz dimension 3 since they are not line graphs, and since each of their proper induced subgraphs is a line graph, they are all critical. Furthermore, any other graph that is of Krausz dimension 3 must contain one of these. Now it is readily seen that L7 can be obtained by joining two vertices in L4 and L2 can be obtained by merging the same pair of vertices. It is now a straightforward matter to verify that none of the remaining seven graphs is 3-dependent on any of the others. They therefore constitute the Krausz 3-basis. Theorem 17.16 The Krausz 3-basis is the set of graphs {L1 , L3 , L4 , L5 , L6 , L8 , L9 } in Fig. 17.8. Determining the Krausz 4-basis appears to be quite difficult. One place to start is with the 3-basis. Clearly, if a new end edge is added to any graph G of dimension d, the resulting graph Go (called the corona of G) has dimension d + 1, but it may not be critical even if G itself is critical. This technique was used to find the first eight graphs in Fig. 17.9. The other four were discovered by other means, and it can be verified that all twelve are basic. We do not know whether there any others, but we think it likely that there are. For higher dimensions, very little is known beyond using these methods to get larger graphs.

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17 The Krausz Dimension of a Graph

L3 :

L1 :

L5 :

L8 :

Fig. 17.8 The Krausz 3-basis

Fig. 17.9 Some 4-basic graphs

L4 :

L6 :

L9 :

Glossary

I. Graphs Graph: consists of vertices and edges, with edges being unordered pairs of vertices (written as juxtaposition e = vw) Incidence, Adjacency, Degrees Incidence: for e = vw, e and v are incident Adjacent vertices: vertices incident with a common edge Adjacent edges: edges incident with a common vertex Degree of vertex v: the number of edges incident with v Walks, Trails, Circuits, Cycles, Paths Walk: an alternating sequence of vertices and edges, beginning and ending with vertices, each edge incident with the vertices before and after Open walk: the first and the last vertices different Closed walk: the first and the last vertices the same Trail: a nontrivial walk with all edges different Circuit: a closed trail Cycle: a circuit with all vertices different except the first and the last Path: an open trail with all vertices different Graph Types Regular: all vertex degrees equal Connected: there is a walk between any two vertices Eulerian: has a circuit containing every edge Hamiltonian: has a cycle containing every vertex Graph Operations, Binary Sum G + H : the disjoint union of a copy of G and a copy of H Join G ∗ H : the sum G + H and each vertex of G adjacent to all those in H Product: G × H : replacement of each vertex of H with a copy of G and in H corresponding vertices adjacent according as vertices in H are adjacent © Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4

283

284

Glossary

Attachment (complete graphs) Kr · Ks : the two graphs have one vertex in common Graph Operations, Unary Complement G: vertices are the same as G with two adjacent if and only if not adjacent in G Multiple copies kG: the union of k disjoint copies of G Square G2 : G augmented by edges joining those vertices at distance 2 Subdivision S(G): replacement of each edge with a new path of length 2 II. Digraphs Digraph: consists of vertices and arcs, with arcs being ordered pairs of vertices (the same as a relation on a finite set) Arcs Loop: an arc with the two elements of its ordered pair equal Head and tail of an arc: w is the head of arc a = vw and v is its tail Opposite arcs: arcs a = vw and b = wv are opposite arcs Underlying graph: remove loops, replace each arc with an edge Vertex Degrees In-degree: the number of arcs of which a vertex is the head Out-degree: the number of arcs of which a vertex is the tail Source: a vertex with out-degree positive and in-degree 0 Sink: a vertex with in-degree positive and out-degree 0 Digraph Types Connected: the underlying graph is connected Strongly connected (strong): has a path from each vertex to each of the others Eulerian: has a circuit containing every arc Hamiltonian: has a cycle containing every vertex Orientation: assignment of a direction to each edge of a graph Oriented: there is at most one arc between any two vertices and no loops Tournament: an orientation of a complete graph Bipartite tournament: an orientation of a complete bipartite graph

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Index of Names

A Acharya, M., 263 Aigner, M., 39, 150, 151, 253 Akka, D.G., 76, 79, 81, 85 Alon, N., 125

B Bacsó, G., 123 Bagga, J.S., 234, 238, 244, 248, 255 Balbuena, C., 156 Bauer, D., 9, 91 Behzad, M., 204, 208, 210–212, 262, 265 Beineke, L.W., 26, 49, 76, 113, 118, 184, 234, 238, 244, 248, 276 Bernshteyn, A., 124 Bertossi, A.A., 106 Blanuša, D., 116 Bondy, J.A., 102 Broere, I., 276 Broersma, H.J., 103, 107, 220, 224, 225, 227, 230, 241 Brualdi, R.A., 156

C Cameron, P.J., 58 Carroll, L., 116 Cartwright, D., 264 Castagna, F., 115 Chang, L., 7 Chartrand, G., 9, 39, 74, 88, 92, 101, 105, 213, 215, 262, 265 Chetwynd, A., 119

Choudum, S.A., 121, 122 Chvátal, V., 102 Coulson, C.A., 53 Cvetkovi´c, D., 41, 56

D Deogun, J.S., 67 Ding, Z., 81 Doob, M., 52, 56, 58

E Ellis, R.B., 255 Erd˝os, P., 118 Esperet, L., 124

F Ferrero, D., 253, 255, 261 Fiamˇcík, I., 124 Fiol, M.A., 156, 158 Fiorini, S., 118

G Ghebleh, M., 82, 84 Glebova, O., 271 Goethals, J.M., 58 Goldberg, M.K., 120 Greenwell, D.L., 63, 79, 101 Grunewald, S., 118 Gupta, R.P., 118 Gutiérrez, A., 248

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293

294 H Hall, P., 235 Hamelink, R.C., 230 Harary, F., 21, 22, 74, 79, 99, 145, 151, 154, 159, 177, 178, 182, 211, 257, 260, 264 Hartke, S.G., 15 Hedetniemi, S.T., 36 Hemminger, R.L., 63, 79, 152, 153, 177, 193 Heuchenne, C., 159 Higgins, A.W., 15 Hilton, A.J.W., 119, 120 Hlinˇený, P., 271 Hoede, C., 220, 224, 225 Hoffman, A.J., 8 Holyer, I., 113, 118 Huang, Y., 81 I Isaacs, R., 116 J Jackson, B., 104 Jakobsen, I., 120, 121 Jendro´l, S., 79, 80 Jensen, T.R., 119 Johnson, P.D., 120 Jung, H.A., 17 K Kainen, P.C., 79 Kaiser, T., 104 Karp, R.M., 211 Khatirinejad, M., 82, 84 King, A.D., 124 Kirchhoff , G., 51 Klešˇc, M., 79, 80 König, D., 117 Klešˇc, M., 79 Knor, M., 15, 93, 94, 128, 130, 132, 133 Kratochvíl, J., 271 Krausz, J., 26, 36, 98, 223 Kriesell, M., 104 Krylov, E.V., 104 Kulli, V.R., 76, 81, 85 Kuratowski, C., 62, 74 Kureethara, J.V., 247

L Lai, H.-J., 104 Li, H., 225, 243

Index of Names Li, X., 227, 230, 243 Lin, Y., 225 Liu, J., 101, 216 Liu, Y., 230 Lladó, A.S., 158, 248

M Mathews, M.M., 102 McDonald, J., 120 Meng, J., 101, 102, 216 Menger, K., 88, 91 Metelsky, Y., 271 Mitjana, M., 156, 158 Moon, J.W., 7

N Nash-Williams, C.St.J.A., 99 Nebeský, L., 101, 104 Niepel, L'., 15, 93, 94, 128, 130, 132, 133 Norman, R.Z., 145, 151, 154, 159, 177, 178, 182

O Ore, O., 81

P Palmer, E.M., 21, 22 Panshetty, S., 79 Parreau, A., 124 Petkovšek, 133, 138 Plummer, M.D., 103 Prins, G., 115 Prisner, E., 256

R Radjavi, H., 208 Rao, A.R., 38 Reed, B.A., 124 Richards, P.I., 159 Roberts, F.S., 230, 231 Robertson, N., 116 Rowlinson, P., 56 Rushbrooke, G.S., 53 Ryjáˇcek, Z., 102, 103, 107, 123, 241

S Sabidussi, G., 21 Sachs, H., 55

Index of Names Sampathkumar, E., 81 Sanders, D., 116 Saražin, M., 105, 107 Sato, I., 232 Schiermeyer, I., 241 Schwenk, A.J., 79 Sebastian, M., 247 Sedláˇcek, J., 63, 68, 76 Seidel, J.J., 58 Seymour, P., 116 Shrikhande, S.S., 7 Shult, E.E., 58 Simi´c, S., 41, 56 Sinha, D., 263 Skowro´nska, M., 216 Skums, P., 271 Slater, P.J., 36 Šoltés, L'., 15, 31, 128, 130, 132, 133 Spencer, J.H., 230, 231 Stewart, M.J., 9, 88, 92, 213, 215 Sudakov, B., 125 Sumner, D.P., 102 Sysło, M., 216 Szekeres, G., 116 T Tait, P.G., 110 Tapadia, S.A., 247 Thomas, R., 116 Thomassen, C., 102 Tindell, R., 91 Toft, B., 119 Tutte, W.T., 116, 211 Tuza, Z., 123

295 V van Rooij, A.C.M., 12, 26, 69 Varma, B.N., 234, 238, 244, 248 Vasquez, M.R., 238 Vetta, A., 124 Vizing, V.G., 112, 117, 118 Vrána, P., 103, 104

W Wagner, K., 62 Wall, C.E., 105 Waphare, B.N., 247 Watkins, M.E., 115 Whitney, H., 17–19, 88, 209 Wilf, H.S., 12, 26, 69 Wilson, R.J., 113, 118 Wu, B., 101, 102

X Xiong, L., 107

Z Zaks, A., 125 Zamfirescu, C.M., 184, 216 Zamfirescu, T., 89, 90 Zaslavsky, T., 267 Zhan, S., 104 Zhang, H., 243 Zhao, B., 230 Zhao, T., 81

Index of Definitions

Symbols 0-edge, 262 1-edge, 262 -path graph, 134 d-basic, 281 d-dependent graph, 280 k-centerable, 133 k-connected, 87 l-edge-connected, 87 NP-complete, 106, 113, 118, 123 d-operation type-A, 280 type-M, 280 2-tree, 274 3-critical, 120

A Adjacency matrix digraph, 156 graph, 51 signed graph, 258 Affine plane, 275 Arborescence, 177 Arc, 146 head, 146 Heuchenne, 166 opposite, 161, 228 parallel, 146 tail, 146

B Binomial set (bin), 224 Biplanar graph, 210

Bisection number, 248 Boolean coloring, 264 labeling, 257 line graph, 262 signed graph, 262 Bristle, 135 Bristly cycle, 135 C Cactus, 75, 107 Carrier, 215 Caterpillar, 135 Center, 131 Chain breakable, 107 type-1, 14 type-2, 14 Characteristic polynomial digraph, 157 graph, 52 Chromatic index, 110 Chromatic number, 110 edge-, 110 Circuit digraph, 150 graph, 99 Claw, 12, 73, 105 Clique, 26 number, 121, 124, 125, 231 Cluster, 269 number, 269 co-line graph, 43 Coloring, 110 edge-, 110

© Springer Nature Switzerland AG 2021 L. W. Beineke, J. S. Bagga, Line Graphs and Line Digraphs, Developments in Mathematics 68, https://doi.org/10.1007/978-3-030-81386-4

297

298

Index of Definitions

strong, 124 weak, 123 Column-orthogonality property, 160 Common intersection property, 230 Connectivity, 87 Contraction, 61 Converse, 134 Core digraph, 151 Counter-arborescence, 177 Cover, 99 Crossing number, 76 Cycle-separated, 183

line digraph, 151 line graph, 212 total graph, 212 Even triangle, 26

D Degree matrix, 54 Diameter digraph, 146, 150 graph, 128, 130, 131 Diamond, 133 Digraph bicomplete, 149, 162 component, 146 connected, 146 connection, 160 contrafunction, 154 converse, 146 eddy, 180 function, 154, 177 fundamental, 153 strong component, 146 strongly connected, 146 unicyclic, 178 Dimension-critical, 278 Distance digraph, 146 graph, 127 Double partition, 160 Double--path graph, 134

G Graph biregular, 67 bow-tie, 104 Class 1, 118 Class 2, 118 claw-free, 70, 71 cocktail party, 55 comparability, 133 floral, 56 hourglass, 104 outerplanar, 74 pancyclic, 104 planar, 62 strongly regular, 6 theta, 74 unicyclic, 21 Group-identical, 21

E Eccentricity, 128 Edge-connectivity, 87 Edge-covering walk, 99 Edge-independence number, 236 Edge-isomorphism, 17, 20 Edge-pancyclic, 241 Eigenvalue, 52 Eigenvector, 52 Elementary contraction, 61 Eulerian digraph, 151 graph, 70, 98

F Fan, 246 Flower, 56 Foundation of a digraph, 178 Four color theorem, 111 Fundamental crossing, 80 Fundamental theorem on edge colorings, 117

H Hall’s marriage theorem, 235 Hamilton-connected, 104, 241 Hamiltonian digraph, 151 line digraph, 151 line graph, 99 total graph, 212 Hamiltonian index, 105 Heuchenne condition, 177 first, 184 second, 184 History of a vertex, 93 Homeomorphic, 61 Hooked walks, 186 Hook-related walks, 186 I In-arc, 146 In-incidence matrix, 156

Index of Definitions Incidence matrix, 54 Independence number, 236 In-neighbor, 146 In-radius, 146 In-tree, 177 In-tree index, 181

299 Orientation, 133 Out-arc, 146 Out-incidence matrix, 156 Out-neighbor, 146 Out-radius, 146 Out-tree, 177 Out-tree index, 181 Overfull, 118

J jump graph, 101

K König’s theorem, 117 Krausz dimension, 270 partition, 269 Krausz d-basis, 281 Kuratowski graphs, 62

L Line completion number, 244 Line-degree, 5 Line digraph, 147 iterated, 173 partial, 158 Line-forbidden graph, 30 Line graph, 4 characterizations, 26 floral, 56 iterated, 12 of a multigraph, 124 of a signed graph, 260 line graph generalized, 56 Line-graphical sequence, 11 Locally connected, 102

M Multidigraph, 159, 161

N Negative cycle, 259 Neighborhood-connected, 102

O Octahedron, 133 d-dimensional, 55 Odd triangle, 26

P Panconnected, 241 Pancyclic, 241 Pancyclicity, 104 Path digraph, 227 graph, 220 Path-comprehensive, 241 Period, 178 Periodic, 178 Perron-Frobenius theorem, 53 Petal, 56 Petersen graph, 62, 114–116, 119 generalized, 115 Planarity index, 84 Positive cycle, 259 Prolific, 12, 105, 131

R Radius digraph, 151 graph, 128, 130 Root graph, 4, 98 Root system, 58 Row-orthogonality property, 160 Ryjáˇcek closure, 103

S Self-centered, 132 Self-converse, 134 Separating set, 88 Signed graph, 257, 258 balanced, 258 underlying graph, 258 Signed-partition property, 259 Signed-path property, 259 Snark, 116 Spectrum, 52 Splitting a vertex, 236 Square of a digraph, 215 Square of a graph, 205, 206, 211, 215

300 Star, 17 Star-closed, 58 Star-preserving, 17 Subdivision, 226 digraph, 215 double, 226 graph, 61, 204 Super line digraph, 252 graph, 234 multigraph, 254 T Total digraph, 213 graph, 204 Tournament, 154 Trail digraph, 150 graph, 99 Transitive orientation, 133 Transitive triple, 216 hypotenuse, 216 leg, 216

Index of Definitions Triangular prism, 75 Tri-level subdigraph, 186

V Vertex end-sink, 186 end-source, 186 in-degree, 146 isolated, 146 juncture, 146 opposite, 228 out-degree, 146 sink, 146 source, 146 Vertex-critical, 119 Vertex-isomorphism, 17, 20 Vertex-pancyclic, 241 Vizing’s theorem, 117, 121

W Wheel, 246 Windmill, 246