Surveys in Combinatorics, 2001 0521002702, 9780521002707

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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor N.J. Hitchin. Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OXI 3LB, United Kingdom The titles below are available from booksellers, or, in case of difficulty, from Cambridge University Press. 90 96 97 99 100 104 105 107

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144 146

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Polytopes and symmetry, S.A. ROBERTSON Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics. D.S. BRIOOES & F. RICHMAN Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Elliptic structures on 3-manifolds, C.B. TIIOMAS A local spectral theory for closed operators, I. ERDELYI & WANG SHENGWANG Compactification of Siegel moduli schemes, C.-L. CHAI Diophantine analysis, J. LOXTON & A. VANDER POORTEN (eds) Lectures on the asymptotic theory of ideals, D. REES Representations of algebras, P.J. WEBB (ed) Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups- St Andrews 1985, E. ROBERTSON & C. CAMPBEll (eds) Descriptive set theory and the structure of sets of uniqueness, A.S. KECHRIS & A. LOUVEAU Model theory and modules, M. PREST Algebraic. extremal & metric combinatorics, M.-M. DEZA, P. FRANKL & I.G. ROSENBERG (cds) Analysis at Urbana, il, E. BERKSON, T. PECK, & J. UHL (eds) Advances in homotopy theory, S. SALAMON. B. STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces. E.M. PElNADOR & A. RODES (eds) Surveys in combinatorics 1989. J. SIEMONS (ed) Introduction to unifonn spaces, I.M. JAMES Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSHINO Helices and vector bundles, A.N. RUDAKOV el al Solitons, nonlinear evolution equations and inverse scattering, M. ABLOWITZ & P. CLARKSON Geometry of low-dimensional manifolds 1. S. DONALDSON & C.B. TIIOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & MJ. TAYLOR (eds) Classification theories of polarized varieties, TAKAO FUJITA Geometry of Banach spaces, P.F.X. MOLLER & W. SCHACHERMA YER (eds) Groups St Andrews 1989 volume 1. C.M. CAMPBELL & E.F. ROBERTSON (eds) Groups St Andrews 1989 volume 2. C.M. CAMPBELL & E.F. ROBERTSON (eds) Lectures on block theory, BURKHARD KOLsHAMMER Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SHR.lKANDE & S.S. SANE Surveys in combinatorics, 1991, A.D. K.EEDWELL (ed) Representations of algebras, H. TACHIKA WA & S. BRENNER (eds) Boolean function complexity, M.S. PATERSON (ed) Manifolds with singulariti~ and the Adams-Novikov spectral sequence, B. BOTVINNIK Squares. A.R. RA1W ADE Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry, W.J. HARVEY&. C. MACLACHLAN (eds) Lectures on mechanics, J.E. MARSDEN Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE & A. PITIS (eds) Lower K- and L-theory, A. RANICKI Complex projective geometry, G. ELLINGSRUD et al Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLUCOTI Geometric group theory I. G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory ll, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, 0. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D.J.A. WELSH Surveys in combinatorics, 1993, K. WALKER (ed) Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY Polynomial invariants of finite groups, DJ. BENSON Finite geometry and combinatorics, F. DECLERCK et al Symplectic geometry. D. SALAMON (ed) Independent random variables and rearrangement invariant spaces, M. BRA YERMAN Arithmetic of blowup algebras, WOu.ffiR VASCONCELOS Microlocal analysis for differential operators, A. GRIGIS & J. SJOSTRAND Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI tt al The algebraic characterization of geometric 4-manifolds, J.A.lill..I.MAN Invariant potential theory in the unit ball of MANFRED STOLL The Grothendieck theory of dessins d'enfant. L. SCHNEPS (ed) Singularities. JEAN-PAUL BRASSELET (cd) The technique of pseudodifferential operators, H.O. CORDES Hochschild cohomology of von Neumann algebras, A. SINCLAIR &. R. SMlTii Combinatorial and geometric group theory, A.J. DUNCAN, N.D. Gll..BERT & J. HOWIE (eds)

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255 256 257 258 259 260 261 263 264 265 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287

Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds) Groups of Lie type and their geometries, W.M. KANTOR & L. DIMARTINO (eds) Vector bundles in algebraic geometry, N.J. HITCHIN. P. NEWSTEAD & W.M. OXBURY (eds) Arithmetic of diagonal hypersurfaces over finite fields, F.Q. GO~A & N. YUI Hilbert C*-modules, E.C. LANCE Groups 93 Galway I St Andrews I. C.M. CAMPBELL ~r al (eds) Groups 93 Galway I St Andrews II. C.M. CAMPBELL et al (eds) Generalised Euler-Jacobi invmion formula and asymptotics beyond all orders, V. KOW ALENKO et al Number theory 1992-93, S. DAVID (ed) Stochastic partial differential equations, A. ETHERIDGE (ed) Quadratic forms with applications to algebraic geometry and topology, A. PFISTER Surveys in combinatorics, 1995, PETER ROWLINSON (ed) Algebraic set theory. A. JOYAL &. I. MOERDUK Harmonic approximation, S.J. GARDINER Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds) Analytic semigroups and sem.ilinear initial boundary value problems, KAZUAKI TAIRA Computability. enumerability, unsolvability, S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds) A mathematical introduction to string theory, S. ALBEVERIO, J. JOST, S. PAYCHA, S. SCARLATII Novikov conjectures, index theorems and rigidity 1, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Novikov conjectures, index theorems and rigidity n. S. FERRY, A. RANICKI &. J. ROSENBERG (eds) Ergodic theory of zd actions. M. POLLICOTI & K. SCHMIDT (eds) Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN Semigroup theory and its applications. K.H. HOFMANN & M.W. MISLOVE (eds) The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS Finite fields and applications, S. COHEN & H. NIEDERREITER (eds) Introduction to subfactors, V. JONES & V .S. SUNDER Number theory 1993-94, S. DAVID (ed) The James forest. H. FElTER & B. GAMBOA DE BUEN Sieve methods, exponential sums, and their applications in number theory, G.R.H. GREAVES tt a/ Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TOOOROV (eds) Clifford algebras and spinors, P. LOUNESTO Stable groups, FRANK 0. WAGNER Surveys in combinatorics, 1997, R.A. BAILEY (ed) Geometric Galois actions I, L. SCHNEPS & P. LOCHAK (eds) Geometric Galois actions II, L. SCHNEPS & P. LOCHAK (eds) Model theory of groups and automorphism groups, D. EVANS (ed) Geometry, combinatorial designs and related structures, J.W.P. HIRSCHFELD et al p-Automorphisms of finite p-groups. E.I. KHUKHRO Analytic number theory. Y. MOTOHASHI (ed) Tame topology and a-minimal sbllctures, LOU VAN DEN DRIES The atlas of finite groups: ten years on, ROBERT CURTIS & ROBERT WILSON (eds) Characters and blocks of finite groups. G. NAVARRO Grt\bner bases and applications, B. BUCHBERGER & F. WINKLER (eds) Geometry and cohomology in group theory, P. KROPHOLLER. G. NIBLO, R. STOHR (eds) The q-Schur algebra, S. DONKIN Galois representations in arithmetic algebraic geometry, A.J. SCHOLL & R.L. TAYLOR (eds) Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds) Aspects of Galois theory. HELMUT VOLKLEIN tt al An introduction to noncommutative differential geometry and its physical applications 2ed, J. MAOORE Sets and proofs, S.B. COOPER & J. TRUSS (eels) Models and computability, S.B. COOPER & J. TRUSS (eds) Groups St Andrews 1997 in Bath, I, C.M. CAMPBELL tt a/ Groups St Andrews 1997 in Bath, II, C.M. CAMPBELL tt al Singularity theory, BILL BRUCE & DAVID MOND (eds) New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds) Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI &. N. SMART Surveys in combinatorics. 1999, J.D. LAMB & D.A. PREECE (eds) Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJOSTRAND Ergodic theory and topological dynamfcs. M.B. BEKKA & M. MAYER Analysis on Lie Groups, N.T. VAROPOULOS & S. MUST APHA Singular perturbations of differential operators, S. ALBEVERJO &. P. KURASOV Character theory for the odd order function, T. PETERFALVI Spectral theory and geometry, E.B. DAVIES&. Y. SAFAROV (eds) The Mandelbrot set, theme and variations, TAN LEI (ed) Computatoinal and geometric aspects of modem algebra, M.D. ATKINSON eta/ (eds) Singularities of plane curves, E. CASAS-AL VERO Descriptive set theory and dynamical systems, M. FOREMAN et ai (eds) Global attractors in abstract parabolic problems, J.W. CHOLEWA&. T. DLOTKO Topics in symbolic dynamics and applications, F. BLANCHARD. A. MAASS &t A. NOGUEIRA (eds) Characters and Automorphism Groups of Compact Riemann Surfaces, T. BREUER Explicit birational geometry of 3-fold.s, ALESSIO CORTI & MaES REID (eds) Auslander-Buchweitz approximatioos of equivariant modules, M. HASHIM:OTO Nonlinear elasticity, R. OGDEN & Y. FU (eds) Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SUU (eds) Rational Points on Curves over Finite Fields: Theory and Applications, H. NIEDERREITER &. C. XING Clifford Algebras and spinors 2nd edn, P. LOUNESTO Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE. A. F. COSTA &t E. MARTINEZ (eds)

London Mathematical Society Lecture Note Series. 288

Surveys in Combinatorics, 2001

Edited by

J. W. P. Hirschfeld University of Sussex

CAMBlliDGE •

UNIVERSITY PRESS

PUBUSHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Stree~ Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge, CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, VIC 3166, Australia Ruiz de Alarc6n 13,28014 Madrid, Spain, Dock House, The Waterfront, Cape Town 8001, South Africa http1/www .cambridge.org @ Cambridge

University Press 2001

This book is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written pennission of Cambridge University Press. First published 2001 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this book is available from the British Library

ISBN 0 521 00270 2 paperback

This book is dedicated to Crispin Nash-Williams 19 December 1932 - 20 January 2001

Contents

•

Preface

lX-X

J. Sheehan 1-9

Crispin Nash- Williams

M. Aigner The Penrose polynomial of graphs and matroids

11-46

I. Anderson Some cyclic and 1-rotational designs

47-73

A.R. Calderbank and A.F. Naguib Orthogonal designs and third generation wireless communication

75-107

L.A. Goldberg Computation in pennutation groups: counting and randomly sampling orbits

109-143

B. Mohar

Graph minors and graphs on surfaces

145-163

M.S.O. Molloy

Thresholds for colourability and satisfiability in random graphs and boolean formulae

165-197

J.G. Oxley On the interplay between graphs and matroids

199-239

J.A. Thas Ovoids, spreads and m-systems of finite classical polar spaces

241-267

D. R. Woodall List colourings of graphs

269-301

••

Vll

Preface

On the occasion of the 18th British Combinatorial Conference at the University of Sussex, 1 to 6 July, 2001, this book comprises the survey papers by the nine invited speakers and a memoire of Crispin Nash-Williams, past chairman of the British Combinatorial Committee. The survey papers range across many parts of modern combinatorics. Martin Aigner discusses the ideas of Penrose on the 4-colour problem, as well as the application of Penrose polynomials to other combinatorial structures.

Ian Anderson surveys some of the key ideas in the study of cyclic designs, including some of the classical results of the past 150 years as well as some very recent developments. Robert Calderbank and Ayman Naguib show the connection between the practice of wireless communication with the mathematics of quadratic forms developed by Radon and Hurwitz about a hundred years ago. This occurs through orthogonal designs, known as space-time block codes in the communications literature. Leslie Goldberg surveys the computational problems of randomly sampling unlabelled combinatorial structures, and of counting and approximately counting unlabelled structures. Bojan Mohar considers the interplay between graph minors and graphs embedded in surfaces. Michael Molloy surveys the progress on two fundamental problems in random graphs and random boolean formulae. The first is the question of how many edges must be added to a random graph until it is not almost surely k-colourable. James Oxley considers aspects of the interplay between graphs and mar troids, and shows the fruitfulness for both fields of applying results from the other.

•

IX

Preface

X

Joseph Thas considers the geometrical structures fundamental to finite simple groups, namely finite classical polar spaces, and the properties of their

substructures. Douglas Woodall discusses two problems of graph theory associated to colourings of a graph in which each vertex receives a colour from a prescribed list of colours. The conference is grateful for the support of the London Mathematical Society, the Institute of Combinatorics and its Applications, Hewlett Packard,

and AT&T. James Hirschfeld University of Sussex 4 March 2001

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t h e

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[4]:

t h o u g h t s

s t e r e o t y p i c a l G e n t l e m a n

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m e a s u r e

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r e m a r k e d

[1]:

o f

o n

s e e m

p h e n o m e n o n

A l v a h ,

s t a n d i n g ,

grill

Y o u

r e t i r e m e n t ,

F u n .

g r e a t

t h i n k

w a s

n e a r i n g

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t h e s e

is

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i n

a l s o

t h a t

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D e s c a r t e s

I

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g r a p h

H u m a n i t y

M a t h e m a t i c s

h i s

o f

T h e o r y

M a t h e m a t i c s ?

t h a t

l o g i c .

I

m i n o r

a m o u n t

a p p r e c i a t e d ;

W h a t

t o

a

a l o n e . P r o f e s s o r

A s

b e i n g

C o m b i n a t o r i a l

N a s h - W i l l i a m s .

n o t

3

W h i t e

S t - J a n d

A l v a h

n o b l e

o f

s p i r i t . O u r a n d

C o m b i n a t o r i a l

p r i d e

t h a t

h e

t o o k

C o m m u n i t y s u c h

a

w i l l

d e l i g h t

i n

a l w a y s o u r

r e m e m b e r

s u b j e c t

a n d

h i m

o u r

w i t h

a f f e c t i o n

c o m p a n y .

4

J o h n

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T h e

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Y o r k

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Conf.,

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circuits

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Applications,

Combinatorial

B a l a t o n f u r e d ,

Combinatorics,

C a l g a r y

sets,

Combinatorial

I n t e r n a t i o n a l

s t r u c C o n f e r e n c e ,

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w h o s e and

N o r t h - H o l l a n d ,

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large

va-

Colloq.,

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6

J o h n

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N e w

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31.

arcs

a n d

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=