Deletion-Contraction Techniques for the Chromatic Symmetric Function of a Graph

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D e le tio n -C o n tr a c tio n T e c h n iq u e s for t h e C h r o m a tic S y m m e tr ic F u n c tio n o f a G r a p h

By D avid D. Gebhard

A Dissertation Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY

Department of Mathematics

1998

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UMI Number:

9909305

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R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

ABSTRACT D e le tio n -C o n tr a c tio n T e c h n iq u e s for t h e C h r o m a tic S y m m e tr ic F u n c tio n o f a G r a p h By David D. Gebhard

Recently. R. P. S tanley defined and stu d ie d a sym m etric function. A+■ which generalizes th e ch ro m atic polynom ial of a g ra p h . G. T h is generalization has b o th ad­ vantages and d isadvantages. The m ain ad v an tag e is th a t it gives us m ore inform ation about th e colorings of G th an the chrom atic p o ly n o m ial. However, one disadvantage is th a t this new sy m m etric function does not sa tisfy a d eletio n -co n tractio n recurrence sim ilar to th e one for th e chrom atic polynom ial. In this th esis, we define a sim ilar g raph in v a ria n t called Yc - T h is invariant is defined using n o n com m utativ e variables, and from it we can recover A'c by allowing the variables to com m ute. This new invariant is also a sym m etric function.

More

im portantly, by using noncom m utative variables we will be able to o b ta in a deletioncontraction recu rren ce for Yc- We m ay then o b ta in som e of S tan ley 's resu lts for A'c in a uniform m a n n e r by using induction. In a d d itio n , th is will allow us to m ake some progress on th e 3 + 1 C onjecture of Stanley a n d Stem bridge.

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

ACKNOWLEDGMENTS

I would like to th a n k my advisor. B ruce Sagan, for ju st ab o u t ev ery th in g you can im agine. T h e patience he showed as he went over the first d raft o f th is thesis was nothing sh o rt of rem arkable. His help an d su p p o rt on my way to finishing this work were w onderful, and for th a t I give him my thanks. T h e way he went over this m anuscript w ith a fine-toothed com b an d m ade it a readable d o cu m en t was impressive, and for th a t you should give him thanks! I would also like to thank M ark M cC orm ick for all his help w ith my BTjrX files, for his friendship a n d for helping to keep m e in shape. Similarly. I would like to th an k Kevin a n d M elissa Dennis for th eir frien d sh ip , th e ir food, and fun and gam es. Finally. I need to th a n k my family, especially my parents and my b ro th e r for their love, su p p o rt, a n d encouragem ent th ro u g h this (to them ) seem ingly endless journey towards my P h.D .

iii

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

TA BLE OF C O N T E N T S

IN T R O D U C T IO N

1

1

7

2

3

4

5

P re lim in a rie s 1 .1

S ym m etric F u n c t i o n s ...........................................................................................

7

1.2

T he C h ro m a tic Sym m etric Function. A ' c .....................................................

10

T h e N o n c o m m u ta tiv e C a se

15

2 .1

S ym m etric Functions in N oncom m uting V a r i a b l e s ...................................

15

2.2

D evelopm ent and R esults for Yq

20

...................................................................

O r i e n t a t i o n s a n d S in k s

29

3.1

Acyclic O rien tatio n s

...........................................................................................

29

3.2

T he M odified B lass-Sagan A l g o r i t h m ............................................................

34

R e s u lts o n e - p o s i t i v i t y

41

4.1

Inducing e „ ............................................................................................................

41

4.2

Some e -p o sitiv ity R e s u lts ....................................................................................

48

4.3

T he (3-F1 )-free C o n je c tu r e ................................................................................

56

O p e n P r o b l e m s a n d C o n je c t u r e s

67

5.1

P a rtitio n in g Acyclic O r i e n t a t i o n s ..................................................................

67

5.2

A'g an d T r e e s .........................................................................................................

70

B IB L IO G R A P H Y

74

iv

R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

INTRODUCTION As early as 1912. W h itn ey [19] b egan to stu d y g ra p h colorings from a m ath em atical point of view. Today th e th e o ry o f graph coloring h a s m an y applications to b o th scheduling problem s an d efficient netw ork design. [11]. H ere we will use sym m etric functions to enum erate g ra p h colorings.

W hile th is sectio n co n tain s much o f the

background leading up to o u r stu d y , we will try to in tro d u c e n o ta tio n and definitions as they are needed th ro u g h o u t th e te x t, ra th e r th a n all a t once. We will generally follow Stanley [15. 14] for co m b in a to ria l n o tatio n o r a n y th in g specifically related to the sym m etric function of a g ra p h . A'c- and M acD o n ald [10] for sym m etric functions in general. To begin, let G be a finite g ra p h w ith vertex set V ( G ) a n d edge set E( G) . where th e edges consist of u n o rd ered p a irs of th e vertices. W e m en tio n here th a t if the edge set consisted of ordered, pairs o f vertices we w ould have h ad a g ra p h w ith directed edges, referred to as a digraph. A i'i-u n walk in a g ra p h is a sequence of vertices. t’i. V2 , . . . . vn such th a t Vi-iVi is an edge for all

2

< i < n. A g ra p h , G. is connected

if there is a u ,v walk for every p a ir of vertices, u a n d v in T (G ).

The connected

components of G are ju s t th e m ax im al connected su b g ra p h s o f G . Finally. H is a spanning subgraph of G if V ( H ) = \ ’(G) and E { H ) C E ( G ) . In o ur study we will actually consider m ultigraphs, in w hich m ultiple edges a n d loops are allowed. T he oth er definitions above e x te n d in th e n a tu ra l way to m u ltig ra p h s. Since our m ain in terest h ere is in coloring g rap h s, we define a coloring of G to be

1

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

2

F igure

1 A V

2

t ’l

V-2

1

3 W o a

a

t'l

Figure

a W

w

t'3

. A coloring (not p ro p er) o f P 3.

2 W Vo

1 A w

2

2

1 A V

t-h

2 W I’’)

1 • 1'3

. Two proper colorings o f P 3

a m ap a : I"(G ) — > C . w here C is the color set. In p a rtic u la r, a proper coloring o f G is a coloring such th a t no two adjacent vertices are th e sam e color, i.e.. a ( i \ ) # o (c j) if I'iVj is an edge of th e graph. For an exam ple, we show a coloring for the p a th on three vertices. P 3. w hich is not a proper coloring in F ig u re

1

an d two pro p er colorings

for P 3 in Figure 2. W hitney's o b je ct of stu d y was the chrom atic p o lyn o m ia l of a graph. X G{n). which is defined to be th e n u m b er of ways to properly color G using the color set C = {1.2

n}

[n]. For P3. since there are n ways to color t’i from a set of n colors.

and n — 1 ways to color each of th e rem aining vertices, we see th a t Xp 3 (n) = n ( n — l ) 2. It is som ew hat su rp risin g th a t X G(n) is always going to be a polynom ial in n. O ne easy way to see th is is to use induction along w ith th e D eletion-C ontraction L em m a. which we will now discuss. Given a g rap h G a n d an edge e € E( G) . we can define th e graph G — e to b e th e

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

3 g ra p h G w ith th e edge e deleted from its edge set. T he contraction o f G by e. G / e . is o b ta in e d from G by contracting e (in th e topological sense) to a single vertex. Given these definitions, th e D eletion-C ontraction L em m a states th a t

X c ( n ) = XG- e( n) ~ XG/ e i n).

T his gives us a recursive way to co m p u te th e ch ro m atic polynom ial of a g ra p h , as well as to estab lish various properties of X c ( n ) by induction. Two of W h itn e y 's results th a t can be proven using this m eth o d are s ta te d here. T h e o re m

1

[19] For a fin ite graph, G.

Xc ( n ) =

( - l ) i5i” C(5)S C E {G )

where c(S ) is the number o f connected com ponents o f the spanning subgraph o f G with edge set S , which by abuse o f notation we ju s t denote by S .

■

As an illu stra tio n , we will use this th eo rem to again calculate X Pz(n). If we let the edge set of P 3 be { e i,e 2}, where e x = ui i ’2 a n d e 2 = t’2 f.’3. then we can m ake the following table.

„ c ( 5)

S C E(G)

( - 1 ) |S|

0

1

n3

ei

- 1

n2

e2

- 1

n~

1

nl

e\. e 2

T h is shows us th a t according to th e T h eo rem . Xp 3 (n) = n 3 — 2nr + n = n( n — l ) 2. w hich agrees w ith our previous calcu latio n .

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

4

T he o th e r theorem of W h itn ey 's in w hich we will be in terested is th e one known as the B roken C ircuit Theorem. A cycle o r circuit is a closed walk w ith d istin ct vertices and edges. v i , c 2, . . . . v m, c l? for m >

1

. If we fix a to ta l o rd er on E ( G ) . a broken

circuit is a circuit w ith its largest edge (w ith respect to th e to ta l order) removed. Let the broken circuit complex B e of G denote th e set of all S C E ( G ) which do not contain a broken circuit in our fixed ordering on the edges. T h e Broken Circuit Theorem th e n asserts: T h e o r e m 2 [19] For any fin ite graph. G . on d vertices we have

Xc(n) =

( - 1

[ S' nd- SK

S€BC

■ If we ag ain calcu late Xp f i n ) using th is theorem , we will com e out w ith exactly w hat we h a d before, only w ith n 3 a n d n l reversing positions in th e table, since P 3 contains no circuits and hence no broken circuits. As a less triv ia l exam ple, we will use this th eo rem to verify th a t the ch ro m atic polynom ial for A 3. th e com plete graph on 3 vertices is indeed given by n( n — l ) ( n — 2). which can be o b ta in e d by noticing th a t th ere are n ways to color the first v ertex, n — 1 colors left available for the second vertex, a n d n — 2 colors allowed for th e last vertex. We label E ( h \ ) = {e 1 .e 2 .e 3 }. where the fixed o rd er on the edges is th e obvious one induced by th e subscripts. Since th e only circuit in K 3 is { e i.e 2 , e 3}, the only broken c irc u it will be { e ^ e , } . This gives us th e following table, w here we notice th a t here d = 3.

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

(- I)'* !

n d-\s\

1

n3

ei

- 1

n2

e2

- 1

n2

e3

- 1

n2

e i,e 3

1

nl

C‘2 ? ^3

1

nx

s e bg 0

T his gives us X ^

3

= n 3 — 3 n 2 + 2n . which again agrees w ith th e previous calculation.

Following these e a rly resu lts, som e of the m ore in te re stin g applications are those of Zaslavsky in [20. 21.

2 2

],

In th a t series of p ap ers he in tro d u ces the notion of

colorings for certain g e n e ra liz a tio n s o f graphs called sig n ed g rap h s. These colorings have very nice conn ectio n s to ch aracteristic polynom ials o f c e rta in types of hvperplane arrangem ents.

A re la te d resu lt by Zaslavsky and G reen e [7] concerns the sinks of

acyclic o rien tatio n s for G . A n orientation of G is a d ig ra p h D o b tain ed by assigning a unique direction to each edge o f G. An o rien tatio n is acyclic if it has no directed cycles. VVe also define a s in k o f D to be a vertex v € V { D ) such th a t

£ B ( D ) for

all x £ V ( D ) . Also, for n o ta tio n a l convenience we a d o p t th e convention th a t

X c (n) = Go -F a \n + aon~ -F • • • + a ^ n ^ .

T h e o r e m 3 ([7] T heorem 7.3) Let vq be any vertex o f G .

The num ber o f acyclic

orientations o f G w ith a unique sin k at t'o is |a t |.

■

T his theorem is re la te d to one o f Stanley, which sta te s: T h e o r e m 4 [13] The n u m b e r o f acyclic orientations o f G is

|a t |.

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■

6

All of these theorem s axe a c tu a lly specializations o f resu lts which can be o b ta in e d from S tan ley 's sym m etric fu n ctio n generalization o f th e ch ro m atic polynom ial. T h e first three theorem s listed previously can all easily be derived from th e recurrence relation for th e chrom atic polynom ial. However, th is sy m m etric function does not satisfy any sim ilar deletio n -co n tractio n recursion, w hich elim inates induction as a tool for these proofs.

In w h a t follows we will ex ten d th e Stanley's definition by

using sym m etric functions in noncom m utative variables. T h is settin g will allow us to establish a recurrence and ag ain allow induction as a valid approach to o ur proofs.

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CHAPTER 1 Preliminaries 1.1

S y m m e tr ic F u n ctio n s

Here we will review th e basic facts about sym m etric fu n ctio n s in com m uting variables. Our developm ent will closely m irror th a t found in S a g a n 's book [12]. The in terested reader should consult eith e r M acDonald [10] or Sagan [12] for a more com prehensive discussion. We will begin w ith th e m onomial sym m etric functions. Let x = {xi.x- 2 . x ^ . . . . } be a countably infinite set of com m utative variables, a n d let A = (A[.A_>.... .A*) be an integer p a rtitio n of n. denoted A I- n. where th e A, form a weakly decreasing = n - ^ we allow r, to be the nu m b er of

sequence of positive integers such th at 5 Zf=i

parts of A equal to i. th e n we may also express A = ( l r i . 2r2. . . . . nTn) as an a lte rn a te notation. T he m o n o m ia l sym m etric fun ctio n corresponding to A is given by

m xA =

«2

' • • -ci>Jfck ■

where th e sum is over all distinct monomials having ex p o n en ts Ai

i

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

Afc. As an

8 exam ple we can see th a t

^ ( 2 , 1 ) — X\X 2

X^X$ + • • • + x \ x X + X->X3 + • • • -+- XjX^ + • ■•

.

We then define th e ring o f sym m e tric fu n c tio n s as the vector space over C spanned by the m onom ial sym m etric functions. It is an elem entary fact th a t th e m onom ial sym m etric fu nctions are actually lin early independent over C a n d so form a basis for the vector space of sym m etric functions. It is im p o rtan t to note here th a t while we will usually consider th e sym m etric fu n ctio n s as a vector space in th is thesis, the fact th a t they form a closed set under m u ltip licatio n also makes the sy m m etric functions a ring. W hile it is clear from our developm ent th a t the m onom ial sy m m etric functions form a s ta n d a rd basis for the sy m m etric functions, there are o th e r nice bases for this vector space which are ro u tin ely used.

These include th e elem entary, power

sum . and com plete hom ogeneous sy m m etric functions as well as th e Schur functions. We will define th e power sum sy m m etric functions and th e elem en tary sym m etric functions here, as th ey will be relevant to th e rest of this thesis. For a description of the com plete hom ogeneous sym m etric functions and the Schur functions, please see either [1 0 ]. [1 2 ], or [4] The r th pow er su m sym m etric fu n c tio n is

t>i and th e r th elem entary sym m etric fu n c tio n is

i i < —< i r

W hile these a re seem ingly n atu ral definitions based on the m onom ial sy m m etric func-

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

9 tions, they obviously do not form bases since th ey are n o t even indexed by integer p a rtitio n s. Hence we m u st extend these definitions to p \ an d e x where A f- n. We will do this m ultiplicativelv. by defining

P(xux2

These are th e

def Afc) — P X i P x - i ' ' ' P x k

pow er su m

an d

,

def a*) =

a,i d

■- e \ k -

elem e n ta ry sy m m e tric fu n c tio n s,

respectively. T hey

also form bases for th e space o f sym m etric functions. To illu s tra te these definitions, two sim ple exam ples are com p u ted . For the integer p a rtitio n (2 .1 ) we have

P(

2.1)= P 2P \

—

(•£? +

+ ' ’' )(xi + x2 + • • •)

= X^ + X^ + • • • + X J X 2 d” X ^ X 1 4" ’ ‘ ‘

= ni(3) + m (2.1)

and

e{2 .i)

= e2^l

=

(-^lJ-2 d~ -t'iJ .3 +

•

• • ~r X2 X3

-r

• • • ) ( x t + X2 ■+■X3 -+-•••)

= XJX -2 + xi]Xi + • • • + 3x 1 X2 X3 + • - • = m (2. \) +

These functions are referred to as sym m etric functions for th e following reason. Given a p e rm u ta tio n in th e sym m etric group on n elem ents.

8

E «5„. we define a

n a tu ra l action on th e set of functions in C[x] by

S f ( X i . X 2 . X 3. . . . ) = / ( X ^ ! ) . X,j(2). X,J(3). . . . ).

It should be clear th a t, for any p e rm u ta tio n

8

. th e elem ents in o u r space of sym m etric

functions will rem ain in v arian t u nder this action. G iven this in fo rm atio n a b o u t th e space of sy m m etric functions, we now have

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

10

sufficient background to introduce S ta n le y ’s ch ro m atic sym m etric function.

1.2

T h e C h r o m a tic S y m m e tr ic F u n c tio n , X

q

In "A Sym m etric F unction G eneralization of th e C hrom atic Polynom ial of a G ra p h ” [15] (see also [16]). R. P. Stanley in tro d u ce d a sym m etric function. A g - which generalizes the chrom atic polynom ial asso ciated w ith a labeled graph on d vertices. D e f in itio n 1 .2 .1 Let G have vertex set \ ' ( G ) = {cl- vo

AG

A f;(X j . X-2 . .

. . ) — ^

] X K( ^ j . . . -f K(

q } . We define

i.

K where the sum ranges over all proper colorings, k : \ ’{G) —> P. and P is the set of positive integers. N ote th a t X q is hom ogeneous of degree d = |l'( G ) |, where | * j denotes cardinality. We also notice th a t if G has loops this su m is em pty, giving A'c = 0. To illu stra te this definition, we will com pu te the ch ro m atic sy m m etric function for our sta n d a rd exam ple of the p a th on th ree vertices. P 3 . We can see th a t any proper coloring of this graph will have one of two possible types: th e coloring could have uy and

1*3

one color

w ith vo a different color, or it could have all th ree vertices different colors. Since there are

6

different ways to color the three vertices w ith the sam e set of three different

colors, we obtain

X p 3 =

x \ x 2

+

x \ x i

+

• • • +

6 X 1 X 2 X 3

+

6 X \ X 2X A +

■■ ■

It should be clear from th e definition th a t X q is a sym m etric function, since any p e rm u ta tio n of th e su b scrip ts sim ply p e rm u te s th e colors and doesn’t affect th e set of colorings. We can also see it more explicitly in th is case, since the previous expression

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

11 clearly shows th a t X p 3 = rri( 2 , i )

For A

+ 6 m ( u . 1) .

n having r, parts of size i. we can also use the n o ta tio n A = . n rn). a n d define |A| = rj.! - *- r„!.

( l r'. 2 r 2

We say th a t a p a rtitio n of the ver­

tex set of G is stable if no block of the p a rtitio n contains adjacent vertices. Then it is not h ard to see [15] th a t X g — £[]AaA|A|mAT w here a \ is the n u m b er of stable p artitions w hose block sizes correspond ex actly to th e p a rts of A. We are also easily able to see th a t for th e disjoint union of two g ra p h s. G = H\±)I. we have X g = X p X [ . We can verify th a t th is sym m etric function is a generalization of th e chrom atic polynom ial. X c (n). since settin g j j = x 2 = .. ■ = x n = 1 and x t = 0 for all i > n in A'c. denote by A 'c fl" ). yields Xg{ti). To see this, n o te th a t th is s u b stitu tio n will produce a te rm equal to

1

for each m onom ial in A'c which comes from a proper

coloring of th e g ra p h using the first n colors, an d a term equal to zero for each monom ial arising from a proper coloring which uses a color not in [n], Hence the sum of all these m onom ial term s after this s u b stitu tio n will ju s t be the nu m b er of proper colorings of G w hich only use the first n colors. T his is precisely Xc { n ) . Once we are assured th a t this is a g en eralizatio n of the ch ro m atic polynom ial, one might expect th a t previous results a b o u t th e chrom atic polynom ial should also generalize.

It is also n a tu ra l to study the ex p an sio n of this ch ro m atic sym m etric

function in term s of th e different sym m etric fu n ctio n bases. T h e calcu latio n of X G for various specific grap h s is also of interest.

S tan ley pursues all of these lines of

inquiry in his p ap er. Several of S ta n le y ’s results for A'c are extensions o f W h itn ey 's [19] theorem s for the chrom atic polynom ial. For example. S ta n le y ’s sym m etric function extension of Theorem

1

utilizes th e power sum sym m etric functions.

R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

12 T h e o r e m 1 .2 .2 [15, Theorem 2.5] Let G be a fin ite graph o f order d. We have

Xg — ^

( ~ l ) |5 |PA(5)r

SCE(G )

where A(S) = (A j.A j, . . . . A*) is the integer partition o f d w ith A, being the number o f vertices in the ith com ponent o f S .

■

We can see th a t this resu lt d irectly implies W h itn e y 's first th eo rem by noticing th a t p r( 1 ") = n for any r. a n d so p ,\(ln ) = n l{X). where l{A) is th e n u m b er of p arts of A. Hence pA(s)(ln ) = n c{SK com pleting th e reduction. Not surprisingly. S ta n le y also has a generalization o f T h e o re m 2. T h e o r e m 1 .2 .3 [15, Theorem 2.9] For any fin ite graph G . we have

Xg =

m

seBc

In this thesis, we will be stu d y in g an analogue of S ta n le y 's ch ro m atic sym m et­ ric function X c - called Y c , w hich is defined using n o n co m m u ta tiv e variables. We wish to consider this analo g u e because we know th a t m any re su lts for th e chrom atic polynom ial can be proven easily using induction an d th e d e letio n -c o n tra ctio n recur­ rence. U nfortunately. S ta n le y 's sym m etric function has no su ch deletion-contraction property, which deprives him of in d u ctio n

as a tool for his proofs.

To see where th e problem lies, note th a t X g is hom ogeneous of degree d. while X G/e is homogeneous of degree d — 1 . In order to find a recu rren ce, we would need to a d d another variable to each m onom ial in X c / e■ B u t which v ariab le? In th e proof of th e deletion-contraction ru le for th e chrom atic polynom ial, we have p ro p er colorings of G [ e corresponding to colorings o f G — e w ith u an d v th e sam e color, where e = uv. However, while X c / e gives us m ore inform ation a b o u t th e colorings of G / e th an the chrom atic polynom ial, it does not give us th e explicit in fo rm atio n we need to fix the

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

13 problem : namely, w hat color was assigned to th e v ertex o b ta in e d bv co n tractin g uv. We lost th a t inform ation w hen we allowed the variab les to com m ute. To correct this difficulty, in C h a p te r 2 we in tro d u ce an analogue of A'c which is a sym m etric function in non co m m u ta tive variables. T h a t is. for any m u ltigraph G w ith vertices labeled tq. u2. . . . . t'd in a fixed order we define th e analogue of X G as

K w here again the sum is over all proper colorings of G . b u t th e x t are now no ncom ­ m uting variables. T h e reason for using no n co m m u tin g variables is so th a t we can keep track o f th e color which k assigns to each vertex. Since we still have th e hom ogeneity problem in YG/e. we define an o p e ra tio n on th e non -co m m u tativ e sy m m etric functions which will allow us to use d eletio n -co n tractio n techniques for co m p u tin g YG■ In this c h ap te r we also provide som e basic expansions of Yc which closely resem ble W h itn ey ’s and S tan ley 's theorem s. In C h ap ter 3 we will fu rth e r explore th e in te rre la tio n sh ip s between ch ro m atic polynom ials, chrom atic sy m m etric functions, acyclic o rie n ta tio n s and sinks. T here is an interesting connection betw een Theorem 2 and th e resu lt of G reen and Zaslavsky. T heorem 3. From T heorem 2 we can interpret th e coefficients of th e chrom atic poly­ nom ial as the num ber of sets 5 € B G of a certain size. From T heorem 3. we see th e coefficient of n in X G[n) is th e num ber of acyclic o rie n ta tio n s of G w ith a unique sink a t any fixed vertex o f G. It follows th a t th e n u m b e r of acyclic o rien tatio n s of G w ith a unique sink at th e fixed vertex is the sam e as th e num ber of sets S € B G w ith |S | = d — 1 . E lem en tary g rap h theory tells us th a t th is is also be th e num ber of spanning trees of G which co n tain no broken circu its. We will provide a bijective pro o f of this fact by m odifying an algorithm due to B lass an d Sagan [1 ], Finally.

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

14 we will also e x tra c t som e of th e inform ation th a t th e non-com m utative chrom atic sym m etric function can give us about acyclic o rie n ta tio n s an d sinks. In C h a p te r 4 we will consider a conjecture ab o u t th e coefficients of X q , when it is expanded in term s o f th e elem entary' sym m etric fu n ctio n basis. We will make some progress here on th e (3 -F l)-fre e conjecture o f S tan ley a n d Stem bridge. proving it in some special cases. We finish in C h a p te r 5 w ith some other p a rtia l re su lts a b o u t acyclic orientations and sinks, as they re la te to the (3 4- l)-free co n jectu re.

We conclude w ith some

open problem s, as well as a few ideas on how th ey m ight be approached using our techniques. Before we begin, however, we will need to discuss sym m etric functions in noncom m uting variables and o u r analogue of A'c in th a t settin g . This is the focus of our next ch ap ter.

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

CHAPTER 2 The Noncommutative Case 2.1

S y m m e tr ic F u n c tio n s in N o n c o m m u tin g V ari­ a b les

We begin w ith some background on sy m m etric functions in noncom m uting variables. Much of this follows from the work of D oubilet [4]. although he does not explicitly m ention these functions in his work. T hese n o ncom m utative sym m etric functions will be indexed by set p artitions, which form a la ttic e u n d er a certain p a rtia l o rd er, which we will define here. A lattice is a poset (partially ordered set) C such th a t every p air x . y € C has a least upper bound (or join) denoted by x V y an d a g reatest lower b ound (or m eet) denoted x A y. Any finite lattice has a unique m inim al elem ent den o ted by 0 and a unique m axim al elem ent denoted by L

We let Elj

denote the set p a rtitio n s of

{ 1 .2 .... . d} = [d], T his forms the set p a rtitio n lattice, where the p a rtia l o rd er is defined as follows. If a = A \ / A 2/ • • • / A k a n d r = B i / B 2/ ■■■/ B m. th en a < r if and only if for all 1 < i < k there exists som e j w ith 1 < j < rn such th a t .4, C B } . T h a t this partial order on IT* actu ally form s a la ttic e is an elem entary resu lt. As an exam ple, we have included the Hasse d ia g ra m for

in Figure 2.1. G iven a poset.

15

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

16

i = 1234

123/4 « ^ T 2 4 / 3 « C l 2 / 3 4 ^ C l 4 / 2 3

1 3 /2 4 ^ * 1 3 4 / 2 1 / 2 3 4

1 2 /3 /4 < < 1 3 /‘2 / - h < T 4 / 2 / 3

1 / 2 3 / 4 ^ 1 / 2 4 / j p * 1 /2 /3 4

0 = 1 /2 /3 /4 F igure 2.1. The p a rtitio n la ttic e

.

P. we also recursively define th e Mobius fu n c tio n . //. of P on intervals [x. y\ in P by

fi(x. x ) =

1

an d

/2 ( x . y ) = —

/i(x. z) for all x. y € P . T ri^ e } would c o n ta in a circuit of G. So S '(J { e } w ould contain a broken circu it of G. since e < em . T h is contradicts 5 € Be- To c o n stru c t th e inverse, we sim ply m ap

-

I S

\

if S € B C-e

5(J{e} if S e B c / e -

It is clear th a t this is th e inverse, provided again th a t th e m ap is well-defined. An argum ent sim ilar to th e one given above shows th a t th is is indeed the case.

■

We can now o b ta in a characterizatio n of Yq in term s o f th e broken circuit com plex of G for any fixed to ta l ordering on the edges.

T h eo re m 2 .2 .7 We have

*c =

( - 1 )|5IP t(S). S€0c

where again ir(S) denotes the partition of { 1 , 2 . . . . . d ) with blocks corresponding to the connected components of S .

m

P ro o f. We again in d u ct on the num ber of non-loops in E ( G ). If the edge set consists only of n loops, it should be clear th a t for n > 0 we will have every edge

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

28

being a circuit, a n d so th e em pty set is a broken c irc u it. Thus we have

E s e o M P /M s ) = 0

if n > 0.

E se { « } ( “ 1)'[SlP * ( S )

if n = 0.

= Pi/2/.../d

T his again m atch es eq u atio n (2.6). For n > 0 a n d e a non-loop, we again co n sid er Y g = V c-e —Tc/eT- w here G —e and G /e both have one less edge th an G. an d so in d u c tio n applies. From the preceding lem m a and a rg u m e n ts as in Proposition 2.2.5. we have

s ~B 4.(G)| has at least one sink. P r o o f . W hile th is is a well-known g rap h th eo ry result, we prove it here for com plete­ ness. by way o f contradiction. C onsider th e finite set of d ire c ted walks in -4(G) given by

S = {*-’,[ —y

—> . . . —> L\k : vlt £ I (G) for 1 < / < £. a n d k < |I (G )| -I- 1}.

C learly S ^ o. since for any vertex u € G. the triv ial walk given by v will be an elem ent o f S . So we may consider a walk, IT, in S w ith m ax im u m k. If k = |T (G )|-t-l. th en I V co n tain s a cycle, c o n tra d ictin g D 6 A{ G ) . If k < |C (G )|. then we claim th a t i\k

w ill be a sink of

D.

If it is n o t a sink, th en there is a d ire c te d edge

e

=

v lk i f .

for som e w € E ( G ) . Adding this d irected edge to W will a g ain give us a walk in S. B ut th is c o n tra d icts our choice of IT . Hence v ik m ust be a sin k o f D , com pleting the proof. As a n im m ed iate corollary we have th e following result.

R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

■

31 C o r o lla r y 3 .1 .3 For an y D € |*4(G)|, the number o f sinks is greater than or equal to the number o f components o f G.

■

Using L em m a 3.1.1 a n d C orollary 3.1.3. we m ay now prove T heorem 3 by showing th at the b oundary condition s and recurrence relatio n s for |a i| and the nu m b er of acyclic orientations w ith a unique sink at a fixed vertex. c0. are th e same. T h e o r e m 3 .1 .4 [7] For any fixed vertex v0, the n u m be r o f acyclic orientations o f G with a unique sink at vq is |a x|.

P r o o f . We prove this theorem by showing th a t th e b o u n d a ry conditions an d recur­ rence relations for |a t | a n d \A{G. i’o)| both m atch. O u r recurrence will only be valid when there is a non-loop incident w ith t’o. and so o u r b o u n d a ry condition will occur in the case where only loops are incident w ith c0. If d = 1. th en

n

if G = K \ .

0

if G has loops.

x G{n) = ;

So in this case.

{1 0

if G = A'1: ^ = IA { G . f 0)|. if G has loops

If d > 1. then having only loops incident w ith t’o is eq u iv alen t to having a t least two com ponents in G. In this case we see from T h eo rem 1 th a t |a xj = 0 and from Corollary 3.1.3 th a t |.4 (G . i’o)| = 0 as well. T hus th e b o u n d a ry conditions m atch. Now if there is a non-loop e incident w ith t’o, we can see th a t |a t | is the sum of the absolute values of th e coefficients of n from X c ~ e an d X c / e since the signs on the coefficients of the ch ro m atic polynom ial a lte rn a te . H ence it follows from Lem m a 3.1.1 th a t the recurrence relatio n s are also the sam e a n d so th e theorem is proven. ■ Stanley has a stro n g er version of this result.

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

32 T h e o r e m 3 .1 .5 [15] I f X q = 53a c*e *- then the nu m be r o f acyclic orientations o f G with j sinks is given by £

cx .

m

l(X )= j

W e can prove an an alo g u e o f th is th is theorem in th e noncom m utative se ttin g by using techniques sim ila r to his. have not been ab le to do so using induction. We can inductively d e m o n stra te th e w eaker versions w hich follow.

T h e o r e m 3 .1 .6 Let Y q = ) ' c^ew. Then f o r a n y fixed vertex,

vq.

the nu m be r of

:rend

acyclic orientations o f G with a unique sink at vq is (d — 1) Icr^.

P r o o f . We again in d u ct on th e num ber of non-loops in G. In the base case, if all the edges of G are loops, th e n

I e i/ 2 /.../ u p a th into a walk containing a w —» t/o p a th in Dk. If th e arc a = wi, in D k -\ was d e leted for the second reason, ag ain we need only consider th e possibility th a t for th e vertex tv, there is no tv —*■ t'o p a th in Dk. But then th ere is no oriented arc wu' w ith a ^ u' . since otherw ise a ll u' —r cq paths must also use th e arc a. as there are no w

c0 p a th s in D k . T hus D k -

1

would have a cycle

contain in g w. C ontracting all u n o rien ted arcs from iv and re p e a tin g this argum ent as necessary, we see th a t w would th e n be a sink of c{Dk~\) — a, which contradicts our reason for deleting a. (d) Suppose for the sake of c o n tra d ictio n th a t the u n o rien ted p a rt of Dk contains a broken circu it. C —x, where x is th e g reatest element of th e cycle C . Since the un­ oriented p a rt of D k - 1 d id n 't contain an y broken circuits, and since th e only difference betw een D k - i and D k is at the kth. edge a. we see th a t a m ust be unoriented in Dk and th a t a € C — x. B ut then x is g re a te r th a n a. and so x is p resen t in Dk in one of its o rien tatio n s. B ut all the oth er edges in C are also present a n d unoriented. Hence. C forms a cycle in D k , contradicting th e previously verified fact th a t Dk is acyclic. ■

L e m m a 3 .2 .3 .4* is one-to-one.

P r o o f. Suppose D\ and D 2 are tw o d istin ct elem ents of D k -

1

which are both

m apped to D by th e algorithm . Since th e algorithm only affects th e k t h edge, we

R e p r o d u c e d w ith p e r m is s io n o f th e co p y rig h t o w n e r . F u rth er re p ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

39

note th a t D x and D 2 (an d consequently c ( D x) an d c ( D 2)) m u st only differ in th e arc a. W ith o u t loss o f generality, vve may assum e th a t a has an ab n o rm al o rien tatio n in D 2 an d a norm al o rie n ta tio n in D x. We note th a t D was not o b tain ed from D x a n d D 2 by th e deletion of th e arc a: we know th a t a could n o t have been deleted from e ith e r D x o r D2 for form ing a cycle, as th e o th er w ould then have contained th a t cycle. Also, if a was deleted by the algorithm from b o th D x an d D 2 for th e second reason, th e n a was abnorm ally oriented in both D x a n d D 2, which is not possible. If a was not deleted by th e algorithm , th en c ( D 2) — a has an ad d itio n al sink. So if a = xBi. is the arc in c ( D 2). th e n w

c0 since v0 is a sink, a n d so w m ust be the

additional sink in c ( D 2) — a . B ut this m eans th a t w was a lre ad y a sink in c ( D x). co ntradicting D x G V k - 1 -

L e m m a 3 .2 .4 Ak m aps V k -

P ro o f.

■

1

onto Vk-

Given Dk € V k we m ust construct D k - 1 € V k - i which m aps onto it.

Hence for any d ig rap h , Dk G V k . we m ust c o n stru ct a d ig ra p h D k - i and verify th a t the algorithm does indeed m ap D k - 1 onto D k . an d th a t D k - i satisfies properties (a)(d). For all of the following cases, it will be im m ed iate th a t th e D k -

1

we co n stru ct will

satisfy properties (a), (b). and (d). so we will only show th e verification of p ro p erty (c). Let a be the A:th edge of G. T here are two cases. T he first case is when a £ Dk- If there exists a u n iq u e o rien tatio n of a in which D k w ould rem ain acyclic, we give a th a t o rien tatio n in D k - 1 - If b o th o rien tatio n s of a would preserve th e acyclicity o f Dk, then we choose th e a b n o rm a l o rientation for a in D k - 1 - We note th a t a t least one of the o rie n ta tio n s o f a m ust preserve acyclicity, since otherw ise a com pletes two different cycles in D k - 1 - T hese two cycles to g eth er would contain a cycle in Dk, which is a co n trad ictio n .

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

40 T h a t the alg o rith m m aps th e digraph D k - i o b ta in e d in the previous p a ra g ra p h to Dk is obvious w hen only one orientation o f a p ro d u ces an acyclic o rien tatio n o f D k - iHowever, if b o th p ro d u ce acyclic orientations, we need to check th a t c(Dk~i ) — a has a unique sink a t

v 0 .

T h is is tru e , since it is easy to see th a t c (D k -i) —a = c(D k~i —a) =

c(D jt). To verify th a t c ( D k - i ) co n stru cted above still satisfies property (c). we note th a t adding an arc c an n o t destroy any existing p a th s. For the o th e r case, we suppose th a t a is p resen t in Dk as an u n o rien ted edge e. an d so neither o rie n ta tio n can produce a cycle in D k - i • We note th a t th ere m u st be a t least one o rie n ta tio n o f e = wu such th a t th e re rem ain s an x —> c0 p a th for every x G D k - i • If all x —> Vq p ath s p use the arc a = ur& for som e x. and if all y —►c0 p a th s q use a' = m l \ th e n th e x —> w p ortion o f P to g e th e r w ith the w —►v0 p o rtio n of Q contains an x —> v0 p a th avoiding a. which c o n tra d ic ts o u r assum ption a b o u t x. If there is a unique o rien tatio n of e = w u so th a t th e re rem ains an x —> v0 p a th for every x € D k - 1 we choose th a t one to m a in ta in p ro p e rty (c) for D k - 1 - say a = u7&. U sing the sam e arg u m e n t we used to prove th e second case o f (c) in L em m a 3.2.2. it is easy to verify th a t th e algorithm will take th e D k - i so constructed and m ap it to D k by unorienting a since c ( Dk - i ) — a has an a d d itio n a l sink at w . In the subcase w here e = uw is present in Dk as a n unoriented edge and we would still re ta in property (c) w ith eith er o rien tatio n o f e. we will consider the d ig ra p h D k - i o b ta in e d from D by giving e the norm al o rie n ta tio n , say a = 1F&. It is clear th a t the alg o rith m m aps D k - i to Dk- since Dk -

1

+ a' = Dk is acyclic and a has th e norm al

o rie n ta tio n .

R e p r o d u c e d w ith p e r m issio n o f th e c o p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

■

CHAPTER 4 Results on e-positivity 4.1

I n d u c in g

We now tu rn o u r a tte n tio n to the expansion o f Y g in term s of th e elem en tary sym ­ metric function basis. We recall th a t for any fixed " € I w e use

1) to denote

the p artitio n of [d + 1] form ed by inserting th e elem ent (d + 1) into th e block of rr which contains d. We will denote the block o f 7r which contains d by B T. We also let ~ / d + 1 be th e p a rtitio n of [d + 1] formed by ad d in g th e block {d + 1} to rr. In order o b ta in inform ation about the coefficients for the expansion o f Yc in non­ com m utative elem en tary sym m etric functions using o ur d eletion-contraction results, it is necessary for us to understand the coefficients arising in e* T- We have seen th a t the expression for e^-T is rath er com plicated (see equation (3.1)). However, if the term s in the expression of eTt are grouped properly, the coefficients in m any of the groups will su m to zero. To see th a t such a grouping should exist, we use the following lem m a.

L e m m a 4 .1 .1 Fo r rr € n ^ , let evt =

cTeT. Then

41

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F urth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

42

1 /|£ M

i f A (r) = A(tt/c/ + 1).

I ] cT = J ren^+i

_ i / | 5 rr|

,/A (r)= A (7 T + ( r f + l ) ) ,

A(r) fixed

0

else.

P r o o f . Let 7r = B \ / B 2/ . .. / B T/ . . . / B k G

n„.

We also let 6* d en o te the size of

the block B t .We m ay now consider the g ra p h s on d + 1 vertices given by

G tt =

K si

tu As, tri • • •(±f (Aht

+

e)

y •••y Asfc.

where K Bl is th e com plete graph on 6, vertices labeled w ith th e elem ents where K B, + e is the com plete g raph on of

of B ,.and

= b vertices lab eled w ith th e elements

which also contains an ad d itio n al vertex labeled d + 1. a n d an additional edge

from d to d -+- 1. By the recurrence relation for \ G. we can see th a t \ G, = l 'c , - e — ^G,/et- Equiv­ alently. this gives us th a t YG, / e

YCv- e ~ ^ c , - It is easy to see th a t Y'c , /e = eT. and

so this gives us th a t enf = YG, - e — I c T- If we let C be the o p e ra to r which allows the variables to com m ute, we get C (e TT) = A 'c^-e — A'G t. We now proceed to calculate X G, - e a n d X Gr. Since it is easy to see th a t G * —e = I \ Bl y A'e2 y • • • y K b , W K ^ + i} t±J - - - y K s k - we m ay use th e p ro d u c t rule for A'c to show th a t X G, - e = 7rte(A(;r),i)- To calculate A'Gt we again will use th e product rule, together w ith th e fact that

Arti-l

where a \ is th e num ber of stable p a rtitio n s o f th e vertex set o f K b , o f type A. Letting H = X ( B \ / B 2/ ■■• / B ^ j ■• • /B k ) , we can easily count the n u m b er o f stab le partitions of K b, to o b ta in the equation

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

43

X g* =

[(b - 1)(6 - l)!m (l6-i.2) + (b + l)!m (1fc+i)] .

We m ay th e n use th e fact th a t m (16-i 2) = e(6il) —(6 + l)e(6^i) an d

= e(6-i-i)-

com bining this w ith th e previous results to obtain:

C (e„f)

= Tr!e(A(ir).i) — M-eti

[(& — l ) ( b — 1 )!

(e(6.i) — ( b

+

1 )e (6—i ) ) +

(6

-F

l ) ! e (£,_uj .

Following sim plification, th is leads to

C { e i r t ) — --------- 7--------- e ( A ( T ) , l ) ------------------7------------- CA(T-r{d-rl))-

(- 1. 1)

cTeT. th e n

This essentially com pletes th e proof, as we know th a t if e - t = r €rij1.I

C (eA )=

cr C ( e r). r€

Hence we can see th a t

(

E *

r -nj-rl

\ A ( r ) fixed

/

is the coefficient of e\(T) in CE(e^T)- P u ttin g this to g eth er w ith eq u atio n (4.1) gives the result.

■

This lem m a show us th a t th e coefficients of various term s from e _ | can be com ­ bined in a nice way a n d even in d icates exactly how to do so. We need to sum to g e th e r th e coefficients from set p a rtitio n s which are of the sam e ty p e (as integer p a rtitio n s ), and whose block co n tain in g d have the sam e size.

If we do this, th en alm o st all

these am alg am ated coefficients will drop out. We need to know, however, if th ere is a p a tte rn to these co m b in atio n s which will allow us to rep eated ly use d eletio n -

R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

44 contraction techniques. We see th a t the c o n trib u tin g coefficients o f eTT will have type A(tt + ( d + 1)) o r A(7r/ d + 1 ) . If vve w ant to be able to re p e at th is process, though, it will be necessary to know the size o f th e block of eTf which co n tain s d -f 1. YY'e want all those term s o f e ^ f w hich do not have ty p e A(7t -}- { d + 1)) or A(tt/ d + i). and whose block containing d + 1 is not the sam e size as th e block co n tain in g d + 1 in rr 4- (d + 1) or rr/d 4- 1 to d ro p o u t. YY’ith one m ore b it o f n o ta tio n , we m ay show this is indeed the way th e coefficients will behave. Let P ( a ) = P ( a l : a 2

a*) be the set of all

partitio n s of [d 4- 1] w hich are less th a n o r equal to tc 4- (d 4- 1). have blocks of size Qi, a 2, . . . . Qj. a n d for which d + 1 is in a block of size a^. T he proper grouping of the term s of e , t is given by the following lem m a.

L e m m a 4 .1 .2 I f e* f

=

^ cTe T. then cr = 0 unless r < ~ + (d 4- 1). and fo r ren 1. Finally let J denote the p a rtitio n o b ta in e d from r by m erging the blocks of r

r .

which contain d and d + 1. allowing J = r if d an d d + 1 are in th e sam e block of R eplacing a + (d + 1) by a € fld+i in eq u atio n (4.2). we see th a t

Now for any B C [d 4- 1] we will consider th e sets

L ( B ) = {cr E n dTl : {d.d + 1} C B £ cr. where J < a < r r + (d - i-l)} .

The nonem pty L ( B ) p artitio n the interval [J. - + (d + 1)] according to the content of the block co n tain in g { d . d + 1 } and so we m ay express

To com pute th e inner sum . we need to consider th e following 2 cases.

C ase 1) For some k > q + 2.

is stric tly contained in a block of tt + (d + 1).

In this case, we see th a t each non-em pty L ( B ) form s a n o n-trivial cross-section of a

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

46

product of p a rtitio n lattices, a n d so for th is case

T hus p a rtitio n s in this case will not co n trib u te to

^ r € P( a)

C a s e 2) For all k > q + '2.

is a block of ir + (d + 1). In this case, we have q > 0.

since otherw ise we m ust have r = tt + ( d + 1) which we have already considered. Then we can show

m

Y2

if B — B - , 1)

M r -cr) =