Matroid Theory and Its Applications (C.I.M.E. Summer Schools, 83)

A. Barlotti ( E d.)

Matroid Theory and Its Applications Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, August 24 - September 2, 1980

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11109-9 e-ISBN: 978-3-642-11110-5 DOI:10.1007/978-3-642-11110-5 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 st Reprint of the 1 ed. C.I.M.E., Ed. Liguori, Napoli & Birkhäuser 1982 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

C O N T E N T S

M. BARNABEI, A. BRINI, G. C. ROTA Un'introduzione alla teoria delle funzioni di Miibius A.

BRINI Some remarks on the critical probl'em

T. BRYLAWSKI

The Tutte polynomial

.......... pag.

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

110

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

125

J. G. OXLEY On 3-connected matroids and graphs

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

R. PEELE The poset of subpartitions and Cayley's formula for the complexity of a complete graph

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

A.

RECSKI Engineering applications of matroids

D. J. A.

7

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

WELSH

Voltage-Graphic Matroids

289 301

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

323

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

417

Matroids and combinatorial optimisation T. ZASLAVSKI

177

CENTRO INTERNAZIONALE MATEMATICO E S T I V O (c.I.M.E.)

UN~INTRODUZIONE ALLA TEORIA DELLE FUNZIONI

M. BARNABEI

-

A.BRINI

-

G.C.ROTA

DI M ~ B I U S

Marilena Barnabei (~niversit;di ~errara) Andrea Brini (~niversitidi Bologna) Gian-Carlo Rota (~assachusettsInstitute of Technology)

Le idee principali che ci sono servite da guida e che vengono sviluppate in queste note sono le seguenti. In primo luogo si adotta pienamente la dualit; tra il concetto di insieme parzialmen te ordinato e quello di reticolo distributivo, idea che risale a Garrett Birkhoff e che 8 stata ulteriormente sviluppata da M.H. Stone, fino ad ottenere la versione definitiva nella tesi oxfordiana di Ann Priestley. Nel caso pi6 semplice degli insiemi parzialmente ordinati finiti, questa dualit; si riduce all'osservazione che ogni famiglia finita di sottoinsiemi di un insieme qua2

-

siasi, chiusa per intersezione e unione ma non sempre per c m plementazione 6 funtorialmente isomorfa alla famiglia di tutti

-

i sottoinsiemi decrescenti di un insieme parzialmente ordinato.

Questo fatto si esprime in mod0 naturale nelltequivalenza tra la categoria degli insiemi parzialmente ordinati finiti e la categoria dei reticoli distributivi finiti.

In linea di massima, si

pu6 pensare che ogni propriet3 canbinatoria di insiemi parzialmen_ te ordinati sia esprimibile in mod0 equivalente mediante i reticg li distributivi.

In generale, ltespressionedi tali propriet; in

termini di reticoli distributivi 6 preferibile, non solo perch6 permette a volte generalizzazioni a1 caso infinito, ma soprattutto perch& si inserisce pi6 agevolmente nella problematica dellla& gebra e della logica di oggi. In second0 luogo, sviluppiamo il concetto di anello di valutg zione di un reticolo distributivo, concetto che esprime in forma algebrica un processo di linearizzaziane noto da tempo in analisi funzionale, cio& il passaggio da una misura su una famiglia di in_ siemi alllintegralesulltanellodi funzioni semplici ad essa ass? ciate. questo processo di linearizzazione ci permette di studiare le valutaziani su un reticolo distributivo come funzionali lineari sulltanellodi valutazione, in analogia con lo studio della caratteristica di Eulero per le unioni finite di canvessi

- e pi5

-

generalmente con i ~uartermassintegralidi Minkowski fatto da Hadwiger e dalla scuola di Blaschke per la gemetria integrale. Infatti, ancora sulle orme della geometria integrale, riuscig mo a definire un analog0 combinatorio della caratteristica di Eulero per i reticoli distributivi finiti (e quindi per gli insiemi parzialmente ordinati), come la valutazione, evidentemente unica, che prende il valore unit: sugli elementi sup-irriducibili (0 tlco ni") non zero. Ltespressionedi questa caratteristica di Eulero mediante la funzione di ~8biusnon 6 che unlestrema generalizzazione della nota formula di ~ulero-~chlxfli per i poliedri. La teoria delle funzioni di ~Zbiusdegli insiemi parzialmente ordinati viene quindi sviluppata in base a questo legame fondamen tale con la caratteristica di Eulero. ~iusciamocosi a ritrovare

in forma semplice e, vorremmo credere, definitiva, le identit; scoperte Einora per le funzioni di ~8bius,nonch6 varie disuguaglianze profonde dovute a C. Greene, e le eleganti applicazioni geometriche dovute a T. Zaslavsky. Nei paragrafi 3 e 4 sviluppiamo dettagliatamente la struttura algebrica dell'anello aumentato di valutazione, introdotto da uno di noi e poi studiato da L. Geissinger in tre eleganti lavori. Troviamo cosi che con l'uso sistenaatico del1,aumentazione si semplificano varie dimostrazioni. Si pub affermare che, con il concetto di anello di valutazione aumentato, la classica dualith insiemistico-booleana viene linearizzata. I1 presente materiale, con IIeccezione del paragrafo conclusL vo riguardante le applicabioni della teoria delle funzioni di ~ 8 bius a1 problema classico del1,enumerazione delle regioni determL nate da un sistema di iperpiani non in posizione generica nello spazio affine o proiettivo, 4 stato oggettp di alcune delle lezig ni del Corso CIME tenuto a.Varenna nelllagosto 1980. La-letturadi queste note non richiede particolari conoscenze preliminari,a1 di fuori di alcune elementari nozioni di algebra cammutativa.

2.

Reticoli distributivi ed insiemi parzialmente ordinati Nel seguito, L indicheri un reticolo distributivo finito.

si dice sup-irriducibile se p = avb implica p = a oppure p = b. Ltinsieme J ( L ) degli elementi sup-irriducibili di L , con l1ordine indotto, 6 un insieme parzialmente ordinato dotato di minimo. Indicheremo con (L) 11 insieme parzialmente ordinato Un elemento peL

ottenuto da

J (L)

togliendo il minimo.

2.1.

PROPOSIZIONE Ogni elemento del reticolo distributivo L pua essere espresso in uno ed in un solo mod0 come sup di elementi sup-irriducibili a due a due non confrontabili.

DIMOSTRAZIONE Essendo L finito, si riconosce imediatamente che ogni elemento di L si pu?~esprimere come sup di elementi sup-irriducibili; ci limiteremo percia a dimostrare ltuniciti del la rappresentazione. A questo scopo, ricordiamo che, se p 6 un elemento supirriducibile in L e p 2 avb , allora p 2 a oppg

f

..

f

..

re p z b . supponiamo ora che p, ,p2,. , pnf e ql,q2.. , siano insiemi di elementi sup-irriducibili a due a due non

f

confrontabili, tali che

-

Grazie alltosservazioneprecedente, si ottiene, ad esempiov 9, < pl ; d'altra parte

da cui si deduce p = ql 1

.

Iterando questo ragionamento, si pro

va la tesi., Sia P un insieme parzialmente ordinato finito. Un sottoinsieme I di P si dice ideale se x 2 y e I implica x e I

.

La famiglia Y ( P ) degli ideali di P , con le operazioni di unione e intersezione, risulta essere u n sottoreticolo delltalgebra di Boole su P

, ed

8 quindi un reticolo distributivo.

un sottoinsieme F di P si dice filtro se- x 2 ca

y e F

impli-

X C F .

Un ideale I di P si dice principale se esiste pe P tale

f

che I = x e P; x 5 pf

.

Analogamente, un filtm F di P si di-

ce principale se esiste p e P 2.2.

tale che F =

x e P; x 2 pf

.

PROPOSIZIONE Sia L un reticolo distributivo finito, sia (L ) 11insieme parzialmente ordinato dei sup-irriducibili di L privato del minimo, e sia ~ ( J ( L ) ) il reticolo degli A

ideali di

j(L).

Allora, L e Y ( ~ ( L ) ) Son0 isomorfi.

& biiettiva e presemra l'ordine, quindi 6 un isomorfismo di reti-

coli..

2.3.

PROPOSIZIONE Sia P un insieme parzialmente ordinato finito, e sia Y ( P ) il reticolo distributivo degli ideali di P Allora P e i ( . f l ( ~ ) ) son0 isomorfi.

.

DlPIOSTRAZIONE j(g (P))

con pep.

E'

sufficiente osservare che gli elementi di

sono tutti e soli i sottoinsiemi di P della forma

Si verifica immediatamente che lcapplicazione

k biiettiva e preserva l'ordine.

2.4.

Sia P un insieme parzialmente ordinato fi-

PROPOSIZIONE nito, e P"

il suo duale dlordine.

Allora 4(pX)

2 il

duale dlordine di g(P). DIMOSTRAZIONE Osserviamo che 9 (P*) 6 il reticolo dei filtri di P. ~oich&l~insiemecomplementare di un ideale 6 un f i l m e viceversa, llapplicazione

risulta un antiisomorfismo di reticoli.. Siano L?,L2 reticoli finiti.

Un*applicazione

si dirh, u-z morfismo di reticoli se 2'un morfismo reticolare e fa corrispondere il massimo di L a1 massimo di L2 e il minimo di 1

L, 2.5.

a1 minimo di L2. TEOREMA

La categoria dei reticpli distributivi finiti con

gli u-z morfismi & funtorialmente equivalente alla categoria degli insiemi parzialmente ordinati finiti con i morfismi dt ordine. DP~OSTRAZIONE Siano Plt P2 due insiemi parzialmente ordinati finiti, e sia

un morfismo dtordine.

Siano 9(P 1 )

e f(P 2 )

i reticoli degli

ideali di P

1

e P rispettivamente. 2' a* : 9(P2)

Definiamo unlapplicazione

+ 4 ( P1 )

ponendo

dove 1ey(p2);

a* risulta evidentemente un u-z morfismo di re-

ticoli. Viceversa, siano L

L2 reticoli distributivi finiti, e sia

un u-z morfismo; definiamo unrapplicazione

ponendo p*(p) = min (xe j(Ll)i P(X) = pf dove p e J(L~)

9

.

*

& ben posta, in quanto, se x, y e J(L,) sono tali che $(x)-= p = $(y) e sono minimali rispetto a tale condizione, allora P ( x ~ y )= p ; sia X F I Y =p1vp2v vPn , con p sup-irriducibile per ogni i ~oich6 p & a sua volta i sup-irriducibile, deve esistere un indice j tale che @(p 3. ) = p; dato che x,y sono minimali, si ha necessariamente x = p j = y si verifica poi immediatamente che f3* k un morfismo dtordiLa definizione di

$1

.

ne

...

.

Utilizzando le costruzioni precedenti si completa la dimostrg zione..

3.

Coni di valutazione Sia A un anello, e sia

una aumentazione per A , cio6

&

un morfismo di anelli da A alllanello Z degli interi. Definiamo ora una nuova operazione su A , che chiameremo tiplicazione di Geissinger, e indicheremo con x , ponendo

e-

a n b = &(a) b+a~(b)-ab per ogni a,bcA. 3.1.

PROPOSIZIONE (A,+, n) 3 un anello; inoltre, (A, +, x) commutativo se e solo se A & commutativo.

DIMOSTRAZIONE i)

Per ogni a, b, c e A

si ha:

a n (bnc) = a x (&(b)c + bE (c) - bc) = = &(a) E (b) c

+

E

(a) b E(C)

-

t

(a) bc + a ~(b)E ( c ) +

.

+ ~ F ( ~ ) E ( c ) - a & ( b ) & ( ~ ) - a & ( b ) c - a babc &(c)+ L'espressione cosi ottenuta C una funzione simmetrica in a, b, c, quindi

ii)

ax(b+c) = ~(a)(b+c)+a(& (b)+&(c))-a(b+c) = ~(ab ) + ~ ( a )C + as(b)

=

+ ae(c)- ab-ac ;

Analogamente si dimostra l'altra legge distributiva.

C

L'ultima affemazione dell'enunciato segue direttamente dalla definizione dell'operazione x.. 3.2.

COROLLARIO l'anello singer di

L1aumentazione E di (A,+ , r )

A

& unlaumentazioneper

. Inoltre, la moltiplicazione di Geis-

(A,+,r) 4 la moltiplicazione di A.,

Sia A un anello con aumentazione

E

. Un elemento

zcA

si dirs integrale se risulta:

i

>

ii)

3.3.

E(Z)

= 1

~(a)z= az

per ogni a e A

.

PROPOSIZIONE Sia z un integrale di A ; allora, z f elemento neutro dell'operazione x Viceversa, se A possiede elemento neutro moltiplicativo , questo 6 un

.

integrale per

.

(A,+ ,'x)

In particolare 1'integrale, se

esiste, & unico.. un semianello s(+,

0

sara nel seguito una struttura dotata

)

di due operazioni tali che i) s(+)

ed

ii) in s(+)

s(.)

siano semigruppi;

valga la legge di cancellazione;

iii) a(b+c) = ab + ac (a+b)c = ac+ bc

per ogni a,b, c e S

.

che risulti un morfismo di semianelli. On con0 di valutazione & un semianello commutative unitario

s con una apentazione E , dotato di integrale, e tale che sia definita i n s un'operazione x che soddisfi ltidentita: a x b + ab = a & (b) + e(a)b

.

per ogni a, b e S Osserviamo che, nelle precedenti ipotesi, la struttura (S,+ ,x) risulta anchlessaun con0 di valutazione. siano SI * S2 coni di valutazione; unlapplicazione Ip:

S1

-

s*

si dirh morfismo di coni di valutazione se mianelli, ed inoltre: i> it)

Ip(awb) = Ip(a) x~(b) ; &(~)(a)) =

E

(a)

per ogni a,b eS1

3.4.

4 un morfismo di se

PROPOSIZIONE

. (mincipio di inclusione-esclusione)

...,an es.

Sia S un con0 di valutazione, e siano al,a2, Allora:

DPIOSTRAZIONE

Segue per induzione su n

.,

Sia L un reticolo distributive finito.

Consideriamo il semigruppo abeliano libero su L e definiamo su questo semigruppo una struttura di semianello ponendo

se x,yeL

.

, ed

estendendo il prodotto cosi definito per lineari-

Tale semianello sara indicato con N b,

4.

Consideriamo ora le congruenze

per -x,yeL. Osserviamo che le congruenze (r) sono compatibili con la struttura di semianello, in quanto, per ogni a e L e per ogni x,ye L:

~h.4

riQuindi, & ben definito il semianello quoziente di spetto alle congruenze (x). Tale semianello si indichera con v(L 1. Diremo elementi puri di v(L)

quelli che sono imagine di

elementi di L nell*immersionecanonica L + V(L) Definiamo ora un'applicazione:

.

nel mod0 seguente: i) ii)

&(x) = I

se x

xi =

E

i

E

puro;

x i con xi p u m per ogni i

i

risulta percid unlaumentazione per V(L)

.

.

Si ha poi immediatamente che il massimo u del reticolo corrL sponde alltunits del semianello, e che il minimo z del reticolo corrisponde all'integrale del semianello, cio& z-x = E (X)Z per ogni xt v ( L ) .

t(z) = 1

e

~efiniamoora unloperazione su v(L), che indicheremo con

X,

nel modo seguente:

se x,y

x x y = xv y

sono puri

.

dove xi,yj sono puri per ogni i ,j Questa definizione non dipende dalla rappresentazione mediante elementi puri che & stata scelta, poich6, se a,b,c sono puri, si ha

3.5.

PROPOSIZIONE

DIMOSTRAZIW

Se f,g c V(L)

siano f = i

si ha:

xi , g =

J

yj , xi.yj

puri.

~llo-

Di conseguenza, V ( L )

risulta un con0 di valutazione.

Diremo con0 ridotto del reticolo distributivo L il smianello v,(L) quoziente del con0 di valutazione V ( L ) rispetto a1 semiideale generato dall'integrale z :

3.6.

PROPOSIzIONE

Sia P un insieme parzialmente ordinato fi-

nito, e Y ( P ) dine di P Una funzione

il reticolo distributivo degli ideali d'or

.

f : F -+ IN

2 decrescente se e solo se esistono xl,x2 e c,,

..., cn.

eN

,...,xn

e Y(P)

tali che

dove Ixi 4 la funzione caratteristica delltideale xi

.

DIMOSTRAZIONE Sia f : P -t N Qecrescente. Dato che P 4 finito, f assume un numero finito di valori non nulli; siano v

1

< v <...
n

questivalori.

Poniamo

Ciascuno degli insiemi ne di P

.

A,,

Al

,.

..,An,1

6 un ideale dcordi-

La funzione

C anchtessa decrescente, ed inoltre

f

,

(p)

- vl

n- I se pc

U

i=I

fl(p) = altrimenti

\O

Per induzione si ha allora

Dato che ciascun

un elemento di

Q Vera.

.

Viceversa, 6 owio che ogni IA zione decrescente su P

.

JZ(P)

, con

, l*affermaeione

A e Y(P)

,2

UM

fun-

Da questo risultato si deduce il seguente teorema di StrutQ ra:

3.7.

Per ogni reticolo distributive finito L , il con0 ridotto V,(L) 4 isomorfo a1 semianello delle funzioni d= crescenti sull*ordineparziale j(L) , a valori in N TEOREMA

..

4, Anello di valutazione di un reticolo distributivo

Sia L un reticolo distributivo finito ed M un semigruppo abeliano, nel quale valga la legge di cancellazione. Una valutazione su L -

k una funzione

tale che f(a~b)+ f(a~b)= f(a) + f(b) per ogni a,b nel reticolo. Se poi f: L + M 6 una valutazione che associa a1 minimo z di L l*elemento neutro del semigruppo M , f si dice misura. Sia A un anello unitario; una valutazione f:L + A si di ce moltiplicativa se, per ogni x , y c L , risulta

4.1.

PROWSIZIONE

Ltimmersione canonica

del reticolo distributivo L nel suo con0 di valutazione la valutazione universale su L , ciok, per ogni semigruppo M

e per ogni valutazione

esiste un morfismo di semigruppi

tale che

DIMOSTRAZIONE

Definiamo y : V(L)

2) se x =

2

ak

k

.

M

nel mod0 seguente:

se x Q puro;

sp (XI = f(i-l (x))

1)

-*.

con ak puro per ogni k

.

poniamo

Dato che f h una valutazione, questa definizione non dipende dalla rappresentazione di x mediante elementi puri.. 4.2.

PROWSIZIONE

Un morfismo di reticoli distributivi finiti

induce un (unico) morfismo di coni di valutazione

tale che poi = i 09 1 2 ' dove i1' i2 sono le immersioni di L1 in v(L~) e di L2 in v(L~) , rispettivamente. DIMOSTRAZIONE E ' sufficiente osservare che, prolungando per 1= nearits la 9 a , si ottiene una funzione che rispetta

INfi.4

le congruenze a v b i a ~ b= a + b

e che quindi & ben definita su V ( L )

4.3.

PROPOSIZIONE

Siano L1, L2

.

.

reticoli distributivi, e sia

un morfismo di coni di valutazione.

Allora esiste un uni-

co morfismo di reticoli

tale che

dove i denotano le immersioni naturali di L1 v(L?) e di L2 in v ( L ~ ) , rispettivamente. DIMOSTRAZIONE

in

E' owio che gli elementi puri di v ( L 1 ) e v(L2)

sono tutti e soli gli elementi di aumentazione 1 . E l quindi sufficiente provare che 9' muta elementi puri in elementi puri. Ma, se x s v ( L ~ ) , ed x & puro, si ha

che implica 9l ( X I

da cui:

9 8

(XI = 9 ' ( x )

Sia L un reticolo distributivo finito; consideriamo il gruppo abeliano libero su L. Tale gruppo si pu6 dotare di una strur tura di Z-algebra mediante l'operazione di prodotto indotta dall'inf~ del reticolo. Questa algebra si dice algebra di semigruppo di L , e si indica con L&, rrL( Sia I(L) l'ideale di Z E , ~generato dagli elementi del tipo

.

con a,beL

.

L'anello quoziente

si diri'anello di valutazione di L .

Gli elementi di w ( L )

im-

magine degli elementi di L si diranno elementi puri.

4.4.

PROPOSIZIONE

sia L un reticolo distributivo finito, e sia M un gruppo abeliano. Per ogni valutazione

esiste un morfismo di gruppi

tale che

d w e i 6 l~immersionecanonica di L in w ( L ) . DIMOSTRAZIONE

Analoga a quella della Proposizione 4.1,.

Analogamente a quanto fatto per il con0 di valutazione V(L) definiamo l'aumentazione

nel mod0 seguente: i) ii)

se x Q puro;

t(x) = I

L ~ X ~ ) = ~ L ( X ~se ) xi 6 puro per ogni i. i i

Dal momento che W(L) 8 un anello con aumentazione, si pub definire in esso la moltiplicazione di Geissinger X :

per ogni a,bc#(L)

e, se a = r x i i

.

.

Owiamente, se a,b sono puri, risulta

b=ryj

, dove

xiPyj sono puri, si ha:

j

(xivyj)

axb =

i,j

4.5.

PROPOSIZIONE

.

Lfimmersionecanonica j : v(L) -+ w(L)

h un morfismo di coni di valutazione..

,

Osserviamo che la costruzione di W ( L )

pu6 essere ripetuta

utilizzando lloperazione sup di L in luogo dell'operazione inf; X

si ottiene cos? un anello W ( L ) , che gode evidentemente delle stesse propriet3 di w ( L ) pih precisamente:

.

4.6.

PROPOSIZIONE

L l applicaeione

tale che 7 ( x ) = u+ 2- X

6 un isomorfismo involutorio di anelli. DIMOSTRAZIONE

4.7.

El sufficiente osservare che

PROPOSIZIONE Gli elementi di W ( L ) che corrispondono agli elementi sup-irriducibili di L costituiscono una base per w ( L ) .

DIMOSTRAZIONE i) gli elementi supirriducibili di L sono owiamente linearmen te indipendenti in W ( L ) ; ii) sia x s L , non supirriducibile, e sia

con P,,

..., pn

supirriducibili e non confrontabili; grazie

a1 principio di inclusione-esclusione, in W(L)

ciascuno degli addendi a1 second0 membro

, si ha:

strettamente rnino-

re di x ; ripetendo il procedimento per ogni addendo che non sia supirriducibile, dato che il reticolo L i finito, otteniamo

...,qk

Con ql,

sup-irriducibili;

iii)dato che gli elementi puri generano w(L), ltaffermazione b Vera..

4.8.

PROPOSIZIONE

Sia L un reticolo distributive finito. O g n i

valutazione su L Q determinata dai suoi valori su J(L)

,

e qaesti valori possono essere assegnati arbitrariamente.

Segue dal fatto che J(L) una base per W(L) e che ogni valutazione f :. L + A si pud esprimere nella forma !Poi, dove rp: W ( L ) + A 6 un morfismo di gruppi.

D~sTRAzIONE

4.9.

COROLLARI~ se J(L)

un inf-semireticolo, allora J(L)

Q un inf-sottosemireticolo di L , e W(L) semigruppo di

che

Q Italgebra di

(J(L), A ) .

DIMOSTRAZIONE Indichiamo con A lroperazione di inf in J(L)

J

,

A

11 operazione di inf in L

. Siano

, con

p,qc~(L); supponiamo

D'altra p a r t e s i avr;

con a l r a 2

,..., an r J(L) ; questo implica

per qualche i

, il

che b assurdo.

D i conseguenza

La seconda affermazione segue d a l f a t t o che g l i elementi d i J ( L ) costituiscono una base per

4.10.

TEOREMA

w(L)

.

Sia L un r e t i c o l o d i s t r i b u t i v o f i n i t o ;

pua immergere i n untalgebra d i Boole f i n i t a , rango DIMOSTRAZIONE

L

.

B(L)

L si

, di

I~(L)I E'

s u f f i c i e n t e o s s e m a r e che, per il Teorema 2.5,

5 i s m o r f o a un s o t t o r e t i c o l o dell'algebra d i Boole generata

dagli elementi d i 4.11.

..

j(~)

PROWSIZIONE Sia L un r e t i c o l o d i s t r i b u t i v o f i n i t o , e B(L) l ' a l g e b r a d i Boole generata da e

A

J(L) ; allora

w(L)

w ( B ( L ) ) sono isomorfi.

DIMOSTRAZIONE

Segue dal f a t t o che

e quindi i due a n e l l i d i valutazione sono generati d a l l o s t e s s o nwnero d i elementi..

5.

La funzione di ~bbius Sia L un reticolo distributive finito, e sia J(L)

l'insie-

me parzialmente ordinato degli elementi sup-irriducibili di L. Per ogni pef(~) , l'insieme

ha an massimo, che indicheremo con dp. Osserviamo inoltre che, se p1,p , , pn sono gli elementi sup-irriducibili di L tali 2

che piCp

...

per ogni i , allora

Nell*anello di valutazione W(L)

Poniamo in01tre ez ='z 5. I.

PROFQS IZIONE

, definiamo

.

L * insieme

& unit base di idempotenti ortogonali per w(L) per ogni x e L , risulta

. Inoltre,

DIMOSTRAZIONE

Innanzi tutto osserviamo che, per ogni pe~(L),

si ha:

e, per ogni p,qsJ(L)

, Pf

q

, risulta

da cui

. 9= (p-dp)(q-dq)

= p ~ q d+p ~ d q - d ~ ~ q =~ O~ d q

P

Inoltre, per ogni x e L

, risulta

infatti, supponiamo Vera lfaffennazioneper ogni y c L se x non a sup-irriducibile, avremo x = arb; se

allora

poich6 gli e sono idempotenti ortogonali; quindi P

Se invece x 4 sup-irriducibile, si ha:

,

y<x;

dx < x

dal momento che

, la

Dato che g l i elementi d i

t e s i 6 Vera.. J(L)

son0 una base per

W ( L ), sa-

r; in particolare

per ogni

pe J(L)

numeri i n t e r i .

.

I coefficienti

~ ( q , p ) sono evidentemente

Per convenzione, poniamo

Abbiamo quindi: 5.2.

PROPOSIZIONE

DIMOSTRAZIONE f inizione d i

5.3.

Per ogni

xe L

s i ha

.

Segue d a l l a Proposizione precedente e dalla de~ ( pq),

PROPOSIZIONE

.

Per ogni

a , b e ~ ( L ) si ha:

DIMOSTRAZIONE

Senza perdita di generalith, supponiamo che a

sia il minimo del reticolo L.

Ma eb0a = &(eb)-a = 0 se b segue l(affermazione.,

5.4.

PROPOSIZIONE

DIMOSTRAZIONE

sia q e J(L)

Abbiamo quindi:

mentre ea0 a = a ; da qui

Per ogni a,be J(L)

Per ogni pc J(L)

.

.

#a

si ha:

si ha

Allora

la tesi segue dal fatto che x e q

ti per ogni x~J(L), x f q

.,

sono linearmente indipenden

Sia P un insieme parzialmente ordinato finito; in base alfe considerazioni precedenti resta definita una funzione p: PxP -22

che diremo funzione di ~ljbiusdi P. Osserviamo che la funzione di ~8bius& 18unicafunzione soddisfacente le seguenti condizioni:

2)

P(X~X>= 1

M 3)

) p(x,z)

per ogni x e P ; = 0

per ogni x , y c ~ ,x

#

y

X
suo duale d*-ordine,indichiamo con p x

di P

5.5.

DIMOSTRAZIONE

Per ogni x;y c P

b il

la funzione di ~8bius

.

PROWSIZIONE

pK

.

, si ha:

Definiamo una funzione %

%

v: P X P

+

a

ponendo

Abbiamo allora che v verifica le condizioni MI ), M2), M3), e qindi & la funzione di ~8biusdi P*

Ricordiamo che, in un reticolo distributivo finito L , si definisce rango la funzione che ad ogni x e L associa la lunghezza di una catena massimale tra il minimo di L ed x Ossemriamo che, dato che ogni reticolo distributivo r2 modulare, la funzione

.

rango 5.6.

una valutazione su L

.

PROWSIZIONE Sia L un reticolo distributivo finito, ed r la sua funzione rango. Allora, per ogni x c L , si ha:

DP~OSTRAZIONE Consideriamo l*applicazione

e proviamo che k una valutazione. Infatti, se a,b e L e p e ?(L) 4 tale che p 5 av b , allora P 2 a oppure P 5 b , da cui f(av b) + f(a~b)= f(a) + f(b)

.

questo punto & sufficiente dimostrare che f ed r assuL8affermazione& Vera per gli mono gli stessi valori su J(L) elementi di rango zero ed uno; per ipotesi di induzione, supponig A

.

mo lgaffermazione Vera per gli elementi di J(L) Sia a e J(L)

,

r(a) = h+l

,e

sia b < a

Allora

quindi la tesi segue per induzione.,

di rango 2 h

, tale che

r(b) = h

.

.

La valutazione )( definita sul reticolo distributive Einito L come segue:

dove z & il minimo di L , si dice valutazione caratteristica di L. Osserviamo esplicitamente che, se j(~)& un complesso simpliciale, in base alle considerazioni precedenti si ha che )((u) (dwe u 6 il massimo di L ) 6 la caratteristica di Eulero del complesso simpliciale

S(L)

5.7.

PROPOSIZIONE

DIMOSTRAZIONE

.

Per ogni y e L , si ha:

Applicando )( ad ambo i membri dellluguaglianza

Se poi y = dp :

5.8.

TEOREMA (Identit3 di lee) Sia S un inf-semireticolo, e siano x, a , a2,. an, b,, b ,bm elementi di S ta 1 li che x 2 air bJ. per ogni i,j Allora, nell'algebra di semigruppo % si ha:

..,

*,...

.

&, 4,

n i( r a i ) + m ( r b . ) J J

- (x- iyj (ainbj))

=

DIMOSTRAZIONE Senza perdita di generalit2, possiamo supporre S finito. Per il Corollario 4.9 abbiamo che, identificando un elemento di S con llideale (principale) da esso generato, si ottig ne un isomorfismo tra Z E , e~ w(.Y(s)) In quest*ultimaalgg bra, l*affermazionedel teorema si scrive come

.

A

( x - ai) + (x= x-

dove v dentit:

(y

j

bj)

-

(x- .V. (aihbJ. ) ) =

ai)v(?/ b.) J J

indica 190perazionedi sup in

s vera, in quanto, in

>(s)

V (ai4b.) =

i,j

J

(y

$(s)

.

~uest~ultima i-

:

ai)~(Y J b.) J

'm

5.9.

PROWSIZIONE che

J(L)

Sia L un r e t i c o l o d i s t r i b u t i v o f i n i t o t a l e

r i s u l t i a sua v o l t a un r e t i c o l o ; per ogni A

p2,...,

p,sJ(L),

sia

Y

il sup i n L d i

p1,p2,

pl,

...,'r

Allora

ck

dwe

il numero d i sottoinsiemi con k elementi d i

a v e n t i per inf il minimo z d i L .

DIMOSTRAZIONE

5.10.

S i ha:

COROLLARIO

...,pr

t a l i che

c

q, p l , p 2 , e , per ogni

Sia P un r e t i c o l o f i n i t o e siano pi 5 q

per ogni

i

x 5 q , x m a s s h a l e r i s p e t t o a questa proprieta, e s i s t e i t a l e che x = pi ; a l l o r a

dove

6

8 il minim0 d i P

insiemi con in

5

DMOSTBAZIONE

.

k

,e

elementi d i

ck

& il nwnero d i sotto-

f P,, ...,'F f p2,

Nel r e t i c o l o d i s t r i b u t i v o

r

2 ( ~ s) i

aventi i n f

ha

d q = x Pi;

allora, per la Proposizione 5.7, si ha:

Sfruttando il risultato precedente si ha la tesi. rispetti Sia L un reticolo finito; indichiamo con 6 ed vamente il minimo e il massimo di L. Un insieme T = al,a f 2 an di elementi di L si dice taglio (cross-cut) se sono soddi-

,..,,

)

sfatte le seguenti condizioni: i)

6

? non appartengono a

ed

T;

ii) T & untanticatena; iii) ogni catena massirnale tra vuota con T

.

ed

?

ha intersezione non

11 Corollario 5.10 pu6 essere generalizzato come segue: 5-11. TEOREMA (del cross-cut) Sia L un reticolo finito, e sia T un taglio di L Per ogni intero k >_ 2 sia qk il numero dei sottoinsiemi S di T aventi k elementi e tali

.

che

Allora

DMOSTRAZIONE mento

Definiamo la distanza d(x)

di x c L dalltele-

come la massima cardinalita di una catena di L avente se T 6 un taglio di L , definiamo la sua per estremi x ed i

.

A

distanza da

1

ponendo:

.

d ( ~ )= m a x d(x)

xcT

Procederemo per induzione su d(~). Per il Corollario precedente, la tesi 8 Vera per d(~) = 2. Sia ora B un sottoinsieme di L ; per ogni x e L , scrivere mo

x

5B

(x > B)

se esiste y e B tale che y 2 x (y < x)

.

Notiamo che, se B 6 un taglio, per ogni x e L risulta x f B

.

oppure x > B Sia Lm il reticolo i cui elementi sono tutti e soli gli elementi x c L tali che x 5 T (con T un taglio tale che * d(~) > 2) , uniti allvelemento 1 , con lvordine indotto da L.

In L * , il taglio T ha distanza 2 , per cui, denotando con p* la funzione di ~8biusdi L1 , il Corollario precedente implica

dove pk indica il numero di sottoinsiemi ti e tali che, in L v ,

A

di T con k elemen

D1altra parte, avremo:

~oich& , x ) = p 06

) per ogni x 5 T , si ha:

*a.

dato che gli insiemi

f x e L;

x 2 T)

e j x e L; x > T

disgiunti, si pu6 scrivere:

Sia ora

qk(x)

il numero dei sottoinsiemi A di T aventi k elg

menti e tali che

In particolare,

*

q k ( l ) = qk

percia, l'identith (x)

a

.

N e segue che

equivalente all'identith:

Sia ora x tale che T < x < in L. sia ~ ( x = )

p,Xf

?

; consideriamo l@intervallo

~nm,a; dal momento che

~ ( x )4

un taglio dell'intervallo @,d , e che la sua distanza in questo intervallo 6 strettamente minore di d(~) , per l'ipotesi di induzione si ha:

Sostituendo nella (xx), si ha 1' identits voluta.

5.12,

COROLLARIO Se L 6 un reticolo finito tale che il suo massimo $ risulti sup-irriducibile, allora

5.13.

COROLLARIO

Sia .L un reticolo finito; allora:

(a) se L ha un taglio di cardinalit; uno, allora ,A(b,F) = 0 ; (b) se L ha un taglio di cardinalits due, allora

p(6.i) =

o oppure ~(6,:) =

I ;

( c ) se L .ha un taglio di cardinalith tre, allora

(6.7 5.14.

)

pub assumere solo i valori -I, 0 , I , 2.

PROPOSIZIONE finito.

Sia P un insieme parzialmente ordinato

Per ogni a , b e P , con a < b , si ha

dove ck ( a , b ) indica il nwnero di catene con k elementi tra a e b

.

Procediamo per induzione sul nwnero di elementi dell~intervallo b,bZJ se llintervallo 6 costituito solo dai DIMOSTRAZIONE

punti a e b Vera.

.

, risulta

p(a,b) =

-1

,e

quindi l'affermazicne

In generale, si ha:

e quindi, per l'ipotesi di induzione:

in quanto, per ogni k 2 3 :

5.15.

TEOReMA

Siano P,Q insiemi parzialmente ordinati, e

sia f:P+Q un morfismo d'ordine.

.

Siano a,bcP tali che a < b e

f(a) < f(b) Allora, indicando con p la funzione di ~8biusdi P , si ha:

a(a.b) =

vf (a.x) ~(x,b)

xc P f (x)=f (b)

,

dove, per ogni

p, q c P

s i pone

DPIOSTRAZIONE

Per l a moposizione 5 . 1 4

dove ck (a,b) indica il numero d i catene con k elementi tra e b Sia A una variabile formale; poniamo

.

f c k (a,b)

a n a l o g ~ e n t e ,definiamo

come il numero d e l l e catene

t a l i che

per

i=

1, 2,.

..,

k-7

.

Poniamo

Osserviamo che, per ogni catena tra a = t

1

< t2 <...

a

e

b :

< tk = b

a

esiste j 5 k tale che

ne segue che, se a < b e f (a) < f(b) :

xep f(x)=f (b) Ponendo A = - 1

si ha la tesi.

~icordiamoche, se P

un insieme parzialmente ordinato fi-

nito, si dice funzione zeta di P la funzione

5:

PXP

z

definita nel mod0 seguente:

5.16.

COROLLARIO Siano P,Q due insiemi parzialmente ordinati finiti, e sia

Siano p e 5 la funzione di M& bius e la funzione zeta di P , rispettivamente. Per o m a,b e P con a < b e f (a) < f (b) , risulta: un morfism0 dcordine.

DIMOSTRAZIONE

0=

Segue dal teorema precedente, osservando che:

) v(a,x)

=

xzb

-- -

v(a,x) xz b f(x)
vf(a.b) +

v(a.x)

XIb

+

v(a,x> = xzb f( x ) = f(b)

.,

f(x)=f (b)

5.17.

TEOREMA (di complementazione) to, con minimo 0 e massimo 1 A

con sL

DIMOSTRAZIONE

A

.

Sia P un reticolo finiSe s e P , indichiamo

lvinsiemedei complementi di s in P

Sia f:P+

b,fl

f(x) = x v s

;

f 6 un morfismo d'ordine; per il Teorema 5.15 si ha:

. Allora

Sia y e P tale che y v s = ?

Lfinsieme s

.

Consideriamo 1'insieme

, con l'ordine indotto da P , risulta owiamente ur~

reticolo. Per ogni te S si ha t rl (s A y ) e s ; di conseguenza, f'elemento S A y $ un estremo in£eriore per gli elementi massimali di S\ y puesto implica che 4s A yf 6 un taglio di S

f ).

.

In questo caso, per il Teorema del Cross-cut, purchQ s A y # 6 Pf ( 6 , Y) = 0 Da qui, sfruttando il Corollario precedente, si of tiene la tesi.,

.

5. I 8. COROLLARIO

se P

+ un reticolo finito non complelilentato,

allora: ~(6,;) =

o

..

Sia P un insieme parzialmente ordinato finito. Si definisce algebra di ~gbius M(P) di P lo Z -modulo libero su P do tat0 dell'operazione di prodotto definita nel mod0 seguente:

.

per ogni q,r e P Osserviamo esplicitamente che, se P 6 un' in£-semireticolo:

cio&, ltalgebradi ~8biu.sdi P coincide con llalgebradi semigruppo su P rispetto all'operazione di inf. Come abbiamo visto, P 6 isomorfo all 9 insieme parzialmente

.

ordinato i(9(~) ) Se P & dotato di minimo 5 , allora anche & un reticolo distributivo, il cui minimo & 3 )= ( P ) l'ideale costituito dal solo elemento 6. In questo caso

)

A

e quindi P 6 isomorfo a J@(P)). Ne segue che, dato che gli elementi di J(~(P)) sono una base per llanello di valutazione di g ( ~ ,) sussiste 11isomorfismo di moduli

z di y(P) , c i d ltinsieme vuoto, a sup-irriducibile in Y(P) , e quindi .appartiene ad una base di W f4 (P)); z genera percia un sottomodulo < z > di W ( Y(P)) avente rango 1 , e sussiste 1' isomorfismo di moduli: Se invece P non ha minimo, osserviamo che il minimo

Gli isomorfismi ( x ) e ( x x ) risultano inoltre isomorfismi di algebre.

Infatti, in w(~(P))

, si ha:

perci6. lpoperazionedi prodotto in w(~(P))

.

coincide con il pro

dotto in M(P) Si ha di conseguenza: 6.1.

Sia P un insieme parzialmente ordinato finito, e sia M(P) la sua algebra di ~8bius. Per ogni PROWSIZIONE

pc P

, poniamo

Allora, l 1 insieme

ortogonali per M(P)

6.2.

..

f epip e P f

una base cii idempotenti

PROPOSIZIONE siano P,Q insiemi parzialmente ordinati finiti. Ogni morfismo dcordine

induce i morfismi di algebre

definiti nel mod0 seguente:

DIMOSTRAZIONE

6.3.

Owia.,

PROPOSIZIONE (Lemma d i Weisner) t o , e siano

DIMOSTRAZIONE

Allora

Nell'algebra d i ~ 8 b i u s M ( P ) abbiamo

Quindi

Ricordando poi che

abbiamo

a,b,csP.

Sia P un r e t i c o l o f i n i -

da qui si ha la tesi..

Sia P un insieme parzialmente ordinato finito

TEOREMA

6.4.

dotato di minimo

6 e di massimo

:, e sia

per ogni p c P esista p v t in P

.

t e P tale che Allora, nell'algebra

di ~bbiusdi P , sussiste l'identita:

DIMOSTRAZIONE Per t = 'i 1' affermazione & Vera. Sia allora t # 7 Indichiamo con K It insieme degli elementi di P che so

.

n

no coperti da I

,e

sia

Sia L = .? (P) ; lsalgebra di ~8bius M(P) si pua identificare con W(L) , in quanto P dotato di minimo. In w(L) avremo percid

e, dcaltra parte

Inoltre si ha:

(17-c)eccC

(i-c)

=

csC

(7-k)

kcK

=

Per ogni c e C e per ogni r c P , r 5 c

, si ha:

C

A

(I-c)r = ~ * r - c = ~0 r

.

Allora

A

e percia, posto d = v c ceC

=

5

(?-c)(s . ,u(r,?)r)

ccC se r

(in Y(P)):

d in >(P)

r d

, allora

rvt =

; in

P , e viceversa.

Osserviamo che, nell 'algebra di ~Bbiusdell1intervallo [It, dellfinsiemeparzialmente ordinato P

, sussiste l'identith:

I(?-~)=e-.=>~(r,?)r ccC rlt

,

poichk la funzione di ~8biusdi un qualunque intervallo di P la restrizione di p

.

Di conseguenza, otteniamo 19identitA

a

6.5.

TEOREMA

Sia P un reticolo finito, e sia Po 19insieme

dei coatomi di P .

Siano A,BcP,

tali che AnB = $

e

AuB = Po. Allora

dove H(A), H(B) sono gli inf-sottoreticoli di P generati da A e B , con funzioni di ~bbius rispettivapA,pB

mente. DIMOSTRAZIONE

Abbiamo, in M(P) :

Da qui, grazie a1 Teorema precedente, segue lluguaglianza.

I

Sia P un insieme parzialmente ordinato finito.

Un operato-

re di chiusura S un1applicazione

tale che: i)

g

6 un morfismo d'ordine;

ii) per ogni xc P, e (~(x)) = Q(x) ; iii) per ogni xeP, x _< @(x) Gli elementi x e P

.

tali che Q(x) = x

si dicono chiusi.

Se P,Q sono insiemi parzialmente ordinati finiti, una nessione di Galois tra P e Q & una coppia di funzioni

con-

t a l i che: i ) g, e o invertono l'ordine, cio& s e

x , y ~ P ,x 5 y , a l l o r a

~ ( x 5) 9 (Y) e analogamente per u ;

i i ) a09 e p a sono operatori d i chiusura su P e Q vamente 6.6,

.

, rispetti

Siano P e Q insiemi parzialmente o r d i n a t i fir&

TEOREMA

ti; untapplicazione

induce un morfismo

s e e solo se: i ) cp Q un morf ismo dl ordine; i i ) l a controimmagine attraverso q d i un f i l t r o principal e i n Q b un f i l t r o principale d i P , oppure l r i n s i e me vuoto. Supponiamo che

DIMOSTRAZIONE

9 :M(P) + M ( Q ). M(P)

, si

ha che

poichi! x 2 y

4 ( x y ) = + ( x ) $(Y) , quindi

i ) Q verificata. S i a ora

qcQ

,e

sia

x (y implica

cp : P + Q

induca un morf ismo

i n P s e e solo s e xy = x

in

$(xy) = ( ( x ) ; d ' a l t r a parte,

q(x) 5 q(y)

.

~iconseguenza, l a

allora, esiste

s i ha che

e

peP

t a l e che q ( p ) 2 q

.

poich6 i n M ( Q ) :

4 uno d e i termini che compaiono n e l l a somma

q

x 2 p t a l e che e compaia come addendo n e l l * 4 espressione d i +(ex) ; percia questo x 4 t a l e che ~ ( x >_ ) q I n o l t r e , t a l e x 6 unico, poich& s i a g l i et , t c Q , s i a i $(ey) , y c P sono idempotenti ortogonali d i M(Q) ed M(P) r& e quindi e s i s t e

.

spettivamente.

e quindi l a i i )

.

D i conseguenza,

xl p

verif icata.

Viceversa, supponiamo che

s o d d i s f i l e condizioni i ) e i i ) . Poniamo

Per

q a Q,

,

sia

per ogni

pe P

t a l e che

si ha allora che g, e $, costituiscono una connessione di Galois, e che le funzioni

sono operatori di chiusura.

Definiamo ora un ornomorfisrno

ponendo

ed estendendo per linearits. E' immediato verificare che

$ coincide con g, su P , e quindi

& il morfismo voluto. B 6.7.

COROLLARIO

Sia P un insieme parzialmente ordinato fini-

to, e sia Po un sottoinsieme di P

. Lgalgebra

M(P,)

6

isomorfa alla sottoalgebra di M(P) generata da Po se e solo se la restrizione a Po di ogni filtro principale di P

& ancora un filtro principale,

0,

equivalentemente, se

e solo se esiste un operatore di chiusura C? su P tale the Po coincida con llinsiemeparzialmente ordinato dei chiusi di P rispetto a 6.8.

PROPOSIZIONE

@

.

Sia P un insieme parzialmente ordinato fi-

56

nito, e sia

un operatore di chiusura su P.

si ha, indicando con p

-

e p

Posto

le funzioni di ~8biusdi P e

P

rispettivamente:

per ogni x,yeP

. -

DIMOSTR~ZIONE Per il Teorema 6.6, dato che la restrizione a P di ogni filtro principale di P 4 un filtro principale di P , g,

-

si pud estendere ad un morfismo di algebre

Per ogni x c P si ha:

dove Gx 8 l9idempotentecorrispondente ad x in in M(P) , risulta:

M(P).

Ma.

confrontando i c o e f f i c i e n t i d i

7

in

1' uguaglianza voluta.

Se poi

6.9.

-

x < x,

e

X

ed i n

((ex)

s i ha

$ ( eX ) = 0 fornisce l a seconda uguaglianza.. Siano

COROLLARIO

-

P,Q

insierni parzialmente o r d i n a t i f i -

n i t i ; supponiamo che siano date due funzioni

che formino una connessione d i Galois f r a P e Q. per ogni

peP

dove

e

pp

rispettivarnente.

ed ogni

qc Q

,

risulta

sono le funzioni d i ~ 8 b i u sd i 8

Allora,

P

e

Q

,

7.

La funzione di ~8biusnei reticoli semimodulari

.

Sia P un insieme parzialmente ordinato finito dotato di mi* nimo 0 e tale che, per ogni x,yeP, x 2 y tutte le catene massimali tra x e y abbiano lo stesso numero di elementi; si dg finisce rango di un elemento x e P l'intero

A

dove l(x) 6 la cardinalita di una catena massimale tra 0 ed x. Inoltre, se P & dotato di massimo i , si dice rango di P A

1' intero

Per ogni a.b~:P, a 5 b , si dice polinomio caratteristico dell 'intervallo b , q il polinomio nella variabile formale A :

Osserviamo che

7.1.

TEOREMA

Sia P un inf-semireticolo dotato di rango; per

ogni a,b,ccP, con

c 5 a h b , si ha:

DIMOSTRAZIONE

Si ha:

=

7.2.

A-~(Y) p(c,t) E(f,x)p(x,y) ~ f b xeP tfa

a,b,deP

tali che

a ~ =b a ~ d . Allora, per ogni

c c P , si ha:

DMOSTRAZIONE

=

Sia P .un inf-semireticolo dotato di rango, e

COROLLARIO siano

A"~)

Per il Teorema precedente, si ha:

dato che

7.3.

a r b = a ~ ,d s i ha l a tesi..

COROLLARIO

siano

Sia P un inf-semireticolo dotato d i rango, e

a,bfP, a 5 b.

Allora

Un r e t i c o l o f i n i t o L si d i c e semimodulare se gode d e l l a seguente p r o p r i e t i : per ogni

.

a,be L

t a l i che

a

copra

a ~ ,b

a v b copre b Ricordiamo che un r e t i c o l o semimodulare $ dotato d i rango r ; i n o l t r e , per ogni

a,be L

,

risulta:

Se L 6 un r e t i c o l o semimodulare,

a,beL

formano una coppia

modulare s e

o, equivalentemente, s e , per ogni

xcL

t a l e che x < b

, si

ha

Un elemento a di L si dice elemento modulare se, per ogni (a,x) 6 una coppia modulare. Sia L un reticolo semimodulare. si dice k-mo numero di nitney di prima specie di L il numero x ~ L , la coppia

dove

6,

sono rispettivamente il minimo ed il massimo di L

d la funzione di ~8biusdi

.

,

wk risulta quindi il coef ficiente di hk nel polinomio caratteristico P(L;A) Si dice poi k-mo numero di Whitney di seconda specie di L

e p

L

.

il numero

Si dice inoltre invariante di Crapo di L il numero

Un reticolo semimodulare si dice supersolubile se esiste una A

A

catena massimale tra 0 ed I i cui elementi siano tutti modulari. Un reticolo semimodulare L si dice geometric0 se i soli el= A

menti supirriducibili di L sono gli atomi e il minimo 0.

7.4.

TEOREMA

x,yeL

Sia L un reticolo semimodulare. Per ogni si ha:

a ) se

r(x) = r(y)

nulle, allora

e

( , x ) ( 6 )

c.

A

sono entrambe non

p(0,x)

e

p(0,y)

r(x) = r(y) + I e n u l l e , a l l o r a p(0,x)

x e

) , ( y ) sono entrambe non p(GPy) hanno segni opposti;

b) s e

A

c ) se

g(6,x) j 0

a seconda che

,

allora

r(x)

hanno l o s t e s s o segno;

p(6,x)

G positivo o

negative,

s i a p a r i o d i s p a r i , rispettivamen-

te; d ) s e L & geometrico,

DIMOSTRAZIONE

Sia

xeL

p(6,x) j 0

, tale

.

che r ( x ) = k > z

ma d i Weisner, s i ha, per ogni a tomo

ae

G,d :

.

Per il Lem_

e , grazie a l l a semirnodularit~d i L :

~imostriamol a prima affermazione procedendo per induzione su r ( x ) .

Se r ( x ) = 2 ,

e lwinsieme A p(6,y) gue

con y r A

l a t e s i 6 Vera.

Se r ( x ) = k > 2

,

non vuoto, per l t i p o t e s i d i induzione t u t t i i hanno l o s t e s s o segno, e , grazie a l l a ( a ) , s=

l a prima affermazione.

La seconda affermazione s i deduce

d a l l a prima applicando nuovarnente l e i d e n t i t i (x).

La t e r z a af-

fermazione s i dimostra ancora per induzione, a p a r t i r e d a l l a (x). Se poi il r e t i c o l o L 6 geometrico, ricordiamo che:

~ ( 6 ~ >x 01

per ogni

x c ~ t a l e che

.

procedendo per induzione su r ( x )

ha l * u ltima af f ermazione. 7.5.

COROLLARIO

r(x) = 2 ;

e utilizzando ancora ( x ) , s i

Sia L un r e t i c o l o geometrico.

e f f i c i e n t i d e l polinomio c a r a t t e r i s t i c o

Allora, i co-

P(L;A) ham0 se-

gni a l terni..

E' ben noto che ogni r e t i c o l o geometrico L & isomorfo a1

re

t i c o l o d e i c h i u s i d e l l ' a l g e b r a d i Boole B generata dagli atomi di L , r i s p e t t o ad un operatore d i chiusura

t a l e che:

i) cp muta atomi d i B i n atomi d i B ; ii) per ogni a , b , x c B

, con

a,b

atomi,

implica

7.6.

PROPOSIZIONE

Sia L un r e t i c o l o geometrico, e s i a

l'insieme a e g l i atomi d i L

v a-x aeF

.

Allora, per ogni

xeL

A

,s i

Segue dalla Proposizione 6.8..

DP~OSTRAZIONE

7.7.

COROLLARIO

Sia L un reticolo geometrico, e sia A l*in-

sieme degli atomi di L

.

Allora:

dove

DIMOSTRAZIONE

Si ha, per la proposizione precedente:

7.8.

Sia L un reticolo semimodulare, e sia

TEOREMA

lemento modulare di L.

DIMOSTRAZIONE L

si ha:

a

un e-

Allora:

Dal Teorema 6.4 applicato al reticolo duale di

effettuando l e s o s t i t u z i o n i : x

+

A

tvy + A

r(;)-r(x) r(?)-r(tvy)

s i ha:

(a1

ora, poiche

a & un elemento modulare, per ogni y , t c L

ta: r(yva) + r(yna) = r(a) + r(y)

(1)

e r ( t v y v a ) + r ( t v y ) a~) = r ( t v y) + r ( a )

(2

se

t 5 a

, essendo

a

i

un elemento modulare, r i s u l t a :

risul-

66

da cui, utilizzando (I) e (2):

sostituendo quest'ultima identit; nella (x) si ha la tesi..

COROLLARIO Sia L un reticolo semi-modulare. e sia p un atomo di L. Allora

7.9.

..

DIMOSTRAZIONE a = P

7.10.

Segue dalla Proposizione precedente, ponendo

TEOREMA

Sia L un reticolo s~persolubile,e sia 0 = a, < a, <

...

A

< ak-l < a k = ?

una catena massimale di elementi modulari in L.

P(L;A)

= (A-nl)(A-n2)

Allora

.

...(A-nk)

dove n = i per

i = 1, 2,...

DIMOSTRAZIONE

If

x e L; x atomo, x 2 air x

3 ai-lf

I

,k Procediamo per induzione su k.

fermazione 6 Vera.

Sia k > 1.

Se k = 1

, lvaf-

Dalla Proposizione 7.8, si ha:

d i a l t r a parte: { Y ~ L ; ~ * a =~ 6 -) ~=

f 5 )u

=

f y e ~ ;r ( y ) = 1. Y

A ak-lf

9

da cui

7.11.

Se L 6 un r e t i c o l o supersolubile, a l l o r a

COROLLARIO

(

6

= I

k

Sia L un r e t i c o l o geometrico.

nl n2

... 5 .,

I1 polinomio d i Mcbius d i L

6 il polinomio d e f i n i t o nel mod0 seguente:

S i a L un r e t i c o l o geometrico, e s i a

m i d i L.

A

l t i n s i e m e d e g l i at?

Un sottoinsieme I d i A s i d i c e indipendente se, po-

68

sto

y = V

X€ I

, risulta

x

Un c i r c u i t 0 d i L 6 un sottoinsieme d i A non indipendente minimale.

7.12.

Sia L un r e t i c o l o geometric0 d i rango r ; i n d i

TEOREMA

chiamo con e con

n

c

l a c a r d i n a l i t s minima d i un c i r c u i t 0 i n L

il numero d e g l i atomi d i L .

Allora sussistono

l e seguenti disuguaglianze:

1

>

Iwr-k

per ogni

k

I

c-2

f=O

(

n-r+i-1 r-i i )(k-i)

5 r ; 18uguaglianza s u s s i s t e se e solo s e

L

& isomorfo a1 prodotto d i r e t t o dellwalgebra d i Boole d i rango

PC+?

Boole d i rango

dove

q

con il

(c-1)- troncamento d e l l 'algebra d i

n-r+c-1

.

In p a r t i c o l a r e

8 il numero d e i c i r c u i t i d i L aventi cardinali-

th esattamente c .

I n particolare

,

3) Se L 6 connesso e c 5 r

k 5 r

per

- 2;

,

inoltre:

e, infine:

.ove f l ( L )

& lcinvariante di Crapo del reticolo geometric0

-L I!

8.

Laanello di Tutte-Grothendieck Sia L un reticolo geometrico.

Una

base

di L

4 un insieme

di atomi di L tale che:

ii) A

Q minimale rispetto alla propriet; i).

Un istmo di L & un a t m o che appartiene ad ogni base di L

.

Ricordiamo che, se p b un istmo di L , si ha lBisomorfismo:

dove @ indica l'usuale soma diretta di reticoli. Viceversa, se per un atom0 p sussiste l~isomorfismo( n ) , allora p 8 un is-0 di L. Inoltre, se p & un ism0 di L , si ha lBisomorfismo [P,?] dove L-p

*

L ' P

indica il reticolo geometric0 generato dagli atomi di

L diversi da p. E' poi facile verificare che, se p,q

sono atomi di L

, si

ha:

Cs*?I- (pvq)

Cqr

L-p

dove Lq, ;&-P indica 1' intervallo Cq, i] nel reticolo L-p Osserviamo infine che, fissato p e L , llapplicazione

.

.

muta atomi di L in atomi di b,?] Indicheremo in seguito con - 9 lBinsiemedelle classi di is2 morfismo di reticoli geometrici. con x

In particolare, indicheremo

la classe di equivalenza dei reticoli geometrici con due

elementi. Dato un anello comutativo unitario A , un imariante di Tutte-Grothendieck a valori in

tale che:

A

b unBapplicazione

se p 6 un atomo di L e non 6 istmo di L. Indichiamo ora con Z(9) la Z-algebra commutativa unitaria

.

libera generata da Y Sia $ l'ideale di ~ ( 9generato ) dagli elementi della forma :

T2) L - (L-p)-

Cp, ?] , se

p 5 un atomo di L e non 6 un istmo

di L. Si dice .anello di Tutte-Grothendieck l'anello quoziente

8.1.

TEOREMA (di struttura)

Lganellodi Tutte-Grothendieck isomorfo alltanellodei polinomi in ma variabile a coeffi

cienti interi -senzatermine noto. DMOSTRAZIONE

E' sufficiente provare che, per ogni L e y ,

la classe di equivalenza di L in

riel

esiste un unico polinomio a

coefficienti interi (positivi), senza termine noto, nella variabile x , che rappresenta la classe di equivalenza dei reticoli geometrici con due elementi. Con un semplice argomento di induzione 6 facile provare che ogni reticolo L 6 equivalente ad un tale polinomio. Per provare che questo polinomio 6 unico, 4 suf_ ficiente dimostrare che applicare ad L le identiti subordinate da TI e ~ 2 ordinatamente ) rispetto agli atomi p,q 6 equiva-

l e n t e ad applicare l e s t e s s e i d e n t i t a ordinatamente r i s p e t t o a g l i atomi

q,p

.

Supponiamo dapprima che

p,q

non siano i s t m i per L .

striamo prima d i t u t t o che, i n quest0 caso, s e e solo s e per

q

istmo per

, si

L-p

una base B d i

non 6 istmo per L , B

contenendo p L-q

& istmo per

, esiste

L-q

t o che

q

,6

L-p.

una base per

L-p

.

p 6 istmo per

Dimo-

L-q

I n f a t t i , s e p non 6 istmo

L-q

che non contiene p .

Da-

& anche una base d i L e , non

.

D i conseguenza,

q non

6

Allora, n e l l ' i p o t e s i che p non s i a un istmo per

ottengono l e seguenti identit;:

Se invece

p

istmo per

Supponiamo o r a che

p

L-q :

s i a istmo per L , e

q

non l o sia.

Allora: L = (L- p ) =~

((L

- f p,q))

+

Cq,i] L-P

)X

=

Infine, se p e q sono entrambi istmi per L , allora:

8.2.

PROPOS IZIONE

L immersione naturale

4 l-invariante di Tutte-Grothendieck universale, cio8, per ogni invariante di Tutte-Grothendieck

esiste un unico morfismo di anelli h:Y

+ A

tale che

Inoltre, per ogni L s Y , q ( L )

si ottiene da

T(L)

effet

tuando la sostituzione

8.3.

PROPOSIZIONE

ristico:

I1 "valore assoluto" del polinomio caratte

& un invariante di Tutte-Grothendieck, ed il suo valore si

ottiene da T(L)

DIMOSTRAZIONE

effettuando la sostituzione

Ricordiamo che, se a e L A

1

e b e L2 , allora

A

IIL,(o,a) PL2 (Ovb) = PLIBL

ed inoltre, in Ll @L2

( 6 , avb)

, risulta

Da cib si deduce

Proviamo ora che, per ogni L c 2 e per ogni p atmo di L p non istmo, risulta:

Infatti, se

A

b llinsiemedegli atorni di L

, si ha:

,

dove

r indica il rango i n P v b V B = be8

Cp, ;] ,

e

vC=

v

ccc

C

.

Infine, ricordiamo che, s e p non 6 istmo per L

,

Da qui segue l a tesi..

8.4.

COROLLARIO

I1 valore assoluto

d i ~ 6 b i u st r a g l i estremi

6

ed

I

(8 );,

7

1

d e l l a funzione

un invariante d i

Tutte-Grothendieck, DIMOSTRAZIONE

Segue d a l l a Proposizione precedente, osservando

che .Ip(G,?)l

8.5.

COROLLARIO

s i ha:

= (-1) (

Per ogni

L e6P

) L

O

.

e per ogni p atomo d i L

,

a)

IP(P,;)I

5 I p ( ~ ~ ~ );l

l'uguaglianza sussiste .se e solo se p & un istmo;

IP~-~(~.;)I

5

dove p L-P

indica la funzione di ~bbiusdi L-p ;

b)

c)

IB(~.?)I

se x,y e L sono tali che x e yL

6

e inoltre (x,y)

una coppia modulare, allora

lluguaglianza sussiste se e solo se si ha llisomorfismo

8.6.

COROLLARIO

L1invariantedi CraFo soddisfa le seguenti

identitd: i) se p e L

e p non & istmo, allora

B(L) = B(L-P)

+

P(

b,?] ;

ii) B(L, 0 L 2 ) = 0

8.7.

TEOREMA

Sia L un reticolo geometrico, e sia a e L

.

Al-

lora

dove (y,a)

yla

indica che y 6 un complemento di a e che

una coppia modulare.

L'uguaglianza vale se e solo se a

un elemento modulare

di L. DIMOSTRAZIONE

Procediamo per induzione sul nwnero n di atomi

di L che non sono minori o uguali ad a. zione & banale.

Sia n > 1

gni atomo peL, p

a

Se n = I

l'afferma-

. Per lfipotesidi induzione, per o-

, risulta

.

poichC a e L-p e pL-p (;,a) = ~(5,a) Se p 6 un istmo di L il teorena segue immediatamente, in quanto

e y l a in L-p

se e solo se (yvpfla in L.

Se invece p non 6 istmo, sommando bri della (x), otteniamo:

ma, per ltipotesi di induzione, abbiamo:

I p (p,?) )

ad ambo i mem-

,

1

grazie a1 Corollario 8.5, in quanto (av p) y in [p,;] solo se y 2 p

e a 1y

.

Osserviamo ora che, se y

dove lluguaglianzasussiste se e solo se p G,d Di conseguenza:

.

se e p

,

non & istmo per

In mod0 analogo si dimostra la seconda affermazione del Teo-

rema..

8.8.

COROLLARIO Sia L un reticolo geometrico, e sia x un coatomo di L

risulta:

.

Posto

Lquguaglianza s u s s i s t e s e e solo s e dulare

.

DIMOSTRAZIONE to

aeL

x

& un elemento mo-

Dal momento che x 6 un coatomo d i L

t a l e che

complemento d i x

(x,a)

, un

elerne2

s i a una coppia modulare e a s i a un

Q necessariamente un atomo non minore d i x ; l a

disuguaglianza segue a l l o r a dal Teorema precedente..

8.9.

Sia L un r e t i c o l o geometrico d i rango n ; al-

COROLLARIO

lora

dove

wi

d i L.

indica l'i-mo nwnero d i Whitney d i seconda specie Lvuguaglianza s u s s i s t e se e solo s e L 5 modulare.

DIMOSTRAZTONE

Con l e notazioni del c o r o l l a r i o precedente, s i

ha :

per il Lemma d i Weisner, quest~ultimoi n t e r o 6 uguale a

8.10.

COROLLARIO

i = I ,2,.

Sia L un r e t i c o l o geometrico.

.., n-I

risulta:

Per ogni

8.1 I.

COROLLARIO

Sia L un reticolo geometrico, e sia

una catena massimale in L

.

Posto

ik = 1faeL; a atomo, a 5 xk, a

3

,

risulta:

Lquguaglianzasussiste se e solo se ciascun xi 6 modulare. DIMOSTRAZIONE

...,

elementi x ~ - ~ x ~ , - ~ , x1

9.

'.

Si applica il Teorema 8.7 successivamente agli

Unlapplicazione geometrica

Nello spazio afPine Ed consideriamo un insieme finito H di iperpiani. guesfi individuano in $ un nwnero finito di poliedri d-dimensionali (aperti e non necessariamente limitati), chiamati regioni. Gli iperpiahi di H saranno detti tagli; diremo partizione dello spazio mediante H l'insieme di tutte le facce (aperte) k-dimensionali, per k = O,l,...,d , dei poliedri d-dimensionali individuati da H.

La partizione relativa alla famiglia di iperpiani H si dir;

centrale se

per estensione, anche la famiglia H si dira centrale. Data una faniglia finita H di iperpiani in Wd

.

consideria

mo l'insierne parzialmente ordinato S costituito da tutte le intersezioni non vuote degli elementi di H , ordinate per inclusig d ne. Per convenzione, poniamo ll? c: S. Indichiamo con LH il dua_ le dgordine di S iH risulta un inf-semireticolo, e si d i d semireticolo associato ad H Osserviamo che, per costruzione, il semireticolo LH 5 atomico ed 6 dotato di rango r ; in particolare, per ogni x e LH . , si ha:

.

.

r(x) = d-dim x

.

Per definizione, poniamo

f

r ( ~ ~=)max r(x); x e L

Notiamo che, se H 6 centrale, post0

& isomorfo alla restrizione ad H del duale d'ordine dellein LH d tervallo del reticolo dei sottospazi di IR Di conseguenza, LH risulta un reticolo geometrico.

.

&,g

Sia H un insieme finito di iperpiani di W C(H)

= nwnero delle regioni individuate da H

d

.

Poniamo:

,

f (H) = numero delle facce k-dimensionali individuate da H ,

k

per

k = 0, 7 , .

.. d . r

La funzione generatrice degli interi f (H) : k

si d i d f-polinomio di H.

TEOREMA

9.1.

Sia H un insieme finito di iperpiani di

JR

d ,

Allora

DIMOSTRAZIONE Proviamo innanzi tutto che C(H) soddisfa la s= guente recursione: per ogni h iperpiano di R d , h 6 H , poniamo

risulta allora

Infatti, sia P una regione individuata da H.

Se P non 6 in-

tersecata da h , allora k anche una regione relativamente ad Hvh Se invece P & intersecata da h , P si pu6 suddividere d in tre parti disgiunte: due sottoinsiemi aperti di R , che so no regioni relative ad H u h , e Poh , che 6 una regione relati va ad Fh Viceversa, se Q 6 una regione relativa ad Fh , allora esi

.

.

ste una regione P relativa ad H tale che Q = Pnh ; infatti,

se cia non fosse vero, avremmo necessariamente Q = h l per gualche h8 e H

, e quest0 implicherebbe h

= h8 r H

, il che & ass-

do. Quindi, ogni regione di H corrisponde ad una regione di H u h , oppure a due regioni di Huh e ad una di Fh , e questa corrispondenza esaurisce le regioni di Huh e di Fh Quest0

.

prova la ricursione (x). Poniamo ora, per ogni insieme finito H di iperpiani di I?d.

.

e proviamo che 6 soddisfatta la recursione:

Sia

y e LHuh ; l'intervallo @,y]

metric~,per o w i e considerazioni.

in L ~ u h 6 un reticolo geoDi conseguenza si ha:

P~ indicano la funzione di ~8biusdi LHvh e di dove PHvh L~ , rispettivamente. Infatti, se h 1 y , allora

se invece h 5 y , l'identit; segue dal fatto che C6,y] & un reticolo geometric0 e 1 p (6.y) 1 6 un invariante di Tutte-GroHuh thendieck. Sommando le identits ( x x ) per ogni y c LHuh si ot-

tiene la ricursione voluta. Osserviamo ancora che, se l'insieme H 6 costituito da un solo elemento, cio& & la catena di due elementi, si ha LH

Ora, poich& le funzioni V(H)

e C(H)

soddisfano la medesi-

ma ricursione sulla classe ereditaria dei semireticoli associati ad insiemi di iperpiani, ed assumono lo stesso valore sulle catene con due elementi, si ha

per ogni insieme di iperpiani H

9.2.

COROLLARIO

.

Sia H una famiglia centrale; allora

Osserviamo che, se H & una famiglia finita di iperpiani di JRd

tale che

hnk # $

perogni

h,kcH

,

allora si pua completare il semireticolo LH .aggiungendoun elemento massimo, ed il reticolo L;I cosi ottenuto 6 un reticolo geometrico.

85

9.3.

S i a H una famiglia d i i p e r p i a n i t a l e che

COROLLARSO

hnk

#4

per ogni

h, k c H ;

allora

dove

9.4.

p

s i intende r e l a t i v a ad

TEOREMA

Lg H

-.

d Sia H una famiglia f i n i t a d i i p e r p i a n i d i IR

.

Allora

DIMOSTRAZIONE

S i o t t i e n e applicando il r i s u l t a t o precedente

t u t t i i f i l t r i pri.ricipali d i L v e n t i l o s t e s s o rango.

9.5.

COROLLARIO

H'

a

e sommando s u t u t t i i punti a-

a

s e H 8 una famiglia c e n t r a l e , r i s u l t a

d S i a H una famiglia f i n i t a d i i p e r p i a n i d i IR ; l ' i n v a r i a n t e d i Eulero d i Bd

6 d e f i n i t o come l l i n t e r o :

9.6.

Ltinvariante d i Eulero non dipende d a l l a famiglia d d d i iperpiani H I n o l t r e , k(W ) = (-1 )

TEOREMA

.

.

Per ogni famiglia H r i s u l t a :

DIMOSTRAZIONE

S i a H una famiglia f i n i t a d i iperpiani d i

canpatto d i

IRd

IR

d

,e

con interno non vuoto; indichiamo con

s i a K un HI

il

sottoinsieme d i H c o s t i t u i t o dagli iperpiani che intersecano llinterno d i K .

H'

induce a l l o r a una partizione d i K ; indi-

chiamo con

f ' il numero d e l l e facce d i dimensione j d i t a l e j partizione. La c a r a t t e r i s t i c a d i Eulero d i K s i definisce come 1 1

intero

9.7.

TEOREMA

Per ogni cornpatto K d i IRd

con i n t e r n 0 non v x

t o e per ogni famiglia H d i iperpiani r i s u l t a

DIMOSTRAZIONE

Sia H'

il sottoinsieme di H costituito dagli

iperpiani che intersecano l'intero di K ; indichiamo con SH

il

semireticolo ottenuto da LH,

eliminando gli elementi relativi

a sottospazi di ?Rd

Relasi possono ripetere tutti i ragio

che non intersecano llinterno di K

tivamente a1 semireticolo SH

.

namenti fatti in precedenza per LH ; procedendo come nella dim2 strazione del teorema precedente, si ha la tesi.. d Sia H una famiglia finita di iperpiani in IR , e sia L~ il semireticolo ad essa associato; in generale, LH risulta is2 morfo ai semireticoli associati ad altre famiglie di iperpiani in spazi di dimensione diversa. Fra queste famiglie, diremo rap n presentazione minimale di LH una famiglia di iperpiani di IR , con n = r(L H ) , il cui semireticolo associato sia isomorfo ad L~ ' Vogliamo ora esaminare il caso in cui alcune delfe regioni

relative alla Pamiglia H di iperpiani siano limitate; osservis mo che, in questo caso, la famiglia H non 6 centrale, ed d una rappresentazione minimale del semireticolo ad essa associato.

9.8.

TEOREMA

Sia H uxia famiglia di iperpiani di ]lid

che sia

una rappresentazione minimale del semireticolo LH ad es-

sa associato; allora, il numero di regioni limitate indivL duate da H d dato da

DIMOSTRAZIONE

Sia h un iperpiano di IR

d

, h 4H ; poniamo

Con argomenti analoghi a quelli utilizzati nella dimostrazione del Teorema 9.1, si ottiene che 1 soddisfa la seguente ricursione:

Dato che, se

)HI

= 1

, risulta:

6 sufficiente provare la ricursione:

I t P~~~(B*x)I 1 t

PH(~,X)I +

=

XeL~vh

Xt2LH

It

PHuh(h,x)I XCLHuh

; analogamente a quanto fatto nella dimostrazione Huh del Teorema 9.1, si dimostra che

Sia x c L

di conseguenza: p ~ v(6.~1 h = xrLHuh

XeL H

E

~ ~ ( 6 . x- ) ~~,~(h*x) XCLHuh

i reticoli ottenuti da LH ed Indichiamo con iH ed iHvh L , rispettivamente, aggiungendo un massimo 1 ; in questi Huh reticoli l'identita ( x x ) diventa: CI

.

Ax "X Osserviamo ora che i reticoli LH ed LHuh , duali di iH ed A L rispettivamente, sono semimodulari; di conseguenza, grazie Hllh

alllaffermazione d) del Teorema 7.4, abbiamo

che 6 equivalente alla ricursione (x).

Consideriamo ora lo spazio proiettivo reale pd

, di

dimend sione d Se H 6 una famiglia finita di iperpiani di IP , pos d siamo considerare la partizione di JP - H in parti connesse mas-

.

sirnali, che chiameremo regioni, e definire i numeri C(H)

e

f .(H) in modo analog0 a quanto fatto nel caso affine. CostruenJ do analogamente a1 caso affine il semireticolo LH , questo ris* ta un reticolo geometrico.

9.9.

TEOREMA

Sia H una famiglia finita di iperpiani di

4;

allora

DIMOSTRAZIONE Ricordando la corrispondenza tra sottospazi proiettivi di JPd e sottospazi vettoriali di IRdf1 , abbiamo che la famiglia H corrisponde ad una famiglia HI di iperpiani per llorigine in nd+', e risulta owiamente

d Ogni regione di 1P - H corrisponde a due regioni relative ad in Pd+l, sinmetriche rispetto al190rigine; cosi si ottiene

H,

la prima identit;. La seconda affermazione si dimostra vrocedendo in modo anal2 go, ricordando per6 che la faccia n h in pd corrisponde h€H solo alla faccia in Pd+1

.

hzHl

.

Sia H m a famiglia finita di iperpiani dello spazio affine

.

Pd

Consideriano il completamento proiettivo I P ~di IRd ottg

nuto aggiungendo 1 iperpiano all*infinite, ha ; sia L

la re-

4

strizione del reticolo duale di quell0 dei sottospazi di al llinsiemedi atomi H uhm. L~ 6 un reticolo geometric0 e conH tiene come sottosemireticolo L~ ' La famiglia Ha= H u h a o di iperpiani di si dira canpletarnento proiettivo di H Le regioni relative ad Ha in pd sono nello stesso nwnero di quelle relative ad H in IRd ~isultainoltre

4

.

.

9-10. TEOREMA Sia H una famiglia finita di iperpiani in nd , che sia una rappresentazione minimale del semiretim colo associato LH ; detto LH il reticolo del completg mento proiettivo di H , risulta

dove P

6 llinvariante di Crapo.

DIMOSTRAZIONE tic010 : L

Indicando con

la funzione di ~8biusdel re

, per il ~orollario7.9, si ha:

Grazie a1 Teorema 9.8,

6 dunque sufficiente provare che

questo segue immediatamente dal fatto che , nell * intervallo

@,

del reticolo : L , con x h , , non ci sono elementi maggiori di ha, quindi gli intervalli [z,x] in : L e [;,XI in LH sono isomorfi.

9.11.

m

PROWSIZIONE

d sia H una famiglia centrale di IR

,e

sia h$H un iperpiano parallel0 ad uno degli iperpiani in H Allora

.

DIMOSTRAZIONE E' sufficiente osservare che le regioni limitate relative alla famiglia Huh corrispondono biunivocamente a& le regioni limitate indotte su h dagli iperpiani

.

di Ed-'

quindi,

dove ih,?], lo lrbh

.

6 ~lintervallosuperiore con minimo h nel retic? Ora, si verifica facilmente che la funzione

y :x

(dove il sup 8 inteso in

-+ x v h

L ~ u)h

& un isomorfismo di reticoli;

da qui si ha la tesi..

Vogliamo ora mostrare come i problemi affrontati a proposito delle partizioni di uno spazio affine (o proiettivo) mediante fa miglie di iperpiani penettano di provare in mod0 assai semplice alcuni significativi risultati relativi alle orientazioni aciclg che di un grafo. Sia G un grafo non orientato privo di lati paralleli e di loops; unlorientazioneaciclica di G 8 unforientazionedei lati di G tale che il grafo orientato cos? ottenuto non cohtenga circuiti orientati. Dato un grafo non orientato G con vertici p,, p2 pd , associamo ad esso la famiglia H(G) di iperpiani dello spazio d affine $ cosi definita: indicato con hi lliperpianodi I(

,...,

di equazione xi = xj * h..e~(G) se e solo se pi,pj sono vey ij tici adiacenti in G. Essendo H(G) una famiglia centrale, risulta un reticolo. Inoltre, 6 facile verificare che i L H(G) circuiti del grafo G corrispondono biunivocamente ai circuiti Di conseguenza, indicato con gG(h) il del reticolo L H(G) ' polinomio cromatico del grafo G e con c il numero delle compg nenti connesse di G

, applicando la ricursione di Tutte-Grothen-

dieck s i dimostra facilmente l t i d e n t i t ; :

Osserviamo ora che s u s s i s t e una corrispondenza biunivoca t r a l e r e g i o n i r e l a t i v e a l l a famiglia c l i c h e d e l grafo G . ogni l a t o

f pi,

pj)

H ( G ) e l e orientazioni aci-

I n f a t t i , f i s s a t a una regione d i di

G

, tutti

, per

H(G)

i punti d e l l a regione soddisfa-

no una (ed una s o l a ) d e l l e disugwglianze:

orientiamo a l l o r a il l a t o {pi, pj )

scegliendo come sorgente

pi

x i < xj ' E ' f a c i l e v e r i f ic a r e che l t o r i e n t a z i o n e c o s i ottenuta 6 a c i c l i c a .

se & s o d d i s f a t t a l a disuguaglianza

D a queste considerazioni segue: 9.12.

TEOREMA

I1 nwnero d e l l e orientazioni a c i c l i c h e del gra-

f o G 6 dato da

dove L b il r e t i c o l o geometric0 associato a1 grafo G

9.13.

TEOREMA

Fissato un l a t o ( p i , p j i

.,

d e l grafo G , il numg

r o d e l l e orientazioni a c i c l i c h e d i G aventi come m i c a sorgente

pi

e unico pozzo p

j

6 dato da

dove L 8 il r e t i c o l o geometrico associato a1 grafo G , e c ( x ) & il nwnero d e l l e componenti connesse del grafo ass2 c i a t o a1 punto x d i L

.

Sia d il nwnero dei v e r t i c i del grafo G , e s i a

DIMOSTRAZIONE

h

che interseca

H(G) il

. pj .

l'iperpiano avente equazione lato

f pi,pj)

h

x = x. + i Ogni regione d i j 1 d i luogo ad un*orientazione d i G i n c u i

& orientato da

pi

a

I n o l t r e , con cons&

derazioni elementari s i dimostra che, t r a queste regioni, quelle che danno luogo ad orientazioni del t i p o voluto sono t u t t e e sol e quelle che corrispondono a regioni l i m i t a t e i n una rappresentazione minimale del r e t i c o l o

L ~ u h'

Da qui segue l a tesi..

Con metodi analoghi s i dimostra i n f i n e il seguente r i s u l t a t o :

Sia

TEOREMA

9.14.

pi

un v e r t i c e del grafo G

l e orientazioni a c i c l i c h e d i

per c u i

G

.

I1 numero d e l

pi

r i s u l t a l'u-

nica sorgente 6 dato da

1 dove

p

z

XG(0)I =

/( \ O

6

)1

se G

+ connesso

altrimenti,

8 l a funzione d i ~ 8 b i u sd e l r e t i c o l o geometrico a s s o c i s

t o a1 grafo

*I

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CEX TRO IPU'TERNAZI ONALE MATEMATIC0 E S T I V O (c.I.M.E.

1

SOME REMARKS ON THE C R I T I C A L PROBLEM

ANDREA B R I N I

SOME REMARKS ON THE CRITICAL PROBLEM

Andrea B r i n i ( U n i v e r s i t s d i Bologna)

Introduction

I n 1970, Crapo and Rota proposed a reformulation of a c l a s s i c a l extremal problem on f i n i t e v e c t o r spaces, namely, t h e problem of f i n d i n g t h e l a r g e s t dimension o f a subspace having empty i n t e r s e c t i o n w i t h a given s e t of v e c t o r s S. Be3ides achieving an h i g h e r degree of g e n e r a l i t y , t h e i r most p l e a s i n g re-

s u l t was t h a t of showing t h i s problem t o be equivalent t o a problem concerning the l o c a t i o n of zeroes of t h e c h a r a c t e r i s t i c polynomial a s s o c i a t e d t o the mat r o i d s t r u c t u r e induced

on t h e s e t S.

Recently, Kung, Murty and Rota succeeded i n proving t h a t an analogous problem

on f i n i t e a b e l i a n groups admits a s o l u t i o n i n terms of t h e l o c a t i o n of

zeroes of another f u n c t i o n , t h e so-called RSdei z e t a

f u n c t i o n of a r e t of

points i n a Dirichlet l a t t i c e . I n t h i s paper, we deal with an u n i f i e d e x ? o s i t i o n of sone extremal problems which can be solved i n t h i s way. I n s e c t i o n 1,we d e s c r i b e t h e c r i t i c a l problem f o r f i n i t e a b e l i a n groups and i t s connection with a c l a s s of RSdei z e t a

fun-

c t i o n s . I n s e c t i o n 2,we r e c a l l Crapo and Rota's Theorem r e l a t i n g t o the c r i t i c a l problem f o r . f i n i t e v e c t o r s p a c e s , h e r e seen a s a simple consequence of t h e preceding r e s u l t on groups. Section 3 summarizes s p e c i a l i z a t i o n s t o graph col o u r i n g s . I n s e c t i o n 4 , we r e c a l l connecti'ons

with t h e ( l i n e a r ) coding pro-

blem; i n p a r t i c u l a r , we show t h a t some c l a s s i c a l r e s u l t s on bounds can be e a s i l y d e r i v e d from purely matroid t h e o r e t i c f a c t s about Hamning spheres.

-l . A

l a t t i c e of DirichZet type i s a p a i r ( L , v ) , where i ) L i s a l a t t i c e with

8

element

i i ) v: LxL--+Z s a t i s f i e s t h e i d e n t i t i e s v(x,y) = 0

if

xCy

and v(x,y) = v(x,z)v(z,y)>O

f o r every

X6zSy.

Given a s e t E of elements i n L, l e t M(E) be t h e l a t t i c e spanned by E ; t h e s e t E i s c a l l e d a Re'dei s e t i f , f o r every element x i n M(E) and every p o s i t i v e i n t e g e r n , t h e number of elements y i n M(E) such t h a t v(x,y) = n i s f i n i t e . I n p a r t i c u l a r , every f i n i t e s e t i s a R6dei s e t . Given a R6dei s e t E of elements i n L and an element aeL, t h e R6dei

Zeta

function of E based on a i s d e f i n e d a s p(s;a,E)

where

'fi

<-l)lA1v(a,~)-s, ACE denotes t h e supremum i n L of t h e elements belonging t o t h e f i n i t e =

subset A. The following r e s u l t e x h i b i t s t h e connection between t h e REdei

zeta

function of a s e t E and t h e Mijbius f u n c t i o n of t h e l a t t i c e 11(E) spanned by E. (1.1) P r o p o s i t i o n : Let M(E) be t h e l a t t i c e spanned by E , 11 i t s Mijbius function; then ~ ( s ; G , E )=

v(S,X)V(~,X)-~. xr M(E)

Proof. I t i s e a s i l y seen t h a t we can d e l e t e elements from E u n t i l i t i s an a n t i c h a i n , without changing

i t s R6dei z e t a

f u n c t i o n : t h e n , we can suppose

t h a t E i s an a n t i c h a i n . Now, given xc?l(E), l e t us consider i n M(E); t h e s e t ~n[6,x] rem p6],

i s a cross-cut

of [8,x]

-

t h e i n t e r v a l L6.x)

and, by t h e Cross-cut Theo-

we g e t

Z

p ( ~ ; ~ =, ~ ) (-~)IAIV(^O,A)-S= ACE

Let G be a f i n i t e a b e l i a n group and L(G) be t h e modular l a t t i c e of i t s subgroups; by s e t t i n g v(A,B) =

IB(/IAI

f o r every p a i r A,B,ACB of subgroups of G,

t h e l a t t i c e L(G) t u r n s out t o be a l a t t i c e of D i r i c h l e t type.

Let

x = -

A

(xl,

= {A1,

.

.

,A 1 be a c o l l e c t i o n of subgroups of G; we say t h a t a k-tuple

,xk) of c h a r a c t e r s of G distinguishes

xi

i f , f o r every A

j'

there

t h a t X. i s not t r i v i a l on A.. The 3 c r i t i c a l problem i s t o f i n d t h e s m a l l e s t k f o r which such a k-tuple e x i s t s . e x i s t s a t l e a s t a character

(1.2)

Theorem: Let A = {A1,

in

IS. such

. , An }be a

c o l l e c t i o n of subgroups of a f i n i t e

a b e l i a n group G. The number of ordered k-tuples of G d i s t i n g u i s h i n g

A

x=

(xl,

.

,xk) of c h a r a c t e r s

is given by t h e e v a l u a t i o n

.

IGI~P(~;~,&) Proof.

F i r s t of a l l , we r e c a l l t h a t a c h a r a c t e r of G can be seen a s an homo-

morphism of G i n t o t h e complex u n i t c i r c l e ( s e e e.g. [13]).

Given t h e s e t

. ,A n land a k-tuple y = (xl, . ,xk) of c h a r a c t e r s , we d e f i n e t h e "kernel of x r e l a t i v e t o A" a s t h e l a r g e s t subgroup B i n M(A) - t h e l a t t i c e spanned by A - such t h a t X. i s t r i v i a l on B f o r every i; t h u s , B i s t h e sub-

A = {A1, -

group o f G generated by t h e s u b c o l l e c t i o n of a l l subgroups A

j

such t h a t

X. is

t r i v i a l on A. f o r every i. J The proof i s now by ~ g b i u si n v e r s i o n over M(A). L e t us d e f i n e two f u n c t i o n s

i n t h e following way: f(C) = number of k-tuples of c h a r a c t e r s of G whose kernel relative t o

A

is e x a c t l y C.

g(B) = number of k-tuples of c h a r a c t e r s of G whose kernel r e l a t i v e t o

A

c o n t a i n s B.

Thus, we have

f o r every B c If@). We r e c a l l now t h a t a f i n i t e a b e l i a n group G can be s p l i t t e d i n t o a d i r e c t sum

c1@c2$ ' *

.$cr

of c y c l i c groups of prime power o r d e r pei; hence, a c h a r a c t e r of G i s known

.

whenever i t s value on a generator of C . i s given, f o r every i = 1, ,r. Since e. i t s v a l u e must be an p l - t h r o o t of t h e u n i t y i n t h e complex f i e l d , we i n f e r t h a t t h e number of c h a r a c t e r s of a f i n i t e a b e l i a n group equals i t s order. Thus, we have

and, by M6bius i n v e r s i o n , f ( ~ )= f o r every B i n M(A).

2 ~(B,C)(IGI/ICI)~ C>,B

By s e t t i n g B =

6

and r e c a l l i n g ~ r o ~ o s i t i o(1.1), n

we g e t

the assertion. 3y Theorem ( 1 . 2 ) , t h e c r i t i c a l problem i s solved whenever z e r o e s of t h e RSdei

z e t a f u n c t i o n p ( s ; 6 , ~ ) of t h e s e t

&

a r e known.

2.As an i n s t a n c e o f t h e preceding r e s u l t , it i s p o s s i b l e t o e x h i b i t a simple proof of t h e well-known r e s u l t by Crapo and Rota [a] concerning t h e criticaZ

problem f o r f i n i t e v e c t o r spaces. This problem can be s t a t e d a s follows: given a s e t S = {vl,

.

1 of v e c t o r s i n t h e f i n i t e v e c t o r space V o f dimension n n over t h e Galois f i e l d GF(q), we say t h a t a k-tuple of l i n e a r f u n c t i o n a l s

-F =

(F1,

,V

. ,Fk) distinguishes

S if

ker Flnker F2n

..

k e r F k n S = 1;

t h e c r i t i c a l problem i s t o f i n d t h e s m a l l e s t p o s i t i v e i n t e g e r k such t h a t t h e r e e x i s t s a k-tuple of l i n e a r f u n c t i o n a l s d i s t i n g u i s h i n g t h e s e t S. This i n t e g e r i s c a l l e d t h e c r i t i c a z exponent of S and w i l l be denoted by c(S;q). A s s t a t e d 'in t h e i n t r o d u c t i o n , t h e c r i t i c a l problem i s e q u i v a l e n t t o t h a t of f i n d i n g t h e l a r g e s t dimension of a subspace having empty i n t e r s e c t i o n w i t h a given s e t of v e c t o r s . More p r e c i s e l y , we have (2.1) Proposition: Let S be a s e t of v e c t o r s i n t h e v e c t o r space V.

The

l a r g e s t dimension k of a subspace U of V such t h a t U n S = 1 i s given by k = n

- c(S;q).

We come now t o t h e main r e s u l t of t h i s s e c t i o n : (2.2) Theorem: Let S be a s e t of v e c t o r s i n V , M(S) be t h e matroid induced on S by r e s t r i c t i o n of V. Then, the number of ordered k-tuples of l i n e a r function a l s which d i s t i n g u i s h S equals qk(n-r (MI)

P(H(S) ;qk)

,

where P(E!(s)';~) i s t h e c h a r a c t e r i s t i c polynomial of t h e matroid M(S) and r(M) denotes i t s rank.

V a s t h e d i r e c t sum group G = Gl@ G2@..@G m ,where t h e a r e a l l c y c l i c groups of o r d e r p , pr= q; f u r t l ~ e r m o r e , t h e r e is an obvious

Proof. We can regard G.'s

i d e n t i f i c t i o n between l i n e a r f u n c t i o n a l s on V and c h a r a c t e r s of G. By Theorem (1.2),

we have \ c l k p ( k ; 6 , s ) = qnk

5 p@,x)q-r(x)k=

xeM(S)

Let M(S) be a matroid without loops, r e p r e s e n t a b l e over GF(q). The preceding Theorem l e a d s u s t o d e f i n e t h e c r i t i c a l exponent C(M;q) o f M

as the smallest

p o s i t i v e i n t e g e r k such t h a t P ( M ( s ) ; ~ ~ ) > o . Obviously, i f 6: S 3 V is any r e p r e s e n t a t i o n o f M(S) i n V, we g e t

Furthermore, we have (2.3) Wq).

Proposition: L e t M(S) be a matroid without loops, r e p r e s e n t a b l e over Then i ) ~ ( ? 4 ( ~ ) ; q ~ ) f3o0r every non-negative i n t e g e r k i i ) P(M(S); q j ) > 0 f o r every i n t e g e r j*C(M;q).

S e v e r a l bounds f o r c r i t i c a l exponents

of r e p r e s e n t a b l e matroids have been

found i n r e c e n t years; j u s t a s an example, we mention a r e s u l t which we need i n t h e l a s t p a r t of t h i s paper. L e t R(M) denote t h e s e t of simple r e s t r i c t i o n s of My C ( M ) t h e s e t of coc i r c u i t s o f M and ral t h e s m a l l e s t i n t e g e r g r e a t e r o r equal than aaB; we have (2.4)

Theorem: I f M i s r e p r e s e n t a b l e over G F ( ~ ) and i t h a s no loops, then c ( M ; ~ ) s ~(2+max o ~

( min

N€R(M) CcC(P1)

ICI))]

.

3 . A proper k-colouring of a graph G = ( V , & ) i s a mapping y from the v e r t e x s e t -

7 t o a set A = faly

.

,ak? (colours) s a t i s f y i n g t h e following condition: i f

u and v a r e adjacent v e r t i c e s , then y(u)

+ y(v).

The coZow.ing probZem i s t o f i n d t h e s m a l l e s t p o s i t i v e i n t e g e r k such t h a t a

proper k-colouring of t h e graph G e x i s t s ; t h i s i n t e g e r i s c a l l e d t h e c h r o m a t i c

number of G and i s u s u a l l y denoted by x(G). Now, l e t K be any f i e l d . S e t

Let us o r i e n t t h e edges of G i n some fashion; i f t h e edge e = {u,v) i s d i r e c t e d from u t o v , we s e t e-= u and e+= v , r e s p e c t i v e l y . The coboundary operator i s t h e l i n e a r o p e r a t o r

6: RO(K)-R

(K) d e f i n e d

1

a s follows: (6g) ( e )

=

g(e+)-g(e-)

f o r every ecE and g6RO(K). The image space o f 6 i s c a l l e d t h e coboundary space of t h e graph G and w i l l be denoted by C(G,K). One e a s i l y checks t h a t k e r 6 = {f:V&K;

i f u and v a r e a d j a c e n t v e r t i c e s , then f ( u ) = f ( v ) ] ;

then, denoted by k(G) t h e number of connected components of G, we g e t dim ( k e r 6 ) and dim (C(G,K)) =

=

k(G)

IvI

-k(G).

Furthermore, C(G,K) i s independent of t h e choice of t h e o r i e n t a t i o n on G. For every eel?, we d e f i n e a l i n e a r f u n c t i o n a l on C(G,K) a s follows: = f ( e ) f o r every

feC(G,K).

Now, i t i s well-known t h a t t h e c y c l e matroid M ( E ) of t h e graph G

=

(It,S) i s

represented over K by t h e a p p l i c a t i o n

JI: E -+Hom(C(G,K) ,K) such t h a t $ ( e ) = Le f o r every esE. Thus,

M(E) i s r e p l e s e n t a b l e over any f i e l d K and we can s t a t e t h e following

IvI

(3.1) Theorem: S e t n = , K. = GF(q). There i s a b i j e c t i o n between t h e k s e t of a l l proper q -colourings of t h e graph G = ( V , E ) and t h e s e t of a l l n ordered k-tuples of l i n e a r f u n c t i o n a l s on V = K which d i s t i n g u i s h t h e s e t of v e c t o r s corresponding t o

E.

We g i v e two p r o o f s of t h i s r e s u l t .

a) L e t x ( A ) be t h e chromatic polynomial of t h e graph G = (V,E) and l e t r be G t h e rank of i t s cycle matroid M(E). The Tutte-Grothendieck recursion i n t h e h e r e d i t a r y c l a s s of graphic matroids y i e l d s t h e i d e n t i t y

.

~ ~ ( =1 P) ( G ) ~ ( ~; A( )~ ) By Theorem (2.2),

r e c a l l i n g t h a t k(G)=n-r, we g e t t h e a s s e r t i o n .

'la

b) L e t 6 be t h e l i n e a r o p e r a t o r

d e f i n e d a s follows:

f o r every LrHom(C(G,K) ,K) 'L

.

The o p e r a t o r 6 i s one t o one and hence t h e a p p l i c a t i o n

y i e l d s a r e p r e s e n t a t i o n of M(E) i n Hom(R (G,K) ,K). k O We choose B s a c o l o u r s e t A ( I ~ l = q ) t h e s e t of a l l k-tuples given mapping f: V-A Hom(Ro(G,K) JK)

, by

o n K ; hence, any

can be seen a s a k-tuple of l i n e a r f u n c t i o n a l s on

setting

f(v) = (fl(v),

.

,fk(v))

f o r every vsV. Furthermore, we have: f is -

k a proper q -colouring of G=(V,E)

such t h a t P O

W

*

f o r every eeE t h e r e e x i s t s an i

6fi(e) # 0 @ f o r every e6E t h e r e 'L

foreveryecE,(<6Lelfl>,

.

,

This completes t h e proof.'

(3.2)

Corollary: Let G be a graph without loops and l e t M(E) be i t s cycle

matroid. Then, f o r every prime power q, we have 9

CIM;q)-1

<XG 5 9

C(M;q)

4 . L e t V be t h e vector space of dimension n over GF(q). Given a d i s t i n g u i s h e d b a s i s B = Ibl, ,b 1 of V, t h e weight of a v e c t o r veV i s t h e number w(v) of

.

n

i t s non-zero coordinates w i t h r e s p e c t t o B. An [n,k,d]

-linear code i s a subspace C of dimension k of V, such t h a t

f o r every VPC -{01.

The i n t e g e r d i s c a l l e d minimwn distance of t h e code C.

Given n , d c ~ + , t h e ( l i n e a r ) coding problem i s t o f i n d t h e l a r g e s t i n t e g e r k such t h a t an[n,k,d]

- l i n e a r code e x i s t s .

Set S n,q,d-1

= IvcV; v#g

, w(v)$d}

;

we c a l l Hanmring matroid t h e matroid M(S ) o b t a i n e d by r e s t r i c t i o n of V n,q,d-1 on t h e s e t S n,q,d-1' The coding problem i s a s p e c i a l c a s e of t h e c r i t i c a l problem f o r f i n i t e v e c t o r s p a c e s . I n f a c t , by P r o p o s i t i o n (2.1),

we g e t

(4.1) P r o p o s i t i o n : An n k d l i n e a r c o d e o v e r GF(q) e x i s t s i f and o n l y i f t h e 1:s

9 1 -

inequality C(M(Sn,q,d-l)

;q) dn-k

holds. Now, s e t A

r

{(jl,

.

,j n ) c z n ;

moreover, f o r e v e r y ( j l ,

.

jl<j2<

<jn$n, ji i s non-negative for i=l,.,n);

,jn)eA, s e t

The s u b s e t

of V i s a f l a t of t h e m a t r o i d M(S

n,q,d-1

) and w i l l be c a l l e d

coordinate f l a t .

We have (4.2) P r o p o s i t i o n :

I f F i s a c o o r d i n a t e f l a t of M(Sn,q,d-l)

and r(F)6d-1,

then F is a modular f l a t . Proof. By i n d u c t i o n on t h e rank o f F. I f r ( F ) = l , t h e assumption t r i v i a l l y h o l d s . Now, assume t h e s t a t e m e n t t r u e f o r e v e r y c o o r d i n a t e f l a t o f rank small e r t h a n mSd-l.

Let F and H b e c o o r d i n a t e f l a t s , r(F)=m, r(H)=m-l,

HCF. L e t K

be any o t h e r f l a t o f M(S ). We c o n s i d e r two c a s e s : n,q,d-1 i ) Suppose K n H c K n F; t h e n r ( H ) + r ( K ) + l = r(KAH)+r(KVH)+l s r ( F n K ) + r ( F U K ) 6 r ( K ) + r ( F ) = r(K)+r(H)+l. Hence, (F,K) i s a modular p a i r . i i ) Suppose KflH = K n F. Let b t be t h e element o f t h e d i s t i n g u i s h e d b a s i s B

o f V which b e l o n g s t o F b u t n o t t o H. Assume t h a t bt i s spanned by t h e s e t KVH; t h e n t h e r e e x i s t s a minimal s e t {ki}V{h]of bt = L j h j + j Now, i f r ( F ) g d-1,

elements i n K and H such t h a t

5 iiki ,

p j ,Si€GF(q).

i

the vector

is i n FAH b u t n o t i n H n K , and t h i s i s a c o n t r a d i c t i o n . Hence, r(KUH) =

=

r ( K V F ) + l and (F,K) i s a modular p a i r . By P r o p o s i t i o n (4.2) and S t a n l e y ' s Theorem on modular e l e m e n t s i n geometric

lattices

([lj , Theorem

2 ), we g e t

(4.3)

P r o p o s i t i o n : The c h a r a c t e r i s t i c polynomial P(M(S );A) n,q,d-1 d-1 p o s i t i v e r e a l r o o t s ri, namely,

has a t l e a s t

i 'i = q is a root f o r every i = 0,1, ,d-2.

.

Proof. L e t F and H be c o o r d i n a t e f l a t s , H C F , r(H) = r(F)-16d-2.

The number of

which a r e c o n t a i n e d i n F b u t n o t i n H i s given by q r(H) ,q,d-1) By t h e Theorem mentioned above, q r ( H ) i s a r o o t of P(M(Sn ,q,d-l) ;XI* 1 - f l a t s o f M(s,

(4.4) C o r o l l a r y (The s i n g l e t o n bound) : 1 f a n b , k , d )

.

- l i n e a r code e x i s t s , t h e

inequality n-kad-1 holds. The s p e c i a l c a s e M(S a l s o [9]

, pag.

2) h a s been widely s t u d i e d by T.Dowling i n [ll] ( s e e n ,q, 290 f f . ). I n p a r t i c u l a r , h e proved t h a t e v e r y c o o r d i n a t e f l a t

o f M(Sn

i s a modular f l a t and hence i t s geometric l a t t i c e i s a super,q,2) s o l v a b l e l a t t i c e . Thus, we have n-1

and t h i s i m p l i e s (4.5)

P r o p o s i t i o n : ~ n F , k , d - l i n e a r code o v e r GF(q) e x i s t s i f and o n l y i f t h e

inequality l+(q-1) (n-l)
holds. The preceding r e s u l t i s nothing b u t t h e well-known Gilbert-Varshamov bound (

[lg , pag.34

) i n t h e case d=3, h e r e seen t o be a l s o a necessary condition

f o r t h i s c l a s s of codes. Our next aim i s t o d e r i v e t h e g e n e r a l form of t h i s bound a s a s p e c i a l i z a t i o n of t h e g e n e r a l bound on c r i t i c a l exponents given i n Theorem (2.4). (4.6) Theorem: There e x i s t s an[n,k,q - l i n e a r code over GF(q), provided d-2

Proof. Let G(Sn

d-l) be t h e combinatorial geometry a s s o c i a t e d t o t h e Hamming 39s matroid M(Sn d-l). For every c o o r d i n a t e hyperplane H ( j = l , ,n) of ~ q r j M(Sn,q,d-l) , denote by Ti the corresponding hyperplane o f G(Sn,q,d-l). It is j e a s i l y seen t h a t t h e c a r d i n a l i t y of H i s

.

j

a-1

furthermore,

H is IH .I .

if

o r equal than

any hyperplane of G(S n,q,d-1)'

i t s c a r d i n a l i t y i s smaller

J

Choose now a 1 - f l a t ~ G ( s ~ , ~ , we ~ - can ~ ) always ; f i n d an i n t e g e r j such t h a t

- -

P # H ~ . Then, we have max NCR(G(Sn

(min ,q ,d-1

Recalling t h a t P(C(Sn Theorem (2.4)

.

I c I ) =

1) ccC(N)

d-l) ; A ) >q,

= p(M(sn

,q,d-1

) ;A)

, the

statement follows by

References

1. G.D.BIRKHOFF, A determinantal formula for the number of ways of colouring a planar map, Annals of Math. 14 (1912), 42-46. 2. G.BIRKHOFF, "Lattice Theory", 3rd ed., Providence: Amer. Math. Soc. Coll. Publ. 25 (1967)

.

3. R.C.BOSE, Mathematical theory of symmetrical factorial design, Sankhya 8 (1947), 107-166. 4. A.BRIN1, Improving Gilbert-Varshamov bound for linear codes, in preparation. 5. T.H.BRYLAWSK1, Intersection theory for embedding of matroids into uniform geometrhes, Studies i n AppZ. Math. 61 (1979), 211-244. 6. T.H.BRYLAWSK1 and D.G.KELLY, "Matroids and Combinatorial Geometries", Carolina Lectures Series, Univ. of North Carolina, 1980.

7. H.H.CRAF'0,

MEbius inversion in lattices, Archiv.Math. 19 (19681, 595-607.

8. H.H.CRAF'0

and G.C.ROTA, "Combinatorial Geometries", MIT Press, Cambridge,

Massachussets, 1970.

9. P.DOUBILET, G.C.ROTA andR.P.STANLEY, On the foundations of combinatorial theory VI: The idea of generating function, Proc. 6th Berkeley Symp. on Math. Stat. and Prob., vol.11: Probability Theory. Univ. of California, 1972, 267-318. 10. T.A.DOWLING, Codes, packings and the critical problem, Atti del Convegno di Geometria Combinatoria e sue Applicazioni, Perugia, 1971, 210-224. 11. T.A.DOWLING, A q-analog of the partition lattice, A Survey of Combinatorial Theory (J.Srivastava, Ed.), North Holland Publ.Comp.1973, 101-115. 12. J.P.KUNG, M.R.MURTY and G.C.ROTA, On the Rddei zeta function, to appear

13. N.JACOBSON,

"Basic Algebra I", W.H.Freerpan and Comp., San F r a n c i s c o , 1970.

1 4 . F.J.MACWILLIAMS and N.J.A.SLOANE,

"The Theory of E r r o r - C o r r e c t i n g Codes",

North Holland Publ. Comp., 1977. Colouring, packing and t h e c r i t i c a l problem, Quart.J.Math. 29

15. J.G.OXLEY,

(1978), 11-22. 16. G.C.ROTA,

On t h e f o u n d a t i o n s o f c o m b i n a t o r i a l t h e o r y I: Theory of ~ g b i u s

f u n c t i o n s , Z. Warsch. 2 (1964), 340-368. 17. R.P.STANLEY, Modular elements i n geometric l a t t i c e s , Algebra Universalis

1 (1971). 214-217. 18. R.P.STANLEY, 19. W.T.TUTTE,

S u p e r s o l v a b l e l a t t i c e s , Algebra Universalis 2 (1972), 197-217

On t h e a l g e b r a i c t h e o r y o f graph c o l o u r i n g s , J. Comb. e.1,

(19661, 15-50. 20. W.T.TUTTE,

P r o j e c t i v e geometry and t h e 4-colours problem, Recent P r o g r e s s

i n Combinatorics (W.T.Tutte,Ed.) 21. D.J.WELSH,

"Matroid theory",

Academic P r e s s , 1969, 199-207.

Academic P r e s s , 1976.

22. N.WHITE, The c r i t i c a l problem and coding t h e o r y , v o l . 111, s e c t i o n 331, 1973, 37-66.

J e t Prop. Lab. SPS,

CENTRO I N TERNAZION ALE MATEMATICO E S T I V O (c.I.M.E.)

THE TUTTE POLYNOMIAL PART I: GENERAL THEORY

THOMAS BRYLAWSKI

D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of N o r t h C a r o l i n a C h a p e l H i l l , N. C. 27514

* T h i s w o r k w a s s u p p o r t e d b y N.S.F. G r a n t MCS- 7801149 and w a s p r e s e n t e d a t t h e T h i r d C.I.M.E. C o n f e r e n c e on M a t r o i d T h e o r y and i t s A p p l i c a t i o n s h e l d a t V a r e n n a , I t a l y , A u g u s t ,

1980.

1.

Introduction. Matroid theory (sometimes viewed as the theory of combinatorial

geometries or geometric lattices) is reasonably young as a mathematical theory (its traditional birthday is given as 1935 with the appearance of [159]) but has steadily developed over the years and shown accelerated growth recently due, in large part, to two applications. The first is in the field of algorithms. To coin an oversimplification: "when a good algorithm is known, a matroid structure is probably hidden away somewhere."

In any event, many of the standard good

algorithms (such as the greedy algorithm) and many important ones whose complexities are currently being scrutinized (e.g., existence of a Hamiltonian path) can be thought of as matroid algorithms.

In

the accompanying lecture notes of Professor Welsh the connections between matroids and algorithms are presented. Another important application of matroids is the theory of the Tutte polynomial

where aij

is the number of subsets A

and cardinality r(~)-i4j.

of M

with rank r(M)-i

The Tutte polynomial and its chief evalua-

tion, the characteristic polynomial

(the sum being taken over all elements in the geometric lattice associated with M, and the rank 'function and Mtlbius function u(0,x)

b e i n g computed i n t h a t l a t t i c e ) , h a v e come up i n a v a r i e t y of a p p l i c a t i o n s . The c h a r a c t e r i s t i c polynomial a s a l a t t i c e i n v a r i a n t can b e thought of a s a g e n e r a t i n g f u n c t i o n f o r t h e M6bius f u n c t i o n .

It h a s a p p l i c a t i o n s ,

of c o u r s e , t o o t h e r , non-geometric, l a t t i c e s and i t s r i c h g e n e r a l theory is p r e s e n t e d i n this'volume by P r o f e s s o r s Barnabei, B r i n i , and Rota. The T u t t e polynomial, on t h e o t h e r hand, seems t o b e s p e c i a l t o matroids.

I t i s t o i t s g e n e r a l theory and a p p l i c a t i o n s t h a t we

address our notes.

I n t h e p r e s e n t volume, we p r e s e n t t h e f i r s t p a r t ,

c o n c e n t r a t i n g ( a f t e r t h e n e x t , m o t i v a t i o n a l , s e c t i o n ) on t h e s t r u c t u r e of

t(M)

f o r a g e n e r a l m a t r o i d and on t h e n a t u r e of a "Tutte-

Grothendieck i n v a r i a n t . "

These l a t t e r i n v a r i a n t s d e s e r v e a s p e c i a l

t r e a t m e n t , a n d we g i v e i t i n t h e second p a r t t o a p p e a r elsewhere.

For

now, a s j u s t i f i c a t i o n f o r o u r g e n e r a l survey we g i v e a sampling of some of t h e a r e a s i n which c e r t a i n e v a l u a t i o n s o f t h e T u t t e polynomial c o i n c i d e w i t h important i n v a r i a n t s .

-

minimal flow c a l c u l a t i o n s i n networks

11,

27, 81, 82, 95, 103,

124, 136, 139, 140, 151, 153, 1541

-

graph c o l o r i n g [8, 27, 29, 39, 57, 64, 82, 95, 107, 117, 124, 135, 137, 138, 140, 142, 145, 146, 151, 153, 154, 157, 160, 1721

-

p e r c o l a t i o n t h e o r y [47, 71, 115, 116, 1521

-

hyperplane arrangements, convexity, and s e p a r a t i o n i n a f f i n e and p r o j e c t i v e s p a c e (32, 49, 50, 76, 77, 163, 164, 167, 168, 174, 1 7 5 1

-

a c y c l i c , t o t a l l y c y c l i c , and c o h e r e n t o r i e n t a t i o n s of graphs and o r i e n t e d m a t r o i d s [16, 44, 50, 61, 76, 77, 87, 89, 90, 134, 163, 167, 1701

-

zonotopes [77, 128, 163, 1671

- packing

and coding theory 136, 65, 75, 107, 153, 154, 1551

- intersection

numbers f o r s u b s e t s of p o i n t s i n a f i n i t e

p r o j e c t i v e space and t h e c r i t i c a l problem [27, 37, 43, 60, 65, 83, 107, 144, 148, 151, 153, 1541

- electrical

networks [12, 24, 33, 129, 103, 1301

- combinatorial - quantum and

- trees - signed

designs 119, 62, 63, 70, 105, 1621

s t a t i s t i c a l mechanics 19, 71, 1271

[7, 27, 331 and v o l t a g e graphs 166, 67, 168, 169, 170, 1721 -.

- Eulerian paths - covering

[91, 92, 99, 100, 1491

[lo41

-

s c o r i n g i n tournaments 176, 77, 1671

-

t o p o l o g i c a l d i s s e c t i o n s 1164, 1651

- embedding graphs - root

i n s u r f a c e s [91. 991

systems [168]

W e view t h e f a c t t h a t i n v a r i a n t s i n a l l the above a r e a s a r e e v a l u a t i o n s of t h e same polynomial a s ample evidence t h a t a general theory i s merited.

I n a d d i t i o n , i t o f t e n occurs t h a t a p p l i c a t i o n s

i n one a r e a s u g g e s t lnalogous formulas i n another.

A famous example

i s t h e c r i t i c a l exponent of Crapo and Rota applying i d e a s from t h e chromatic t h e o r y of graphs to coding and packing theory i n f i n i t e p r o j e c t i v e spaces.

I n f a c t , t h e T u t t e polynomial f o r matroids was

invented by Crapo [ 5 6 ] a s a g e n e r a l i z a t i o n of T u t t e ' s work i n graph c o l o r i n g 1137, 1381.

Recent examples a r e contained i n t h e work of

Oxley [107, 108, 1091 and i n [83],where t h e Hajos c o n s t r u c t i o n f o r graphs of chromatic number a t l e a s t

q

matroids of c r i t i c a l exponent a t l e a s t

i s analogized t o r e p r e s e n t a b l e k.

Although t h e s e n o t e s s k e t c h ( o r give r e f e r e n c e s t o ) previously published r e s u l t s , much i s new.

I n s e c t i o n t h r e e , we e x p l o r e t h e

n a t u r e of what i s meant by a "Tutte-Grothendieck

invariant," d i s t i n g u i s h -

i n g s e v e r a l types while r e l a t i n g them t o

I n s e c t i o n f o u r , we

show t h a t v a r i o u s o p e r a t i o n s on

M'

t(M).

(such a s adding a p o i n t i n f r e e

p o s i t i o n and "tensoring" ( a new o p e r a t i o n ) ) a f f e c t t h e T u t t e polynomial i n p r e d i c t a b l e ways,while f o r o t h e r s (such a s t h e f r e e e r e c t i o n ) , t h e T u t t e polynomial of t h e r e s u l t i n g matroid general be c a l c u l a t e d from

t(M).

M'

cannot i n

Section f i v e concerns reconstruction:

what p a r t i a l information about a matroid allows one t o c a l c u l a t e i t s T u t t e polynomial?

Conversely, what information can

t(M)

give us

about u s e f u l s t r u c t u r a l i n v a r i a n t s (such a s t h e number of c e r t a i n closed s e t s ) ?

The t o t a l number of closed s e t s (indexed by rank and

c a r d i n a l i t y ) cannot i n general be t a b u l a t e d i f only but formulas f o r i t a r e given f o r c e r t a i n c l a s s e s . of t h e s e c l a s s e s

- near-designswhich

t(M)

i s known,

We explore one

simultaneously g e n e r a l i z e

p r o j e c t i v e spaces and paving matroids. I n t h e f i n a l s e c t i o n , we study some g e n e r a l i d e n t i t i e s and i n e q u a l i t i e s (such a s l o g concavity) s a t i s f i e d by t h e e v a l u a t i o n s and c o e f f i c i e n t s of

t(M).

Many i n e q u a l i t i e s depend on parameters

and a r e s h a r p , becoming e q u a l i t i e s when t h e matroid i s i n a certain class.

Examples of t h e s e extremal c l a s s e s a r e given, a s

we-1 as a g e n e r a l franework t o f i n d o t h e r extremal c l a s s e s and t h e i r r e s p e c t i v e sharp bounds.

A few e x e r c i s e s and many research

problems accompany each chapter. We w i l l assume t h e kind of f a m i l i a r i t y with matroid theory obtained from [151] (or, t o g i v e t h e o t h e r coauthors equal treatment, 1421 o r 1601). W e v i s h t o thank t h e C.I.M.E.

f o r t h e i r sponsorhip of t h e

l e c t u r e s on which t h e s e n o t e s a r e based and f o r a f f o r d i n g an o p p o r t u n i t y f o r a l l of us a t t h e conference t o s h a r e i d e a s i n t h e b e a u t i f u l surroundings of Varenna. I n a d d i t i o n , we thank Hazeline Lewis; Gary Gordon, P r o f e s s o r Rhodes Peele; Hazeline Lewis; and P r o f e s s o r Adriano B a r l o t t i , r e s p e c t i v e l y , f o r t h e i r utmost p a t i e n c e during t h e typing, proofreading, r e t y p i n g , and overdue r e c e i p t of t h i s paper. V o r r e i d e d i c a r e q u e s t i appunti a t u t t i i miei amici i t a l i a n i ed i n p a r t i c o l a r e a Bruna ed a l l a n o s t r a nuova v i t a insieme.

2.

A P r o t o t y p i c a l Example. I n t h i s s e c t i o n , we i l l u s t r a t e t h e idea of a "Tutte-

Grothendieck" theorem:

one which can be ( e a s i l y ) v e r i f i e d on loops

and isthmuses, and which i s then proved i n d u c t i v e l y on matroids of

K p o i n t s by showing a r e l a t i o n s h i p between r e l e v a n t p r o p e r t i e s of G

on t h e one hand and t h e p a i r

(G-p,

G/p)

on t h e o t h e r .

I n the

course of t h e proof, many i d e a s w i l l be presented such a s t h e n a t u r e of d e l e t i o n and c o n t r a c t i o n f o r v a r i o u s s p e c i a l c l a s s e s of matroids (graphic, r e p r e s e n t a b l e , e t c . ) . Theorem 2.1. 1.

For a binary matroid

is affine (i.e.

M

M

t h e following a r e equivalent.

M,

i s isomorphic t o a s u b s e t

of some b i n a r y a f f i n e space

AG(n,2)

w i t h , perhaps,

multiple points). I n a binary v e c t o r r e p r e s e n t a t i o n f o r

2.

exists a linear functional f

3.

(x)+ 0

(i.e.,

A l l c i r c u i t s of

f (v) M

-

f

1)

M,

there

such t h a t

1E M.

for a l l

a r e even ( i . e . ,

have even

cardinality).

4.

A l l hyperplane complements of

5.

M*

C

a r e even.

has a p a r t i t i o n i n t o c i r c u i t s .

For 6 and 7 when

6.

M*

M

is viewed a s a l i n e a r code

contains the vector

(1,1,

...,1 ) .

C:

$, the dual code, is an even-weight code.

7.

If M

is graphic with graphic representation G(M): 8.

If M

G(M)

is two-colorable. of M*

is cographic with a representation G(M*)

as a

connected graph:

9.

G(M*)

10.

M

has an Eulerian cycle.

is loopless,and the associated geometry of My

fi, obeys any (or all) of the above properties. No pair of the above equivalent properties is hard to prove directly. (1-2, ture.

Some are classical 3-4,

5-6,

etc.),

(3-8,

4-91,

some are trivial

and all have appeared in the litera-

Our proof'will be different than the ones usually presented

as it will proceed by induction. In particular, for each i, let

xi

be the "characteristic function" of the property Pi, i.e.

1 if M xi(M)

satisfies Pi

=

0 if M does not. Then each

xi

is an invariant and we will show that each obeys

the following two boundary conditions: B1.

xi(M)

= 0

B2.

xi(M)

=

Further.

xi

if M

1 if M

is a loop. is an isthmus.

obeys the following recursions:

Bg.

xi(M)

=

xi(M-p) -xi(p)

B4.

xi (M)

=

xi(M-p)-x

an isthmus.

i (M/p)

if p is a loop or isthmus. if p 'is neither a loop nor

It then follows that all of the properties are equivalent since xi(M) =

xj (MI

on a one-point matroid (necessarily a loop

or an isthmus), and, by an induction hypothesis,on smaller matroids. If,for example, M has a point p which is neither a loop nor an

- xi(M/p)

isthmus, xi(M) = xi(M-p)

=

- xj (M/P)

xj(M-p)

=

xJ (MI.

The rest of this section will be devoted to showing that

-

...,

xi

obeys the system B = {B .B ,B ,B for i 1,2, 10. A common 1 2 3 4 thread in many of the proofs will be that x (M) counts something i which is positive exactly when Pi holds. The first property is a geometric one and we will use synthetic arguments. Proof of PI.

By considering the affine span of M, it is easy

to see that M

is in some binary affine space if and only if it

is in AG(n-1,2), where n = r(M).

the affine space of dimension n-1 Let

xi

M in an embedding of M one since PG(n-1.2)

-

(rank n),

be the number of hyperplanes which miss in PG(n-1,2).

{H1,H2) = AG(n-1.2)

(Clearly there is at most

- H = AG(n-2,2),

an

affine space of lower rank.) We will show that

xi

obeys B and thus so will

xl.

There are no loops in affine space while an isthmus is isomorphic to AG(0.2).

Hence, B1,BZ,

as well as B3 when p is a loop

are all trivfal. If p is an isthmus and H then clearly H n

is a hyperplane which misses M,

is a hyperplane in the.projective span of

M-p, of

G, which misses M-p. M-p which misses M-p,

PG(n-1,2)

which misses

l i n e containing

p

M, where

misses

M-p.

H' u q q

i s a hyperplane of

M-p.

This r e l a t i o n s h i p

= x;(M-p)*xi(p).

i t w i l l be e a s i e r t o show t h a t

B4,

+ x;(M/p).

For t h i s , l e t

Then,it e i t h e r c o n t a i n s

p

be any hyperplane which

H

o r i t does not.

l a t t e r c a s e i t g i v e s a hyperplane which misses c a s e t h e r e i s an associated hyperplane which misses

M/p.

a hyperplane

HI'

l i n e s through

p, a l l t h e p o i n t s of

M/p,

then

p a s s e s through

in

8' u p

p.

p

p

PG(n-2.2) means picking

and p r o j e c t i n g , v i a

H, PG(n-1,2),

and

M-p,

and i f a hyperplane

is a hyperplane which misses

This exhausts a l l t h e cases.

by

K

P2 and

matrix

f

H'

M-p

and

M

and

P6.

M-p.) A binary representation f o r

N where n = r(M), K = !MI,

such t h a t

M

i s an

and dependency i n

corresponds t o l i n e a r dependency among t h e columns of functional

it

M/p, s i n c e t h i s would lead t o two d i s t i n c t

hyperplanes which m i s s Proof of

M,

(For example,

n o t e t h a t t h e r e cannot be both a hyperplane with misses another which misses

M-p

Conversely, i f a hyperplane misses

H".

g i v e s a hyperplane which misses misses

H' = H/p

which does n o t c o n t a i n

In the

M, and i n t h e former

Here p r o j e c t i o n by t h e point

r e s p e c t i v e l y onto

i s a hyperplane

H'

i s t h e t h i r d p o i n t on some

xi(M) = x;(M-p)

To prove r e c u r s i o n =

then

and another p o i n t of

is bijective so that

x;(M-p)

Conversely, i f

N.

f ( v ) = 1 f o r a l l columns 1 of

corresponds t o a row v e c t o r

w

such t h a t

w.N =

1

where

n M

A linear

N

then

Iis

t h e v e c t o r of a l l ones.

1

the vector of l e n g t h

b e i n g i n t h e row s p a c e of

c o n s i s t i n g of t h e

x2

is equivalent t o

However, t h e code

N.

K and dimension n a s s o c i a t e d w i t h a m a t r o i d

N

r e p r e s e n t e d by a m a t r i x

Thus

w

The e x i s t e n c e of s u c h a

and

x6

IF/"

v e c t o r s spanned by t h e rows of

y such t h a t T N

=

1.

xl.

i s z e r o i n t h a t column f o r any, y.

r e p r e s e n t e d by t h e m a t r i x

111

w-N -

n

matrix,

= f&.P

c o n t a i n s a loop,

M

F u r t h e r , a n isthmus i s

and f o r t h i s r e p r e s e n t a t i o n , t h e s c a l a r

w

such t h a t

row-equivalent r e p r e s e n t a t i o n s of x

X' 2

1 g i v e s t h e d e s i r e d conclusion.

The number of v e c t o r s

n

Let

A loop i s r e p r e s e n t e d ( i n

any dimension) by a column of z e r o s , ' and t h u s , i f

w=

N.

a r e e q u i v a l e n t and we can view o u r proof a s a n

count t h e v e c t o r s

vector

M

i s p r e c i s e l y t h e v e c t o r subspace of

a n a l y t i c a l f o r m u l a t i o n of t h e s y n t h e t i c arguments f o r

w-N -

C

P-N

-1).(P-N).

TN

=

M, s i n c e i f

also represents

M

NOW assume t h a t

1

i s p r e s e r v e d under

P

is a nonsingular

by s t a n d a r d t h e o r y , w h i l e

p E M

i s n o t a l o o p and

( a p p l y i n g a p p r o p r i a t e row o p e r a t i o n s i f n e c e s s a r y ) i s r e p r e s e n t e d \

of

by t h e ( f i r s t ) column v e c t o r

t h e f o l l o w i n g form:

where n-1

x

=

2

i f and o n l y i f

The m a t r i x

N

i s t h e n of

J

i s a row v e c t o r of l e n g t h

K-1,

N.

p

K-1, A

i s a m a t r i x of s i z e

i s a n isthmus,

A

represents

M/P,

[ i]

a d

represents

M-p.

Using t h e s e r e p r e s e n t a t i o n s , i t i s obvious t h a t , i f isthmus, vector If v -

w'-A =

w'

i f and o n l y i f

preceded by a one.

p

Thus,

1,where

=

FN

B3

y is the

is satisfied.

i s n o t an isthmus, t h e n some e n t r y of

i s 1. L e t

w

be a v e c t o r such t h a t

w

i f and o n l y i f

y

[a]

is t h e v e c t o r

w

the invariant

xi,

= 1.

-

Then

h a s a one i n i t s f i r s t coordinate,and

a z e r o i n i t s f i r s t c o o r d i n a t e i f and only i f

Proof o f

is an

p

w'.A

t h i s shows t h a t

xi(M-p)

= x;(M)

=

w

has

= 1, where

with its f i r s t coordinate deleted.

1

TN

w'

I n terms of

+

x;(M/p).

A loop i s a n odd c i r c u i t , a n d an isthmus i s i n no

P,.

c i r c p i t , s o t h a t recursions Now assume t h a t

p

i n some c i r c u i t

C.

If

x 3 (M-p)

then

M

does a l s o

= 0

B1,B;.

and

B

3

are trivial for

x3(M).

i s n o t a n isthmus,so t h a t i t must b e contained

x 3 (M-p)

We prove t h e r e c u s i o n M-p

c o n t a i n s a n odd c i r c u i t

(x3(M) = 0 ) .

I n addition,

of ( a t most two) c i r c u i t s i n

C'

= x3(M)

C'

+

x3(M/p).

so that

i s t h e d i s j o i n t union

M/p. (The subgeometry

C u p

has

n u l l i t y a t most two,so i s g r a p h i c w i t h g r a p h i c a l r e p r e s e n t a t i o n a

8 graph i n t h e n u l l i t y - t w o case.) x3(M/p) = 0. circuit

C

Thus, one of t h e s e c i r c u i t s i s odd, and

On t h e o t h e r hand, assume

containing

p

x3(M-p)

= 1.

If the

i s odd, i t i s an elementary p r o p e r t y of

t h e c i r c u i t e l i m i n a t i o n axiom f o r b i n a r y matroids t h a t a l l c i r c u i t s containing

p

a r e odd,and t h a t

M/p

c o n t a i n s o n l y even c i r c u i t s ,

-

in which case x3(M/p) circuits containing p.

Proof of P4.

1, and x3(M) = 0. Thus

x3 (M)

= 1

If C is even, so are all and x3(M/p) = 0.

P4 is of course a (dual) restatement of P3 since

*.

circuits of M are bonds (complements of hyperplanes) of M

Let

us,however,see what a direct proof (involving bonds) would involve.

*

1 if M has all even bonds

Then,

=

Let x4(M)

0 otherwise.

*

x4(M)

= x4(M

*

r

*

and the recursions in the proof of

x3(M ),

) =

x3

become:

*

B1.

*

*

x3(M ) = 0 if M

*

is a loop

B2.

x3(M*)

=

1 if M*

.:B

x3 (M*) *

=

X (M*-~).X~(P) 3

x,(M*)

=

x3(M*-p)

is an isthmus if p

is a loop or isthmus

of M

*

B4.

-

x3(~*/p)

if p is neither a

loop nor an isthmus. But standard matroid theory shows us that a loop is dual to an isthmus,

* - B4*

and that deletion is dual to contraction. Thus, B1

*

B1.

*

B2.

*

B.

*

x4(M)

*

x4(M)

*

x4(M)

= 0

if M

become:

is an isthmus

=

1 if M

is a loop

=

x~(MIP)-x~(P) = xt(~-p)-x;(p)

if p

is an isthmus

or loop

* B4.

*

x4(M)

=

x;(M/p)

-

x*(M-p) 4

if p

is not a loop or isthmus.

Thus a proof of property

* B1

*

- B4

for

P5.

Proof of

M

If x;(M)

* x4,

i s equivalent t o proving r e c u r s i o n s

P'

and we w i l l u s e t h i s observation i n our proof of P 5'

* - B*

A s o u t l i n e d above, we must prove c o n d i t i o n s

[1

if

CI,

otherwise.

h a s a loop

p,

= x5(M-p). x;(p)

4

for

has a p a r t i t i o n into c i r c u i t s

M

*

B1

clearly

= x;(M-p).

If

contains an isthmus,

M

i t cannot b e p a r t i t i o n e d i n t o c i r c u i t s (an isthmus being i n no

*

c i r c u i t ) , s o we may concentrate our proof on r e c u r s i o n m a t r i x proof.

When

i s represented by t h e m a t r i x

M

elementary t o show t h a t

M

only i f every rob of columns of

has an even number of ones.

and

,I:[

and

w e n rows (i.e. a l l even rows

Proof of w -

P

7'

such t h a t

x ~ ( M= ) x;(M-p) M/p

(x:(M)

is a s i n the

N

Assume

If

w.v_

=

cL

A

has an odd row, s o do

= x:(M/~) = 0.

If

A

has a l l

has a p a r t i t i o n i n t o c i r c u i t s ) , then a.1)

iff

The dual code

of

C'

C

has

i s t h e s e t of a l l v e c t o r s

C forms t h e rows of a m a t r i x

forms t h e rows of

N

has an odd number of ones i f f

0 f o r a l l (row) v e c t o r s 1 i n

t h a t when a b a s i s f o r basis for

(A s e t of

(noting t h a t elementary row o p e r a t i o n s p r e s e r v e t h e

2

property of every row being even). N

it is

is p a r t i t i o n e d i n t o c i r c u i t s i f and

sum of those v e c t o r s 3s zero.) P

N,

i s a d i s j o i n t union of c i r c u i t s i f and only i f the

N

moduJo-two proof of

M

We use a

B4.

N',

N

'and

N'

C.

N

Standard theory shows and when a

r e p r e s e n t dual

matroids.

But l i n e a r combinations (over

have e v e n w e i g h t , s o t h a t

F ) 2

of even-weight v e c t o r s

i s a n even-weight code i f and o n l y i f

C'

i t h a s a r e p r e s e n t i n g m a t r i x a l l of whose rows a r e even. above proof o f

P5

Proof o f

This i s p e r h a p s t h e p r o t o t y p i c a l example o f t h e

P8.

serves equally well f o r

Thus,the

theory s i n c e t h e recursions

f o r graph c o l o r i n g have been

- B4

B1

P7'

well-known s i n c e G. D. B i r k h o f f ' s p i o n e e r i n g work [ l o ] s e v e n t y y e a r s ago and were developed i n t o a t h e o r y by T u t t e [137]which a n t i c i p a t e d f o r g r a p h s t h e p r e s e n t point-of-view..

Let

G

b e a connected graph

and l e t

xX(G)

with

c o l o r s s o t h a t no two v e r t i c e s of t h e same c o l o r a r e

X

b e t h e number o f ways t o c o l o r t h e v e r t i c e s of

connected by a n edge.

( I t w i l l b e a consequence of o u r proof t h a t

?

xX(G) i s a polynomial i n of

G.)

B;,

Bg

W e w i l l show t h a t

and

B4

xXp2(G(M))/2 =

where

x 8 (MI,

G

whose d e g r e e i s t h e number o f v e r t i c e s xX(G)/X

is

B;

B2

obeys t h e c o n d i t i o n s w i t h 1 r e p l a c e d by

we w i l l t h e n b e done.

e i t h e r z e r o o r two two-colorings.)

If

B1'

1-1.

Since

(A connected graph h a s

contains a loop

G

( g r a p h i c a l l y , a n edge j o i n i n g a v e r t e x t o i t s e l f ) , i t c a n n o t b e colored s o t h a t

xX(G) = 0.

If

G

i s an isthmus, i t c o n s i s t s of

two v e r t i c e s j o i n e d by an edge,so t h a t

is s a t i s f i e d .

(3.5),

We w i l l prove

B3

xA(G) = X(X-l),

only f o r trees.

B3.

If

G

is a t r e e with

n

xA(G) = A ( A - ~ ) each ~, edge b e i n g a n isthmus. G

B;

I n t h e following section

we w i l l show t h a t t h i s a p p a r e n t l y weaker r e q u i r e m e n t i n

f a c t implies

of

and

i s t h e d i r e c t sum of

n

edges,then Further, t h e matroid

i s t h m u s e s , and, c o n v e r s e l y , a d i r e c t

sum of

n

isthmuses i s represented by any t r e e of n x ~ ( ~' xP~ )( P ) A = (1-1)" = - A To prove A

edges.

-.

for trees

xA(G) + xX(G/p) = x A ( G p ) .

show t h a t

But,if

p

B4,

Thus,

we must

i s not a n

i s represented by t h e connected graph formed by

isthmus, G-p

d e l e t i n g t h e edge

p, while i f

p

i s represented

i s not a loop, G/p

by t h e graph formed by i d e n t i f y i n g t h e two v e r t i c e s joined by and then d e l e t i n g t h e edge

p.

every (proper) c o l o r i n g of

G-p

t h e two v e r t i c e s joined by

p

It i s then elementary t o show t h a t

i s e i t h e r a c o l o r i n g of

G/p

g e t s t h e same c o l o r i n

p

(when

G

receive different colors) o r

corresponds i n a unique manner t o a coloring of v e r t e x n o t i n c i d e n t with

p

(where each

G/p

and

G,

and t h e new v e r t e x g e t s t h e c o l o r o f t h e two i d e n t i f i e d v e r t i c e s ) .

Proof of

This w i l l be generalized l a t e r i n P a r t I1

Pg.

i n a manner analogous t o

*

s a t i s f i e s - t h e conditions a connected graph v e r t e x of

G

B1

*

- B4

using t h e Euler c o n d i t i o n t h a t

has an E u l e r i a n c y c l e i f and only i f every

has even degree.

G

C e r t a i n l y i f G (= G(M)) h a s an

i s t h m u s , i t cannot have an E u l e r i a n cycle; while, i f

B;

and

in

G, and l e t

If

d(w)

x;(G) and

-

G/p

B;

are trivial. d(w)

i s odd i n X;(GP)

-

Now assume t h a t

p

G

for

w

neither

even f o r each of these v e r t i c e s .

nor

P8).

v

v',

v'

in

G.

then

* x9(G/p)

Then

d(w)

= 1, while

*

If

Gp is

= 1, s i n c e t h e

must a l s o have even degree.

x;(G)

and

w

Assume now t h a t

of t h e v e r t e x degrees of any graph i s even.) have even degree,then

joins

(using t h e r e p r e s e n t a t i o n s f o r

given i n t h e proof of

GIp

v

i s a loop,

p

denote t h e degree of t h e v e r t e x

X;(~/p) = 0

identified vertex i n

*

x9

For now, we w i l l show t h a t

P8.

v

x9(Fp) = 0.

(The sum

and

v'

both

Conversely,

if v

X;(GP)

and v '

both have odd degree,then x*9(G)

= 0

while

= 1.

When Proof of P 10' invariant (see 3.201, Research Problem.

xi P10

is viewed as a (generalized) Tutte-Grothendieck follows as a special instance of (3.15).

Find other properties of a matroid (perhaps

in some special class) whose characteristic functions satisfy a similar recursion.

3.

The T u t t e Polynomial.

xi

The underlying i d e a of t h e i n v a r i a n t s

of t h e previous

s e c t i o n was t h a t t h e r e was a r e c u r s i o n of t h e general form

+ bf(M/p)

f(M) = af(M-p) loop,and

f(M) = f(M-p)-f(p)

a = 1 and than

M

when

b = -1.)

when

p

n o t a n isthmus.

p

was n e i t h e r an isthmus nor a

otherwise.

But we n o t e t h a t

i s not a loop,while Thus, f o r each

( I n t h e previous s e c t i o n , M/p

has rank one l e s s

r(M-p) = r(M)

i, f M i

= -

l

r

when

M

is

p

iobeys

the additive recursion

(*I

fi(M) = fi(M-p)

+ fi(M/p)

a s well a s the multiplicative recursion conditions:

Bj

f . ( l o o p ) = 0, f (isthmus) = -1. i

and t h e boundary Many o t h e r i n v a r i a n t s

can beWfudged up" t o obey t h e same a d d i t i v e recusion (*).

The

abundance of t h e s e i n v a r i a n t s which we w i l l e x h i b i t i n t h e following s e c t i o n s and, e s p e c i a l l y , P a r t I1 motivatesus t o e s t a b l i s h a general theory f o r a l l such invariants.

The fundamental i d e a i s t h a t of a u n i v e r s a l i n v a r i a n t

c a l l e d t h e Tutte poZynmiaZ.

A s we saw i n t h e previous s e c t i o n ,

t h e i d e a i s based on t h e work of Tutte[137] and was generalized t o matroids and given i t s p r e s e n t name by Crapo 1561.

The theorem

below is from 1271 while t h e g e n e r a l c a t e g o r i c a l framework i s presented i n [261.

I n t h e following, we d e f i n e a

Tutte-Grothendieck invariant a s a f u n c t i o n on matroids t a k i n g values i n a commutative r i n g below.

T~s

To we

R

which s a t i s f i e s conditions

T1

simplify t h e statements of such conditions a s use t h e term factor t o denote a p o i n t

loop o r isthmus.

p

and

T2

T1 and

which i s a

The term comes from t h e f a c t t h a t a matroid

M

f a c t o r s a s a d i r e c t sum i n t o a p o i n t M

= @(M-p)

i f and only i f

p

and i t s complement:

p

i s an isthmus o r loop.

If

p

is

n e i t h e r , i t i s c a l l e d a nonfactor. There i s a unique two-variable polynomial w i t h i n t e g e r

Theorem 3.1

c o e f f i c i e n t s a s s o c i a t e d with any matroid nomial then

which i s an isomorphism i n v a r i a n t ( i f

t(M;x,y) t(Ml)

TI.

+ t(M/p)

t(M) = t(M-p)

t (L) = y

M1 = M2,

and which obeys t h e following f o u r p r o p e r t i e s :

= t(M2)),

~ 2 . t(M) = t(M-p). t ( p )

T3.

c a l l e d t h e T u t t e poly-

M

and

if

if

i s a nonfactor

p

p

is a factor

t ( 1 ) = x , where

p o i n t matroid of rank zero),and

i s a loop (one-

L

I

i s an isthmus (one-

p o i n t matroid of rank one). T4.

If

f

i s any Tutte-Grothendieck i n v a r i a n t w i t h v a l u e s

i n a commutative r i n g

f (M) = t(M;f (I), f (L)), where t h e

R, then

polynomial o p e r a t i o n s t a k e p l a c e i n We w i l l p r e s e n t two proofs of t h e theorem.

R.

The f i r s t i s longer and

more t e c h n i c a l b u t can be adapted t o a l g e b r a i c o b j e c t s o t h e r than matroids.

The second uses a more c o n c r e t e matroid i n v a r i a n t .

Abstract c a t e g o r i c a l a l g e b r a techniques s i m i l a r t o t h o s e used by Grothendieck show t h a t t h e r e i s a r i n g with v a l u e s i n

R'

R'

and unique i n v a r i a n t

which has t h e "universal" property of T4

respect t o conditions

T1 and

T2

on a l l matroids.

The e s s e n t i a l

p a r t of t h e proof i s i n showing t h a t t h i s commutative r i n g f r e e (i.e.,

with

R'

is

a polynomial r i n g ) s o t h a t t h e r i n g homomorphism i s evaluation.

We w i l l s e e i n t h e f i r s t proof t h a t t h i s i s s o because an a b s t r a c t

"decomposition" of

M

a l o n g t h e l i n e s of

d e n t o f how we o r d e r t h e p o i n t s o f

M.

F i r s t Proof.

Mo

Mo((pl,

...,p k ) )

Consider t h e c l a s s and t h e f u n c t i o n

r e c u r s i v e l y d e f i n e d by

T'l-T'3

and

T2

i s indepen-

of ordered matroids

to : Mo

+

E [x,y]

which i s

below:

-

T12.

to(Mo) = to(pk) to(Mo-pk)

T13.

to(isthmus) = x, t o ( l o o p ) = y.

This function c l e a r l y s a t i s f i e s

of s h o v i n g t h a t

T1

it s a t i s f i e s

T3

otherwise.

and

TI and

T4, and t h e proof c o n s i s t s T2

( f o r any p o i n t

p).

This

is done by showing t h a t we can i n t e r c h a n g e t h e l a s t two p o i n t s (pk

and

pk-l)

i n t h e o r d e r and g e t t h e same polynomial.

are many c a s e s t o c o n s i d e r . M - - pk

but not i n

M,

then

For example,if p

k-1

and

i n s e r i e s , a n d t h e r e is an automorphism o f pk

and

pk-l.

Then, i f

o'

p

pk

k- 1

There

i s a n isthmus i n

form a p a i r of p o i n t s

M which i n t e r c h a n g e s

is the ordering

( ~ ~ . . . . , p ~ . p ~ -we ~),

have t h a t

To c o n s i d e r one more c a s e ( t h e g e n e r i c one) when n e i t h e r nor

pk-l

i s a l o o p o r isthmus, and they a r e n o t i n s e r i e s o r

pk

146

p a r a l l e l , then, e , g . ,

( M - P ~ - ~ ) /= P ~( M / P ~ ) - P ~ - ~ and . again

Re use s i m i l a r arguments f o r t h e o t h e r cases,

to(Mo) = to(Mo,). and we n o t e t h a t i f

i s n o t t h e l a s t point i n t h e o r d e r , we

pk

can s t i l l transpose i t with

pk-l.

t a k e c a r e of a l l t h e p o i n t s

pi

i n each monomial.)

t(M)

T1

by

and

where

0

Second Proof.

where S,

M

o"

o

n u l l i t y of

A (IAI-r(A)).

T1

and

satisfies

M.

E

X through

to, and we may d e f i n e

S,

and f o r any s u b s e t

The f u n c t i o n

and tl(M)

o r not.

If

of

is the

i s a well-defined

T3.

I f we show t h a t

and t h e theorem.

Now,let

p

be a n

Then,

p

E

A,

a s well a s nullity i n

then t h e s u b s e t M-p

as

A

A-p

does i n

corM(A) = c o r (A) M-P

+ 1.

A

contains

h a s t h e same corank

H.

has t h e same n u l l i t y c a l c u l a t e d i n t h e matroid M-p, whereas

A

then induction shows t h a t

T2, T4

nul(A)

and we break up t h e sum according t o whether t h e s u b s e t p

M.

We d e f i n e t h e f u n c t i o n

a l s o obeys

isthmus of

p

i n which i t i s t h e l a s t point,

i s t h e corank of A (r(S)-r(A))

t(M) = t'(M)

to

T2'

and then transpose

polynomial obeying t h e boundary conditions of t'

and

i s any ordering on t h e p o i n t s of

i s a matroid on t h e s e t

cor(A)

i > k

T2 a r e s a t i s f i e d by

t (?I ) 0

with

TI'

Thus, we can f i l t e r any p o i n t

t h e o r d e r t o give a new o r d e r so that

(We apply

If

M

Therefore:

p i A,

A

a s i n t h e matroid

+ 1

cor (A-p) nu1 (A-p) (x-1) M-p (y-1) M-p

AzS =

1

x

(x-1)

cor (A) nu1 (A) M-p (y-1) M-p

A similar arrangement of the summation proves

of a loop and proves T1 when p

T2 for the case

is neither a loop nor an isthmus.

In

factsit is the latter case which shows why we only use the identity T1 when p

is a nonfactor: p

is not a loop iff

nu1 (A-p) = nulM(A) for all subsets A containing p (corank M/P is preserved for any point) while for all subsets A not containing p, p is not an isthmus iff cor (A) = corM(A) M-P preserved). Definition 3.2.

(nullity is always

For a matroid M of rank n, define the corcmk-

nullity polynomioZ by S(M;u,v) =

1 ucor(A)vnu1(A) AZS

Proposition 3.3.

.

1.

S(M;u,v) = t(M;u+l,v+l)

2.

t(M;O,O)

3.

t(M;1,1)

is the number of bases of M.

4.

t(M;2,1)

is the number of independent sets of M.

5.

t(M;1,2)

6.

=

0.

is the number of spanning sets of M. K t(M;2,2) = 2 , the number of subsets of M (K =

\MI).

Proof.

The first statement follows from the second proof of

Theorem 3.1 (where it is shown that S(M) invariant).

is a Tutte-Grothendieck

An inductive proof based on the recursive definition of

t gives 0.3.2,while 0.3.3) ately evaluating S(M)

- 0.3.6)

are all easily shown by appropri-

using0.3.1).

The final statement,showing that the Tutte polynomial of the dual matroid is obtained by interchanging the two variables,can be proved via the corank-nullity generating function,since cor (A) computed in M

is the same as nul(S-A)

S(M-;u,v) = S(M;v,u)

.

computed in

*

Then,elementary arguments show that

t(M;y,x).

*

, so that

An alternate proof can be formulated ealoit-

ing the universal property T4 of t. Define t (M)

invariant. Hence,

M

t*(~)

=

t(M;ti(l),

* t

to be t(M*).

is a Tutte-Grothendieck t*(~))

= t(~;t(~) ,t(~))

This is essentially the proof of x4

and x5

=

given in

the previous section. Example 3.4.

Let M be the matroid consisting of the six vertices

of a triangular prism with a seventh point on the center of one of the rectangular faces. We illustrate the calculation of t(M) below,where all pictures are to be viewed in the appropriate Euclidean space, decompositions are with respect to circled points, juxtaposed points represent multiple points,

J

\

represents deletion,

for an application of T2.

0

encloses a loop,

represents contraction, and

150

Completing the decomposition we obtain:

In the proof of only f o r t r e e s .

P8

i n Section 2, condition

T2 was v e r i f i e d

The j u s t i f i c a t i o n f o r that apparently weaker

v e r i f i c a t i o n i s given i n the next proposition.

P r o p o s i t i o n 3.5

Let

f

be an i n v a r i a n t which s a t i s f i e s t h e

a d d i t i v e r e c u r s i o n f o r a l l matroids

M

and nonfactors

p

E

M:

Then t h e following a r e equivalent: T2.

f(M) = f(M-p)-f(p)

f o r a l l matroids

and f a c t o r s

M

p E M. T2'.

f(M1@M2) = f(Ml)-f(M2) f o r a l l d i r e c t sums

T2".

f (B) = f (B-p)*f (p)

M = M1@M2.

f o r a l l t o t a l l y s e p a r a b l e matroids

those i n which every p o i n t i s a f a c t o r :

(i.e.,

a boolean

a l g e b r a with loops). Proof.

Clearly

T2'

We show f i r s t t h a t

implies

T2"

implies

Assume t h a t an i n v a r i a n t by i n d u c t i o n on t h e s i z e of

T2.

Properties

so l e t

M

T2

and

follows from not a f a c t o r . of

If

f

T2"

T2", s o assume

T2".

satisfies f

T1

and

T2".

We show,

must, i n a d d i t i o n , s a t i s f y

a r e equivalent on one-point matroids,

K+l M

which i n t u r n i m p l i e s

T2.

M, t h a t

be a matroid on

a l l K-point matroids.

T2

p o i n t s and assume

f. obeys

T2

on

has only loops and isthmuses, T2 p

i s a f a c t o r of

M

and

q

E

M

is

Using arguments s i m i l a r t o those of t h e f i r s t proof

0.I)we o b t a i n : f (M) = f(M-q)

+ f (M/q)

= f (M-p) 'f (p)

.

To show that T2 implies T2', assume f is a Tutte-Grothendieck invariant, azd that M = MI %M2. with

/M21 = 1.

Property T2 is precisely property T2'

The proof that T2 and

T2'

are equivalent in

general uses induction on the cardinality of M2 and is similar to the inductive proof above. The proof rests on the fact that deletions and contractions in a direct sum may be performed upon the whole matroid or within the appropriate direct-sum factor resulting in isomorphic matroids. isthmus of M2 M1@(MZ-p)

For example, if p

c

M2, then p

if and only if it is an isthmus of M = M1@M2. and

= (M1M2)

for all M2 with

- p.

Now assume

/M21 = K.

If M2

lM2I = K+1, and that T2' holds is totally separable, T2'

follows easily from T2, so let p c M2 be a nonfactor.

Corollary 3.6

is an

Then:

If t is the Tutte polynomial, then t(M1%M2)

=

t(Ml)t(M2).

We remark that Proposition 3.5 says that Tutte-Grothendieck invariants satisfy the apparently stronger T2' which is a useful property (say in computations or applications),while in verifying whether a given invariant is a Tutte-Grothendieck invariant, the multiplicative cnndition T2 need only be verified in the very special cases 6f T2".

We w i l l come back t o o t h e r a p p l i c a t i o n s and examples and give more p r o p e r t i e s of t h e T u t t e polynomia1,but f i r s t we d i s c u s s t h e n a t u r e of a Tutte-Grothendieck i n v a r i a n t . The a b s t r a c t a l g e b r a i c i d e a i s represented i n t h e commutative diagram below

-Here, M

f r e e commutative r i n g generated by matroid (isomorphism c l a s s )

KO

M,

i(M)-i(M-q)*i(q). p

Here, q

t h e c a n o n i c a l epimrphism,and

invariant

f

i s t h e map which takes a

1 i s the ideal

i(M)-i(M-p)-i(M/p)

-

t

coi

The map

and

E

:

R

+

R/I

M,

is

assigns t o any matroid i t s

General a l g e b r a i c theory [ 2 6 ] s t a t e s t h a t , f o r any

with values i n a commutative r i n g

i s zero on t h e generators of

1 ( i . e . , obeys

i s a unique f u n c t i o n e : R / 1

.+

R/I i s

i s the

i s a loop o r isthmus of t h e matroid

i s n e i t h e r a loop nor isthmus.

T u t t e polynomial.

i

i t s generator,and

generated by a l l elements of t h e form

and

R

i s the s e t of matroid isomorphism c l a s s e s ,

R

such t h a t

R which T1

and

f = eot.

TZ),

there

The r i n g

c a l l e d t h e Tutte-Grothendieck ring and t h e essence of

Theorem 3.1 i s t h a t

R/1

i n two v a r i a b l e s ) , while a m a t t e r of t a s t e . t h e empty matroid.)

i s f r e e ( a polynomial r i n g over t h e i n t e g e r s

e

i s evaluation.

(Whether

R/I

contains 1 i s

I f i t does, then one can d e f i n e t(E) = 1,where

E

is

In this categorical context, Theorem 3.5 states that the ideal

I

contains all elements of the form i(M)-(i(M1)-i(MZ))

where

M = Ml(bM2, while, on the other hand, 7 is generated by the relations T1 along with only relations of the form i(B)-i(B-p).i(p) separable matroids B.

for totally

We now define some invariants with properties

generalized from T1 and T2. Although each is formally different from the rest,all are related in a fairly natural way to the Tutte polynomial.

It will be the purpose of the rest of this section to

explain these relationships. Definition 3.8 1.

A Tutte-Grothendieck group invariant f is one with-values

in an abelian group which obeys axiom T1. for all nonfactors p 2.

E

That is, f (M) = f (M-p)

+ f(Mlp)

M.

A Tutte-Grothendieck set invariant f is a function (with

values in some set S) for which f(Ml)

=

f(M2)

whenever

t(M1) = t(M2). 3.

A generazized Tutte-Grothendieck invariant is one with

values in an R-module in which axiom T1 is replaced by Tl'.

f(M) = af(M-p) in R p

4.

E

+ bf(M/p)

for elements a and b

(independent of M) and nonfactors

M.

A geometric Tutte-Grothendieck (group) invariant f

is one which is defined for combinatorial geometries (matroids without loops or multiple points)

G and which obeys

f(G) = f (G-p)

+ f (G/p) for -

ans p

E

G which is not

an isthmus, where G/p is the canonical combinatorial geometry associated with

Glp.

In

-

G/p multiple points of G/p are identified,

and it is characterized by its lattice of closed sets which is isomorphic to the interval [p, 11 in the lattice of closed sets of G.

It is the

contraction studied by Crapo and Rota [60].

5.

A geometric Tutte-Grothendieck ring invariant

f is one

which obeys TIG and T2G.

f(G)

=

for any isthmus p

f(G-p).f(p)

G.

t

Combinations of the above may also be defined in the obvious way (e.g.,a

geometric Tutte-Grothendieck set invariant).

following, we will use acronyms such as T-G

for Tutte-Grothendieck,

We now explore these invariants in more detail.

etc.

In the

First, we

give the commutative diagram equivalent to (3.7) for (3.8.1) and (3.8.2). In the following, t(M)

=

1 bijxiyj i,j

Proposition 3.9

Let G be the free abelian group generated by

the set of matroid isomorphism classes M with obvious map.

i :M

Further, let S be the subgroup of G

elements of the form i(~)-i(M-p)-i(M/p). the integer polynomial ring Z[x,y]

+

G

the

generated by

Then G/S = FAG[.

..,x iyj ,...],

viewed as a free abelian group.

The Tutte polynomial t : M

-+

G/S

serves as a universal T-G

group invariant in the following commutative diagram:

G / S = FAG I .

Here, h is any T-G

..,xiyj,...]

group invariant into an abelian group G I

while e is evaluation on totally separable matroids:

where B~'

Proof.

is the rnatroid consisting of i isthmuseS and j Ioops.

See 1261 or 1271.

Proposition 3.10

Let T be the set of all Tutte polynomials.

we have the following commutative diagram

Then

s i s t h e canonical s u r j e c t i o n , and,for any

where

invariant

f :M

f = fees.

Thus, f '

cients

ib

ij f

+

S,

(n) =

f

T-G

T-G

blO,

..)

'(b00~b10,b01.b20,bll,b02..

r i n g i n v a r i a n t is a l s o a

T-G

group i n v a r i a n t o r generalized

set i n v a r i a n t .

ant is

An example of a T-G

t h e c o e f f i c i e n t of

x

in

invariant which we w i l l d i s c u s s l a t e r . group) i n v a r i a n t i s t h e rank of Example 3.11

with

t(M):

C l e a r l y , any and any

f'

set

( i n theory) can be c a l c u l a t e d from t h e c o e f f i -

in

1

t h e r e i s a unique f u n c t i o n

T-G

Let

M'

M,

T-G

T-G group i n v a r i a n t , i n v a r i a n t is a

group (but not r i n g ) i n v a r i t(M)

c a l l e d t h e Crapo beta

An example of a s e t (but not

r(M) , where

r(M) = max(i : b

ij

be t h e matroid c o n s i s t i n g of t h e seven p o i n t s a s

p i c t u r e d placed

When t h e c i r c l e d p o i n t (p) is d e l e t e d and contracted, we g e t t h e two matroids on t h e second l i n e of Example 3.4. M/p = M*/p, where while

M # Ms.

M

# 0).

is rhe matroid of (3.4).

Thus, M-p = MI-p, and Hence, t(M)

=

t(M'),

Therefore,an example of an i n v a r i a n t which i s not a

s e t i n v a r i a n t i s t h e c h a r a c t e r i s t i c function

T-G

XM1

1

if

0

otherwise.

XMl :

G = M'

(G> =

Another example i s t h e number of three-point planes of a matroid. since

M

has f i v e three-point planes while

M'

has s i x .

This example p o i n t s o u t one way of c o n s t r u c t i n g nonisomorphic matroids with t h e same T u t t e polynomial. t h i s f a c t t o another matroid concept: and

M2

t h e s t r o n g map.

a r e two matroids on t h e same s e t such t h a t

and such t h a t each closed s u b s e t of t h e r e i s a matroid M

We r e l a t e

M

with

M-p = M1

and

MI

r(M2) = r(M1)-1,

i s closed i n

M2

If

M/p = M2.

MI,

then

The matroid

i s determined up t o isomorphism not j u s t by t h e s t r u c t u r e of

M1

and

M2, b u t a l s o

t h e a c t i o n of t h e s t r o n g map ( i . e .

l a b e l i n g ) needs t o be taken i n t o account. s t r o n g maps M-p

-+

M/p

and

MI-p

.+

line

{a,dl

For example, consider t h e two

M1/p of (3.4) and(3.11)

The i n v e r s e image of t h e double p o i n t

the

{p1,p2}

respectively.

i s t h e two-point

i n Example 3.4, and t h e two-point l i n e

ib,cl

in

in Example 3.11.

Since no automorphism of MI

takes {a,d)

to

ve have M f M'.

{b,c),

Another perspective on T-G

set invariants is to define two

related classes of invariants (I2 and

I3 below):

Proposition 3.12

Let IO be the class of (generalized) T-G

invariants (3.8.3).

and Il be the class of T-G

Let I2 be the class of invariants f : M there exists a function f2 : S f(M)

=

f2(f(M-p),

x

f(M/p))

+

set invariants. S for which

S + S such that for any nonfactor p.

Let I3 be the class of invariants f for which there exists a function

f3

defined on (isomorvhism classes of) matroid pairs such

that f(M)

=

f (M-p,M/p) for any nonfactor p. 3

Then:

1.

I3

11'

2'

I3 g 12'

3.

I1 n I2.t 2 1 0'

4.

(I1

n 12)

- IO

contains, for example, the cardinality

function f(~) = (MI. 5.

I3

-

(I1 u 12)

contains, for example, the characteristic

function of the desarguesian projective plane of order 9.

I1

6.

-

I2 contains, f o r example, t h e c h a r a c t e r i s t i c f u n c t i o n

of t h e three-point l i n e .

7.

The c h a r a c t e r i s t i c f u n c t i o n of t h e matroid M of (3.4) i s an i n v a r i a n t not i n I ?- . proof. 1. I f f (M) = fl(t(M)), l e t f3(M-p,M/p) = fl(t(M-p) + ~ ( M / P ) ) 2.

Let f 3 ( M-p ,M/p) = f 2 ( f (M-PI, f (M/p)).

3.

Let

f(M) = af(M-p)

3.20 t h a t

is a s e t invariant.

f

define t h e function 4.

+ bf(M/p).

f 2 by

We saw e a r l i e r (3.3.6)

a s e t invariant f2(r,s) = r

We w i l l prove below i n Proposition Further,

+ bs.

f2(r,s) = a r

t h a t the cardinality function,

( ( M I = log2(t(M;2,2))).

+ 1.

I2 s i n c e we may

f

I t is i n

It i s c l e a r l y n o t a generalized

We w i l l s e e below ( P r o p o s i t i o n 5.15.3) t h a t t(M1)

M1

and

is

I2 using, e.g., T-G

5.

IMI,

invariant.

= t(M2)

if

a r e rank-three combinatorial geometries w i t h t h e same

M2

number of atoms and i-point l i n e s f o r a l l

i.

Thus, t h e T u t t e

polynomial can n o t d i s t i n g u i s h two p r o j e c t i v e planes of o r d e r nine. However, i f p

M

i s t h e desarguesian plane,

and c o n t a i n s

lines. placing

M

81

ten-point l i n e s a s w e l l as

contraction plicity nine

6.

1 0 nine-point by

on t h e i n t e r s e c t i o n of t h e nine-point l i n e s s i n c e t h e M/p iff

c o n s i s t s of a l i n e with t e n p o i n t s each of multip

i s on

10

ten-point l i n e s i n

forward argument then shows t h a t i f

M' and

is independent of

may be reconstructed up t o isomorphism froni M-p

p

f u n c t i o n of

M-p

M,

then

fM(M') = f

M-P

M.

A straight-

fM is the c h a r a c t e r i s t i c (MI-p).f

M/P

(M'/p)

f o r all

p.

A three-point l i n e

L

i s t h e only matroid w i t h T u t t e polynomial

xL

+ x + y.

I2 since

Its characteristic function, %,is certainly not in (xL(L-p),

xL(L/p))

almost all matroids M.

=

(0.0) = (xL(M-p),

xL(M/p))

for

(Similarly, of course, the invariant fM in

3.12.5 is not in 12.)

7.

This was shown in (3.11).

An open problem is whether I2 is contained in I1.

As a means

of attacking the problem, we formulate it in a different manner. Define the equivalence relation

-

on matroids recursively

generated by the following relations:

(3.12')

El.

M-M'

E2.

M

if M - M '

- M'

if there is a nonfactor p

nonfactor q M/p

-

M'/q

Note that since both El and

E

M'

with

(M-p)

-

E

M

(MI-q)

and a and

. E2 are symmetric and reflexive, we

need-only take the transitive closure of the relations El and E2. Also, it is inductively well-defined,since we may determine how the equivalence relation behaves on matroids of cardinality K by knowing how it behaves on matroids of cardinality K-I. Further, if G

-

H

then t(G) = t(H)

since the latter

equivalence relation ("having the same Tutte polynomial") satisfies El and

E2.

Proposition 3.13

The class of invariants I2 is contained in the

class of invariants I1 if and only if M pairs such that

t(M) = t(M')

.

-

M'

for all matroid

Proof.

On

define

M,

-.

under t h e r e l a t i o n fact, define such t h a t

f (A,B) 2

f(M"-p)

I

2-

We claim t h a t

and

= A

i s an

f

M

f(MW/p) = B.

we have

M

I invariant. 2-

In

containing

C

M"

Conditions E l and E2

f2(f(M-p),f(M/p))

+

M + M',

for

t o b e t h e equivalence c l a s s of

t o be t h e equivalence c l a s s

guarantee t h a t t h e map t(M) = t(M')

f(M)

*

f(M)

is well defined. f(M1), and

f

i n v a r i a n t which i s n o t an I - i n v a r i a n t . 1 Conversely, we w i l l show t h a t i f , f o r a l l M and M',

If

is an

t(M)

=

t ( ~ ' ) implies

t h e n t(M) = t(M1) i m p l i e s f(M) =.f(M1) f o r any 1 2 - i n v a r i a n t

M-M',

f.

(The proof e s s e n t i a l l y rests on a n argument t h a t t h e f u n c t i o n d e f i n e d above which t a k e s

I invariant.) 2implies

M

f(M) = f(M')

a chain by

E2.

+

K

f o r a l l m a t r o i d s on

1 with

t(M) = t(M1).

M = MI-M2

-

-M3-...

f2.

M

and M

-

f

is an

M' have c a r d i M',

s o there is

(The chain i s f i n i t e s i n c e t h e r e a r e o n l y a f i n i t e number of

+ 1 points.)

nonfactors

such t h a t

pi

and

- Mi+l/qi+l. 3

qi+l Thus,

t(Mi+l/qi+l).

f ( M ~ - P ~= ) f (Mi+l-qi+lX

t(Mi-pi)

f 2(f (Mi+l-qi+l), E x e r c i s e s 3.14

For each

(Mi-pI)

i, t h e r e e x i s t

" (Mi+l-~i+l)

' t(Mi+l-9i+l)

and

and

Using t h e i n d u c t i o n h y p o t h e s i s , and

f (Milpi)

f 2 ( f (Mi-pi),

for all

f (Mi+l/qi+l))

Thus, s i n c e t h e

= f (Mi+l/qi+l).

arguments a r e t h e same i n both c a s e s ,

i.

f(M~/P~))

Hence,

f (M) = f (MI).

1. We i l l u s t r a t e a c h a i n of E2-equivalences f o r

two r a n k - t h r e e matroids fy that

Let

t ( ~ )= t(M1)

w i t h each e q u i v a l e n c e given

M',

Mt

K

t(Mi/p i)

K points,where

By assumption.

nonisomorphic matroids on

Mi/pi

is a universal

For a d i r e c t , i n d u c t i v e , proof, assume t h a t

I -invariant with associated function 2 nality

-

t o i t s e q u i v a l e n c e c l a s s under

(Mi-pi)

M1

= (Mi+l-9i+l)

and

M4 and

with Milpi

t(M1)

= t(M4).

= Mi+l/qi+l'

I n each c a s e v e r i -

2.

The reader may verify that the rank-three matroids M

and M' below have the same Tutte polynomials by exibiting an E2-chain between them.

3.

M1

+

M2

Each point is labeled with its multiplicity:

Show that there are no other essentially different strong maps in (3.11V), and, in fact, no other matroid MI' with t(HV') = t(M)

and M" f M,MV.

We now turn our attention to geometric T-G and (3.8.5).

invariants (3.8.4)

We show that, in fact, they are a special case of T-G

(matroid) invariants. Consider any invariant f on combinatorial

-

geometries and define f on matroids by:

\

0 r if (3.15')

?(MI =

f (2)

1

M

has a loop

o t h e w i s a where

2 is

geometry a s s o c i a t e d with

M

t h e canonical (with m u l t i p l e

(points identified). Similarly, l e t

be any i n v a r i a n t which i s z e r o on matroids

g

with loops, and such t h a t

g(M) = g(M')

same canonical geometry.

T h e n , c l e a r l y , t h e r e i s a unique i n v a r i a n t

f

on geometries such t h a t

Proposition 3.15

when

and

M

have t h e

M'

-f = g.

The following a r e equivalent,where

f

7

and

a r e two i n v a r i a n t s r e l a t e d by (3.15'1, 1. satisfies

T1

(and

G

i

2. if

i s a geometric

f

is a

T-G

group ( r i n g ) i n v a r i a n t .

f

Thus,

T2G). group ( r i n g ) i n v a r i a n t with

T-G

Z(M) = 0

c o n t a i n s a loop.

M

3.

(T is

? a

T-G

is a

T-G

group i n v a r i a n t with

ring invariant with

Proof. --

We f i r s t show t h a t f o r a

whenever

M

l o c p l e s s matroid

where

M'

T-G

invariant F(M') =

d i r e c t i o n t h i s follows from t h e f a c t t h a t i f

i s a nonfactor,and

q

q

i s a loop of

j .> 0.

I , f (M) =

?(s')

p

I n one

i s a p o i n t of

(but i s not a l o o p ) , then M/p.

0

f o r any

i s t h e canonical geometry.

M'

whtch depends on another p o i n t

for

?(loop) = 0).

has a loop i f and only i f

-

f (B~') = 0

Therefore

?(M'/p)

M'

p = 0

Continuing i n t h i s m a t t e r w i t h any m u l t i p l e p o i n t , we e v e n t u a l l y obtain

f(M') = ? ( f i t ) .

matroids

M',

loop

let

p,

T(M') = ?(fi;).

*

{q,q').

-

= M"-q,

For any matroid

and

Then

q

M"/q = M.

with a single

M

(M-p) @ M'

be t h e m a t r o i d

M"

multiple point

But

On t h e o t h e r hand, assume f o r a l l l o o p l e s s

where

is the

M'

i s a n o n f a c t o r , and

Thus,

?(M) = 0.

For

k

loops,

we d i r e c t sum w i t h a m u l t i p l e ( k + l ) - p o i n t and proceed by i n d u c t i o n u s i n g t h e above argument.

Now, assume (3.15').

If p

M

f

satisfies

TIG and d e f i n e

7

on m a t r o i d s from

M

is loopless.

We must show t h a t f o r any m a t r o i d ,

h a s a l o o p a l l terms a r e

is i n a multiple point,

not a multiple point,

0,

s o assume

T ( M / ~ )= 0

while

is loopless,

M/p

-

g

M-p =

=

G.

E-P,

and

If

If

is

p

Elp

=

M/p.

Thus,

=

Conversely, i f by (3.15').

?

satisfies

7( M - ~ )+ ? ( M / ~ ) . T1 and i s z e r o on l o o p s , d e f i n e f from

Then,for.any geometry

G

and n o n f a c t o r

p, we have

7

Thus, f

satisfies

i s equivalent t o (3.15.2),

TIG and (3.15.1)

in

t h e group case. We now show t h a t (3.15.2) and (3.15.3) group case. Bij

a r e equivalent i n t h e

The case of r i n g i n v a r i a n t s i s t r e a t e d s i m i l a r l y .

has a loop i f and only i f

j > 0,

any

T-G

i n v a r i a n t which i s

B

zero on matroids with loops must be zero on a l l Conversely, assume for a l l

j > 0.

If

g

is a

M

t(M) a r e zero whenever

j = 0.

g

group i n v a r i a n t w i t h

j > 0.

g ( ~ i j )= 0 b

iJ

in

Thus

1 biOxi i

i s a universal invariant f o r

Bi"

with

has a l o o p , then t h e c o e f f i c i e n t s

?(G) =

where

ij

The geometric Tutte poZymkZ:

Corollary 3.16

any geometric

T-G

Since

T-G

-

TIG

t(G;x,O)

(and

group i n v a r i a n t

In particular, for

f:

i s t h e boolean a l g e b r a w i t h

i s a geometric

TZG).

i

isthmuses.

Further, i f

T-G r i n g i n v a r i a n t , then g(G) = t(G;g(I).O)

where

I

i s an isthmus.

(For s i m p l i c i t y , we w i l l henceforth use

t o denote t h e boolean algebra

Bi70,

and

bi

Bi

t o be t h e c o e f f i c i e n t

b

i0')

A consequence of (3.1) and (3.2) is that the corank-nullity polynomial

S(M)

is a universal T-G

By (3.16),

t(G;x,O)

group or ring invariant.

is a universal T-G

invariant and thus any geometric T-G from S(M;u,-1).

geometric (group or ring)

invariant can be evaluated

However, there is a more intrinsically appealing the characteristic

universal geometric invariant derived from S(M),

We also give a

lor Poincarg) polynomial of a matroid, x(M). two-variable generalization i(M)

called the coboundary poZynomiaZ

by Crapo when he introduced it (561 as a generalization of Tutte's work on graphs 11421. We also define the cardinality-corank polynomial as a connection between S(M)

Definition 3.17 with L(M) 1.

and

?(MI.

Let M be a matroid of rank n

on the set S

the geometric lattice of flats of M.

?'he cardinality-comnk polynovial

SKC(M)

is given by

(M) is the number of subsets A of S, with ij and corank j (so that r(A) = n-j). where a

2.

The characteristic poZynomiaZ

0 if M

x(M)

is defined as:

contains a loop

otherwise i=O

i points

where

v(0.x)

[o.xl

i s t h e M6bius f u n c t i o n i n

(0,

t h e zero of

The c o e f f i c i e n t s

{wi1

o f the first kind of 3.

M

of

L(M),

i s t h e empty (closed) s e t ) .

and w i l l be s t u d i e d i n s e c t i o n s i x . M

i s given by

1

u'XIhCOr(y)~(x,y) x,yaL(M) : xsy

Here, t h e i n t e r v a l

[x,l]

of rank

Proof.

.

i s an upper i n t e r v a l i n

i s t h e geometric l a t t i c e of t h e c o n t r a c t i o n P r o p o s i t i o n 3.18

of t h e i n t e r v a l

a r e c a l l e d t h e Whitney nwnbers

x(M)

The Poincar6 poZynomiaZ of

=

L(M)

L(M)-.and

MIX.

We have t h e following r e l a t i o n s f o r a matroid

M

n:

The f i r s t i d e n t i f y i s immediate.

t h e f a c t t h a t i n any l a t t i c e , ~ ( 0 . 1 ) = X(-1)'*! over a l l s u b s e t s

The second follows from where t h e sum i s

A of atoms whose supremum is 1 ( s e e [I181 1,

and,hence,in a l o o p l e s s matroid,

y(0,x) = Z(-l)IBI

where t h e sum

is over all sets of points B found in [56] or 1371.1 from (3.18.1)

5

= x.

(A

complete proof can be

The third identity follows

and (3.3.1).

Identity (3.18.4) follows from (3.18.2)

in turn implies (3.18.5) by setting u = 0.

and (3.18.3),and

From (3.18.5)

such that

and (3.16), we may deduce the following.

Corollary 3.19

1. If f is a geometric T-G

ring invariant,

then : f (G) = (-l)r(G)~(~;l-f (I)) 2.

If f is a geometric T-G

where n = r(G).,

where 1 is an isthmus.

group invariant, then

and B~ is the i-point boolean algebra.

So far,in this section, we have shown that from the Tutte polynomial of a matroid we may evaluate any invariant which obeys either of the additive recursions T1 or TIG. The reader may ask himself why we went into such detail for invariants satisfying T1 and have thus far

neglected invariants which satisfy Tl',

the

generalized additive recursion, especially in view of the fact that the invariants

xi

of section two all satisfied a recursion of this

type (with a = 1, b = -1).

The reason is that the Tutte polynomial

allows us to evaluate these invariants also without developing a more general theory. The following proposition (3.20.2) first appeared explicitly (for ring invariants) in [I161 but is implicit in 1271 and 1751.

P r o p o s i t i o n 3.20 R-module

P

Tl'.

Let

f

b e an i n v a r i a n t w i t h v a l u e s i n a n

such t h a t f o r a l l m a t r o i d s f (M) = a f (M-p)

b

of

+ bf (M/p)

M

and n o n f a c t o r s

f o r f i x e d elements

a

p

(E

M,

and

R.

Then,

1.

f

(M) =

1 i,j

b 2.

i s t h e c o e f f i c i e n t of

ij

If

-

If

f

TIVG.

a nu1 (M)+,r(M)., (,

(Bij)

xiy'

in

;*

f, i n additioqsatisfies

f (M) 3.

b . :anul(M)-j.br(M)-?f 13

T2

(where, a s u s u a l , t(M)).

(here

R=P), t h e n

, 2a- U )

i s a geometric r i n g i n v a r i a n t which s a t i s f i e s f (G) = f (G-p) + bf (G/p), t h e n f ( G ) = b r ( G ) . t ( ~ ; v , 0).

Proof.

1.

Let

fV(M) d e n o t e t h e right-hand s i d e of (3.20.1).

Since t ( B i j ) = xiyj, f o r t o t a l l y s e p a r a b l e matroids,we have o n l y one nonzero term i n t h e summation: f ' ( ~ ~ j= ) a j - j ~ ~ ~ - ~ =f (f ~( ~~ ~j )j ) ~t . now s u f f i c e s f o r an i n d u c t i v e proof t o show t h a t

f'

obeys

t(H-p) = ~ b ; ~ xand~ ~f (Mfp) ~ , = by riy j 15

.

TI'. Then:

Let

2.

If f obeys T2, then f (Bij) = (f (~))~(f (L))',

=

3.

Define

-f

(3.15),

so that

anu1 (MIbr (M) (M; b ' a

from f as in (3.15').

Then, as in the proof of

is a matroid invariant which obeys T1'

with a = 1.

The rest of the proof imitates (3.15).

We note that there is no more general geometric analog for TI' when a

2

1. One reason for this is that nul(G/p)

cannot be

calculated from nul(G).

Thus, the Tutte-Grothendieck ring for the

recursion .f(G) = af (Gp)

+ bf (G) is not

a polynomial ring.

The reader may easily verify this by decomposing the matroid

in two different ways.

Starting with p, we obtain

First deleting and contracting q, we obtain: f(G) = a2f (B3)

+ 2abf (B2) + b2f (B1) .

We also note that every step in the proofs of section two have been verified since,for all i,

In those arguments, M was always binary but this affords no problem

M

since the entire theory of this section holds when by any hereditary subcZass and M is in

M',

of matroids (where if M = M',

so are M', M-p, and Hip).

Research Problems 3.21. posed as in (3.4),

M'

is replaced

1. When a matroid M on K points is decom-

what is the expected number of nonisomorphic matroids

which appear in the decomposition? (An upper bound is 2K , but this is toohigh. As our example shows, partial decompositions can be combined.)

Note that if this number can be shown to be always

for graphic matroids and a polynomial p,

p(K)

then results in Part I1 will

show that all nondeterministic polynomial algorithms have a polynomial counterpart if there is a polynomial check for graph isomorphism (i.e., the graph isomorphism problem is NP-complete). 2.

Is the equivalence relation (3.12')

the same as Tutte equivalence?

In particular, are all projective planes of order n equivalent? Ifwe modify

-

(3.12') forgeometriesspecifying that M/p

-M'fq,

does this correspond

to having the same chromatic polynomial? 3.

Characterize algebraically the "Tutte-Grothendieck ring" for

invariants which satisfy TleG in (3.20.3) for a

*

1.

4.

Matroid Constructions In this section, we will review some basic matroid operations

and show how they affect the Tutte polynomial. Of course, the prototypical operations for which the Tutte polynomial can be computed

*

are duality:

(t(G ;x,y) = t(G;y,x)),

and direct sum:

(t(G(BH) =

We will see that for other operations (such as trun-

t(G)-t(H)). cation).the

Tutte polynomial can be calculated directly from the

polynomial(s) of the operand(s) or from related polynomials. First, we treat the operation of adding a point in general position. Definition 4.1 a point p, M

For a matroid M(S)

+ p,

the free extension of M by

is the matroid on the set S u p whose indepen-

dent sets are the independent sets of M along with subsets consisting of p, along with any independent nonbasis of M. to the truncation of M in the following way: while M+p = T(M@).

T(M)

It is related

T(M) = (M+p)/p,

is most easily described by its closed

sets which consist of the closed sets of M except for the hyperplanes.

It is not defined if r(M) = 0. The free coextension of M by

p, M u p ,

is defined by M x p =

Proposition 4 . 2 formulas:

(M*+~)*.

If t(M) = Zb .xiy', i3

we have the following

Proof.

1.

It should be familiar by now that many identities

involving the Tutte polynomial may be proved by showing that the identity holds for totally separable matroids M

and are "linear"

in that they are preserved under deletion-contraction decomposition by nonfactors.

If M = B~',

then M+p

is an (i+l)-element circuit

along with j loops. Thus, its Tutte polynomial is (xi

... + x

+ xi-I +

any case (4.2.1) so in M+p.

+ y)yj

holds.

if i > 0 and yj+'

otherwise.

If q is a nonfactor of M, it remains

Further, (M+p)-q = (M-q)+p, and

(M+p)/q = (M/q)+p.

It is then an easy matter to show that (4.2.1) holds for M+p it does for 2.

(M+p)-q t(T(M))

and

Thus, we may subtract t(M)

1.

t(M+p)

=

-

= t(W)-t(M)-

t((Mfp)-P)

from the right-hand side of (4.2.1).

follows from (4.1.1) by duality (see 3.3.7),ao

a formula for the free l i f t , Example 4 . 3

when

(M+p)/q.

t((M+p)/p)

Formula (4.1.3)

In

Let M1

FL(M)

=

docs

(T(M*))*.

be the matroid consisting of two

intersecting three-point lines.

Then t(Ml)

= x3

+ 2x2 + x + ~ W + Y

and

t(Ml+p) = t (MI)

Here, L5 = T(M1)

+ t(L5)

=

t(M1)

+

is a five-point line.

that t(~~+p)= t(~~), where M2

2 x

+ 3 x + 3y +

2 2y

+

+Y 3 y

2 I

.

The reader easily checks

is the six-point rank-three matroid consisting

of two (~arallel)three-point lines. In fact, they are the unique smallest pair of nonisomorphic matroids with the same Tutte polynomial. These examples have the following consequences:

-

One cannot tell from t(M)

free position.

whether M

contains a point in

-

t(E(M))

cannot be c a l c u l a t e d from

"adjoint" o p e r a t i o n of t r u n c a t i o n :

t(M), where

E(M)

i s the

t h e f r e e e r e c t i o n of Crapo [58].

I n f a c t , t ( E ( 5 i - p ) ) = t ( y @ ) = xt(M ), which i s n o t equal t o 1 2 2 t ( E ( 5 ) ) = t(L3*L3) = (x +x+Y)

.

2.

We cannot even determine from

U has a n o n - t r i v i a l e r e c t i o n . and l e t

and

q(S)

M (S)

2

t(M)

whether

For example, l e t

S = (a,b,c,d,A,B,C,Dl

be two rank-four combinatorial geometries

both o f which have as dependent f l a t s : S and hyperplanes cCdD, dDaA,

and

aAcC.

dependent hyperplane hyperplane

abcd.

F u r t h e r , assume

bBdD, while

Then, M1

M2

has t h e a d d i t i o n a l 1 has t h e a d d i t i o n a l dependent M

i s e r e c t a b l e , while

results frpm t h e n e x t s e c t i o n (5.15.3)

aAbB, bBcC,

i s n o t , but

M2

show t h a t

t(M1) = t(M2).

Another operation which we have studied i s t h e forming of t h e canonical geometry

E, where

eliminated,and m u l t i p l e p o i n t s i d e n t i f i e d .

loops of a matroid

Information about

t(G)

i s contained i n t h e following p r o p o s i t i o n and example. P r o p o s i t i o n 4.4

1.

r (M) = max b

2.

If

r(M) = n,

loops. where

3.

Let

>O ij

M'

.

(i). (Dually, nu1 (M) = max (j ) ) b >O ij

then

(Dually, i f M

contains

b = 6(j,m), where nj nul(M) = k , i

be t h e matroid

t(M1) = t ( ~ ) /where ~ ~ and a l l

j.

then

b

M

jk

contains m = 6(j,i),

isthmuses.) M

brim

with i t s loops removed. =landb.

lj

Then,

= O f o r a l l i > n

M

are

5.

-t(M)-

Although

c a n b e c a l c u l a t e d from

f a c t , from

when

is loopless),

M

be determined i n g e n e r a l from

Proof.

t

(and, i n

(E)

cannot

t(M).

The f i r s t t h r e e s t a t e m e n t s a r e e a s i l y proved by d e l e t i o n -

contraction. (4.4.3),

t(M)

(4.4.4)

(4.4.4),

Example 4.5

is Corollary

3.16, w h i l e (4.4.5)

f o l l o w s from

and Example 4.5 below.

1.

The d u a l s

M

*

and

M'

*

of t h e m a t r o i d s i n (3.11)

They have t h e same T u t t e polynomial and

-7 -

T(M'*) = t(M ) = x(x+l)(x+2).

However, t h e r e a d e r may r e a d i l y check t h a t 2.

t(~'*)

*

t(M*).

The above m a t r o i d s may b e modified t o o b t a i n o t h e r matroid

p a i r s w i t h i d e n t i c a l T u t t e polynomials. we c a n s e e t h a t

t(M1)

= t(M2)

D e l e t i n g and c o n t r a c t i n g

f o r t h e following matroids:

p

Another important o p e r a t i o n i s t h e matroid union of Edmonds and F u l k e r s o n [69] and Nash-Williams I1061 : M1(S) where t h e independent s e t s of

M 1

V

M2

M2(S) = (M1VM2) (S)

V

a r e s u b s e t s of

can b e p a r t i t i o n e d i n t o independent s u b s e t s of respectively. g e n e r a l from of

M2

C l e a r l y , n o t h i n g can b e s a i d about t(M ) and 1

d o n ' t change

unions.

t ( 5

which

S

and

MI

,

M2 V

M2)

in

t(M ) , s i n c e d i f f e r e n t o r d e r i n g s on t h e p o i n t s 2

t ( M 2 ) b u t & r e s u l t i n v a s t l y d i f f e r e n t matroid

We might, however, a s k about

i n g e n e r a l b e recovered from

t(M)

M v M.

That

t(MvM)

cannot

can b e checked by t h e r e a d e r by

c o u n t i n g t h e number of s u b s e t s of s i z e f i v e and rank f o u r i n M

1

v M1

and

M

2

r e s p e c t i v e l y i n (4.5.2).

v M2

c a n b e recovered from

t(M).

However,

r(MvM)

The r e s u l t was motivated by rem&ks

of Las Vergnas [92].

P r o p o s i t i o n 4.6

Let

The r a n k . o f t h e union of

M(S)

b e a matroid w i t h

t(M) = Zb

xiy'.

ij

w i t h i t s e l f i s given by:

M

r(MvM) = IS1

+

r(M)

-

max ( i + j ) b >O ij

= 2

max ( i ) b

F i r s t Proof.

Note t h a t

number o f s u b s e t s of

M

ij

+

>O

max ( j ) b >O ij

-

max ( i + j ) . b

>O ij

max ( i + j ) = max ( i + j ) , where a >O b .>O i~ ij of corank

i

and n u l l i t y

j

aij

is the

(3.2, 3.3).

It is t h e n an easy a p p l i c a t i o n of t h e formula f o r t h e rank f u n c t i o n

of

M v M

t o show t h a t

max ( i + j ) = max (IAI

-

+ r(M) - r(A))

r(A)

S A '

aij'O

A second method of proof i s by our s t a n d a r d method

Second Proof.

of deletion-contraction.

Note t h a t one way of i n t e r p r e t i n g both

proofs i s t h a t , t o g e t h e r , t h e y g i v e a d e l e t i o n - c o n t r a c t i o n proof t h a t

+ I S-A1 ) .

r (MVM) = min ( 2 r (A) AZS

MVM =

and

B~',

s i d e of (4.6). the t r i p l e some

r.

M = B~',

If

-

r(MvM) = i = 2 i + j Now,let

(fl(M-q),

Similarly, i f

fl(M))

f2(M)

i s equal t o

ij

in

t(M-p)

f o r some m.

q,

(r*r-l*r+l) (fg(M-q),

.O

f 2 ( ~ f q ) ,f 2 ( ~ ) ) w i l l t a k e one of t h e forms (m,m,m)

For any nonfactor

max ( i + j ) , t h e t r i p l e b

or

t (M) = xiyj.,

which i s t h e right-hand

+ r(M).

fl(M) = (SI

fl(M/q),

(i+j)

then

(m,mfl,el)

The reason f o r t h i s i s t h a t

can never d i f f e r by more than one from

(m+lpm*el)* max ( i + j ) b! .>O 13

max ( i + j )

b" *O

in

ij

t(M/p).

I n f a c t , f o r any s u b s e t

always d i f f e r s by one i n A

h a s t h e same corank i n both

more i n A

M-p

M/p.

If

i s one l e s s i n

p

4 ?i,

MIp.

A 5 S-p, £(A) = cor(A)

and

MIp

M/p. and

If

p E

+ null(A)

( i n M), then

M-p, w h i l e i t s n u l l i t y i s one

t h e n u l l i t i e s a r e equal, b u t t h e corank of Combining t h e s e two i n v a r i a n t s , we s e e t h a t

fl

-

obeys one of t h e following t r i p l e s :

f2

o r (k,k-l,k+l) show t h a t

f o r a p p r o p r i a t e k (=r-m).

(k,k-2,k),

(k-1,k-l,k),

We w i l l b e done when we

r(MvM) obeys t h e same recursion.

Note t h a t , i n any case,

+

r(M/q v M/q) s r(M-q v M-q) r r(M/q v M/q)

2.

By d e f i n i t i o n ,

r(MvM)

e q u a l s t h e s i z e of the union of two maximally d i s j o i n t bases

of

C a l l them

M.

B and

B'.

Consider t h e t h r e e p o s s i b l e t r i p l e s

i n turn. This is t h e c a s e when t h e r e e x i s t such bases

(k,k-2,k). and

B1

such t h a t

with

E

B and

B'

qcBnB'.

(k-1,k-2,k). q

q # B u B'.

This i s when t h e r e e x i s t such

(k-1.k-1,k).

B

For a l l such

B and

B'

,

q c B-B'

or

B1-B.

Details a r e l e f t t o the reader. a r e mutually exclusive: i f q

E

Note t h a t , f o r example, t h e f i r s t two c a s e s B1 n B2 and q

d B; u B;,

then, by b a s i s

exchange, we could f i n d two bases more d i s j o i n t ( s e e P r o p o s i t i o n 7 i n (251). Certainly,if

t(M)

i s known and

g e n e r a l , compute e i t h e r known).

p

E

M,

t(M-p) o r t(M/p)

then one cannot, i n (although t h e i r sum i s

One way t o remedy t h i s s i t u a t i o n ( s e e (4.8.2)

and (4.8.3)

below) i s t o be found i n [ 2 4 ] where basepointed n a t r o i d s were introduced.

D e f i n i t i o n 4.7 where

M E 9 p o i n t of M.

Let

Mp

if

M

Mp

denote t h e c l a s s of pointed rnatroids ,

i s a matroid,and

q

i s a (distinguished)

An i n v a r i a n t f o r t h e c l a s s of such pointed matroids i s

a four-variable version of the Tutte polynomial in which the point

t (M ;x',x,yl,y), P q q is saved for last in the decomposition ordering

used in the first proof of Theorem 3.1.

We then may mdify (3.1)

to show that the pointed Tutte p o t y d a l

tp(Mq;x',x,y',y)

may be

defined such that:

Tip.

t (M ) = t (M -p) + t (M /p) p q P q p q nonfactor and p * q.

TZp.

t (M ) = tp(Mq-p)-t(p) p q and p * q.

T3p.

tp(q) = x'

if q

if p

if p is a

is a factor

is an isthmus, and

t (q) = y 1 P --

if q is a loop

In the above rules, M -p

is the matroid M-p 9 Similarly for M /p = (M/pIq P 9

point q.

= (M-p)

with distinguished

Two important operations on pointed matroids are the series

connection

,M',) and the parazzel connection P(Mq,Mi,). The parallel 4 4 connection is defined (followins 1241) as the matroid pointed by on the set S(M

(M-q)

(M'-q') (;l q,

A u A'

or B u B' u I ; }

while A'

and

whose closed sets are all sets of the form

B' u Iq')

series connection dually:

where A

and B u {qj are closed in M,

are closed in M'. S(Mq,Mi,)

*

We may then define the

* *

(P(M .M',))

.

A related 9 9 operation is the two-sum of Seymour [125]. For our purposes, ve m y

define this operation as P'(M

M',) 9' 9

=

= P(M

,M',) 9 9

- q.

We now l i s t some u s e f u l f a c t s , a l l of whose p r o o f s may b e found i n 1241 (except f o r ( 4 . 8 . 7 )

P r o p o s i t i o n 4.8 sets,where

q

E

Let M

and

which f o l l o w s from (4.8.2)

M

and 9 q ' c M'

and (4.8.5)).

M' be p o i n t e d m a t r o i d s on d i s j o i n t q' a r e both n o n f a c t o r s . F u r t h e r , l e t

+ y V g ( x , y ) , t (M',) = x' f T ( x , y ) + y ' g V ( x , y ) , L d e n o t e a l o o p , p q P 9 and I d e n o t e a n isthmus. We have t h e f o l l o w i n g ~ f o m u l a s : 1. t(M) xf (x,Y) + Y ~ ( x , Y ) .

t (M ) = xlf(x,y.)

y" (t(M-q) - (x-1). t(M/q)). (Formulas'5,6, and 7 below a r e v a l i d w i t h q = L o r I , where, a p p r o p r i a t e l y , f o r g=O) 5. tp(P(Mq,Mh,) = x ' - f - f ' + y q ( ( y - l ) . g * g l + f - g ' + gmf').

.

-6.

t (S(M , M I , ) = x l ( ( x - 1 ) a f . f '

P

Conversely, i f T(S1)

M9

- -q

fi-

P

q

fi-

q

+ f-g' +

g-f')

i s any connected matroid such t h a t

$ M (S ) , then 2 2

Conversely, i f

-

q

5-9

-

= ~ ( 6 -

9

S2,

fi-9

-

S1).

i s any connected matroid such t h a t

= 3 ( S 1 ) % M (S ) , t h e n 2 2

fi-

9

= s(%-/S fi-/S ) . 9 2'q 1

+ yl.g.g'.

ii/q

=

We remark t h a t t h e r e i s no p a r t i c u l a r r e a s o n t o s a v e j u s t one p o i n t f o r l a s t , a n d a t h e o r y could a l s o b e developed where a m a t r o i d M(S) i s "pointed"by

a subset

A

5 S.

The r e a d e r i n t e r e s t e d i n t h i s

generalization i s referred t o [93]. From t h e formula i n (4.8.5), ip(~(~q,M;l)) =x*
we s e e t h a t i p ( ~) ' f p ( ~ ; )

i s independent o f t h e c h o i c e of p o i n t s

q

-

.

and

I n p a r t i c u l a r , t(P(M

q ' , and we have

t h e f o l l o w i n g formulas :

h a s a g e n e r a l i z a t i o n which i s e x p l o r e d i n

Formula (4.8.10) 1291.

on d i s j o i n t sets,assume t h a t a geometry

H(T) a flat G.

Given two c o m b i n a t o r i a l geometries

H(T')

of

Then,the generalized paraZZeZ connectia

closed s e t s a r e a l l t h e s u b s e t s (here.

S'

and

x

A

(S-S')

such t h a t

a r e i d e n t i f i e d ) , and

Px(Gx,Hx) (;I (T-T') A

and

i s isomorphic t o

H and isomorphic t o a modular f l a t

t o b e t h e m a t r o i d o n t h e s e t of p o i n t s

G

x

G(S)

G(S')

of

i s defined

x

whose

n S is closed i n

A n T

is closed i n

H.

We t h e n have t h e following:

W e now d e f i n e a new o p e r a t i o n on m a t r o i d s which encompasses some

of t h e i d e a s o f Bixby 1121, Brylawski 1241, and Seymour 11251.

M',) q' q

Definition 4.9 let

Let

be a m a t r o i d on t h e s e t

M

be a pointed matroid.

M' 9

The tensor product

t h e m a t r o i d formed by making a two-sum of Formally,

M

M' 9

M )

k = M

.

where

{ p1,...,pk3,

M

0

= M,

M

@

and is

M' 9

M' a t each p o i n t of M. 9 and f o r a l l i : 1 5 i 5 k ,

The r e a d e r may r e a d i l y convince himself t h a t

t h i s d e f i n i t i o n i s independent of t h e o r d e r i n g on t h e p o i n t s o f and i n c l u d e s a s s p e c i a l c a s e s : with multiple points

comultiple points ipoints i n s e r i e s ) .

The t e n s o r product i s only d e f i n e d when P r o p o s i t i o n 4.10

If

M

r e p l a c i n q a l l t h e p o i n t s of

o r with

M

q

is a n o n f a c t o r .

t(M) = f ( x , y ) , and

tp(Mh)

= X1f'(x,y)

t h e n t h e T u t t e polynomial o f t h e t e n s o r p r o d u c t of

M. and

+ ylg'(x.y), is

M' q.

given by: t(M

@

M') = ( f ' h , ~ ) ) " ~ ' ~.(g1(x.y)) ' 9

Proof. -

Fix

and l e t

M'

9'

s a t i s f i e s t h e (weighted) and

I

f (L) = t ( L

@

by (4.8.2).

If

p

Finally, l e t

ously i n

M' M'

recursions

T2 and T1' w i t h

if

/q)

% (M-p)

if

9'

9

M

M

@

M'

'

1f

a = f'

by (4.8.3). = (x-1) . f '

,

p

i s a n isthmus

p

is a loop.

be a nonfactor,and l e t

q'

-1)

is a loop

L

S U q

D e l e t i n g and c o n t r a c t i n g t h e p o i n t s o f and

21

If + g'

M , then

% (M-p)

E

First, i f

M ' ) = t(~:-9) 9

(Mi-9)

p

' , f '+

We w i l l show t h a t

f ' + (y-l)-g'

f ( I ) = t(I

i s a f a c t o r of

M',) P' 9

p o i n t s of

T-G

M') = t ( ~ l / q ) 9

i s a n isthmus, t h e n

P'(M

f(M) = t(M a M'). 9

We may t h e n invoke (3.20.2).

b = g'.

then

r(M) (x-1)f '+ .f( gI

we s e e t h a t when

q

denote t h e S

simultane-

becomes a n isthmus

giving becomes giving becomes

I %

f o r t h e decomposition i n

((M-p)

L %

2

@

0

M' t h e t e n s o r product 9' Thus,when a l l t h e p o i n t s of S a r e

M' ) 8) M. 9 i n t h e decomposition of

((M/p)

t h e t e n s o r product

M' 9' S i m i l a r l y , when

Mi)

@

B.

q

becomes a loop

t(M a M') , t h e T u t t e 9 m u l t i p l i e d by f ' ( t h e c o e f f i c i e n t of

d e l e t e d and/or c o n t r a c t e d , we g e t , f o r (M-p) e M' q

polynomial of x'

in

by

g'.

t(M1 ) ) 9

p l u s t h e T u t t e polynomial of

This i s p r e c i s e l y r e c u r s i o n

Example 4.11

If

M:

@

specifying a particular point

f.

M O M'

q c M

t(M1-q) = t(M1-q')

unambiguously without

a s t h e d i s t i n g u i s h e d one., for a l l

q, q'

M',

E

s e e i n t h e n e x t s e c t i o n on r e c o n s t r u c t i o n (5.2) t h a t t(M1/q)

may be computed from

computed from

t(M)

For now, l e t

and

Mk

= x'

t(M').

Thus,

t(M

O

we w i l l

t(M'-q)

M ):

In

and

may be

t(M1).

be a p o i n t of m u l t i p l i c i t y

rank-one l o o p l e s s matroid). tP(n:)

multiplied

has a t r a n s i t i v e automorphism group,then

Mi

we may d e f i n e t h e t e n s o r product

f a c t , even when

T1' f o r

(Mlp)

+

k-2 (Y

M,

let

k

(a k-point

Then: +

yk-3

+

... + y + l ) y l ,

and Given a matroid defined above. multiplicity

Then k.

M ( ~ =) M

O

Mk+',

where

M ( ~ )r e p l a c e s each point of

Further,

M

M~+'

is

by a p o i n t of

t ( d k ) ) = (yk-l

... + y + l ) r ( M ) t ( ~ ; yk-l+. b l + ..+y+x

+

y I n p a r t i c u l a r , f o r example, t h e number of bases of

k

1

...-by+

, Y 1.

M ( ~ )i s

t ( ~ ( ~ ) ; l . l )= k r ( M ) t ( ~ ; l , l ) , i t s number of independent s e t s i s k k and t h e number of spanning s e t s i s ( 2 - 1 ) r ( M ) t ( ~ ; 1 , 2 ).

lcrMt(M;~,l),

Even though we cannot, i n g e n e r a l , c a l c u l a t e t(M-p) from t(M),we c e r t a i n l y

know t h a t ( f o r nonfactors)

t(M-p) 5 t(M)

coefficient

in

of

b'

corresponding

ij

b

ij

P r o p o s i t i o n 4.12

xiy' in

t(M).

If

M'

i n t h e sense t h a t t h e

i s l e s s than o r equal t o t h e

t(M-p)

This remark g e n e r a l i z e s a s follows:

is a minor of t h e connected matroid

M,

then M' can b e obtained from M by a sequence of d e l e t i o n s and c o n t t a c t i o n s by n o n f a c t o r s , and t(M1)

5

t(M).

This w a s shown i n [27].

Proof.

t h a t any connected minor

M'

The proof involves showing

can be obtained by a s e r i e s of d e l e t i o n s

and c o n t r a c t i o n s by p o i n t s such t h a t a t every s t e p a connected matroid (and t h u s one without loops o r isthmuses) i s obtained. t-(M) 2 t(M-p) 2 t(M-p/q) s e p a r a b l e , say

2 t(M-p/q/r)

M' = M' $ 1

Mi,

2

... 2 t(M1).

a minor of

M

i:

' en M'

is

then somewhere i n t h e decomposition

a d e l e t i o n (or c o n t r a c t i o n ) g i v e s a s e p a r a b l e matroid

Mi

Hence,

M

1

$

M2

with

Then, i n t h e s t e p immediately preceding t h a t

d e l e t i o n ( o r c o n t r a c t i o n ) t h e r e was a s e r i e s connection ~ ( M 1 . i ~by ) (4.8.9) ( o r a p a r a l l e l connection by (4.8.8)). t(M;)

S

t(Gi)

by induction, and formula (4.8.6)

I n any event,

gives the desired result.

Another operation f o r which something can be s a i d about t h e Tutte polynom i a l i s t h a t of t h e

a c t i o n of a rank-preserving b i j e c t i v e weak map.

This v i l l b e explored l a t e r i n Proposition 6.16.

5.

Reconstruction. Two celebrated open problems in graph theory are whether any graph

G may be reconstructed up to isomorphism from either its multiset

of isomorphism classes of vertex-deleted subgraphs or from its edgedeleted subgraphs.

In an attempt to solve these problems,there is

a wealth of literature reconstructing various invariants of G from the subgraphs. Both vertex and edge deletions have analogs in matroid theory (bond and point deletions respectively), and while a general matroid cannot be reconstructed, its Tutte polynomial can be computed from selected minors as we shall see below.

In the

following,we will often use the cardinality-corank generating function and the Poincare polynomial along with the formulas of Proposition 3.18 relating each to the Tutte polynomial. First, we give an invertible formula for the Tutte polynomial of a matroid from the weighted sum of the polynomials of its point deletions. Proposition 5.1

Let M be a matroid of rank n and cardinality

K with Tutte polynomial t(M;x,y)

S (M;z,u). KC 1.

We then have the following formulas.

1 t(M/p) + (y-1) 1 t(M/p) p:p is a loop p:p is not a loop = [n

2.

and cardinality-corank polynomial

t(M-P) p:p is not an isthmus

+

(y-1)-

+ (x-1)

a -

ay

a

(x-l)~] t(M;x,~).

1 t(M-P) p:p is an isthmus

= [K-n

+

(x-1)-

a ax

a t(M;x,y). - (y-l)~]

SKC (M;z,u) =

4.

11

SKC(M/p)]dz

T(M;x,y) =

where

T1(M;x,y)=

Proof.

A 5 S

1

t(M/p) p:p i s n o t a l o o p

+ (y-1)

A,

t(M/p).

A

-

p

-

1

t(M/p) p:p i s a loop

1

1

t(M-p) + (x-1) t(M-p) p:p i s n o t an isthmus p:p i s a n isthmus

of c a r d i n a l i t y i j z u

t h e c o e f f i c i e n t of in

i s t h e (formal)

jdz

Note t h a t any

We could proceed by f i r s t v e r i f y i n g (5.1.3).

subset

p

where

n+l m I zn u nd z = z u n+l

i n t e g r a l operator:

where

+ un,

in

i

and corank

SKc(M).

j

c o n t r i b u t e s one t o

Then, f o r e a c h of t h e

c o n t r i b u t e s one t o t h e c o e f f i c i e n t of

i

z'-'u'

points in

The o t h e r formulas f o l l o w from t h e i d e n t i t i e s of (3.18) and

duality. An a l t e r n a t e proof i s based on r e c u r s i o n .

1.

If

t(M) = xnyK-", n-1 K-n nx y

+

t h e n both s i d e s of (5.1.1) (y-1)

equal

n K-n-1 (K-n) - x y

Now, assume t h a t q i s a n o n f a c t o r of M, and t h a t q i s i n a m u l t i p l e point with

m

o t h e r p o i n t s (m

Z

0).

We p a r t i t i o n t h e p o i n t s of

S

-

q

a s follows:

P1

i s t h e s e t of l o o p s of

m

( p o s s i b l y empty) s e t of

points i n p a r a l l e l with

t h e s e t o f a l l o t h e r p o i n t s ( t h e nonloops o f M/q

with

P2

On,

removed, and l e t n'

+

M,

(y-1)-

a aY

M/q).

P2 q, Let

is the

and M'

P3

denote

be the d i f f e r e n t i a l operator

-

(x-1)-

a ax

'

We t h e n have t h e f o l l o w i n g :

= (Y-1)

2.

1

t (M/P) p:p i s a l o o p o f M

When ( 5 ~ 1 . 2 ) and (5.1..1)

K.t(M).

+

is

1

~(M/P) p:p i s n o t a l o o p

a r e added t o g e t h e r , b o t h s i d e s e q u a l

a s (2,~)= -(az az KC az

3.

-

By the chain rule,

x=-

n

a t az

z+u

z+l))

Z+U

+ zn z)ta(T,

= (nzn-' -U

a

z+u

a

( , z+1) = (T ax + -1 t(x,y), z 2 ay

and g = z+l (so that z = y-1 and

z+l).

where

u = (y-l)(x-1)).

Under

this substitution:

= (y-l)n-l[

t(M/p) nonloops

= -

t(M/p;x,y) + ( ~ - 1 ) ~1 ~(M/P;~,Y) loops

.zn-1t ( M / z+u p ; ~, z+l)

=

nonloops

1

(y-1). t(M/p)] loops

.

)

nodloops

4.

+

+

1

z n t ( ~ / p ; e 9z+l)

loops

We integrate both sides of (5.1.3) with respect to

that the constant of integration is a function of

u.

z and note

Thus,

(the term corresponding to the empty set of cardinality zero and corank n = r(H)). 5.

This formula comes from converting the Tutte polynomial to the

cardinality-corank polynomial and applying (5.1.4).

6.

Formula (5.1.6)

i s dual t o formula (5.1.5).

proved i n analogy with (5.1.4) n a l i t y - n u l l i t y polynomial.) homogeneous matroids. Corollary 5.2 0 < n < K,

with

and (5.1.5) by means of t h e c a r d i -

W e now apply t h e above formulas t o

I n (5.2.3),

Assume M

we use (4.8.4).

i s a matroid on

t(M/p) = t(M/q)

for a l l

a l s o have t h e same Tutte polynomial).

3.

K

p, q

p o i n t s of rank E

n:

S. (Thus, a l l d e l e t i o n s

This occurs, i n p a r t i c u l a r , when

a l l c o n t r a c t i o r s a r e isomorphic,or when group.

( I t can a l s o be

M

has a t r a n s i t i v e automorphism

Then:

The pointed T u t t e polynomial of (4.7) i s given by: t (M ;x'.Y'.x,Y) P 4'

K-n

=

1 X'Y) + $x'Y-xY')

a

(1-XI- ax

Examples 5.3 1.

I n t h e matroid N

isomorphic t o a l i n e Equation

5.2.1

2

of Example 4.3.1,

a l l s i x contractions a r e

L with t h r e e s i n g l e p o i n t s and one double point.

then becomes:

The polynomial 6t(L) is also the sum of the Tutte polynomials of the contractions of the matroid X 1

+p

the same Tutte polynomial.

in (4.3.1),

but these contractions do not all have

Thus, t(M) cannot, in general, determine whether

all contractions have the same Tutte polynomial (i.e.,whether t(M/p) = t(M/q) 2.

Assume M(S)

for all p and and M'(Sf)

are two matroids such that, for some ordering

S' respectively, t(M/pi) = t(M1/p;)

on the points of S and all i.

q).

Then, a consequence of (5.1) is that

t(M)

trivial exception is when M has all loops,and M'

=

t(M').

for

(A

is a multiple

point,since only in that case can we not determine from the set {t(~/p)? whether a particular contraction was by a loop.) From the study of nondesarguesian projective planes and Steiner systems,examples abound of (of the same rank) such that MI-pi = M2-pi 2 (see, e.g., 1371). 'IJe exhibit the smallest pair

nonisomorphic matroids M for all i

1

and M

below in "M5bius representation," i.e., both are rank-four paving matroids,each with five four-point planes represented below by lines and circles. In particular,the dependent planes of MI

are 1237, 1245,

1268, 3456,and 3478; while those of M2 are 1234, 1237, 1268, 3456,

The respective duals of these matroids are minimum examples of nonisomorphic matroids with isomorphic contractions.

3.

Surprisingly, the geometric Tutte polynomial cannot be recon-

structed by geometric contraction (upper intervals of corank one). In particular, let G be the planar geometry AG(2.3); be the nine-point combinatorial geometry PG(2,3)-Q, quadrilateral. any point p,

and let G' where Q

Then all rank-two upper intervals are isomorphic:

G/p = G'/p =

-

L4,

12 three-point lines; while G'

has 6 two-point lines, 4 three-point

= I g(G)l

z

= 15.

lu(G')I

We now turn to the hyperplane reconstruction of t(M).

Details may

In the following, let the set

[40].

for

a four-point line. However, G has

lines,and 3 four-point lines. Thus, 16

be found in

is a

\

= {ak)

Thus, F stands a for a rank-three matroid isomorphism class,and it has 1 atoms. a index isomorphism classes of matroids of rank k.

1 7

Now, for a fixed matroid M, let n(F k.M) denote the number of flats a and let n(F i,Fai+l) denote the number of of M isomorphic to F a a flats isomorphic to F in any matroid isomorphic to i+l. The a a hyperplane reconstruction theorem asserts that,for r(M) t(M)

fo m n F M

for all a n

E

A

.

=

n, we may calculate

We first

a

give a formula for n(F k.M) (k < n). a Lemma 5.4

The number of flats x of M

given for all k < n = r(M)

k and a

E

%

isomorphic to F a by the formula:

is

where

m =

1i1

i s t h e unique i n t e g e r ( g r e a t e r t h a n t h e s i z e of any

h y p e r p l a n e ) which s a t i s f i e s t h e e q u a t i o n :

E s s e n t i a l l y , t h e f i r s t formula c o u n t s , i n two ways, independent sequences of atoms

.i

(ak+l,...,a

" F a i -=~Fmi. The

a unique f l a t i n

M

n-1

)

in

M

such t h a t f o r a l l

i,

second formula comes from t h e f a c t t h a t t h e r e i s

of r a n k z e r o .

We may now r e c o n s t r u c t

SKC(M):

P r o p o s i t i o n 5.5

SKC(M;z,u) = ( z + l )

Here,

S (Mt,z) K

k :kcn

[tun-'-1) - 1

k k a EA

i s t h e spanning-set polynomial of

i s t h e number o f i-element s u b s e t s of SK(Mt, z ) =

1 aiz i . i

n ( F k,M)SK(F a

F u r t h e r , n = r(M)

S =

M',

vhich span r(F

n-1

)

+

M',

1, and

so,if then

a

i

=

IMI

=

Q

+

1 n(F 1

a

(1 F

1/ - I ) ,

2 being the number of loops i n

M

( o r , e q u i v a l e n t l y , i n any hyperplane). Knowing t h e Tutte polynomial of

M

a l l parameters, t h e number of a l l s e t s i n

i s equivalent t o knowing.for M

of f i x e d c a r d i n a l i t y

It i s r e l a t e d , though n o t a s s t r o n g l y , t o t h e nwmber of

and corank.

a l l closed s e t s of f i x e d c a r d i n a l i t y and corank.

In fact,from

i t i s not

we have seen how t o compute t h e number of loops (4.4.2),and hard t o g e t t h e number of k-point atoms f o r a l l

xn-l

t h e polynomial c o e f f i c i e n t of

k

t(G)

( e s s e n t i a l l y from

i n t h e Poincar6 polynomial).

However, i n general, t h e number of f l a t s of l a r g e r rank cannot be deduced from t(M). This was seen i n Example 3.11 where f o r two matroids and M', with t h e same T u t t e polynomial, M had 5 three-point planes,while M' had 6 such three-point planes. I n

M

M*

Example 4.5.1, lines.

had 2 two-point l i n e s , u h i l e

MI*

had 3 two-point

Conversely, even i f we know t h e c a r d i n a l i t y and rank of a l l

f l a t s , we s t i l l cannot i n general recover t h e T u t t e polynomial a s the following example shows: Example *

5.6

and i e t

B~

Pi'

and

* M

Let

3

and

M2

be t h e matroids of (4.5.2),

be a s given i n (4.5.11,

let

L3

Then, l e t t i n g

T-~(M)

t o t h e plane, t h e reader may e a s i l y v e r i f y t h a t " " " 3 @ ~M @ M @ W @ B ) = T - ~ ( M ~M I * @ M I * @ M I * @ L3) and M4 = T - ~ ( M M

have t h e same number of f l a t s of c a r d i n a l i t y and

let

be a three-point l i n e ,

be a three-point boolean algebra.

be t r u n c a t i o n of M

M1

j.

However,

t(M3)

*

i

and rank

j

for a l l

t(M4).

The Poincarg polynomial i s a summation of polynomials over closed s e t s , and thus can count c e r t a i n of them.

The reason one cannot

i

recover all the closed sets from x(M)

polynomials

x([x,l],X)

is that certain characteristic

"get lost" when summed with characteristic

polynomials of flats x '

of the same cardinality but greater corank.

This is precisely what happens in (4.5.l),where the characteristic polynomial for the double point "shadows" those for the two-point lines. What can be determined from the number of certain closed sets is the "border polynomial" of the Tutte polynomial. Definition 5.7

Let M be a matroid of cardinality K and rank n.

The Tutte-Grothendieck border poZynomiaZ is the polynomial:

i,j:b. .>O, and biIj,=O 1J

for iq>i,j'>j We similarly have the corank-nullit; border poZynomiaZ:

i,j:aij>O, and airj*

=

0

for it>i,j'>j and the closez-set b o r d ~ rpolynomiaz

is the number of closed sets of M of corank i and ij nullity j, and the sum is again over all indices (i,j) such that

where

f

f > 0 , while ij

f = 0 for i' > i, j' > j. i'j'

The corners of a polynomial suchthat

Cij

Two c o m e r s

c

c

ij

and

i,j

but

* O

t h e r e is no c o m e r c

c

i'',jl'

i',j'

i j lcijx y

c.,.=c I J

ij'

a r e indexed c o e f f i c i e n t s

= O

with ( n - i , j )

for

i 1 > i , j ' > j .

< ( n - i ' , j ') a r e c o n s e c ~ t i v ei f

with (n-i, j ) < (n-i", j") < (n-i', j ' )

.

It i s c l e a r

t h a t terms i n t h e border polynomial a r e given below (with n o t a l l c o e f f i c i e n t s n e c e s s a r i l y nonzero):

where

c

a r e consecutive corners. Some elementary remarks i'j ' about t h e border polynomials a r e contained i n t h e following proposition. ij

and

c

Proposition 5.8 cardinality

K,

1.

For a matroid

M

of rank

on a s e t

n

t h e corank-nullity border polynomial has

p o s i t i v e terms beginning w i t h

n x

yK-n.

and ending with

K

S

of

+1

A coefficient

S counts t h e s u b s e t s A 5 S such t h a t IAI = n-i + j , r ( A ) = n - i , B no subset of t h e same c a r d i n a l i t y h a s l a r g e r corank ( o r , e q u i v a l e n t l y ,

z

-J

in

larger nullity). 2.

The t h r e e border polynomials d e f i n e d above a l l have t h e same corners

(a. = b lj i j = fij)' corank

i

These numbers count t h e s u b s e t s

and n u l l i t y

j

A

of

M

of

such t h a t every smaller s u b s e t has l e s s

n u l l i t y and every bigger subset has g r e a t e r rank.

Each such s u b s e t

A

closed and c y c l i c ( a union of c i r c u i t s ) .

3.

The closed-set border polynomial h a s a t most

K

+

1 terms and

has exactly K + 1 ( p o s i t i v e ) terms i f and only i f M has closed s e t s of every c a r d i n a l i t y (and, thus, h a s an isthmus and no'loops). The Tutte-Grothendieck border polynomial has a t most K + 1 terms and has e x a c t l y

K

+ 1 terms

is

and

M

whenever TK-n

i s connected and n o t t h e t r u n c a t e d boolean a l g e b r a

K (B ) .

1.

Proof.

The empty s e t i s t h e unique s u b s e t of n u l l i t y

(maximal) corank nullity

K-n

and l e t

i(k)

IA~

k.

n, while

and corank

Then,

i ( k ) y j (k) k

2. A

Let

0.

has n u l l i t y

A

and

i s t h e unique s u b s e t o f (maximal)

k

be a n i n t e g e r between

b e t h e maximal corank of a s u b s e t

A

of

0

and

S with

j ( k ) = k - n + i ( k ) , and

gives a d i f f e r e n t index pair.

Let of

aij

S

be a corner of

SmB(M). Then, t h e r e a r e

each of c a r d i n a l i t y

n-i+j

and rank n-i.

aij

Since a

Subsets

i + l , j 5ai,j+l

any s e t s m a l l e r t h a n

A

has l e s s nullity,and every l a r g e r s e t has

g r e a t e r rank.

A

i s c l o s e d and ( a s a subgeometry) h a s no

Thus,

i s t h m u s e s , and

these s e t s

a

i

fi.j

fi,j

S

K,

i s a term i n t h e c o r a n k - n u l l i t y b o r d e r polynomial.

ai(k) , j (klx Each

S

0

for a l l

i.j

i s a c o m e r of

t(M;x,y)

and

=

in

j

FB(M).

= S(M;x-1.y-1)

a l l contribute t o

A

If

1

F(M)

and

f i , j.

S(I1)

i s a c o m e r of

aij

ailjl(x-l)"(y-l)j'

Since, c l e a r l y ,

respectively, then

S(M),

= a i j x iy j

il,j'

+ b

bill

i j = 'ij

c o m e r of

3.

Xi'j ' y ,where

either

i l < io r

Thus,

j l < j.

i',jl

i s a c o r n e r of t (M)

t(M), and a s i m i l a r argument shows t h a t each

i s a c o m e r of

We have s e e n t h a t

f.. 1 1 1

S(M) r e s p e c t i v e l y . a l l c o e f f i c i e n t s of

Since t(M)

have t h e same c o r n e r s ,

S(M)

.

a.. f o r a l l c o e f f i c i e n t s of F(M) and 13 S(M;u,v) = t ( ~ ; u + l , v + l ) , b 5 a for 5

ij

and

S(M).

Since

FB(M) i SB(M) and

ij

F(M), S(M), and t(M)

tg(M) r SB(M) ( w i t h

O9

term-wise

K

+1

ordering).

terms.

If

Thus. FB(M)

of t h e next s e c t i o n

b

i-l,j

(see

t(M)

'O

and

Results

(6.5))

show t h a t i f

f o r a connected > 0. bi,j-l

M

with

> 0.

But (6.5) shows t h a t

bll

i

b.. is a 11 0 and j > 0,

,

Thus, t h e border is a connected

polygonal p a t h of p o s i t i v e c o e f f i c i e n t s i n bll

each have a t most

Conversely, c o n t r i b u t i o n s t o d i f f e r e n t border

terms have d i f f e r e n t c a r d i n a l i t i e s .

then

tg(M)

has a closed s e t of s i z e k , then t h e r e i s a

M

minimal-rank such s e t .

nonzero term i n

and

> 0

if

t(M)

a s long a s

M

i s a connected

nntroidwhich i s n o t f r e e . Using t h e above proposition, we d e f i n e t h e p a i r be t h e index of t h e term of

+ j(k)

i s indexed by t h e p a i r

= k.

f o r some

The essence of t h e following p r o p o s i t i o n i s t h a t

F (M) B

to

Thus,

tB(M)

SB(M), tB(M), and

FB(M)

n-i(k)

each term of

k.

and

SB(M) w i t h

( i ( k ) ,j ( k ) )

( i ( k ) , j (k))

can a l l be derived from each o t h e r .

In

p a r t i c u l a r , n o t e t h a t any c o e f f i c i e n t appearing below i s i n t h e border. We a l s o remark t h a t arguments could be based on the Poincard polynomial. where

f.

(k),

Proposition 5.9

i f nonzero, i s t h e c o e f f i c i e n t of

uk X i ( k )

.

Proof.

1,2.

of c a r d i n a l i t y tabulated i n F

of corank

The c o e f f i c i e n t k

and corank

fi(k),j i(k)

with

a

i(k).

k.

i s t h e number of s u b s e t s

For each such s u b s e t

j 2 j(k).

and n u l l i t y

s u b s e t s of c a r d i n a l i t y

i ( k ) ,j(k)

j'

with

t h e d e f i n i t i o n of

i' > i ( k )

a

-n-i (k)+j (k) ]

Each such s u b s e t h a s corank

i t had g r e a t e r corank, t h e r e would be a s u b s e t

nullity

and

i ( k ) ,j(k) '

j'

7\ i s

A,

Conversely, f o r each closed s e t

there are

j,

A'

> j 2 j(k).

Formula (5.9.2)

= p;j(k)]

i ( k ) , since i f

of corank

1'

and

This c o n t r a d i c t s

i n v e r t s (5.9.1)

by

s t a n d a r d techniques. 3,4.

These formulas come from t h e r e c i p r o c a l e v a l u a t i o n s

t(M;x,y) = S(M;x-1,y-l),

S(M;u,v) = t(M;u+l,v+l),

no o t h e r terms than t h e ones l i s t e d c o n t r i b u t e t o

5.6.

Combining (5.9.3)

A

and(5.9.1),

we o b t a i n :

and t h e f a c t t h a t b i ( k ) , j (k)

.

me

(-111-J (x)

identity j :j '2 j z j (k)

[ ] j

j(k)

comes from c a l c u l a t i n g t h e c o e f f i c i e n t of

in

(l+x)n-l-i(k)+j

'-jk)

-

["-i(k)+j j'-j

'1

= [n-l-i(k)+j I - j(k) j '-j (k)

,xj-j(k).xj '-1

xj'-j(k)

(l+~)~-~(~)+j' (5.9.6) (l+x)j (k)+l

i n v e r t s (5.9.5)

by s i m i l a r i d e n t i t i e s . P r o p o s i t i o n 5.10

*

Let

FB(M ) =

s e t border polynomial of

M*.

Ih

*

f it(k), j * ( k ) ~ i t J k=O

Then. ( i * ( k ) , j*(k))

be t h e closed-

= ( j (K-k) ,i(K-k)),

and

=

Here,

Proof.

c

ij

1

i r i (K-k)

i s t h e number of c y c l e s of

A set

i s closed i n

A

(union of c i r c u i t s ) i n to the nullity in and o n l y i f

(-1) i-i (K-k)

M

of

M.

of corank

S-A.

Thus,

i s a c o r n e r of j,i (dual-closed) s e t s of c a r d i n a l i t y

O (K-k),i(K-k)).

i

i f and only i f

Also, t h e corank i n

f

(i*(k>,j*(k)) =

M*

M

* fi,j

* M

of

A

i s a c o r n e r of

= K-(n-j+i)

But

.

j.

is a cycle

S-A

FB (MI (see (5.8.2)). K-n-i+j

and n u l l i t y

i s equal

*

FB(M ) f

Thus,

*

i.j

if counts

1

W e have a l r e a d y s e e n t h a t i f

7

s(H*) =

a;)

ui v j ,

then

i,j

*

( s e e (3.3.7)).By

a i j = 'ji

(5.9.2),

Thus, i-i (K-k) k+i-i (K-k)

f;(K-k),

Here a

-

a

and (5.10)

[

i (K-k) = iLi fX-k)

i

,

j

follows.

-

k

)

j

j

k

[

(Note t h a t i f

]

lc

* aj(K-k),i

K-k+jK-k - j (K-k) ] f i , j by (5.9.1).

( i , j ) 2 (i(K-k), j (K-k)),

then a t

l e a s t one o f t h e c o o r d i n a t e s i s e q u a l . ) Example 5.11

1. L e t

M

*

b e a s i n (4.5.1).

It t h e n h a s t h e closed-

s e t polynomial: s3

The c o m e r s a r e

+

5s

2

+ 2s

t o closed s e t s of cardinality 0 Thus, F

B

* (M)

= s

3

2

+ 5s +

p o s i t i v e term i n M'*

= 1 corresponding 0,4 and 7 (M*) r e s p e c t i v e l y .

f g , O = I, f2,1 ' 1, f1,2 = 2, and f

F(M*)

-

(0).

2 s t +3st FB(d)

a )$ 4 ( f; , + 2 s t 2 + 4't . Note 2

is

2s.

F M * = F

t h a t the only ~

M where *

i s as i n (4.5.1), s i n c e b o t h have t h e same T u t t e polynomial.

However, From

* -

F(M' )

*

FB(M' ) = 3 s .

*

F (M*), one computes: t (M ) = x B B

3

+2xy

+ 2

2 3x +4y

+ x2y + 2

+3y

3

5xy

+ 4

+ y ,

7 02

which a g r e e s w i t h t h e ( d u a l ) c a l c u l a t i o n i n (3.4) f o r

The b o r d e r polynomial, FB(M) = Zf F (M*) = z f ? . s i t j B 13

1

=

2.

-

= 1

with

K = 7.

4+2-1 4 ]f;,l

[

+

-s

ii

[

M3

and

M4

M4

has

37

]

f*

1,2

*

F(M )

= i(7-4)

= 1, and

= 3-5+8.

from

F(M).

For example,

of (5.6) have t h e same c l o s e d - s e t p o l y n o m i a l s ,

b u t t h e r e a d e r r e a d i l y checks t h a t while

Further,

i j t , can t h e n b e c a l c u l a t e d from

For example, i n N*, j(7-4)

3+2-1 3

I n g e n e r a l , one cannot compute

the matroids

t(M).

M3

three-point c i r c u i t s .

*

F(M ) = 3 The above s u g g e s t s t h a t

38

has

circuits,

Thus,

... + 3 8 s t + ... * t(M)

three-point

*

F(M4).

c a n b e r e c o n s t r u c t e d from

i f t h e c l o s e d s e t s a r e a l l enumerated i n

FB(M).

F(M)

This is t h e case of

a s p e c i a l c l a s s o f m a t r o i d s c a l l e d near-designs. D e f i n i t i o n 5.12 rank

A matroid i s a near-design i f a l l c l o s e d sets o f

m have t h e same c a r d i n a l i t y

a l l atoms, l i n e s ,

..., and

k(m)

for

m = O,l,

...,n-2.

Thus,

c o l i n e s r e s p e c t i v e l y have t h e same s i z e .

S p e c i a l c a s e s o f near-designs a r e r a n k - t h r e e c o m b i n a t o r i a l g e o m e t r i e s (where

k(0) = 0, k(1)

=

I ) , paving m a t r o i d s (where f o r a l l

m

above,

k(m) = m), and homogeneous m a t r o i d s ( s e e [ 3 7 ] ) l i k e p r o j e c t i v e and a f f i n e geometries,where, i n a d d i t i o n , a l l h y p e r p l a n e s have t h e same c a r d i n a l i t y .

Operations which preserve near-designs where

a r e t r u n c a t i o n , minors

I s a f l a t , and t e n s o r i n g with a m u l t i p l e p o i n t ( s e e 4.11).

S'

W e parametrize

.....

(k(O).k(l)

near-designs with t h e v e c t o r

..,fl,K-n, .k(n)

k(n-2);f1,0.fl,lsf1,2,.

i s t h e number of hyperplanes of n u l l i t y

n-1

M(Sf)/T

i

= K), where

f

1,i (and, hence, c a r d i n a l i t y

+ i). Note that we may g e n e r a l i z e t h e notion of a near-design t o a matroid

vhich h a s no f l a t s

F

and

with

F'

r ( F ) > r ( F V ) and

(a necessary and s u f f i c i e n t condition f o r matroids,

IF1 S I F V I For these

FB(M) = F(M)).

t ( f f )and F(M) s h o u l d , b e a l s o mutually d e r i v a b l e . Examples of

such =re g e n e r a l matroids a r e p r o j e c t i v e geometries v a r i o u s m u l t i p l i c i t i e s (up t o

q)

PG(n,q)

with

on the p o i n t s .

When all t h e f l a t s of f i x e d rank (including t h e hyperplanes) a r e e q u i c a r d i n a l , we term t h e matroid a perfect matmid design (design f o r

W e now g i v e some known p r o p e r t i e s f o r designs.

short).

Lemma 5.13

M be a design with parameters

....k(n-l),k(n)

(k(O).k(l).

1.

Let

Everyinterval

of r a n k

s-r

vith

Thus, f ( r , i , s )

(Here

r

S

i

S

[x,y]

f(r,i,s)

r ( x ) = r and

with

f l a t s of rank

is t h e number of f l a t s i n

equals a fixed f l a t t o a fixed f l a t

= K).

x

of rank

y (y 5 x).

r

r(y) = s

i-r M

is adesign

i n its l a t t i c e .

of rank

i

which contains or

and i s contained i n o r equal

Further,

s, and empty products a r e equal t o one.)

In particular, the number of closed sets of M of corank n-i and nullity k(i)-i

(*I 2.

is given by:

fn-i,k(i)-i

3.

f(O,i,n)

=

2

i-l K-k (m) k(i)-k(d

The value of the K6bius function v(x,y)

lattice of flats of M y.

=

(with x

depends only on the rank of x

In particular, if r = r(x),

and

s = r(y),

'

5

y)

in the

and the rank

then

The characteristic polynomial of the contraction of M by a flat

x of corank i is independent of the choice of x and is given by:

4.

The coefficients in the Poincarc polynomial,

are given by:

I:

U(X,Y) x,Y: r (x)=i r(y)=j

Proof.

1.

The identify for f(r,i,s)

can be found in the papers by

Edmonds, Murty, and Young which introduced perfect matroid designs ( 1701 and [162]).

An interval of a design is itself a

design whose parameters can be given by k ' ( j ) = K' = k(i+j).

kl(0) = k(i),

kl(l) = k(i+l),

...,

Thus, we need only prove the formula for f (0,i.n).

This comes from counting,in two different ways, the i-tuples of independent points much as was done for atoms in (5.4). in this context,(5.4)

(Note that,

could be thought of as a generalization of the

Young-Murty-Edmonds formula.) 2.

That u(x,y)

depends only on ranks appears in [66] and 1371.

The first formula above for p(r,s) of the identity:

appears in [19],and a simplification

gives t h e second formula. 3.

When t h e elements

x(i,j)

a r e t a b u l a t e d i n t o an

m a t r i x , i t i s t h e i n v e r s e of t h e m a t r i x (See .[661 o r [37].)

The formula f o r

follows by i n v e r t i n g proof of (5.13.3)

4.

2

W (i,j).

x

(n+l)

W ( i , j ) = f(n-i,n-j,n).

x(i,j)

then

Note t h a t we could g e t an a l t e r n a t e

from (5.13.2);

o r , e q u i v a l e n t l y , could prove (5.13.2)

from

The Poincare' polynomial i s given by

X(M) Thus, t h e c o e f f i c i e n t of r(x) = i

When

(n+l)

2

and

i = j,

=

.

1

xsy

k ( i ) hn-j u

r(y) = j ( i

S

j).

is

Zu(x,y)

over a l l p a i r s

x

Using t h e above formulas we obtain:

t h e formula i s obvious ( i t i s

f ( O , i , n ) ) ; and when

i < n,

terms i n t h e above product cancel t o g i v e t h e d e s i r e d r e s u l t . We now g e n e r a l i z e t h e s e r e s u l t s t o near-designs. Proposition 5.14 polynomial.

1.

A near-design can be recognized by i t s T u t t e

I n particular, i f

Iy

r(M) = n,

then

M

i s a near-design

with

if and only if,in the Poincari. polynomial,

there are precisely n-1 polynomials

{pi (A), pi (a), o 1

...,P

(A)

: iO < il <

in-2

--.

<

in-21

of degree at least two. Further, in this case,the parameters of M

are recoverable from

k(j)

f

1 9 1

K 2.

(...,k(j),...;...,

(M) :

= ij

is the coefficient of

= the u-degree of

Equivalently, M

X(M)

n-l+i u A (when ~

= i0

+

h

~ = +0) ,~ and , ~

(i -i )-c'~,n-f l o 1

(n > 2).

is a near-design if and only if there are precisely

n-1 nonzero terms of s-degree greater than one in FB(M)

F (M) B

fl,i,-• • ;K)

is computed from t(M)

by (5.9.6).

(see (5.7)), where

(The parameters can then

be obtained from the degrees of the terms and from the nonzero coefficients

3.

If M

is a near-design with parameters (...,k(j),...;...,f

then the nonzero coefficients ifij i

in F(M) = FB(M)

-l,js--.;K),

are given by:

4.

If the parameters of a near-design

Poincar6 polynomial of

where, for a l l

5.

M

M

are given a s i n (5.12). then the

i s given by:

i:

The Tutte polynomial of a near-design i s equal to:

t(M;x, y) = ~ a ~ ~ ( x - l ) ~k-nti, ~ ( ~ where - l )

i-1

,

i-1

K-k (m)

I,

K-k(m)

m= 0 I

i

Proof.

1.

x(M;u,h)

is, by d e f i n i t i o n , a sum of c h a r a c t e r i s t i c polynomials of f l a t s

-

of s i z e

i.

such a f l a t .

The A-polynomial

coefficient, p.(A),

The degree of a nonzero

p.(A)

of

u

in

i s t h e maximal corank of

Thus, t h e number o f d i s t i n c t polynomials o f d e g r e e a t

l e a s t two i s t h e number of s i z e s of f l a t s of corank a t l e a s t two. number i s c l e a r l y a t l e a s t p(h)

r(M)

-

1 ( t h e r e i s a t l e a s t one polynomial

o f e v e r y degree corresponding t o t h e l a r g e s t f l a t of t h a t corank).

For n e a r - d e s i g n s ,

t h i s number e q u a l s

r(M)-1.

Further, i f the f l a t s

o f f i x e d corank were n o t a l l t h e same s i z e , l e t corank f o r which t h e r e were would b e

k

b e more t h a n two.

polynomials r(M)

-

be t h e maximum

j

k > 1 different f l a t sizes.

Ipi(h)l

1 polynomials

of degree IPi(A)l

j,

Theqthere

and t h u s t h e r e would

of degree g r e a t e r than

The formulas f o r t h e p a r a m e t e r s f o l l o w from t h e d e f i n i t i o n of

X(M;U,A) where

K,

t h e number o f p o i n t s o f

of l o o p s ( t h e exponent

M,

i0 o f t h e u - c o - f f i c i e n t

i s given by t h e number of

t h e sum o f t h e (nonloop) m u l t i p l i c i t i e s o f a l l atoms. all

This

IEl

atoms have t h e same nonloop m u l t i p l i c i t i e s ,

An

i)

in But,if

(il

-

plus

n > 2. io).

Here, u

i s t h e exponent of

il

il p

(A)

of A-degree

n-1,

u and

il

coefficient

c;

,n-l 1

i n t h e unique polynomial

1fi1,

t h e number of atoms, i s t h e

of t h e l e a d i n g term of t h i s polynomial

(corresponding t o t h e number of c h a r a c t e r i s t i c polynomials summed). By t h e d e f i n i t i o n of a n e a r - d e s i g n , a l l nonzero terms i n

2.

F(M)

appear on t h e border. 3.

These formulas come from t h e f a c t t h a t t h e t r u n c a t i o n o f

M, T(M), i s

a design,so t h a t we may use t h e i d e n t i t y (*) developed i n (5.13.1). 4.

If

The formulas f o r

cor(y)

t

pi(A)

come from t h e f a c t t h a t

0, then t h e i n t e r v a l

t h e formula i n (5.13.4).

and thus 5.

p (A) i

i s a design and we recognize

Thus, t h e formula f o r

t o a c o n s t a n t ( t h e c o e f f i c i e n t of also.since

[x,y]

i

i s c o r r e c t up

AcOr(M) ). But t h i s term i s c o r r e c t

i s a sum of ( n o n t r i v i a l ) c h a r a c t e r i s t i c polynomials,

p i ( l ) = 0.

We could prove t h e s e i d e n t i t i e s from t h e polynomial formula i n However, we w i l l argue d i r e c t l y , a s it shows a t y p i c a l

(3.18.4).

MLIbius i n v e r s i o n argument and p r e f i g u r e s methods From formula (3.3.1), aik

p (A)

we s e e t h a t (5.14.5)

i s t h e number of s u b s e t s

Thus:

a

=

ik

s e t s of

x.

of rank

i:

x:r(x>=i

Sp(x,k)

t o appear i n P a r t 11.

is equivalent t o sharing t h a t and c a r d i n a l i t y

k.

A 5 S

of rank

i

where

Sp(x,k)

counts t h e size-k spanning

We now i n v e r t t h e following formula f o r a f i x e d f l a t

y

t o obtain:

Hence,

i 'j

where

c

i 'j

i s given

Various i n s t a n c e s of (5.13) and (5.14) have appeared i n t h e l i t e r a t u r e and we l i s t some below. Formula (5.15.5)

i s i n [37] and

N o t e , t h a t i f some t r u n c a t i o n of a

the r e s t a r e i n [27]. matroid i s a near-design,

t h e n t h e parameters of t h e t r u n c a t i o n may b e

recovered from t(M); and, c o n v e r s e l y , some knowledge of t(M) may b e o b t a i n e d from those parameters by u s i n g (5.14) a l o n g w i t h (4.2). C o r o l l a r y 5.15 K, with

1.

Let

M

b e a matroid of rank

n

and c a r d i n a l i t y

t(M) = Eb..xiyj. 13

A l l p-element

s u b s e t s of

M

a r e independent i f and o n l y i f

= 0 f o r a l l q < p, j > 0. I f a l l p-element s u b s e t s of M n-s 9 j independent ( s o t h a t 'In-'-' (M) i s a near-design w i t h parameters

are

b

k ( i ) = i ) , then t h e number of f l a t s of corank counted f o r

k

e n-p

and a l l n

fkPj

=

n-s

I I s=k t = O

j

k

by:

t n-s ' (-1) [ t ] [ ~ ] b s , j + t

and n u l l i t y

j

is

is then given (for n-ksp)

The Uhitney number of the second kind

2. s

2

3.

If all p-element subsets of M are independent, then for all n-p,

M is a paving matroid (a near-design with k(n-2)

and only if b

ij

biO =

= 0 for all i 2 2 and

(K-i-1)

In particular, M b = O ij

4.

2

1.

n-2)

if

Further,

for all i 1 2

is a truncated boolean algebra if and only if

forall i-j S O .

In this case,

M is a combinatorial geometry if and only if bn-l,j ' 0 for

all j

and

j

=

2

1. In this case:

The number of p o i n t s , number of l i n e s ,

5.

M

If

WI,

i s a design with parameters

S m ~t h e

K = n+bn-l,O,

= (x-1)

n

Sm

k(i)

for a l l

i,

then

1

(M;x_l, (x-1) (Y-1))

c a r d i n a l i t y - n u l l i t y polynomia1,is given by:

where t h e "ithexcess n u l l i t y " n ( i ) , equals k ( i ) and

and t h e

i s given by:

W2,

t(M;x,y) where

i s given by

- k(i-1) - 1; E ( i ) = K-k(i-l),

i s t h e formal i n t e g r a l o p e r a t o r defined i n (5.1).

Idy

1.

Exercises 5.16

Prove t h e formulas i n (5.15) from t h e

g e n e r a l formulas i n (5.14). 2.

Extend Example 5.6

geometries t(G )

1

*

G1

t(G2)

and or

G2

Y(G;)

by f i n d i n g an example of two combinatorial such t h a t i

F(G;).

F(G1)

= F(GZ), b u t e i t h e r

What i s the s m a l l e s t such p a i r ?

Research Problems 5.17

1.

Find, i f p o s s i b l e , two matroids

M1

and

M2

which a r e not

isomorphic,but such t h a t with a s u i t a b l e r e l a b e l i n g of t h e i r p o i n t s ,

M1-pi = M2-pi for all 2.

and

M1/pi = M2/pi

i.

What relationships other than (5.10) exist between the closed

set-cardinality numbers (i.e..

Ifij] and those for the dual

the cycle-cardinality numbers

*

if } i,j

I)?

Ic i s j

3.

Find examples of near-designs (without loops or mltiple points)

which are not designs or paving matroids.

4. rank

Extend the formulas in (5.14) to matroids all of whose flats of i

+1

are larger than the flats of rank

example, show

that for such matroids M

can be reconstructed from

5.

i

(where

for all i. F O = FB(Xj),

For t(U)

F(M).

Can the Tutte polynomial of a matroid be reconstructed from the

Tutte polynomials of its hyperplanes?

6.

I d e n t i t i e s , I n e q u a l i t i e s , and E x t r e n a l Matroid Theory.

1.

"Research Problems" 6 . 1

Is t h e f o l l o w i n g t h e T u t t e polynomial

of a m a t r o i d ?

By (5.15.4),

we s e e t h a t i f such a n

K = 111 p o i n t s ,

p l a n a r geometry w i t h j t 9.

M

could b e found, i t would be a f1,9

= 111,

and

Thus, i t must be a p r o j e c t i v e p l a n e of o r d e r 10.

fl,j

= 0

for

C l e a r l y such

a problem w i l l n o t b e r e s o l v e d u s i n g Tutte-Grothendieck r e c u r s i o n ! R e l a t e d problems i n c l u d e t h e following:

2.

t(M) : bk,O = 1, bk-l,O

Assume t h a t i n

b2,0 = 2b1,0 consider

M

*

f

.

0.

K = k+3. geometry,

a prime power?

bk,O = 1, bk-l,O

From

h a r d t o show t h a t

m-l 2

Is

*

M

Since

biPl

M*,

with

-

= 3 , blF2

= 3

and

0,

and

To answer t h i s q u e s t i o n , b10

f

0,

i t is not

i s connected, h a s rank t h r e e , and h a s c a r d i n a l i t y 0,

K

i t i s a p l a n a r geometry.

But f o r a p l a n a r

p o i n t s , (5.15.G) g i v e s t h e number of l i n e s ,

W2

-

*

W = b 1 l,o

-

~b:,~

b:,2.

* + b* - b1,l 2,o

which by (6.4 ) below i s e q u a l t o

T h i s i s z e r o i f and o n l y i f

M*

h a s a n e q u a l number

of p o i n t s and l i n e s ( i . e . ,

i s a connected p r o j e c t i v e p l a n e ) .

such a p l a n e o f o r d e r

we would have

m 1 n=2

3.

n,

n2

+ n + 1=

The above i n i t i a l c o n d i t i o n s mean t h a t matroid o f r a n k a t l e a s t seven.

M, fl,O = 0.

the f a c t t h a t i n

Thus,the s e t o f h y p e r p l a n e s

hyperplane.

k+3, s o t h a t

.

Can we have a T u t t e polynomial where, f o r

has a t l e a s t

For

n

H

t r i v i a l blocks) f o r

b

n,O

The e q u a t i o n , by (5.15.1), Thus,

H of

M

forms a t - d e s i g n

= 1,

M would be a connected paving reflects

h a s no independent h y p e r p l a n e s .

M

have t h e p r o p e r t y t h a t each

p o i n t s , a n d t h a t any s u b s e t o f

Thus,

n > 6,

n-1

p o i n t s i s i n e x a c t l y one

( w i t h perhaps m u l t i p l e non-

t > 5, and any such t - d e s i g n g i v e s a paving m a t r o i d .

The e x i s t e n c e of t - d e s i g n s w i t h

t > 5

i s a t p r e s e n t a famous open

problem. The examples above show t h a t i t i s a s i m p o s s i b l e t o c h a r a c t e r i z e t h e image o f t h e map M + Z [ x , y ]

o f (3.7)

(i.e.,

determine a l l

p o s s i b l e T u t t e polynomials) a s it i s t o c h a r a c t e r i z e its domain (determine a l l matroids).

The k e r n e l of t h e map i s e q u a l l y i n t r a c t a b l e .

We have s e e n many examples (3.11,

3.14. 4 . 3 , 4 . 5 ) of two ( o r more)

m a t r o i d s w i t h t h e same T u t t e polynomial.

K

argument shows t h a t f o r l a r g e

An elementary c o u n t i n g

t h i s s i t u a t i o n o n l y g e t s worse:

t h e r e a r e many more m a t r o i d s t h a n T u t t e polynomials. Proposition 6.2.

For any

> 0, t h e r e i s a

E

s o t h a t v h i l e t h e r e a r e more t h a n

K

sufficiently large,

,K(l-€) 2L

nonisomorphic m a t r o i d s 3 a l l w i t h t h e same T u t t e polynomial, t h e r e a r e less t h a n 2K d i s t i n c t T u t t e polynomials o f m a t r o i d s of s i z e Proof.

Let

M

b e a paving matroid of c a r d i n a l i t y

a l l o f whose h y p e r p l a n e s have n u l l i t y

0 or

1.

K and rank

while

Then, s i n c e every

f

l,j

= 0

for

j > 1.

A consequence

i s t h e n t h a t a l l such m a t r o i d s with a f i x e d number

o f dependent hyperplanes have t h e same T u t t e polynomial. o f Knuth ([86], o r s e e 1154) show t h a t , when

o f sire

[$I,

such t h a t any subfamily of

n =

FK

El ,

t h e number o f s u b f a m i l i e s o f

=

[%I

f

1,l

Results

there exists

can b e t h e f a m i l y of

n u l l i t y - o n e h y p e r p l a n e s of such a paving matroid.

fl.l

n

s u b s e t i s i n a unique hyperplane,we have

(n-1)-element

of (5.15.3)

K.

Letting

bK b e

FK each c o n t a i n i n g matroids

members, we o b t a i n

a l l v i t h t h e same T u t t e polynomial.

Dividing by

e l e m e n t a r y e s t i m a t e s we g e t t h e f i r s t s t a t e m e n t .

K!

and u s i n g

To count the number of distinct Tutte polynomials observe that, in the corank-nullity polynomial, there are

(K+112

coefficients

K all of which are nonnegative and which sum to 2

{aij)

[

2K+~?2K K2+2K] <

at most

zK3 choices for coefficients

.

Thus, there are

(and distinct polynomials).

Although the estimate above for the number of distinct Tutte polynomials is small compared with the number of matroids, the actual number of Tutte polynomials is even smaller. Aside from those subtle considerations pointed out in (6.1), there are several identities and equalities which the coefficients fbijf must satigfy. The identities can be characterized completely with reasonable ease,while the search for inequalities leads into many areas of matroid theory and more general mathematics,and seems endless. We begin by characterizing all (affine linear) identities which hold among the coefficients of

{t(M)

:M

has rank n, cardinality K, and is free of loops, multiple points, and isthmuses f. We make these restrictions,since most applications involve combinatorial geometries (no multiple points),while

(4.4.2)

and the dual of (4.4.3) show how to reduce to the isthmus-free case. Let G be the class of all isthmus-free K.n combinatorial geometries of rank n and cardinality K. Further,

Proposition 6.3.

M K. n

let (K+1)

x

be the affine variety spanned by the set of all

(n+l)

such that the (i,j)-th entry of

%.n

in the cardinality-corank polynomial (see 3.17.1) aij (G;z,u) for some G E GKpn.

coefficient

S KC

matrices {%,n)

is the

P.

The dimension of kd

K,n

equals

for the relations which define M K.n

(n-2).(K-n-1).

A basis

follow:

2.

The following identities form a basis for the relations which

hold -g

the coefficients (with tunanegative subscripts) in the T t l t t e

polynolial

Ib.

13

Proof.

1.

to be a basis.

: t(G) = Zb. .xiyJ, G t =J

GK,nl:

These identities are found in [38] where they are shown

2.

One e a s i l y checks t h a t i d e n t i t i e s

(a-f) a r e independent s i n c e each

involves a c o e f f i c i e n t

b.. n o t found i n any previous one. (For 13 can be i n t e r p r e t e d a s a n equation f o r b n-2, k-n+2

example, n-2

I k k 2 K-3, and a s an equation f o r

5

Further, i d e n t i t i e s

i f 0 5 k < n-2.) k,O l e a v e an (n-1) by (K-n) r e c t a n g l e of

(a-e)

b

c o e f f i c i e n t s undetermined, and adding i n i d e n t i t i e s gives t h e dimension of

if

{ I k : 0 c k r K-3)

f o r t h e number of degrees o f freedom.

M

K,n

Thus, by (3.18.3) which can be thought of a s giving a n i n v e r t i b l e a f f i n e map between t h e c o e f f i c i e n t s of

SKC and those of

t,

t h e r e are no

more i d e n t i t i e s . Identities

(a-e)

a r e obvious.

follows from t h e f a c t t h a t

For example,

h a s no isthmuses,and

G

f a c t t h a t t h e l o o p l e s s geometry

IM'~

k.

In fact,

> k.

we w i l l prove

{Ik)

matroid of rank

k

+

and c a r d i n a l i t y

l

K =

b i j + 1 i,j

v = z+l,

v

we g e t :

Kt-k =

k-i

reflects the

so t h a t i+j k-i

z

.

M'

with

Our a l t e r n a t e proof

u = 1. Let Then,

K'.

To prove

and o n l y u s e s K i n a bound

f o r a l l matroids

evaluated a t

k z+l S (M8;z,l) = z t(M';9 z+1), KC

Letting

n

These i d e n t i t i e s f i r s t appeared i n [ 2 7 ] .

below uses equation (3.18.3)

(c)

= 6(i,O)

h a s no m u l t i p l e p o i n t s .

G

( f ) , we f i r s t n o t e t h a t i t i s independent of on

bi,K-n

M'

be any

Considering t h e constant term on t h e r i g h t

(m=j),

we g e t t h a t whenever

K' > k,

This i s p r e c i s e l y i d e n t i t y

M'

general r(M')

= n'

,

Ik i f

r(M') = k.

I r(M1) - kl .

we use induction on

k+l

and rank

n'-1.

M',

number of nonfactors of

Ik on a l l matroids of

Using a second i n d u c t i o n on t h e

we g e t t h a t

Ik c e r t a i n l y holds i f

is t o t a l l y s e p a r a b l e , s i n c e then t h e nonzero c o e f f i c i e n t t(M') t(M')

Ik. B u t , i f

does n o t appear i n

+ t(M'/p),

= t(M'-p)

nonf a c t o r s , i n

Now,assume of rank

n'

p

+1

and size a t l e a s t

bn' ,j

k+l.

t (M1/p).

and

M"-p

have rank

each.

Ik f o r a l l matroids

and t h a t we have proved k+l).

Let

M'

have rank

Then, t(M') = t(M") n'+l

t(M"-p). k+l, so

MI*

obtaining

The two matroids

M"

Ik holds f o r

t(M').

Ikl with

k' < k

t(M)

{I 1 k

t o simplify

The following i d e n t i t i e s

of t h e T u t t e polynomial

-

and s i z e a t l e a s t

We now l i s t t h e f i r s t few i n s t a n c e s of

Corollary 6.4

n'

i s c e r t a i n l y not a boolean a l g e b r a s o t h a t

M'

Thus, i t holds f o r

using i d e n t i t i e s

k+2,

Thus, by

we may make a f r e e coextension, adding a nonfactor p f r e e l y t o M"/p = M'.

2

Ik holds,by induction on the number of

and

(and c a r d i n a l i t y a t l e a s t

with

in

t(M').

k > n',

M"*

M'

i s a n o n f a c t o r , then I M ' I

MI-p, and, by induction on rank, i n

l i n e a r i t y , i t holds i n

Ik f o r a

First, l e t

> k , and assume t h a t we have proved

s i z e at l e a s t

To show

f o r f u t u r e reference, Ik.

{ I k } hold on the c o e f f i c i e n t s

f o r any matroid

M

with

I M l > k.

We now t u r n our a t t e n t i o n t o i n e q u a l i t i e s involving t h e c o e f f i c i e n t s Ib

ij

1.

C l e a r l y , t h e most obvious one i s t h a t

b

2 0

ij

for a l l

(i,j).

However, t h i s can be sharpened. P r o p o s i t i o n 6.5 b

ij

> 0

(i',jV) Proof.

Let

M

be a connected matroid, and assume t h a t

i n i t s T u t t e polynomial.

*

(0,O)

If

M ( s e e (3.1),

b

with

ij

> 0

(iv,j') in

5

t(M),

(3.4)), t h e r e

Then

then,somewhere i n

B ' ~ , and thus Thus, i t has

( i ' , j V ) : (0,O) < ( i ' , j ' )

b10 = bO1 = B(M)

we may f i r s t assume t h a t

(i,j).

M

bij M

B(M)

decomposition of M

has t h i s

Bi ' J '

as a

we may now apply

i s t h e theorem of

w i t h a t l e a s t two p o i n t s ,

i s p o s i t i v e i f and only i f

We w i l l e x p l o r e t h e i n v a r i a n t (6.24.21, (6.26.4)). To determine which

5

(and I1 of (6.4))

Crapo [53] t h a t , f o r any matroid

for a l l

a

..

i s t h e term

A s p e c i a l c a s e of (6.5)

> 0 iV,j'

(i,j).

t o t a l l y s e p a r a b l e matroid a s a minor. minor f o r a l l

b

M

i s connected.

i n more d e t a i l below ((6.15),(6.22.2,3),

can be p o s i t i v e f o r a g e n e r a l matroid

M,

has no loops o r isthmuses,since otherwise we

can reduce t o t h i s c a s e by (4.4).

The number of connected (direct-sum)

components of M

can be determined from t(M)

where (in

bk,O > 0 and bk-l,O = O-

t(M)),

and is equal to k ,

For such matroids with k connected components, an easy application of (3.6) shows that biSk+=c > 0 for all i : 0 bi,j

= 0

for all i+j < k.

for which b..

13

5

i

5

k, and that

Therefore, in theory, the possibilities

are positive can be reduced to the connected case since

the possible

positive b of a matroid with rank n, cardinality ij K, and k components come from all possibilities of nonzero terms in t(T)*t(M2)

-... t(Mk)

where each M is connected, i

1

=

K, and

1

1

r(Mi) = n. In particular, we note that the set of indices i {(i.j) : bij > 01 f o m a rectilinear convex set in the integer lattice L

x 22

in the following sense.

The sequence

b~,O'b~-l,O'""bk,~'

bk-l,l""*bk-j,j7~~~1b0,k~b0,k+l~~~~~bo,K-n gives the "northeast

boundary'' of positive coefficients,and the border sequence of (5.7) gives the "southwest boundary". Further, bij > 0 if and only if it "lies within in the northeast boundary the boundaries" (e.g., if there is a positive b ij ' To complete our and a bij,, in the southwest boundary with j '5 j 5 j")

.

analysis, we give the possibilities for border terms in connected matroids. As

a preliminary, we discuss an interesting class of matroids introduced in

[I141 whichwill also be useful later (see 6.23)).

Definition 6.6

A nested matroid (of cardinality K and rank n) is

one in which the cyclic flats are totally ordered in that if F and F'

are cyclic flats with

IF(

were first defined in [114]. exists an ordered basis B

=

5

IFT(, then F 2 F'.

These matroids

An alternate definition is that there

{bl,.

..,bn1

such that every point

p

n o t i n t h e b a s i s (and n o t a l o o p ) forms a c i r c u i t w i t h a n i n i t i a l

segment

{ bl....,b

k

1 f o r some k .

then s u b s e t s of t h e form where

The c y c l i c f l a t s a r e

u Pi

Ck = I b l , . . . , b k l

i s t h e s e t of p o i n t s which form c i r c u i t s w i t h some ( i n i t i a l o r empty)

PL

s u b s e t of

Bk = {bl,

...,bk 1.

c i r c u i t s p r e c i s e l y when an isthmus i n

zk)

P' k

T h i s i s c l e a r l y c l o s e d and i s a union of

-

i s nonempty ( i . e . ,

P' k-1

when

bk

is not

.

The f o l l o w i n g f a c t s a r e e i t h e r found i n 11141 o r c a n b e e a s i l y demonstrated by t h e r e a d e r .

1.

A n e s t e d m a t r o i d i s determined up t o isomorphism by t h e sequence

(ao,al,

...,a n ) ,

ai = 1 +

IP; -

where P;-ll

a.

=

i s t h e number of l o o p s , a n d , f o r

IBi - Ei-l 1 . a

0

r e f e r t o t h e (unique) n e s t e d m a t r o i d sequence a

i

2.

> 0

(ao,

...,a

for a l l

N(ao,

K

= 0

and

N(ao,

al = 1.

...,a n ) .

....a

)

Note t h a t any

a. 2 0

i s connected i f and only i f i t h a s no l o o p s

and

(ao = 0 )

(an > 1 ) . N(a o , . . . , a

)

i s d e t e r m i n e d by t h e sequence: ( (io.jo),

with

We w i l l h e r e a f t e r

g i v e s a n e s t e d m a t r o i d of rank n i f

)

The connected n e s t e d matroid

(*)

and

i > 0.

and no i s t h m u s e s

3.

ai = K,

Here, of c o u r s e , 1

i s a geometry i f and o n l y i f

M

i > 0,

O = i

0

.. .,( i m , j m ) ) , ... < mi = n , and

(il,jl),

< i <

1

of r a n k

n

and n u l l i t y

Here, ik = r(C k) , the rank of the kth largest cyclic flat Ck* and

1 ck( -

jk =

r(Ck),

the nullity of

Ck.

Any such sequence of

pairs parameterizffia connected nested matroid.

...,an),

N(ao,

with . a = 0 and

In particular,

an > 1, has a nonempty cyclic flat

of rank i and nullity j if and only if ai > 1, in which case

-

j =

k=O let ai = 1 k

i.

Conversely, given any sequence of pairs satisfying (*),

+ jk -

jk-l if

(ik,jk)

is in the sequence (*), while

ai = 1 otherwise (conventionally, j-l = 1). 4.

The class of nested matroids is closed under the following

operations:

...,an) )

a.

Truncation: T(N(ao,

b.

Free extension: N(a o,...,a

c.

Deletion:

where pi

E

N(a o,...,an)

-Bi - BiVl. -

-

n

= )

N(ao,

+p

...,an-23an-1

= N(a o,...,a

.

+ 1).

pi = N(a o,...,ai-l,...,a

)

We use the conventions that, for i > 0,

. .,O,ai+l,. ..,a ) = N(ao,. .. ,l,ai+l-l,. ..,an1, N(ao,. ...an-l,O) = N(ao.. . . ,an-l). N(ao,.

K

an)

+

and that

..,an) /pi = N(ao,. ..,ai-Z,ai-l

+ ai-l,

d.

Contraction: N(ao,.

e. f.

ai+l,...,a n). Free coextension: N(ao, a ) x p = N(O,ao + Iral, a ). Duality: If a nested matroidl N of rank n and cardinalFty

...,

...,

is parameterized by its sequence of cyclic flats as in (*) above, then

* *

N* has rank K-n and has the sequence ((iO,jO),

* * (ik, jk)

* * .. . ,(im, jm))

(K-n-j n-i (For this, recall that m-k' m-k) ' flat of M iff S-A is a cyclic flat of M.) =

*

A

where

is a cyclic

We leave as an exercise to the reader the explicit formula for

*

ai when

5.

The hereditary class of nested matroids has,as excluded minors,

the sequence

Thus, M2 of (4.3.1),

is a line with two double points, M3 and, in general, Mi

is the matroid M2

consists of two disjoint i-point

circuits otherwise freely placed in a space of rank i. 6.

To get the Tutte polynomial t(N(a o,...,an)),

we will use the

cardinality-corank polynomial and apply (3.18.3). consisting of

si points in

is given recursively by fi (s) -

=

+

min(fi-l(s)

each point of

-Bi - Bi-l -

- (Bi - Bi-l)

and so adds one to the rank of

Proposition 6.7

= f (s),

n

-

then r(S1)

where, for all i, fo(z) = 0, and

is in free position in rank i,

S' (unless rank i is reached).

Thus:

Necessary and sufficient conditions for possible

sets S of indices

{(i,j)}

a connected matroid

M (IMI > 1)

2.

...,n),

(i = 0,

is a subset

This is easily shown by induction since

si,i).

S; = S' n

. .,sn)

fn(so,.

If S'

For some i, (i,O)

of positive coefficients in t(M)

E

S

for

are that:

and

(i

+ 1, 0) /

S.

In this case,

2.

For any matroid

M

w i t h T u t t e polynomial

t(M) = Zb. .xiy', 13

we have

I

and

-

j.dj'.dj-l.(il.bi'-lbi

i,j;i',jq

,2

0

1J

f o r a l l a , b, c , and d w i t h

0

5

a

5

b, 0 5 c

S

d,

and

(b-l)(d-1)

2 1.

1. This i s t h e l i n e - p o i n t i n e q u a l i t y f o r geometries:

Proof.

Wg 2 W1

i.bi'bi-l 1-bijab.,.

w i t h e q u a l i t y i f and o n l y i f

G

i s a modular p l a n e ( s e e

(5.15.4)).

2.

These, and o t h e r , i n e q u a l i t i e s come from t h e

s t a t i s t i c a l mechanics i n t e r p r e t e d f o r

t(M)

FKG

i n e q u a l i t y of

i n [127].

W e remark t h a t t h e r e a r e two t y p e s of i n e q u a l i t i e s which we have considered.

One type (e.g.,

b10 > 0

f o r connected m a t r o i d s ) can

b e proved i n d u c t i v e l y u s i n g p r o p e r t i e s o f t h e T u t t e decomposition (T1 and T2), w h i l e t h e o t h e r t y p e (such as (6.8.1)) and cannot

u s e o t h e r arguments

be ( d i r e c t l y ) deduced from t h e decomposition.

Obviously.

t h o s e o f t h e l a t t e r type a r e more d i f f i c u l t t o d i s c o v e r and prove.

The c o e f f i c i e n t s of

t(M)

which have r e c e i v e d t h e most a t t e n t i o n

by r e s e a r c h e r s i n matroid t h e o r y a r e t h o s e o f t h e geometric T u t t e polynomial

and t h e r e l a t e d Whitney numbers of t h e f i r s t kind:

the coefficients

Iw;}

of t h e c h a r a c t e r i s t i c polynomial n X(G) =

Here, we assume

1

n-i

W;A

i-0

=

(-~)?E(G;~-X)

i s a geometry, and we w i l l henceforth t r e a t t h e

G

unsigned Wh-ltney nmbers.

Thus, n

(6.9.1)

IxI(G) =

I i=O

n-i =

wih

-t(G;A+l),

so that

The p r i n c i p a l c o n j e c t u r e i s t h a t t h e sequence (wO = 1, w2 = K,

...,wn = lu(G)l)

i s unimodal, l o g a r i t h m i c a l l y concave,

o r s t r o n g l y l o g concave i n a sense which we w i l l d e f i n e below.

First,

n o t e t h a t t h e r e a r e s e v e r a l a n a l o g i e s of t h e two kinds of Whitney numbers (both of which a r e conjectured t o give l o g concave sequences). (Proofs a r e easy a p p l i c a t i o n s of (6.9.2). ) (6.9.3)

W. =

1

w

1 2

r(x)=i

wi = Wi =

(6.9.4)

,I:[

i

iff all

1.

I~(o,x)

= r(x)=i

with e q u a l i t y ,

(i+l)-element s u b s e t s of

C

a r e independent. (6.9.5)

wo

5

wl 5

... 5 w

.

That

W

0

5

W

< 1-

--.

is still

only a c o n j e c t u r e . wn-i

(6.9.6) It Wn-2

2 w

i

for a l l

i 5

]

.

That

Wn-i

I

Vi

i s y e t unproved.

i s n o t even known a t t h e p r e s e n t time whether ( f o r n 2 5) 2 W2,

o r whether,

Wg

>

W2

for

n 2 4.

However, i t i s known t h a t s f o r a l l

K

wi

5

and

K 5 W

d i f f e r e n t from

i

0

and

n,

i'

We now make some d e f i n i t i o n s and elementary remarks about unim o d a l i t y and r e l a t e d c o n c e p t s .

1. A sequence of nonnegative i n t e g e r s

D e f i n i t i o n 6.10

...,a

a,,,al,

a.

5

that

2. a.

al 5

with

...

5

a

a Q 2 a e+l 2

a 2 min(aj ,%) m The sequence

< al <

... 2 an'

j < rn < k .

i s strongly unimodaZ i f f o r some =

=

... = % >

ak+l >

... >

a . n

5

k:

These

sequences a r e c h a r a c t e r i z e d by t h e l o c a l p r o p e r t i e s t h a t f o r a l l

ai

L

and t h a t , i f

min(ai+l,ai-l),

Equivalently, define

"-1 3.

a 0,

and

A

j

aiWl

A j = aj+l-aj.

> 0

a,

This i s e q u i v a l e n t t o t h e p r o p e r t y

for a l l triples

(ai)

.. . < a t

lazimodaZ i f , f o r some

= 1 is said to be

0

=

Then,

Aj-l

ai,

t h e n ai-*

A. t 0 3

<

i,

a i . t ai+l-

implies

> 0.

Stronger conditions

which g u a r a n t e e unimodality a r e a l s o amenable t o l o c a l c o n s i d e r a t i o n s . The most obvious such i s c o n c a v i t y , t h e d i s c r e t e a n a l o g of having a nonnegative second d e r i v a t i v e .

Here

a

ai+l+ai-l i Z 2

U n f o r t u n a t e l y , t h e Whitney numbers o f t h e f i r s t ( o r second) kind w2 a r e seldom concave ( u s u a l l y , v2 Z ) However, s i n c e ( s t r i c t l y )

-$ .

monotonic f u n c t i o n s p r e s e r v e unimodality (and nonunimodality), a way t o prove s t r i c t unimodality would b e t o f i n d a s t r i c t l y monotonic f u n c t i o n whose v a l u e s on t h e sequence were concave.

It is

customary to choose the logarithm function, and a (unimodal) sequence 2

is said to be 10; cmcave if, for all 4.

i, a.

2

1

a i-lai+1'

We say a sequence is strorigiy log concave if

the sequence

(9

(a; = --L )

is log concave.

al = K r n, and

A motivation for this

definition is that it includes all sequences arising from truncations of polynomials p(x)

with all negative integer roots:

n+m (x+c;). Further, for such truncated polynomials, the i=l 12 inequality will be strict (ai > a!-la;+l) unless all c = 1. (See [79].) Qe introduce this concept because it i =

is known to hold for many well-studied sequences of Whitney numbers such as binomial coefficients and Stirling numbers of the first kind.

Further, strong log concavity is preserved by the

truncation operator since

and

n-m

(T~(G))

=

w,-,(G)

-

+

W,-,+~(G)

• - +

-

It is easy to see that Whitney numbers of the first kind for geometric lattices which come from truncations of boolean algebras, partition lattices, or modular geometries are all strongly log concave. n (The polynomials are, respectively, n i ji=O

(x+lln,

(*i), i=l

and

5.

We c o n j e c t u r e t h a t t h e Whitney numbers of t h e f i r s t kind a r e

s t r o n g l y l o g concave. Thus, we c o n j e c t u r e t h a t

Equivalently,

This r e f l e c t s t h e f e e l i n g t h a t trrtncated boolean a l g e b r a s g i v e the " l e a s t concave" Whitney numbers ( s e e [ l o l l f o r analogous conjectures f o r

-t(G)

(Wi)).

The r e l a t e d c o n j e c t u r e f o r t h e c o e f f i c i e n t s of

i s t h a t t h e sequence:

i s l o g concave and we c a l l t h i s c o n d i t i o n s t r o n g l o g concavity f o r

-t(G).

Equivalently, we c o n j e c t u r e t h a t

2 bi'

(1[-i-1 n-i+l

(K - i ] ( n-i I b i - l b i + l .

Thus, a g a i n f o r sequences of T u t t e c o e f f i c i e n t s

(bi),

truncated boolean a l g e b r a s a r e thought t o be the " l e a s t l o g concave." (See (5.15.3), Research -

problems 6.11

of p o s i t i v e 2.

o r use ( 4 . 2 ) . )

b

ij

1. Find t h e e x p l i c i t c h a r a c t e r i z a t i o n f o r t h e i n d i c e s

f o r a matroid with

k

components ( s e e ( 6 . 7 ) ) .

Show t h a t i f , f o r G1 and G p , t h e Whitney numbers o r T u t t e c o e f f i c i e n t s

a r e s t r o n g l y l o g concave, then

so are these numbers for G1 @ G2.

This amounts for Whitney numbers,

essentially, to showing that if

This latter inequality (*) may not be hard to prove and is interesting in its own right.

It is probably true since it holds when the two

i polynomials p(x) = Zaix and

i q(x) = Ccix have real negative roots.

Further, it is true for isthmuses-, since (*) is proved in [ 7 9 ] for the degree-one case. We also note that the analogous result for the log concavity of (di) was shown by Harper (see 1811).

1; m]

by Harper (see 1811). b.

2.

1f

is

=

1 . 0 .

concave, are the coefficients of

K-1-n

x(G)

of

3.

strongly log concave? In particular, show that the log concavity

(b') i

implies the log concavity of

Find classes of geometries (such as dual paving matroids as we

show below) which give unimodal or strongly log concave Tutte coefficients or Whitney numbers.

4.

We cover our bets by offering the problem of finding a geometry

whose Whitney numbers are not unimodal. We will illustrate some techniques with the next two propositions. The first shows that if

(bi)

is log concave, then so is

Again, this is not surprising since if p(x) of

with real negative roots, then p(x+l),

(wi).

(bi) came from a polynomial (wi) would be the coefficients

another polynomial with real negative roots. We begin

with a lemma presenting the combinatorial arguments we will use. Lemma 6.12

1.

Let

(ai : i

2

0)

and

(a; : i 2 0)

negative eventually zero sequences, and let

ir =

f

a,

be two nondenote the

i=O m

rth partial sum (where

m

=

1

a.). j=o J

in the sense that, for all

(a;)

-

-

i, a. 2 a' 1 i -

Further, let

(bi : i

(b; : i

be two nonnegative sequences such that, for all

b.

2

b;

2

0) and

2

b. r bi+l.

0)

(ai) dominates

Assume that

and

Then,

(a.b)_2 a'

2.

If for all i

-

-

3.

Let k and

am r :a

-

2

0, -I ai

i

(-)_

0 (with - = 0), 0

-

, then ai r a'i for all i. i be fixed.

dominates the sequence

.

Then, the sequence

and if

i,

F u r t h e r , t h e sequence

dominates t h e sequence (bI =

k+j+l

( i+l ]

k-j [i-1)

(k+j+l

+

1 i-1 ] [kii] j :

2

1.

Proof.

(%Im =

1

a r ( b -b,cl) (where we can assume t h a t (b.) i s eventually zero)

r=O

2.

Assume t h a t f o r some

t h i s property.

-a

s

< a'

s

for a l l

a

s > r

< a:,

a

-

r

< a'

r

and l e t

and, f o r a l l

s

r

b e minimal w i t h

r r,

while

6- = h l =

bi+l

: bi::b;+l

[:].

Simplifying the proportions : b;,

a

s

Thus,

5 a'. S

which c o n t r a d i c t s t h e h y p o t h e s i s t h a t

An elementary c o m b i n a t o r i a l argument shows t h a t

3.

and

Then,

r,

and applying (6.12.2)

-

-

am = a: ai+l

=

[i~:],

: a t : :a'i+l : a i'

yields the result.

We a r e now ready t o prove t h e l o g concavity of l o g concavity of if

(bi).

(wi)

from t h e

The same proof gives the s t r o n g e r r e s u l t t h a t

(bi : 1 5 i < n) i s log concave, s o i s

(Gi : 0

S

i 5 n-1),

the

sequence of reduce2 Whitney nwnbers. Reduced Whitney numbers a r e motivated by t h e f a c t t h a t x(G;l) = 0, so t h a t

1-1 d i v i d e s t h e c h a r a c t e r i s t i c polynomial of

(Equivalently, t h e r e is no c o n s t a n t term i n

wi = w ~ + + wi~

(6.12.5)

P r o p o s i t i o n 6.13

-t(G)

c i e n t s of

t h e sequence

i s l o g concave.

(Gi)

By ( 6 . 9 2 we have

Therefore,

(bi)

Then,the Whitney numbers

of c o e f f i (wi)

and

a r e a l s o l o g concave. (

w

[

) =~

m=i

expansion of t h e right-hand s i d e , we o b t a i n :

Similarly,

G.

and

Assume t h a t

reduced Whitney numbers

m.

,

-t(G).)

] b ] 2 .

Symmetrizing t h e

Let

ad

k

be fixed.

(bk+j+ibk-j)

By t h e l o g concavity of

(bi),

(bk+jbkmj)

a r e both nonnegative decreasing sequences.

Further,

t h e sequence of binomial c o e f f i c i e n t s under each summation s i g n i n t h e

expansion of

( w ~ - ~ )dominates t h e respec r i v e sequence of binomial

c o e f f i c i e n t s i n t h e expansion of

~ ~ - ~ - ~ - wby~ (6.12.3). - ~ - ~

Therefore, the conditions i n (6.12 . l ) a r e met, and, f o r each k,

The proof f o r P r o p o s i t i o n 6.14

(ji) Let

and i n

i s e x a c t l y t h e same. G

be a paving matroid.

f (G*)

in both

;(G)

Proof.

W e use t h e formulas i n (5.15.3).

a r e ( s t r i c t l y ) unimodal.

Since, f o r a l l

t h e sequence In fact, since

am-1.

bl

Then t h e c o e f f i c i e n t s

[I,

(b.(G))

i

> 2,

is trivially

t h e sequence i s s t r o n g l y l o g

concave. Nov, l e t

b* = b .(G) = b.(G*). j 03 J

Then, by (5.15.3),

for

j

7

0.

where

G

has

men,

ai

Let

j

*

x(G ) = 0

be such t h a t

and, f o r a l l

We use (6.10.2).

[ i ] a i - 2 ] . n-2 iaj+n-2 n- 2

~f

A. 1 0 J

Thus, i f

-

ai > 0, bin

then

a bl).

has

G

and t h e r e i s nothing t o prove.

i s f r e e of isthmuses.

G

i.

1

dj = b ; + l - b l =

an isthmus, then assume

hyperplanes of c a r d i n a l i t y

So,

i S K-2.

men,

3

i,

Thus, a s we pass from

Aj

to

Aj-l,

each term on t h e left-hand

side

i n c r e a s e s i n r a t i o a s l e a s t a s f a s t a s t h e s i n g l e binomial c o e f f i c i e n t on t h e r i g h t (and perhaps new terms become nonzero).

- 1

a

b

j

* ; bj

a

* bjW1

; and, f i n a l l y , t h a t

(b*) j

So we have

i s unimodal.

For f u t u r e r e f e r e n c e , we now review some of t h e important i n v a r i a n t s which, f o r s p e c i a l c l a s s e s of matroids (such a s graphic and o r i e n t e d m a t r o i d s ) , have i n t e r e s t i n g i n t e r p r e t a t i o n s .

W e then r e l a t e

t h e s e i n v a r i a n t s using t h e c o n s t r u c t i o n s presented i n s e c t i o n four. Remarks 6.15

1. The

T-G

group i n v a r i a n t s below have t h e follow-

ing formulas. a.

The number of independent s e t s of rank

r:

b.

The (absolute) rth Whitney number:

c.

The reduced rth Whitney number:

d.

The beta invariant: B(G)

= bl

-- a t(?1;0,0) ax

= (-1)

n+l d =x(M;~)

e.

The (absolute) Miibius function:

f.

The acyclic or alpha Znvariant and reduced alpha invariant:

g.

The c m p z e ~ t y b(M) (number of bases), independence number

i(M) (number of independent s e t s ) , and subset number

s(M):

2.

These i n v a r i a n t s a r e r e l a t e d by t h e f o l l o w i n g i n e q u a l i t i e s

Further,

s (M) > i(M) > a(M) , and b(M) > P(M) unless

M

i s a boolean a l g e b r a ;

i(M) > b(M)

unless

a(M) > &(M) u n l e s s ;(M)

.

p(M) > B(M)

r(M) = 0,

M

has a loop,

unless

M

has a loop o r

r(M) = 1, and B(M) > 0

3.

unless

M

is a loop o r is separable.

These i n v a r i a n t s a r e a l s o r e l a t e d through t h e f o l l o w i n g c o n s t r u c t i o n s .

a.

For t r u n c a t i o n : b ( T " - ~ ( M ) ) = I,(M).

b.

For f r e e e x t e n s i o n : Ir(M+p) = Ir(M)

+

Ir-l(M)

B(M+p) = u(M)

c.

For f r e e coextension: G , ( ~ x p ) = I (M) a ( ~ x p )= i(M) ~ ( M x p ) = b (M)

d.

For d u a l i t y : B(M*) = B(M)

e.

For d i r e c t s u m w i t h a n isthmus: ji(Mep)

= wi(M)

A l l t h e above can b e e a s i l y proved u s i n g t h e f o r m u l a s of (6.15.1)

a l o n g w i t h t h o s e of (3.18)

Some of t h e above formulas have

and 6 . 2 ) .

g e n e r a l i z a t i o n s o r " c o m b i n a t o r i a l proofs" (one-to-one between two f a m i l i e s of s u b s e t s o f identity).

S,

correspondences

each counted by a s i d e o f t h e

For example, r e s u l t s of [35] show t h a t s u b s e t of if

M({1,2,

...,K}) and

6

i s a n independent

(and c o n t r i b u t e s t o

I u I01 i s a "x-independent subset" of

contributes to

I

M x 0.

( 6 . 1 5 . 3 ~ ) follows.

i s t h a t t h e l o g concavity of

(Gr)

Ir)

i f and only Thus,

A consequence of (6.15.3~)

( o r of

(bi)

by (6.13)) f o r a l l

matroids (with a c o f r e e p o i n t ) implies t h e l o g concavity of a l l matroids.

also

I

(I,)

for

Other combinatorial correspondences appear i n P a r t 11.

A s an example of a l a t t i c e - t h e o r e t i c g e n e r a l i z a t i o n of (6.15.3b),

Zaslavsky 11631 showed t h a t , f o r any p o i n t

B(G) =

I

1.

~(0,x) xcL(G) : ~;CP

When

p

S,

E

G = M

+ p,

t h e (signed) right-hand s i d e i s e a s i l y shown t o b e equal t o

1

u(0,x) = - u M ( o . l ) . xeL(M) : xrl (A combinatorial proof of t h e above i d e n t i t y

appears i n 1463.)

A u s e f u l technique f o r proving i n e q u a l i t i e s among T-G i n v a r i a n t s i s t h e use of b i j e c t i v e rank-preserving weak maps ( s e e 1971).

matroid

M (S)

1

i s s a i d t o b e f r e e r than a matroid

a rank-preserving weak map between on

S.

M1

and

M

2

M (S)

2

M

1'

If

M

1

+ M2,

t h a t t h e map i s non-trivial.

M1 w ' M2'

we say t h a t

M1

i f there is

which i s t h e i d e n t i t y

This i s equivalent t o saying t h a t each b a s i s of

b a s i s of

A

M2 i s a

i s strictZy freer and

We denote t h i s c a s e by w r i t i n g

E s s e n t i a l t o t h e proof of a l l t h e i n e q u a l i t i e s below i s t h a t if

5

>w M2,

t h e n , f o r any n o n f a c t o r

p,

M -p 1

M -p w 2

and

r

zw M ~ / P where, i n a t l e a s t one of t h e i n e q u a l i t i e s , t h e

%/P

minor i s s t r i c t l y f r e e r and no new l o o p s a r e c r e a t e d .

5*

* >w M2,

and, f o r any matroid

we have t h a t

of rank

M

T~-"(B ) 5 M 2 B K w w

~

~

~

Thus, f o r any T-G group i n v a r i a n t a l l l o o p l e s s matroids whenever M

of rank

M

i s l o o p l e s s and

1

f o r nonstrict inequality). t h e r e s u l t i n [27] t h a t on

=K-n

(BK)

M1

n

>w M2

M1

Further,

n

and c a r d i n a l i t y

~

.

-

f :M

-+

K,

i f f (M) > f ( B ~ ' ~ -f o~ r )

W,

and c a r d i n a l i t v

K, t h e n

f(Ml)

> f(M2)

(and a s i m i l a r r e s u l t h o l d s

T h i s g i v e s an e a s y , a l t e r n a t e proof t o

u,a,b,

and i

a r e a l l ( s t r i c t l y ) maximized

among m a t r o i d s o f t h e same rank and c a r d i n a l i t y .

We

l i s t some r e l e v a n t r e s u l t s below. P r o p o s i t i o n 6.16

Let

M1

1.

i ( M ) > i(M2) 1

and

b ( N ) > b(M2). 1

2.

a(M1)

r u(M2)

and

p(M1)

5

be s t r i c t l y f r e e r than

M2.

Then,

r p(M ) w i t h s t r i c t i n e q u a l i t y i f 2

h a s no l o o p s .

3.

wr(M1)

and

b .(If ) OJ 1

4.

$(MI) 5 B(M ) 2

Proof.

zwr(M

r b

Oj

2

1, 6r (M l

2

Ir(M2).

biO(M1)

rbiO(M2),

(M2). with s t r i c t i n e q u a l i t y i f

Formulas (6.16.1)

remarks ( s e e [ 9 7 ] ) .

~ G , ( M ~ )Ir(M1) ,

-

(6.16.3)

We prove (6.16.4).

M1

i s connected.

a r e e a s i l y proved by t h e above This i n e q u a l i t y h o l d s

t r i v i a l l y if K 5 2 , then P

E

B(M) > 0 =

M

and we a l s o n o t e t h a t i f

6(BnsK-") .

Now, l e t

i s connected,

be a connected matroid with

M1

M1Case 1.

If

M -p 1

Case 2.

If

N2

Case 3.

If

M /p

M2 / p

and

a r e both connected, we a r e done

by induction.

i s s e p a r a b l e , then

M2 = P(M1,M") 2 2

= B(Mi)S(M;)

Case 4.

M* > M* 1 w 2'

=

2w M;,

M1

M;'

ZW M;',

and

Hence,

and a t l e a s t one of t h e

and t h e i n d u c t i o n hypothesis,

= B(MZ).

i s s e p a r a b l e , then

B (MI),

i s a p a r a l l e l connection

2

By (4.8.10)

MI-p

> B(M2) = 0.

Further, M / P i s a l s o separable.

and .:41

> B(M;)B(M;')

If

8 (<)

M;

Mi

with

inequalities s t r i c t . B(M1)

i s s e p a r a b l e , then

1

of connected matroids

B(M~)

8 (M;) = 8(M2).

*

M1/p

is separable,

Thus, d u a l i t y reduces

t h i s c a s e t o t h e previous one. We end P a r t I with a d i s c u s s i o n of extremal c l a s s e s .

This

concept r e l a t e s t h e i d e a of a T-G recognizable h e r e d i t a r y c l a s s and t h e theory of (parametric) T-G i n e q u a l i t i e s .

D e f i n i t i o n 6.17 ff

1.

Recall t h a t a h e r e d i t a r y c l a s s of matroids

i s one closed under minors ( e q u i v a l e n t l y , under s i n g l e - p o i n t

d e l e t i o n and c o n t r a c t i o n ) . c l a s s of excluded minors minor of

E

is i n

Any h e r e d i t a r y c l a s s can b e defined by i t s

E

where

E

E

E if E # ff

b u t every proper

Conversely, i t is c l e a r t h a t any c l a s s

ff.

C

of matroids which c o n t a i n s no proper minors of any of i t s members

i s t h e c l a s s of excluded minors f o r t h e h e r e d i t a r y c l a s s H = IM : no minor of

M

is i n

Cl.

Since, when c a l c u l a t i n g T-G group i n v a r i a n t s , we f o r b i d d e l e t i o n and c o n t r a q t i o n by loops o r isthmuses, we a r e motivated t o d e f i n e a

H'

T-G hereditary class

a s one which i s closed under d e l e t i o n

and c o n t r a c t i o n only by nonfactors.

E'

by i t s c l a s s

but M-p

and

M/p

The c l a s s

H'

of T-C excluded minors where

M

H'

are i n

i s a l s o defined

E'

E

f o r a l l nonfactors

p

E

if

M

M.

Obviously,

/ H'

every h e r e d i t a r y c l a s s i s a T-G h e r e d i t a r y c l a s s whereas, f o r example,

{ B ~ I forms a T-G h e r e d i t a r y c l a s s which i s not an (ordinary) hereditary class.

The next p r o p o s i t i o n shows how t o g e t t h e T-G

excluded minors f o r a h e r e d i t a r y c l a s s from i t s c l a s s of (ordinary) excluded minors.

A geometric T-G h e r e d i t a r y c l a s s i s defined analogously,

where we do n o t allow d e l e t i o n o r c o n t r a c t i o n by isthmuses.

2.

A hereditary class

i f t(M1)

t(M2)

f

of matroids i s s a i d t o be (T-G) recognizable

C

whenever

M1

E

C,

and

M2

Examples

1 C.

of r e c o g n i z a b l e c l a s s e s a r e boolean algebras, truncated boolean a l g e b r a s , and paving matroids ( a s we saw i n (5.15)). Since, i n (3.11),

i s t r a n s v e r s a l while

M'

of a t r a n s v e r s a l matroid, and s i n c e

M

graphic, unimodular, and b i n a r y ) , while c l a s s e s i s recognizable.

M

i s not the minor

i s planar graphic (and hence M'

i s not, none of t h e above

Further, t h e matroids i n (4.3.2)

show t h a t

t h e c l a s s of r e p r e s e n t a b l e matroids i s not recognizable.

3.

Let

p(M) = (pO(M), pl(M),

invariants.

We then say t h a t

t h e c l a s s of a l l matroids

A4

...) E p(M)

P be a sequence of integer-valued i s a parmetrization of

M,

and

i s p a r t i t i o n e d i n t o parametric f a m i l i e s .

A class

C of m a t r o i d s i s a parametric (T-G) eztrernal class

with respect t o t h e parametrization invariant

f :M

+

p

and a g i v e n T-G group

i f there is a function

72

g :

(6.17.4)

f(M) = g(p(M))

for a l l

M E C,

(6.17.5)

f (M) > g(p(M))

for a l l

M

T h i s s i t u a t i o n c l e a r l y makes

C a

g i v e s a s h a r p lower bound o n v a l u e s of P r o p o s i t i o n 6.18

E

E

and n o n f a c t o r s

Then

E'

=

such t h a t

22

and

T-G r e c o g n i z a b l e c l a s s , and i t

f

a s well.

H b e a h e r e d i t a r y c l a s s o f m a t r o i d s , and

p

E

F u r t h e r , assume f o r a l l

E, (E-p) fB Bij

and

H

E

(E/p) fB Bij

E

ff.

i s a l s o a T-G h e r e d i t a r y c l a s s w i t h T-G excluded minors:

ff

E

-+

d C.

E b e i t s c l a s s o f excluded minors.

let E

Let

P

IJ

IE

@ Bij

: E

separable matroid

B

ij

E

E,

i

+

is i n

I n p a r t i c u l a r , every t o t a l l y

j > 0).

ff

E.

or

S i m i l a r l y , e x c l u d e d m i n o r s f o r geometric T-G h e r e d i t a r y c l a s s e s a r e d i r e c t sums of o r d i n a r y excluded minors and b o o l e a n a l g e b r a s . Proof.

It i s c l e a r t h a t no member o f

is i n

E'

hand, any T-G c o n t r a c t i o n o r d e l e t i o n of a member must b e o f t h e form

(E-p) fB Bij

or

E'

as a n excluded minor.

M # H,

El fB

... fB Ek,,

then

Decomposing

f a c t o r s , we may assume t h a t

-

(E/p) fB ~~j where

of p

E'

is a

Thus, a l l

a r e excluded minors.

Conversely, i f

E

E fB Bij

H by d e f i n i t i o n .

nonfactor,and these matroids a r e i n members of

On t h e o t h e r

ff..

where

must have a member o f

M M

into its

M = M1 fB M2 fB

k 2 k',

and

connected direct-sum

... fB $

Ei

E

and

i s a minor of

Mi

for

all

i s k'.

For

of m a t r o i d s

0 Zi =

e q u a l s 'M

;-

i > k',

for

above w i t h

pj

i s k',

we may apply (4.12) t o f i n d a sequence m i = Ei where, f o r a l l j < m M'" ,?li i' i

5 , 4,.. .

or

bl;fPj

with

pj

a n o n f a c t o r of

Similarly.

.:M

e i t h e r M, i s a l o o p o r we may f i n d a sequence as m Mii a n isthmus. I n any c a s e , M h a s a member of E '

as a T-G minor.

W e now g i v e some examples o f h e r e d i t a r y c l a s s e s which we w i l l later show t o b e extremal.

1.

P r o p o s i t i o n 6.19

The c l a s s of geometries which a r e d i r e c t sums

of a l i n e and a boolean a l g e b r a , forms a

BL

three-point

circuits,

The c l a s s

Cg @ C

C4,

: m r 3 , , i 2 0)

of t r u n c a t e d boolean a l g e b r a s forms a h e r e d i t a r y

T

minors i s then given by The c l a s s

and t h e d i r e c t sum of two

3'

c l a s s w i t h t h e unique excluded minor

3.

i

{B 1 v ILm @ 'B

geometric h e r e d i t a r y c l a s s whose (geometric) excluded

minors are a four-point c i r c u i t ,

2.

=

E'

= { Bij

B'".

Its c l a s s of T-G excluded

: i 7 0 , j >O).

S P of s e r i e s - p a r a l l e l networks forms a h e r e d i t a r y c l a s s

w i t h excluded minors

E' where

= I L 4 , M(K4)I,

i s a four-point l i n e , and

L4

-

complete four-graph

4.

The c l a s s

H

M*

i s t h e geometry of t h e

of Example ( 4 . 5 ) ) .

S of s e p a r a b l e matroids forms a T-G h e r e d i t a r y c l a s s

w i t h excluded minors If

( t h e geometry

M(K4)

{B1",

BO'l).

i s any h e r e d i t a r y c l a s s whose excluded minors,

E,

are

a l l connected, then excluded minors

5.

The c l a s s

i s a T-G h e r e d i t a r y c l a s s w i t h T-G

u S

E. of matroids with loops i s a T-G h e r e d i t a r y c l a s s

L

with T-G excluded minors

If

i s any h e r e d i t a r y c l a s s whose excluded minors

ff

H u L

l o o p l e s s , then

are all

E

i s a T-G h e r e d i t a r y c l a s s w i t h T-G excluded

minors

E'= Proof. -

1.

minor of then

G

i

i E @ B

:

EEE, i2O).

The reader r e a d i l y checks t h a t any proper geometric

C4

and

C3

% Cg

is in

BL.

Conversely, i f

must contain ( a s a subgeometry) a c i r c u i t

G

with

Ci

o r i t must c o n t a i n a t l e a s t two three-point c i r c u i t s

# 8L, i

and

C;

2

4,

C"

3'

I n t h e former case, c o n t r a c t i n g p o i n t s of C g i v e s C4, while, i n t h e i l a t t e r case, l e t

r

d i s t i n c t three-point

S' = C; 8 C"; 3

r = 2, G 2.

E

if

S' = C;

be t h e maximal rank of

r

circuits =

3, S'

C;

and

CI;

of

over a l l

r = 4,

contains a four-point c i r c u i t ; and i f

BL. B1*',

If

M

1 T,

then

Deleting everything except C,

we o b t a i n

B"'

M C

has a c i r c u i t and

p,

is not a

B O ' ~ and

truncated boolean a l g e b r a , while i t s two minors

p o i n t of

3

If

G.

The d i r e c t sum of a loop and an isthmus,

both a r e .

u C"

C

and p o i n t

gl*O p E

M-C.

and c o n t r a c t i n g a l l b u t one

a s a minor of

M.

3.

This i s proved i n 1241.

4.

S i n c e we a r e not allowed t o d e l e t e o r c o n t r a c t isthmuses o r

loops, i f

M'

M'

i s a minor of

is a t l e a s t a s g r e a t a s

Further,

then t h e number of components of

M,

M.

Thus,

i s a T-G h e r e d i t a r y c l a s s .

S

(4.12) o r ( 6 . 5 ) shows t h a t any connected matroid i s e i t h e r

a loop o r h a s a n isthmus a s a T-G minor.

(Note t h a t an elementary

e x t e n s i o n of t h i s argument shows t h a t the c l a s s with a t l e a s t

sk

of matroids

k connected components i s a T-G h e r e d i t a r y c l a s s

with T-G excluded minors

{Eiij : i

+j

<

kl.)

It i s c l e a r t h a t t h e union of two T-G h e r e d i t a r y c l a s s e s i s i t s e l f such a c l a s s .

ff u 5.

S

That

E

i s t h e c l a s s of excluded minors f o r

follows d i r e c t l y from (4.12).

D e l e t i o n o r c o n t r a c t i o n by nonfactors never d e s t r o ~ l o o p s . The

r e a d e r may supply t h e r e s t of t h e proof by modifying t h e proof of (6.18). Two techniques f o r o b t a i n i n g parametric extremal c l a s s e s a r e The f i r s t i s f o r when t h e

contained i n t h e following p r o p o s i t i o n s .

T-G i n v a r i a n t i s given, and t h e second allows t h e c l a s s (and function g) t o d e f i n e t h e i n v a r i a n t . P r o p o s i t i o n 6.20

A T-G h e r e d i t a r y c l a s s

extremal c l a s s f o r t h e p a r a m e t r i z a i t o n invariant

f

i f there is a function

t h r e e p r o p e r t i e s below:

C

p :M

g : P

-+

i s a parametric +

Z

P

and T-G group which s a t i s f i e s t h e

1.

Whenever

M

then

C,

E

f (M) = g(p(M))

.

A s a n e q u i v a l e n t c o n d i t i o n , we have,

1'. M

E

C

If

M

E

C i s t o t a l l y s e p a r a b l e , t h e n f (M)

= g(p(M)).

is not t o t a l l y separable, then there e x i s t s a nonfactor

If q

M

E

such t h a t g(p(M) 2.

For a l l

M

E

Whenever

E

g(p(M-q))

+ g(p(M/q)).

M

of

recursion

T1 on t h e c l a s s

f

on t h e s i z e of M' = E,

M'.

M'

M'

f o r a l l matroids Since

0

p

b o t h obey t h e

f

which o b e y s c o n d i t i o n s 1, 2,

M ' j C,

M'

not i n

C, we u s e i n d u c t i o n

i t h a s a T-G excluded minor

guarantees t h a t there i s a matroid

then (6.20.3)

w i t h p(M) = p(M'),

g

is t h e same as o u r h y p o t h e s i s (6.20.1).

Condition (6.17.4)

To v e r i f y (6.17.5)

and

C.

Now, assume we have an i n v a r i a n t

E

E

w i t h t h e same parameters as

C

of n o n f a c t o r s s i n c e , by h y p o t h e s i s ,

q

C,

Conditions 1 and 1' a r e e q u i v a l e n t by i n d u c t i o n on t h e number

Proof.

If

M,

E

f (E) > f (M).

such t h a t

If

q

i s a T-G excluded minor f o r t h e c l a s s

t h e n t h e r e i s some member

and 3.

g(p(M/q)).

and n o n f a c t o r s

M

g(p(M)) 3.

+

= g(p(M-q))

and w i t h

M

E. E

C

f(M') > f(M) = g(p(M)) = g(p(MV)).

i s n o t a T-G excluded minor, t h e n t h e r e i s a n o n f a c t o r

such t h a t e i t h e r

MI-q

or

~ ' / q is not i n

former ( t h e l a t t e r c a s e f o l l o w s s i m i i a r l y ) .

Then,

C.

Assume t h e

+

f(M1) = f(M1-q)

is, t h e n f(M'/q)

f(Mt/q).

Either

f(M1/q) = g(p(M'/q)), > g(p(M1/q)).

M1/q

is i n

C

o r not.

If it

and i f i t i s n o t , t h e n , by i n d u c t i o n ,

I n any c a s e ,

f(M'-q)

> g(p(M1-q)),

and

f (M) > g(p(M-q)) + g ( ~ ( M / q ) ) 2 ~(P(M)).

P r o p o s i t i o n 6.21

Let

C

of T-G excluded minors. p :M 1. q

E

+

P

S

g(p(M-q))

E

its class

F u r t h e r , assume t h a t t h e r e i s a p a r a m e t r i z a t i o n

and a f u n c t i o n

g(p(M))

b e a T-G h e r e d i t a r y c l a s s w i t h

+

g :P

+

Z. such t h a t

g(p(M/q))

M

for a l l

E

M and n o n f a c t o r s

M. Then,there e x i s t s a T-G group i n v a r i a n t

f

f o r which

C is

e x t r e m a l i f and only i f t h e f o l l o w i n g two p r o p e r t i e s h o l d .

2.

q

Whenever E

M

M

E

is not t o t a l l y separable, t h e r e e x i s t s a nonfactor

C

such t h a t g(p(M)) = g(p(M-q)) + g(p(M/q)),

and

3.

Whenever

q E E

E

E

E

i s not t o t a l l y separable, there is a nonfactor

such t h a t g(p(E)) < g(p(E-q))

Under t h e above c o n d i t i o n s , i n v a r i a n t f o r which

C

+ g(p(E/q)). f (M) =

1

bijcij i,j i s e x t r e m a l i f and only i f

i s a T-G group

Proof. and

The f u n c t i o n

f(M) = g(p(M)) Let

E

f :M for a l l

by (6.21.2)

M E C

be an excluded minor f o r

there is a nonfactor

(6.21.3),

i s a T-G group i n v a r i a n t by (3.9),

?Z

-+

q

C.

and (6.21.4).

Under t h e h y p o t h e s i s of

such t h a t

g(p(E)) < g(p(E-q)) + g(p(E/q)). But

E-q

and

E/q

g(p(E/q) = f ( E / q ) .

a r e both i n

so that

C,

g(p(E-q))

= f(E-q)

and

Hence, f ( E ) = f (E-q)

+ f (E/q)

= g(p(E-q))

+ g(p(E/q))

> g(p(E)).

Otherwise, f(E) = c

is

E = Ilij

d C

s o t h a t by o u r d e f i n i t i o n of

f,

> g(p(E)).

For a g e n e r a l

M'

d C,

h a s a T-G excluded minor

M'

E and t h e

proof proceeds a s i n (6.20). The n e c e s s i t y of (6.21.2) and (6.21.3) f o r t h e e x i s t e n c e of f and of (6.21.4) f o r c i s obvious. ij We n o t e t h a t t h e above p r o p o s i t i o n s can b e e a s i l y modified t o t h e geometric c a s e , and t o t h e c a s e when t h e Tutte-Grothendieck i n v a r i a n t i s maximized on t h e extremal c l a s s . I n t h i s l a t t e r c a s e , a l l i n e q u a l i t i e s a r e

r e v e r s e d , and we w i l l r e f e r t o t h e c o n d i t i o n s by (6.20.2),

etc.

We now s t a t e some of t h e c l a s s i c a l T-G i n e q u a l i t i e s i n terms of parametric extremal classes.

1.

P r o p o s i t i o n 6.22

Let u s p a r a m e t r i z e geometries by r a n k and

cardinality: P(G) Then, u(G) 2 / G I

-

r(G)

+ 1,

=

and

(r(G),

IGI).

a(G) 2 2

r (G) -1

(IGl

e q u a l i t y ( f o r e i t h e r i n v a r i a n t ) on t h e extremal c l a s s

-

r(G)

BL

+

2), w i t h

of d i r e c t

sums of a line and a boolean algebra (see (6.19.1)). 2.

When matroids are parametrized by rank and cardinality, we have

with equality on the extremal class T

of truncated boolean algebras

(see (6.19.2)). 3.

When geometries are parametrized by connectedness: if G is connected p(G) = if G

then

B(G)

2

p(G)

is separable,

with equality on the extremal class SP u S of

geometries which are either separable or are series-parallel networks (see (6.19.3), Proof. -

(6.19.4)).

The proofs for all these are routine calculations using (6.19)

and (6.20).

We mention below only a few hints and the original

reference for the theorem.

1.

This first appeared in [68].

a geometry of rank r(G)-1

For any geometry G, G/q

and cardinality at least r(G)-1.

is Thus,

for example, when q is not an isthmus of G, (6.20.2) follows from the following set of inequalities:

Further, in verifying (6.20.3) for a;

we get the following

calculations: a(C and

4

@

B ~ )= 14-2i

a(C3 @ C3

2.

@

> ' ~ 2 6

Bi+l

= a(L3

1,

Bi) = 3 6 ~ >2 8-2i+2 ~ = ' (L~

@

B

i+2

).

This result,which first appeared in [27],uses standard identities

on binomial coefficients to verify (6.20.2). on T

is given by (6.15.3a).

The exact formula for

To verify (6.20.31, note that all

of the above invariants are zero on the class of T-G excluded minors, El

{B"

=

T'(B~+')

: i > 0, j > 01, whereas the truncated boolean algebra

has the same parameters as

Bij, and gives a positive

value for each invariant. 3.

That

[53]. G

B(G)

= 0

if and only if

G

is separable first appeared in

For connected geometries, the fact that B(G)

is a series-parallel network is in [24].

B(G) = p(G)

=

0

= 1

if and only if

It is obvious that

for separable matroids,while the fact that

for connected series-parallel networks follows from (4.8.10)

B(G)

= 1

(along

If G-q is separable for a connected geometry with induction and duality). G, G is a series connection and G/q is connected, so p(G) 5 p(G-q) + p(G/q), while

B(M(K~))

= 6(L4)

= 2 > 1 = B(G)

for any series-Parallel network G with p(G) the T-G excluded minors for SP u

= 1.

M(K4)

and

S by (6.19.3) and (6.19.4).

L4

are

We now generalize (6.22.2) class N

from truncated boolean algebras to the

of all nested matroids where the parametrization is, for all

r, by the size of the largest flat of rank r. Proposition 6.23

For the parametrization P(M) = (n;ao,al,a2,..-,a

vhere n = r(M)

and, for all r, r

1

ai = max(1

FI

:

F is a flat of rank r) ,

i=O then, v(M) = 0 if .a > 0, and, in general,

-

where a = a +a +...+ar. i 2 3

Further, equality holds precisely on the

class N u L of nested matroids and matroids with loops.

Proof. 1.

We verify the three conditions of (6.20A). p(M)

assume that .a

= =

0 if and only if M has a loop 0.

(ao > O),

so we may

Further, al does not contribute to the formula

for

g

and, on t h e o t h e r hand,

-

matroids,

M

p(M) = (n;0,1,a2,

i s a geometry.

I M ~ = 1, v(M)

u(M) = v(fi),

...,an ) .

Therefore, we may assume t h a t

We use i n d u c t i o n on the s i z e of

p(L ) = (2;0,l,m-1). RL

have s i z e If

G

contains an isthmus u(G-p) = p(G). G-p

gn+l = g(n+l;O,l,.

n = r(G).

):(

p(Lm) = m-1 =

Thus,

m,

while

= g(p(M)). K, and l e t

K+1.

an+l = 1 s i n c e

G

Clearly, i f

M = Lm f o r some

u(M) = g(p(M)) f o r a l l matroids of s i z e

nested o r not,

If

M.

= 1 = g(1;0,1).

For a nested geometry of rank two,

Assume

where, f o r nested

p,

we n o t e t h a t , whether

Further, i f

i s a hyperplane.

.., a n , l )

=

g(n;O,l,.

r(G) = n+l,

q

then

We must v e r i f y t h a t

.., a

) = gn:

does n o t c o n t a i n an isthmus, then Let

G

be a p o i n t i n f r e e p o s i t i o n .

a

n

> 1 where

is

G

BY (6.6.4~) and (6.6.4dj, p(G-q) = (n;O,l,...,an-l,an-l)

Then, g(p)

-

= p',

and

g(pV) =

Note that a similar calculation shows that for all i

2

1,

which also follows from the fact that if gap obeys the T-G recursion T1 for some nonfactor, it obeys T1 for every nonfactor, and we may apply the formulas in (6.6.4) for q The excluded minor Mi

E

Fi+l = Bi+l

(see (6.6.5))

-

Bi.

has parameters (i;0,1,.

and is a nontrivial weak-map image of the nested matroid

..,1,2,i)

Ni with the

same parameters (where t h e weak map i s t h e i d e n t i t y on t h e c y c l i c flat

Fi-l

of

and sends a l l t h e p o i n t s i n f r e e p o s i t i o n t o t h e

N

complementary f l a t

N u L

of

is

Fiel

of

Mi).

{Mi = i 2 21

Since t h e T-G excluded minor c l a s s

by (6.6.5)

and (6.19.5).

we have v e r i f i e d

(6.20.3). It remains t o check (6.20.2).

nonfactor of (n-l;ai,

by

..., a L l ) .

t h e c a s e when p(M-q)

q

p(M)).

5

Denote

M.

To prove t h i s , l e t

p(M-q)

., a n' ) ,

by (n;a;),ai,..

f; = f i

-

i s i n a m u l t i p l e p o i n t (so t h a t Let

q e F -q i

or

fi,

f;,

is a f l a t i n E

Fi,

rank

M/q

and

f;,

and s i z e z f

M

then,in addition, Let

p

i'

E

2 fi

f;

i

Fi,

d

i+l fy-l

w i t h rank since i f

p(M/q)

-

F.up

q

E

= f.-1.

)

...,2m,

...,2m+l, f' i

j

and

q

i f and o n l y i f

i

and s i z e

f;

fi+l.

fi,

is

Fiuq

-q

while i f M/q

-

fi+l

of

1, and.

f o r some f l a t f !1 = f i-1,

Hence, i f

f n' = f "n-1 = f n -1.

g(n;fO,fl-fO,

- fij' and

fi.

-

then

is a f l a t i n

Fi+l-q

f' i j

and M/q, r e s p e c t i v e l y

q j Fi,

Further,

equal

in

p(M/q) = 0

and s i z e a t l e a s t

-

and s i z e

j = 1,3,5,

j = 2.4,

M

-1

fi

An upper bound f o r

h(n;fo,f l,...,f

assume t h a t f o r while f o r

F i

whenever

w i t h rank

and

respectively, denote t h e

1, and e q u a l s

of rank

then f o r any i

E

-

i < n,

t h i s i s achieved Fi+l

i

f o r every f l a t

Similarly, f o r

q

f

be a

I t i s a r o u t i n e m a t t e r t o check (6.20.2)

s i z e o f t h e l a r g e s t c l o s e d s e t of rank i i n M, M-q, Then

q

=

fij-1,

...,f n-f and

and

f;

= f i -l;

j

f;

-1. j

J

j (Thus,

-1.Oforodd

j, : a

=

j

otherwise.

Further, i1

1

ai +1

for even j, and

a; = a1

j

2.)

Multiple applications of (6.23.1) then yield:

where, for all j,

Let A

(6.23.4)

A

= h(n-l;fo,fl~...~fn-l)

hZe2

fi-l = f -1

where

for all i : i2k-l or i 2 i 2m+l'

S

i

5

iZk-l,

and

fi-l = fi-l otherwise.

It is easy to show that h2nr+2 5 h(n-l;fS,f; =

f,-l,

and

,..., .)l_nf

since

"

f" > f i i for all i.

Therefore, using (6.23.2) and (6.23.4),

2

hl

-

h2 + h 3 - h4

+

we get

...

+

h2m+l

-

h2m+2'

and (6.20.2) will follow from the nonnegativity of this alternating

This last inequality is left to the reader who may verify it by a term-by-term comparison.

Corollary 6.24.

1.

Let M

be a loopless matroid of size K

Further, for all r > 0, assume that M

rank n.

and

has a flat of

size at least fr. Then, p(M)

5

g(n;0,fl,f2-fl.

with equality if and only if M N(0,fl,. 2.

...,fi-fi-l,. ..,K-fn-l)

is the nested matroid

..,fi-fi-l,...,K-fn-l).

For matroids parameterized as in (6.23), $(M)=O

if a o > O or a = 1 (n>l), n

and otherwise B(M) The extremal class is

g(n;0,al,a2

5

N

u L u L*

,...,ai, ...,an-1).

where L*

is the class of all

matroids with an isthmus. 3.

For matroids parameterized as in (6.23), a(M)

S

gn

... + g1

+

with equality on the extremal class N u L

+

gi = g(i;O,al,...,ai-l,a.+ai+l Proof.

1. When n

and

K

fixed, then g(n;O,a l,...,an) . a

+ ... +

ai < bo

+ ... + bi

where, for all i,

... + an). + ... + an

=

a1

>

g(n;O,bl,

= bl

....bn)

+

... + bn

are

where, for all i,

with strict inequality for at least

one

i.

( T h i s i s a consequence of t h e f a c t t h a t f o r t h e two r e s p e c t i v e

nested matroids,

and

Na

to

t h e r e i s a n o n t r i v i a l rank-preserving

Nb,

-

weak map from

Na

2.Amatroid M

h a s an isthmus i f and only i f , i n

-

Nb. )

-

any such m a t r o i d (of rank Otherwise,

B(M)

5

an > 1,

v(M-p)

i t i s s e p a r a b l e and

n > I),

and we use (6.15.3.b)and

f o r any

p

in

is a point i n f r e e position.

M

with parameters 3.

(ao,

(6.16) t o o b t a i n : p

The r e s t of t h e proof f o l l o w s i n a

( ao . . . . , a

p )

i s i n f r e e position, i f and only i f

M-p

M

is

i s nested

...,an -1).

1. P m i t a t e C o r o l l a r y (6.24) t o g e t maximal v a l u e s and

e x t r e n a l c l a s s e s of m a t r o i d s f o r and t h e Whitney number

w

i

r 1,

all

t h e independence number

f o r t h e upper bound of

ai = 1, e x c e p t

i = n

and

i = r.

a bound w i l l t h e n b e o b t a i n e d f o r a l l m a t r o i d s of r a n k and w i t h a f l a t

Fr

Ir

r'

2. Develop a s i m p l e r formula

K,

B(M) = 0.

T h i s i s an easy a p p l i c a t i o n of (6.15.3.a).

E x e r c i s e s 6.25

for

For

with s t r i c t inequality unless

s t r a i g h t f o r w a r d way,noting t h a t i f n e s t e d w i t h parameters

= 1.

p(M), a

of r a n k

r

and s i z e a t l e a s t

v(M)

when,

(By (6.24.1), n, k.)

cardinality

Research Problems 6.26

1.

Find combinatorial proofs (i.e.,

bijections or injections among two appropriate families of subsets) for the equalities and inequalities of (6.15).

2.

Develop an extremal theory for minimizinc T-G invariants on

connected matroids.

In particular, try to extend, in the connected

case, the idea of an extremal class to include the matroids on which the bounds of [34] are sharp.

3.

The most extensive results thus far obtained for parameterized

lower bounds for T-G invariants are found in [IS]. BjBrner obtains the minimum for p(M)

when M

In particular,

is parameterized by

rank, cardinality, and size of smallest circuit.

This result can be

thought of as dual to the upper bound to be calculated in (6.25.2). Can (6.23) be dualized in a similar manner to obtain lower bounds for a finer parametrization (e.g., by the minimum size, c j' cycle of nullity j)?

4.

of a

In [112], Oxley characterizes the classes of matroids on which

B equals two, three, and four respectively. Several parametrizations are implicit in his paper. For exanple, parametrize a loopless,connected matroid M by the invariant p(M), x(M;X)

5

0.

the maximum positive integer

Then, for p(M)

5

5, B(M)

2

p(M)-1,

X

for which

and Oxley's results

give the extremal matroids for these values as well as a class of matroids

({Ld2])

for which

B(M)

=

p(M)-1

=

m.

Is this inequality

true in general? If so, what are the extremal matroids? Similar questions may be asked for a parametrization in terms of connectivity:

2 63

What i s the sharp lower bound f o r connectivity

g(n) =

(2z1;]

n

(with a t l e a s t and shows that

B(M) 211-2

among a l l matroids of points)?

g ( n ) 2 2n-2'

Oxley conjectures that

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CENTRO IIVTERXAZIONALE MATEMATICO E S T I V O (c.I.M.E.)

ON 3-CONNECTED

MATROIDS AND GRAPHS

JAMES G.

OXLEY

ON 3-CONNECTED MATROIDS AND GRAPHS

James G. Oxley University of North Carolina, Chapel Hill, USA and Australian National University, Canberra, Australia

Introduction

This expository paper will be concerned with Tutte's notion of nconnectedness for matroids.

In particular, we shall show how various results

for 3-connected graphs can be extended to matroids.

The terminology used here for matroids and graphs will in general follow Welsh [I91 and Bondy and Murty [I] respectively. In particular, if T is a subset of a matroid

M, then rkT will denote the rank of T, while

M\T and M/T will denote respectively the deletion and contraction of T from M. denoted

The uniform matroid of rank r on a set of Ur,k.

If G

is a graph and

n

k

elements will be

is a positive integer, G will be

called n-connected if one needs to delete at least n vertices from G

in

order to obtain a disconnected or single-vertex graph.

Motivated by this

graph-theoretic concept, Tutte [17,18] introduced the following definition If M

of n-connectedness for matroids. and

E(M)

k is a positive integer, M is said to be k-separated if there

is a subset T

If n

is a matroid having ground set

of

E(M)

such that

IT1 2 k,

is a positive integer, the matroid

is no k

less than n for which M

M

IE(M)\T~

2

k and

is n-connected provided there

is k-separated.

The notions of n-connectedness of a graph G and n-connectedness of the corresponding cycle matroid

M(G)

do not, in general, coincide. How-

ever,it is straightforward to check that (1)

for

n = 2 and n = 3, if

vertices, then G

G

is a simple graph having at least four

is n-connected if and only if

H(G)

is n-connect&.

For larger values of n, the precise relationship between the graph and matroid-theoretic notions of n-connectedness is discussed in [3] and [12]. The matroid concept of n-connectedness generalizes the well-known idea of non-separability or connectivity for matroids. 2-connected if and only if it is non-separable.

In fact, a matroid is

A further appealing property

of this concept is that (2)

n matroid is n-connected if and only if its JuaZ is n-connected.

In [15], Seymour has characterized 3-connected matroids in terms of a decomposition operation which is closely related to Brylawski's operation of parallel connection of matroids [ Z ] .

For i

=

1,2,

let M1

where IS. 1 2 3 and suppose that S1 n S2 = i neither a loop nor a coloop of M1 or M2. Then the 2-sum of on a set

is the matroid on the set

be a matroid where p

5

and M

is 2

S1 u S2 having as its collection of circuits all

circuits of

5

not containing p, all circuits of M2 not containing (C1\p) u (C2\p) where Ci is a circuit of

and all sets of the form

Mi

containing p. (3)

A rnatroid i s

PROPOSITION [IS, (2.10)(b)].

it is non-separabte and cannot be obtained as a

3-connected i f and only i f 2-swn

o f two rnatroids.

One very important class of 3-connected matroids is the class of cycle matroids of wheels. Suppose that m graph having m

+1

vertices, m

3.

The wheel

Wm of order m

is a

of which lie on a cycle (the r i m ) ; the

remaining vertex is joined by a stngle edge (a spoke) to each of the vertices of the rim. Evidently Wm

M(Wm) Wm

is simple and 3-connected. Hence, by (11,

is a 3-connected matroid.

However, whenever an edge is deleted from

the resulting graph has a vertex of degree two and so is not 3-connected.

It follows on applying (1) again that M(Wm)

mztroid,

That is, H(W m )

is a minirnazzy

3-connected

is a 3-connected matroid for which no single-

element deletion is also 3-connected. is isomorphic to its dual.

Now it is easy to check that M(Wm)

Therefore, as M(W ) m

is minimally 3-connected,

it follovs from (2) that no single-element contraction of M(Wm)

is 3-

connected. Another class of minimally 3-connected matroids whose members share this property with the cycle matroids of wheels is the class of whirls. The w h i r 2

kP

is the matroid on E(Wm) having as its collection of circuits,

all those circuits of M([tlm)

other than the rim, together with all sets of

edges formed by adding a single spoke to the set of edges of the rim. Figure 1, the wheel [U3 the whirl

In

is shown along with a Euclidean representation for

3. For comparison, a Euclidean representation for

also shown. We note that W3

is just the complete graph K4-

M(W3)

ig

w3 Figure 1 The following theorem underlies most of the results in the remainder of this paper.

It is a matroid generalization of an earlier graph-theoretic

theorem 116, (4.1)] and, as such, typifies many of the results in this area of research.

(4) THEOREM (Tutte 117, 8.31).

Let

M be a minimally

for which no single-element contraction i s

3-connected.

3-connected matroid Then M i s isomor-

phic t o a whirl or the cycle matroid of a wheel. From this theorem one can deduce a recursive construction for all 3connected matroids of rank at least three. A non-trivial extension of e matroid

M

is an extension N

of M

by a single element

is neither a loop nor a coloop of N and element of N.

e such that

e is not in parallel with any

is a non-trivial l i f t of M

On the other hand, N

.

is a non-trivial extension of

M*

if there is an element e of N

such that N/e = M

Thus N

*

if N

is a non-trivial lift of M where e is neither

is not in series with any element of M.

a loop nor a coloop of N and

e

(5) THEOREM [lO,Theorem 4.11.

A matroid of rank a t least three i s

3-

connected i f and only i f it i s a whirl, the cycle matroid of a whee2,or "3,s

e

or i s obtainable from such a matroid by a sequence of the following

operations: (i) non-trivial extensions; and

(ii)

non-trivial l i f t s

.

The remainder of this paper will concentrate on minimally 3-connected matroids.

In particular, we shall show that a number of results of Halin

for minimally 3-connected graphs have matroid generalizations or analogues. Many of Halin's results have been extended in another- direction by Mader who has generalized them to minimally n-connected graphs for arbitrary n (see, for example, 171 and the survey paper [8]). (6)

Let

THEOREM (Halin 14, Satz 7.61).

having m

vertices.

G be a minimaZ2y

3-connected graph

Then

MoreoiJer, the following are the only graphs attaining equality i n these bounds :

W6

and

i6

Wm-l for

4 (m

K for 394

m=7;

K3,m-3 f o r

m z 8

:

and

.

In the case oi' arbitrary minimally 3-connected matroids, precisely the same bounds hold as in the graph case.

(7)

THEOREM 19, Theorem 4.71.

of rank a t lcast three.

Then

Let M be a minimally

3-connected matroid

From Theorem 6, we know that the bounds in the preceding result are best-possible.

Moreover, those matroids attaining equality in Theorem 7 have

been characterized [9, Theorem 5.2 and Corollary 5.111.

However, this re-

sult is rather cumbersome and, instead of stating it, we merely note that if

M

is binary and minimally 3-connected, and

in the preceding theorem, then M

M

attains the relevant bound

is graphic [9, Theorem 5.121.

Thus M

is

isomorphic to the cycle matroid of a wheel or a complete bipartite graph K3,m-3' A

lower bound on the number of elements in a minimally n-connected

matroid of given rank is considerably easier to obtain than Theorem 7. Moreover, whereas no analogue of Theorem 7 is known for n > 3, the following lower bound holds for all n

2

(8) THEOREM [9, Theorem 3.21.

rank

r where

Eloreover,

r,n

2

2.

2.

Let

M

be a rninimalzy n-connected matroid of

Then

only r' ,r+n-1 and

'r,2r-1

a t t a i n e q u a l i t y here.

We now turn to another property of minimally 3-connected graphs and the corresponding property of minimally 3-connected matroids.

(9) THEOREM (Halin [5, Satz 61).

then G has a t Zeast

)'(vI

lt6

If

G i s a rninirnaZZy

3-connected graph,

v e r t i c e s of degree t h r e e .

It is well-known that in a 2-connected loopless graph G, the set of edges meeting at a vertex forms a cocircuit in M(G).

Thus one possible

matroid analogue of a vertex of degree three is a 3-element cocircuit. Theorem 9 now prompts the question as to what one can say about the number of 3-element cocircuits in an arbitrary minimally 3-connected matroid. The

folloving lemma extends a result of Seymour [14, (2.311 for minimally 2connected matroids.

If C i s a c i r c u i t of a minimally

(10) LEMHA f 9 , Theorem 2.51.

m t r o i d M and

!E(H)

cocircuits meetZng

I 2 4,

then M

has a t l e a s t t u o d i s t i n c t

3-connected 3-element

C.

This lemma, which is proved by induction on the cardinality of

C, con-

tains the core of the proof of the following result.

(11) T B E O m 111, 521. Let M b a s t four e h e n t s .

be a minimal%

Then H has a t l e a s t

3-connected matroid k m i n g a t 1/2(1~(~)I

-

rkM ) + 1 3-element

cocridts. It is straightforward to check that for all k

24,

the matroid M(K .

is minimilly 3-connected and attains the bound in the preceding result.

3,k We

)

n w sketch the main idea of the proof of Theorem 11. Proof outline.

Let X be the set of elements of M

which are contained in

some 3-element cocircuit. By the lemma, X meets every circuit of M, hence

X contains a cobasis B*

of M.

any 3-element cocircuit of M. least 1 1 2 1 ~ ~=1 1/2(1~(~)

I-

But B*

contains at most two elements of

Hence, the number of such cocircuits is at rkM)

.

If one uses the full force of Lemma 10,

then it is not difficult to improve this bound by one and thereby obtain the theorem.

The details may be found in the proof of Proposition 2.20 of [ll].

On applying Theorem 11 to graphs, one obtains that

(12)

a minimaZZy 3-connected graph G has a t l e a s t

I/~((E(G)

(-

I

~ v ( G )+ 1)

+

1 minimal c u t s e t s o f s i z e three.

Now,in a linimally 3-connected graph, one is more interested in vertices of degree three than in arbitrary minimal cutsets of size three. This leads oneto ask whether one can strengthen (12) by replacing "minimal cutsets of size

three" by "vertices of degree three." The next theorem answers this question. (13)

THEOREM [ll, Proposition 2.201.

Zeast 1/2(1E(G)I

- Iv(G)I

+

3)

A rninirnaZZy

3-connected graph has at

vertices of degree three.

The proof of this result is similar to the proof of Theorem 11 and uses instead of Lemma 10 its graph-theoretic analogue (see Halin 15, Satz 51). Both Theorems 9 and 13 give lower bounds on the number of vertices of degree three in a minimally 3-connected graph G.

For certain values of

I E(G) I , the bound in Theorem 9 is the sharper, while for other values of I E(G) I, Theorem 13 gives the sharper bound. The following result is obtained by combining the two theorems and for each value of IE(G) I choosing the better of the two bounds. A small amount of additional argument enables Halin's bound to be sharpened within the given range.

We observe that for

a minimally 3-connected graph G, one can easily check that

IE(G)[

L~/Z~V(G)[. Moreover, by Theorem 6,

IE(G)~

(~IV(G)~

-

number of vertices of degree three in G will be denoted by THEOREM [ll, Proposition 2-20]. Let G be a min5maZZy

(14) graph.

9. The

v3(G). 3-connected

Then

21V(G) 1+7

v3(G)

2

for

31v(G)I

2

< IE(G)I -

91v(z)I-3

5

<

91v(G)I-3 5

IE(G)I

L

;

31V(G)I

-

9.

As another application of Theorem 13 one can characterize all minimally 3-connected graphs attaining equality in Halin's bound [13, Theorem 4.51. We close by briefly considering minimally n-connected matroids and graphs for n

2

4. The proofs of most of the results stated above rely on Tutte's

characterization, in the case n = 3, of those minimally n-connected matroids

for which no single-element contraction remains n-connected (Theorem 4). characterization of such matroids when n

2

~h~

4 is an open problem. Mader

17, Satz 11 has shown that every circuit in a minimally n-connected graph meets a vertex of degree n

and, from this, using the proof method of

Theorem 11, one can deduce a new lower bound on the number of vertices of degree

n

in a minimally n-connected graph 111, Proposition 2-19].

The links between graph theory and matroid theory have always been close (see, for example, 120, 61).

The results above show that in the study

of n-connectedness these links can be successfully exploited in both directions to obtain not only new matroid results but also new graph results.

Acknowledgement.

The author gratefully acknowledges that this paper was

written with the partial support of a Fulbright Postdoctoral Fellowship while he was visiting UNC on leave from ANU.

References 1.

J. A. Bondy and U.S.R. Murty, Graph Theory w i t h AppZications (Macmillan, London; American Elswier, New York, 1976).

2.

T. H. Brylawski, A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971), 1-22.

3. W. H. Cunningham, On matroid connectivity, J. Combin. Theomj Seer. E (to appear). 4. R. Halin, Zur Theorie der n-fach zusammenhgngenden Graphen, Abh. Math. Sem. Univ. Hamburg 33 (1969), 133-164. 5. R. Halin, Untersuchungen iiber minimale n-fach zusammenh;ingendeGraphen, Ekrth. Ann. 182 (1969), 175-188. 6. F. Harary and D. Welsh, Eatroids versus graphs, The Many Facets o f Graph Theory (Lecture Notes in Mathematics Vol, 110, Springer-Verlag, Berlin, Heidelberg, New York, 1969) pp. 155-170.

7.

W. Mader, Ecken vorn Grad n in minimalen n-fach zusammenh;ingenden Graphen, Arch. Math. ( B a s e l ) 23 (1972), 219-224.

8.

Mader, Connectivity and egge-connectivity in finite graphs, Surveys i n Combinatorics, Ed. B. Bollobas (London Math. Soc. Lecture Notes No. 38, Cambridge University Press, 1979) pp. 66-95.

1.J.

9. J. G. Oxley, On matroid connectivity, Quart. J. Math. Oxford (2) (to appear).

10. J. G. Oxley, On 3-connected matroids, Canad. J. Math. (to appear). 11. J. G. Oxley, On connectivity in matroids and graphs, Trans. Amer. Math. Soc. (to appear). 12. J. G. Oxley, On a matroid generalization of graph connectivity (submitted) 13. J. G. Oxley, On some extremal connectivity results for graphs and matroids (submitted). 14. P. D. Seymour, Packing and covering with matroid circuits, J. Combin. Theory Scr. B 28 (1980), 237-242. 15. P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305-359. 16. W. T. Tutte, A theory of 3-connected graphs, NederZ. Akad. Wetensch. Proc. Ser. A 64 (1961), 441-455. 17. W. T. Tutte, Connectivity in matroids, Cmurd. J. Math. 18 (1966), 13011324. 18. W. T. Tutte, Connectivity in matroids, Graph Theory and i t s ApppZications, Ed. B. Harris (Academic Press, New York, 1970) pp. 113-119. 19. D.J.A. Welsh, Matroid Theory (London Math. Soc. Monographs No. 8, Academic Press, London, 1976). 20. H. Whitney, On the abstract properties of linear dependence, Amer. J . Math. 57 (P935), 509-533.

c ~ T R OI N T E W A L I O N A L E MATEMATICO E S T I V O ~c.I.M.E.)

THE P O S E T OF S U B P A R T I T I O N S AND C A Y L E Y ' S FORMULA FOR THE COMPLEXITY O F A COMPLETE GRAPH

RHODES P E E L E

THE POSET OF SUBPARTITIONS AND CAYLEY'S FORMULA FOR THE COMPLEXITY OF A COMPLETE GRAPH

Rhodes Peele Pembroke S t a t e U n i v e r s i t y Pembroke NC 28372 USA

I.

Introduction.

A v e r y w e l l known theorem o f Cayley a s s e r t s t h a t t h e r e are nW2 d i s t i n c t trees w i t h vertex set

(1,2,

... , n3.

We can express t h i s r e s u l t i n t h e

language o f m a t r o i d theory by saying, "The complexity o f the complete graph on n v e r t i c e s i s

nn-'."

Below we o f f e r a p r o o f o f Cayley's Theorem t h a t i s based on Mijbius I n v e r s i o n i n t h e generalized sense o f Rota

121.

Proofs o f Cayley's Theorem

a r e l e g i o n , b u t the present one has the unusual f e a t u r e t h a t the i n v e r s i o n takes p l a c e over a poset t h a t i s n o t a l a t t i c e .

The poset i s an i n t e r e s t i n g

c m b i n a t o r i a l o b j e c t t h a t probably has o t h e r a p p l i c a t i o n s .

11.

The Poset of Subpartitions.

a i s an element of subset of (1,2,

...

, nl

The elements of Thus, a

5

B iff B

E

P =

n

Define a poset

P

n

as follows :

i f f a i s a subset of some (possibly improper)

n.

are ordered by containment (@ refinement). n a (read "B i s a m o f a") implies t h a t B E 6. P

Further notation : U(a) denotes the union of the constituent blocks of a ; la1

denotes the number of blocks of a

;

I$ denotes the minimum element of P

(note that161 = 0 and U ( @ ) = I$ ) ; n v denotes the element of P whose only block i s the singleton {n); n a dot below a l e t t e r in a summation formula indicates the sumnation variable. Observe t h a t i f n > 1 (which we henceforth assume), then the f i n i t e has no maximum element and i s therefore not a l a t t i c e . B u t i f n B ), then [ a , ~ ] i s a Boolean and 6 are comparable elements of P (say,a n algebra, and consequently,

poset

where

P

i s the Mobius function of P

.

n The Hasse diagram bf [v,=) f o r P i s shown in Figure 1. 4 [ v , ~ ) and P are isomorphic. n-1

The Subpartition of:

Given a function f : ", k those j 6 2 f o r which there e x i s t s a k > 0 such t h a t f ( j ) = j are called k recurrent. The relation iRj i f f f ( i ) = j f o r some k > 0 i s t r a n s i t i v e 11.

Function.

Note that

on

1

and becomes r e f l e x i v e and symmetric when r e s t r i c t e d t o t h e setof

r e c u r r e n t elements of f. The induced s u b p a r t i t i o n of

Now l e t T be a labeled t r e e w i t h vertex s e t and only one f u n c t i o n

f :

n -.

n.

nis

denoted by p ( f ) .

Then there i s one

such t h a t f ( n ) = n and f o r j > n, f ( j ) = k This i s i n t u i t i v e l y c l e a r since a f t e r we

i f f an edge o f T j o i n s j and k.

add a d i r e c t e d loop t o T a t vertex n, there i s o n l y one way t o p r o p e r l y o r i e n t t h e edges o f T i n order t o obtain a " f u n c t i o n a l digraph". f, we have P ( f ) = v

, and

For t h i s

conversely, any f t h a t s a t i s f i e s P ( f ) = v comes

from some t r e e T v i a t h e above construction (view f as a f u n c t i o n a l digraph, and erase the loop and arrows). We may

therefore i d e n t i f y the c o l l e c t i o n Tree(n) o f labeled t r e e s

w i t h vertex set

1w i t h

the s e t o f functions f o r which P ( f ) = v

.

(This

i d e n t i f i c a t i o n i s the s t a r t i n g p o i n t o f an already published p r o o f o f (see [I]), b u t now t h e proofs diverge).

Cayley's Theorem due t o Katz

111. functions

Cayley's Formula f o r f :

n

+

#.

Tree(n)

.

Denote the c o l l e c t i o n o f a l l

g(a) =

I

P , define n -n i f : f r! andP(f) = a ]

h(a) =

I

{ f : f €>-and

2 by

For a

E

I

and

Example. I 1 3 and I41.

n P(f) r a 1

I

L e t n = 4 and l e t the blocks o f a be p r e c i s e l y the singletons 4 Then Present any f E 5- as a vector ( f ( l ) , f ( Z ) , f ( 3 ) , f ( 4 ) ) .

g(a) = 8 since only these functions s a t i s f y P ( f ) = a :

Further, h ( a ) = 16 since o n l y these a d d i t i o n a l f u n c t i o n s s a t i s f y P ( f )

> a

:

F i g u r e 1 e x h i b i t s the values o f g and h on t h e i n t e r v a l [v,-).

m. I f a

i s a nonempty s u b p a r t i t i o n o f 5, then h ( a ) = t ( a ) n

n

-

lU(a)l

where t ( a ) i s t h e number o f permutations o f U(a) whose c y c l e s a r e p r e c i s e l y t h e blocks o f U(a). Proof.

I f a f u n c t i o n f i s counted by h(a), then every b l o c k B o f a

Conversely, any f u n c t i o n f t h a t meets n IU(a)l t h i s requirement, i s counted by h ( 3 ) . There are c l e a r l y t ( a ) n

must be c y c l i c a l l y permuted by f .

-

such f u n c t i o n s , since the elements o f

n that

do n o t belong t o U(a) may be

3

assigned a r b i t r a r y values.

(An e x p l i c i t formula f o r t ( a ) i s easy t o w r i t e down, b u t i t w i l l n o t be needed. ) Theorem

Proof.

(cayley). For a l l

L e t n > 1.

a c

P

we have n

and so by M6bius i n v e r s i o n ,

Then !Tree(n)

1

n-2

= n

.

Taking a = v we get

We now f i x an a r b i t r a r y subset S i n n e r sum.

c n such

that n

E

S and evaluate the

There are three cases, and a l l are q u i t e simple ifFigure 1

i s kept i n mind : Case 1 :

IS1 1-1

u(v,v)h(v)

= (-1)

Case 2 :

IS I

There i s only one term, and i t contributes n-1 n-1 t(v)n = n t o the o u t e r sum. =

1.

= 2.

Again there i s o n l y one term, and i t has the

form p ( v , ~ ) h ( 6 ) where 6 has o n l y the two blocks i n 1 and { j l , f o r some 2-1 n-2 n-2 j L n. Such a term c o n t r i b u t e s (-1) t(8)n = -n t o the outer sum. Note however t h a t the s e t S can be selected i n n-1 ways since j < n. Case 3 :

ISI

>

2.

Here,

7

The bracketed f a c t o r vanishes, since e x a c t l y ha1 f o f the permutations of

S\(n)

have an even number o f cycles.

We are now ready t o perform the

outer sum

:

REFERENCES

[I] Moon, J.

Various proofs o f Cayley's Formula f o r counting trees.

Chapter 11 i n A Seminar on Graph Theory, F. Harary, ed. [2]

Rota, G.-C.

On t h e foundations o f cornbinatorial theory I :

1.Wahrscheinl i c h k e i t s t h e o r i e und 340 - 368.

Theory o f M6bius f i n c t i o n s . Verw. Gebiete -

2

(1964)

Figure 1 Hasse Diagram o f [ v , ~ ) f o r P

4

CEKTRO INTEIZKAZIONALE MATEMATICO E S T I V O

(c.I.M.E.

ENGINEEMKG A P P L I C A T I O N S OF MATROIDS ANDRAS RECSKI

-

A SURVEY

ENGINEERING APPLICATIONS OF MATROIDC

RESEARCH INSTITUTE

FOR

-

A SURVEY

ANDRAS RECSKI TELECOMMUNICATION, 1026 EUDAPEST, HUNGARY

INTRODUCTION T h e a i m o f t h e p r e s e n t 2 3 n t r i b u t i o n i s t w o f o l d . F i r s t , we s k e t c h some e n g i n e e r i n g p r o b l e m s w h e r e m a t r o i d s c a n b e a p p l i e d t o otkain nontrivial resiilts.

Next,

one a p p l i c a t i o n i n e l e c t r i c

n e t w o r k t h e o r y I s a e s c r i b e d i n some d e t a i l s .

E f f o r t was made t o

g i v e a more o r l e s s c o m p l e t e l i s t of r e f e r e n c e s . intended f o r mathematicians

-

The p a p e r i s

no p r e v i o u s k n o w l e d g e i n e n g i n e e r

ing is required.

1,

For the problem of rigid bodies we refer to [3,4,7,12,13, 36,40,42,58,65,66]. See also [64] in the present volume. For example, if the planar systems of rigid braces, like on Fig. 1,

are considered and fixed to the plane at points A and B then system is rlgid while system E is not rigid ithe points C and D can simultaneously move around A and B respectively) and a further "pin" at C or D or a further brace between, say, A and D is required. System C is rigid with respect to mechanical motions but not to "infinitesimal" ones, attacking vertically at point C. Hence a further "pin" is required at C. All what we wish to emphasize here is that there are essentially two types of questions considered: Question & Is the given system rigid? Question 2 If not, determine the minimal number of extra braces or pins to make the given system rigid (and give such a system). Some of the quoted sources treat more general problems. Not only "pinned braces" but "slides", "rotors" etc. are considered, and in higher dimensions as well.

2, Strongly related is the activity of [60,61] concerning the 'man-machine communication via a graphical display. If the designer draws the 2-dimensional image of a 3-dimensional body, the computer must "recognize" the original body. If the recognition of vertices, edges and faces (and the incidence relations among them) is solved, still uncertainities, caused by "wrong" or "misunderstandable" drawings can arise. For example, if the second drawing of Fig. 2 is 0 P fig.2 I '1'. the input, the corn1 1 I 8' t 8 ' ,' puter should probI \ ,'t\ t ' ably "ask" the user whether the "nonexistence of the point P" was intentional. Again, two types of questions can be quite natural: Question Is the given input understandable for the computer? Question 2 If not, determine the minimal number of further questions to be answered (and give such a system of questions).

,," ,'

a

3 , Consider the fundamental problem of network analysis. Details will be given in Chapter 111, so only the questions are formulated now: Question 2 Is the given network uniquely solvable? Question 2 If yes, determine the "degree of freedom" or "complexity" of the network (and give a system of as many independent state variables).

1. The list of problems in Chapter I is by no means complete. We are not considering here the applications of matroids in information theory [21-24,321 or in control theory [31]. Matroidal concepts and results are also widely applied to operations research, linear programming etc. , see e. g. [6,19,37]. Even such seemingly esoteric fields as behaviourial linguistics have already applied nontrivial matroidal tools [59,62]. 2. Questions J& above, for i=1,2,3, can be answered by "yes" or "no" while Questions 2 by a non-negative number and a subset of a suitable set. The common feature of all these problems is that qualitative questions are posed, so discrete mathematical tools are hopefully applicable. Questions 2 will turn out to be, in a sense, special cases of the corresponding Questions g . This will be made somewhat more precise in Chapter IX. 3. At the first glance all these questions (at least those of form 2 )seem to be routine tasks of linear algebra. In practical applications this is not the case, for the following reason. A necessary and sufficient condition to these problems in terms of linear algebra would be the nonsingularity of a usually large matrix with real entries. Checking this nonsingularity by numerical methods can lead to qualitatively wrong answers, due, for example, to roundoff errors in arithmetical operations among real numbers (which are represented by decimals of a finite length in a computer). We shall return to this and related questions in Chapter VII. 4. Speaking quite generally, if an arbitrary phenomenon should be described by a possibly simple mathematical model then

finiteness and linearity of the model are perhaps the most frequent assumptions in many fields of science and technology. Hence tools of discrete mathematics and linear algebra are widely applied. Matroid theory, being a branch of combinatorics, still containing linear algebra as a special case, can therefore serve as a unifying tool. (If a reader find this remark somewhat too general and of little practical value, he/she is requested to return to this point after having finished Chapter V. Both matroids G and A contained many important information, reflecting properties by graphs and by matrices respectively. However, only their union G v A gave the answer to Question %.I

A network is an interconnection of devices. The devices are given by their physical properties, in form of linear equations among voltages and currents, see Remark 3 below, while the interconnection is described by a graph. 2 &. 3 Example 1. Consider the network of Fig. 3. The devices are as follows: A voltage sourcefll I is defined by ul=u(t), i.e. a given function of time, while its current i, is arbitrary; a current s o u r c e f 5 ) is defined by is=i(t) while its voltage US is arbitrary; a resistor(2) is defined by u2=Ri2 and an idea2 transformerf3,4i by us=ku4 and i4=-ki3. The interconnection of these devices is represented by the network g r a p h of Fig. 4 , where edges correspond either to 2-terminal devices (sources and resistors) or to individual ports of the multiports (see below); and two edges are incident to the same vertex if the devices or ports are joined to the same node.

The problem (cf. Remark 3 below) of network analysis is: determine all the voltages and currents of every device (as unique functions of the voltages of the voltage sources and currents of the current sources). For example, in the above network iz=k-'is or u3=u1-rk-li5 etc.

AS a common generalization of the concepts of resistor- tone linear equation relating a voltage and a current) and that of the ideal transformer (two linear equations relating two voltages and two currents) let us define[5] the concept of n-port as a system

of n linear, homogeneous algebraic equations relating n voltages and n currents. Each port of such an abstract device has one voltage and one current; and every such port is represented by an edge of the network graph. The expression rnultiport will also be used, if the number of ports need not be emphasized. Remark l._The class of networks defined above is usually called linear, memoryless and time-invariant. The second condition will be dropped in Chapter IX, see also the next remark. The first (meaning that the multiport equations are linear) and the third (meaning that the structure of the network graph does not change - e.g. by switches) are essential in what follows. However, the concept of multiports is so general that even this class of networks contains a great variety of practically important ones. ~ d r 2k. Linear devices "with memory", like capacitors and inductors, are excluded in Chapters 111-VIII. Hence, if the answer to Question 2 is positive, the system of network equations is algebraic and Question 2 does not arise before Chapter IX. Remark 3. Some devices may be defined in a more convenient way in terms of other physical quantities, e.g. as in Example 5 in Chapter X below, rather than by voltages and currents. Hence the definition of a multiport and the problem of network analysis can be formulated in a more general way. The voltages and currents in the network must satisfy not only the defining equations of the devices but also the Kirchhoff Voltage Laws and Current Laws (KVL and KCL respectively), posed by the structure of the interconnection. KVL states that the sum of the voltages along any circuit of the network graph must be zero. KCL states that the sum of the currents along any cut set (i.e. minimal set of edges whose removal disconnects the graph) must be zero. E.g. u7+u3+u+=O a ~ diz+i~=Oare examples for KVL and KCL respectively, in the network of Example 1. (The question of reference directions is of no particular importance and will be neglected here. Hence Question 2 sounds like that: Is the system of equa-

tions for the voltages and currents of the network (obtained from the device equations, from KVL's and KCL's) uniquely solvable?

I V , THE

CLASSICAL

RESULT

If the network graph contains a circuit so that every edge of the circuit corresponds to a voltage source then the currents of these edges cannot be uniquely determined. Similarly, a cut set formed by current sources leads to uncertainity of voltages. Hence t h e subgraph GU o f t h e network graph, f o r m e d by t h e e d g e s of t h e voltage sources, must not c o n t a i n any circuit, a n d

(C1) t h e subgraph Gi, f o r m e d by the current sources, must not c o n t a i n any cut set (i. e. the c o m p l e m e n t of Gi must be connected).

This necessary condition can be proved to be sufficient if the network contains voltage and current sources and resistors (u=Ri, R>O) only. The proof of the sufficiency (essentially [33], see also [ 4 4 , 8 ] and some recent textbooks on circuit theory, like [ 5 7 ] ) is by no means trivial. If the network contains arbitrary multiports, too, then (Cl) is not sufficient any more. CI fig.5 Example 2. If the equations of the 2-port on Fig. 5 are u3=ku4, i ~ = 0then the network will have no unique solution. (No solution at all, if i5#0 while infinitely many if is=O since then u4 and us may be arbitrary.) However, observe that the network g r a p h of Example 2 is the same as that of Example 1. Hence we can also conclude that a necessary and sufficient c o n d i t i o n of u n i q u e solvability, i n t e r m s

of t h e network g r a p h only, c a n n o t be g i v e n a t all.

V,

STRONGER NECESSARY C O N D I T I O N

Let the network graph contain k edges and let uV and iv denote the voltage and current of the vth edge, respectively. If we

formally generate all the network equations, the system to be solved is Nx=3 where x=(u,,u2 ,...,uk,il,i2 ,ik)*. N can be de-

,...

composed like

[z 1 i],

where 1 is the collection of the matrices

A

of the multiport-equations, 0 denote the zero matrices and 6 and Q are the edge-circuit and the edge-cutset matrices, respectively. The edges 1,2,. ..,k can be labelled in such a way that 1,2, a correspond to the voltage sources and B,$+l, k correspond to the current sources (see the remark below). Let uo= Then Nx=O can also be (u,,uZ,. ,U I* and io=(i6,16+l,...,ik)*. '

...,

...,

..

I: [

written as [ N ' I N ~ ~ Nxo ~ ] =O. Hence the network is uniquely solvable if and only if No is nonsingular, or if and only if det No#O, and then, of course, x o = - N z 7 (N'uo+Nr1io 1. Generally speaking, if M is a square matrix of the form then, by the Laplace expansion, det M arises in form E +det Pl det P2 where P1 is formed by the rows of M1 and'by some columns of M, P2 is formed by the rows of M2 and by the rest of the columns of MI and the summation is performed over every choice of the columns. Hence a necessary condition of det M #O is that there must exist at least one decomposition of the coiumn set of M so that both det Pl and det P2, belonging to this decomposition, should be nonzero. Let the column space matroids of the matrices A , B etc be denoted by the corresponding script letters A , B etc respectively. Then det M#O means that the full column set T of M i,s independent in M and, by the above reasoning, this implies that there exists a decomposition T=T1UT2 so that Ti is independent in Mi. But it exactly means that T is independent in M,vM2 where v denotes matroid union. Let S=Eu,,u2, uk,i~,i2,...,ik}denote the full column set of N , let furthermore U= lul ,u2,. .,ual and I= Ci B f lB+lf.. ' ,ikl denote those of n' '. and A'" respectively (see the remark below). Remark: Certainly a
I21

...,

.

.

respond to voltage sources and deleting the edges which correspond to current sources" if only solvability is concerned. This makes everything simpler since solvability does not change if the corresponding homogeneous system is considered only. If we denote the matrix by 8 , we could finally formulate the following necessary condition for unique solvability:

[z g]

S-(UUI)

(C2 )

i s a base

ifi

GvA

. O

Before presenting some examples we recollect that if K= 10 K 2 1 then K = K l + K 2 where + denotes direct sum. (Here the matrices need not be squares.) Also recollect that if B and Q are the circuit and the cut set matrices of a graph H respectively then B=M*(H) and Q=!=I! ( H )where M (H) denotes the cycle matroid of H. In order to visualize the above concepts, let us return to Example 1. Both matroids G and A happen to be graphic - they are my.6 illustrated as the cycle matroids of the graphs on Fig. 6. The drawing for G is obvious by the above remarks, while to check the drawing for A let us explicitly describe the matrix A :

The matroii GvA also happens to be is graphic in this example and can be il1 lustrated as the cycle matroid of the graph of Fig. 7. The set S-{u,,i5} is clearly a base of this matroid, hence ( C 2 ) is met. On the other hand, the matroid '4,3,5

4

I; \iJ

fi*. 9

A of the second example can be drawn as the cycle matroid of the graph of Fig. 8 and then the result GvA will be illustrated by the graph of Fig. 9. The set S-{uq ,is} will not be a base now, hence ( C 2 ) is violated. U3

U,,5

V I , HOW CAN

ONE CHECK CONDITIONS

(C1)

AND

(C2) ?

Checking condition (Cl) in a graph is certainly an easy task. The reader is suggested to prepare an algorithm or is referred to [25, pp.41-42]. On the other hand, (C2) poses some questions. A large number of algorithms have been published in the last 12 years to decide whether a given set is a base in the union of

..

the matroids M, , M z , . ,Mnl see e.g. [2,15,17,18,20,34,35,37,38]. These have polynomial complexity provided we have an "independence oracle" for each Mv (see [64] in this volume). Roughly speaking, we can obtain an algorithm, polynomial in the size m of the common underlying set S of the Mv's and in their number n, if the information "whether the given subset of S is independent in Mv or not" can be obtained in a time polynomial in m. An independence test in G is essentially as easy as checking (C1). For testing independence in A observe that A always arises as a direct sum, with the summands corresponding to the individual devices. Hence a subset of S is independent in A if and only if, for every p, the intersection of this subset with the set of voltages and currents of a multiport device N is independent in lJ the summand A . lJ Still, we have to check whether a subset is independent in A,,, which is given as the colurnn'set matroid of a matrix with real entries. However, the difficulty described in Remark 3 of Chapter I1 does not exist any more, for the following reason. In the practical realizations of a computer program for network analysis the user has to specify only the devices to be interconnected - and, of course, the way of the interconnection. The models of the devices are stored in the data bank of the computer, as subnetworks, matrices e t ~ .Once a new device becomes available at the market, its model should be prepared and filled into the data bank. At this stage the corresponding matroid A p can also be prepared and stored in a suitable way. Even if this task is "difficult" (requires long time or high precision in numerical calculations), this should be done only once, when.establishing the model to be stored in the data bank. Then this information is stored for ever and each time when the device is required in an actual analysis (which can be the case many times),

3 101 we use only the result of these "difficult" calculations.

VII,

Is (C2>

SUFFICIENT?

The proof of the necessity of Condition fC2) clearly indicates that it cannot be sufficient in the mathematical sense. It only implies that we have a nonzero term in the Laplace expansion but it can easily be cancelled out by other nonzero terms, leading to det M= 0. Example 3. Consider the trivial example on Fig. 10 where the sum of the two resistances is zero. The network is clearly singular but (C2) is met. However, one can argue that situations like that of Example 3 are artificial and can be disregarded. After all, one can never buy such two resistors Rq and R2 in the shop that the a p r i o r i given relation R4+R2=0 is exactly met, since the physical parameters of the devices are subject to technological constraints etc. Putting it slightly more exact, if we suppose that the numerical values of the device parameters are, say, algebraically independent transcendentals over the field from which the entries of B and Q are chosen, then (C2) turns out to be sufficient, too. This additional assumption is usually called "generality" [30] and also arises in some form in many other mathematical and engineering considerations, see e.g. [16,42]. However, there are nontrivial problems of its correct definition. The interested reader is referred to [55]. Although we are not going into details of the rigidity problems, let us return now to Fig. 1 for a moment. The reason of the differences between A and C is that the distance between the pins A and B is equal to the sum of the distances of the braces AC and BC in case of system C. Roughly speaking, if such "algebraic relations" are forbidden, then the rigidity problems become purely combinatorial, e.g. the answers to Questions 2 , will depend on the "graph" of the braces only, and the only "infinitesimal" motions of the system will be the velocities of the mechanical ones [3]

.

V I I I , SHOULD

WE ALWAYS ACCEPT "GENERALITY"

?

The above argument (about the technological uncertainities) is acceptable only as long as a priori given relations among parameters of different devices are concerned. But if such relations are given among different parameters of the same device, one should be more careful. E.g. the two k's in the defining equations of an ideal transformer (see Example l) are equal, and this equality is not a coincidence but it reflects the physical feature of the device. Such "internal relations" might cause difficulties, as shown by the following example: Example 4. Let the 2-port of the network of Fig. 11 (the gyrator) be defined by up=Ri3, u3=-Ri2. It is not difficult to verify that the network is singular but (C2) is met. This phenomenon as well as some other similar ones have been well known by network theorists since long time. [1,45-49,571 are typical examples for avoiding such difficulties by describing that the independent subsets from G and A (to meet (C2)) must satisfy certain additional conditions. E.g. if {x,y) is the edge set of a 2-port N then only those trees T of the network graph are accepted for which I {x,y]nT I=l if N is an ideal transformer and I {x,ySnTI#l if N is a gyrator. This approach was in a sense generalized for arbitrary 2-ports in [53] where we proved that, among the theoretically possible infinite variety of algebraic relations among the 2-port parameters, only five relations can really cause such problems. The essence of this result is that these 5 critical relations are independent of the way how the 2-port is embedded into the network, hence they can be checked in stage of establishing the model of the 2-port for the data bank (c.f. the end of Chapter VI). The above examples indicate that the additional conditions to be checked are at least as hard as the matroid 2-parity problem for represented matroids. Anyhow, this is still a polynomial problem[39-41]. The reader is also referred to 1641 in this volume but should observe that generalizing the above questions to k-ports (k>2) will generalize the matroid 2-parity problem in a different way. For this latter we refer to [56] and mention only

...,

one interesting question: Let S 1 , S ~ , St be disjoint k-element subsets of S and let M be a represented matroid over S. Decide whether there exists a subset X with cardinality at least q so that lS.nXl#l for every i=1,2,...,t (observe that this reduces 1 to the 2-parity problem for k=2 1, and X is independent in M.

IX,

SOME

REMARKS ON QUESTION

3B

Only memoryless n-ports were considered till now. However, essentially the same approach works in case of devices "with memory" as well. This concept means that the device can store energy. For example, the c a p a c i t o r or the inductor with the defining du and u=+ di respectively, can store (electric resp. equations i=Cdt dt magnetic) energy. In fact we may suppose without loss of generality that only these two kinds of new devices are introduced (i.e. a linear n-port with memory can always be substituted by a subnetwork, containing, for a suitable k, a linear memoryless (n+k)port and k capacitors and inductors.) The order a of the system of differential equations to be solved is clearly bounded from above by the number nC of capacitors plus the number nL of inductors. However, the case o
and the one which contains the minimal number kL of inductors. (It is not difficult to prove that both values kc and kL can be reached by the same tree as well.) Then o=kC+kL. Observe that to determine kc and kL (for answering Question 3B) we need Kruskal's algorithm with gradually decreasing weights for voltage sources, capacitors, resistors, inductors and current sources, while to check (Cl) itself, for answering Question 2 only, we need simple independence tests only. We refer to [30,50-521 for determining o in networks containing voltage and current sources, capacitors, inductors and linear memoryless n-ports, but wish to emphasize that the algorithms require weighted matroid.partition algorithms, unlike in case of memoryless networks, where cardinality matroid partition algorithms could already do. Generally speaking, Questions seem to require weighted versions of those algorithms which were used to answer Questions i A (for i=1,2,3).

-

-

X,

HPPLICATI ONS TO NETWORK SYNTHESIS

Until now we considered the problem of network analysis, i.e. given the network, determine its properties. In this last chapter we give some remarks on the inverse problem (the real engineering problem: given some specifications, design a network which meets them ) Whether such a problem is solvable depends on the set of devices we are allowed to use as building blocks for our realization process. Negative results like "a particular specification cannot be realized from a given set of building blocks'' are usually considered as parts of "realizability theory" while "synthesis" in the narrow sense means rather realization processes, canonical configurations etc. Here we intend to show how matroids can help to obtain new realizability criteria. For more detailed results the reader is referred to [28,54]. Suppose we wish to interconnect the multiports N,,N2,...,Nk to form a new multipart No. Their matroids A?,A,, Ak and Ao, respectively, are defined in the same way as in Chapter V - sim-

.

...,

ply the column set matroids of their describing matrices. The way how they are interconnected is described by the network graph H again -edges correspond to ports- and G=M* (H)+M (H ), as before. If the edge set E of the graph H is of the form EoU(E-Eo) where Eo is the set of ports of No then [54] A, can be obtained from A,,A2, Ak and G by forming Gv(A1+A2+...+ Ak) and then contracting the elements, corresponding to voltages or currents of E-Eo Example 5. Consider the 3-port c i r c u l a t o r defined by x,=yor xn=y3 and x3=yqrwhere xv=uv+iv and yv =uV-iv (the physical meaning of xv and yv are incident and reflected waves at port v). It is not difficult to verify [28,52,54] that the matroid of the circulator is just the cycle matroid U3 of the graph on Fig. 13. If we interconnect the circulator with a resistor in the way shown on Fig. 14a, the network graph will look like Fig. 14b, and a not quite obvious calculation shows that the matroid

...,

.

of the resulting 2-port will be the cycle matroid of the graph of Fig. 14c. Of course, a condition, like "generality" in Chapter VIL, is required again: we usually call it q u a Z i t a t i v e r e l i a b i l i t y IQR) in this context[54]. Roughly speaking it means that the realization of a multiport No from the Nv's is QR if and only if small quantitative changes in the N V r s cannot lead to a qualitative change of No, hence, after the interconnection of the NVrs,no "mutual tuning" of these components is required any more for meeting the qualitative specifications of No. (Somewhat similar ideas have already been presented in [9-111, too. The following theorem will be a typical example to show how this approach can be applied: Theorem: The 3-port circulator cannot be QR-synthesized from 1- and 2-ports, using series-parallel topology (i.e. if the network graph is series-parallel [14] ). Sketch of the proof: It requires a straightforward verifica-

tion that the matroids of all the 1- and 2-ports are ganmoids[43]. The matroid G is a ganunoid iff the network graph is series-parallel. Finally, the gammoid property is preserved by direct sum, union and contraction, but the matroid of the circulator is not a gammoid, hence the assertion follows. In addition to the more or less well known matroidal classes (like gammoids, base orderable matroids, representable matroids), which are closed with respect to union and contraction, one can construct new matroidal classes, too, which also lead to new realizability criteria. Some of them have interesting meaning in network theory as well. See [28,54] for further details.

ACKNOWLEDGEMENTS : The author's contributions to these topics could not be obtained without the highly stimulating discussions in the last ten years with many friends and colleagues, most notably with A. Csurgay, A. Frank, M. Iri, L. Lov6sz, T. Roska and V. T. Sbs.

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109. T.Ozawa-Y.Kajitani, Diagnosability of linear active networks, Proc-IEEE 1ntern.Symp.Circuits and Systems,Tokyo,l979.866-9. 110. T.Ozawa-T.Nitta,Some considerations on the state equations of linear active networks and network topology, Mem.Fac.Eng. Z y o t o University, 34 119721 413-424. 111. PBsztor Z., thesis, University of Budapest, 1977. 112. Petersen,B., thesis, Danmarks Tekniske H~jskole,Lyngby,l976. 113. B-Petersen, Investigating solvability and complexity of linear active networks, Proc.1978.European Conf. Circuit Th. and Design, Lausanne, 1978. pp. 508-512. 111. B.Petersen, The qualitative appearance of linear active network transfer functions by means of matroids, Proc. IEEE Intern.Symp. Circuits and Systems, Tokyo, 1979. 992-995. 115. B.Petersen, Algorithms for topological investigation of linear active networks, Proc. IEEE Intern. Symp. Circuits and Systems, New York, 1980. 354-358. 116. B.Petersen, Circuits in the union of matroids: An algorithmic approach. 117. A.Recski, On partitional matroids with applications, CoZZ. Math.Soc.J.BoZyai 10. North-Holland, 1974. 11. 1169-1179. 118. A.Recski, Matroids and state .variables, Proc.1976 Europ.Conf. Circuit Theory and Design,Genova, 1976. I. 44-51. 119. A.Recski, On the sum of matroids 11, Proc. 5th British Combinatorial Conf, Aberdeen, 1975. 515-520. 120. A.Recski, Applications of graph theory to network analysisa survey, B e i t r a g e sur G r a p h e n t h e o r i e u n d d e r e n A n w e n d u n g e n Proc.Conf. Oberhof lG.D.R.1 1977. pp. 193-200. 121. A.Recski, Matroidal structure of n-ports, CoZ2.Math.Soc.J. B o Z y a i 18. North-Holland, 1978. 11. 893-909. 122. A.Recski, Matroids in network theory, Proc.6th 1ntern.Symp. on Microwave Communication, Budapest, 1978. 123. A.Recski, Sufficient conditions for the unique solvability of networks containing linear memoryless 2-ports, Proc. 1978. Europ.Conf.Circuit Theory and Design,Lausanne,l978. 93-97. 124. A.Recski, Terminal solvability and the n-port interconnection problem, Proc. IEEE Intern. Symp. Circuits and Systems, Tokyo, 1979. pp. 988-991. 125. M-Saito-T.Asano-T.Nishizeki, On the 3-terminal matroids, Techn-Research Report of the IECEJ, 1979. 126. M.Sengoku-T.Matsumoto, On maximal circuit-free and cutsetfree subgraphs and hybrid trees in a linear graph, Proc.IEEE Intern Symp. Circuits and Systems, 1975. pp. 104-107. 127. S.Shinoda-H.Sakuma-T.Yasuda, Semimatroids, Proc.IEEE Intern. Symp. Circuits and Systems, 1979. Tokyo, pp. 1004-1007. 128. C.A.B.Smith, Electric currents in regular matroids, Proc.4th British Combinatorial Conference, 1973. 129. C.A.B.Smith, PatroidsIrT.CornbinatoriaZ Th.Ser.B.16 119741 64. 130. J.Tak~cs,thesis,University of Budapest, 1978. 131. N.Tomizawa, A practical criterion for the existence of a linkage pair of bases in a muoid and an algorithm for determining the maximally distant linkage pair of bases, Trans. IECEJ. 59-A 119761 280-286. 132. N.Tornizawa, Strongly irreducible matroids and principal partition of a matroid into strongly irreducible minors, Trans. IECEJ. 59-E 119761 14-15. 133. N.Tomizawa-M.Iri, An algorithm for solving the "independent assignment problem" with application to the problem of de-

134.

135. 136.

137. 138. 139.

termining the order of complexity of a network, T r a n s . 1 ~ ~ ~ ~ . 57-A 119741 627-629. N. ~omlzawa-M. 1ri , An algorithm for determining the rank of a triple matrix product AXB with application to the problem of discerning the existence of the unique solution in a network, T r a n s . I E C S J . 57-A 119741 50-57. V.Toma, thesis, University of Budapest, 1977. Z.unver, thesis, Middle East Technical Univ.,Ankara, 1978. Z-Unver-Y.Ceyhun, On a graphical representation for matroids L.Weinberg, Matroids, generalized networks and electric network synthesis, J . C o m b i n a t o r i a Z T h e o r y S e r . B.23 119771 106-126. L.Weinberg, Duality in networks: Roses, bouquets and cut s e t s ; Trees, forests and polygons, N e t w o r k s , 5 119751 179.

Dr. Andrgs Recski Research Institute for Telecommunication H-1026 Budapest, Ggbor A. u. 65. H u n g a r y

CENTRO IKTERNAZIOKALE MATEMATICO ESTIVO (c.I.M.E.)

MATROIDS AND COMBINATORIAL OPTIMISATION

D.J.A.

WELSH

M e r t o n C o l l e g e , U n i v e r s i t y of O x f o r d

L e c t u r e s given a t t h e C e n t r o Internazionale M a t e m a t i c o E s t i v o , V i l l a M o n a s t e r o , V a r e n n a , f r o m A u g u s t 24 September 2 , 1980

-

Matroids and Combinatorial Optimisation

1.

2.

3.

Computational Complexity

1.

Introduction

2.

Low l e v e l complexity

3.

The c l a s s NP

4.

NP-complete problems

5.

Some examples

6.

P-space

Matroid Theory

1.

Definitions

2.

Classes of matroids

3.

s p e c i a l matroids

4.

Excluded minor theorems

5.

Matroid polyhedra

Matroids and Algorithms

1.

The greedy algorithm

2.

The union of mat'i-oids

3.

The Shannon switching game

4.

A polynomial algorithm f o r k-connectivity

4.

~ o v s s z ' sAttack c n t h e P a r i t y Problem 1.

5.

6.

The p a r i t y problem

2.

2-polymatroids

3.

The ~ a l l a i - ~ o v s si zd e n t i t y

4.

ii minimax theorem f o r l i n e a r 2-polymatroids

5.

On a polynomial algorithm f o r t h e 2-parity problem

6.

Pinning p l a n a r s t r u c t u r e s

Oracle Bounds on Algorithms

1.

The concept of an o r a c l e

2.

Examples a n d f u r t h e r r e s u l t s

3.

The 2-parity problem i s exponential

Seymour's C h a r a c t e r i s a t i o n o f Regular Matroids

1.

Binary and r e g u l a r matroids

2.

Splitters

3.

The decomposition theorem

4.

A polynomial algorithm t o decide whether a matrix i s t o t a l l y

unimodular

7.

8.

Colouring, Flows and Blocking Problems 1.

The blocking problem

2.

Colouring graphs

3.

Flows t a k i n g values i n an a b e l i a n group

4.

Tangential blocks

5.

F u r t h e r problems

Flows i n Matroids 1.

The max flow min c u t theorem

2.

Mu1ticommodity flows

3.

Multicommodity flows i n matroids

4.

Summary o f r e s u l t s

1.

91.

Computational Complexity

Introduction Although the study of the computational complexity of solvable

decision problems has been to a large extent motivated by considerations of their practical computability it has also led to a re-examination of the nature and relationship of classical problems in fields as diverse as number theory, combinatorial optimisation and recursive function theory. Indeed, one of the outstanding open problems in contemporary mathematics is the conjecture (discussed in 83 below) that the class P of languages recoynisable in polynomial time by a deterministic Turing machine is not equal to the class NP of languages recognisable in polynomial time by a non-deterministic Turing machine. In this first lecture I briefly review relevant concepts of complexity theory, in subsequent lectures I concentrate on the interplay between complexity theory and recent results about matroids.

82.

Low level complexity

Consider the problem of multiplying two n x n matrices. The standard 3 algorithm demands n multiplications and n3 - n2 additions. One of the more striking early results in complexity theory was the theorem of Strassen (1969) who proved that a pair of n x n matrices can be multiplied in log 7 5 c n arithmetic operations, where c is some constant.

For l a r g e n, s i n c e l o g 7 = 2.81, 2

t h i s r e p r e s e n t s a considerable

saving over t h e standard algorithm, and it n a t u r a l l y r a i s e d t h e question: by how much could t h e index l o g 7 be reduced?

T h i s q u e s t i o n i s still

unresolved. Even though arguments which reduce t h e upper bound i n t h e above problem a r e complicatedandingenious, it i s a much more d i f f i c u l t problem t o o b t a i n a non t r i v i a l lower bound t o such a computational problem.

A

moment's r e f l e c t i o n suggests t h a t an obvious lower bound on t h e complexity o f t h e above problem i s 0 ( n 2 )

-

a l l t h e d a t a has t o be examined.

(Prove

this!) S i m i l a r arguments apply t o combinatorial problems.

Consider t h e

problem o f f i n d i n g t h e s m a l l e s t c i r c u i t i n a graph on n - v e r t i c e s .

Such

a graph would be p r e s e n t e d t o t h e computer a s a n x n m a t r i x and it i s n o t d i f f i c u l t t o show t h a t i n o r d e r t o decide t h e s i z e o f t h e l a r g e s t c i r c u i t we must examine

each e n t r y i n t h e adjacency matrix, g i v i n g u s a lower

bound o f 0 ( n 2 ) and t h a t t h e r e does e x i s t an algorithm f o r s o l v i n g t h e problem which t a k e s a t most 0 ( n 4 ) a r i t h m e t i c o p e r a t i o n s . Loosely speaking, t h e r e f o r e , t h e problem i s t r a c t a b l e and t h e only i n t e r e s t i s i n deciding t h e e x a c t value o f t h e exponent.

Problems such

a s t h e s e , although o f p r a c t i c a l i n t e r e s t w i l l n o t be o u r main concern here. (For a d i s c u s s i o n o f such problems we r e f e r t o Bollobas C781, Milner and Welsh C761, and Borodin and Munro C751.1

We w i l l be more concerned with

t h e gap between polynomial and exponential complexity.

83.

The c l a s s NP Let rr be a computational problem, t h a t i s a c o l l e c t i o n o f computational

t a s k s each o f which i s c a l l e d an i n s t a n c e o f n.

For example t h e problem o f

prime t e s t i n g i s t o determine f o r any given number n whether it i s prime and a t y p i c a l i n s t a n c e i s a s p e c i f i c i n t e g e r . When studying t h e complexity o f a problem n we a s s o c i a t e with each instance I a

size 111.

The choice o f t h e p a r t i c u l a r measure used t o d e f i n e

s i z e i s n o t unique b u t g e n e r a l l y speaking we t a k e t h e s i z e (11 t o be c o r r e l a t e d t o t h e minimum amount o f memory space r e q u i r e d t o s p e c i f y I . Thus t h e s i z e In1 o f a n i n t e g e r n i s defined a s t h e number of d i g i t s i n t h e b i n a r y r e p r e s e n t a t i o n o f n, t h u s In1 = log n. 2

The s i z e o f an i n s t a n c e

corresponding t o a graph G on n v e r t i c e s i s t h e number o f v e r t i c e s p l u s t h e number o f edges, however f o r most problems it i s s t a n d a r d t o use n, t h e number o f v e r t i c e s t o s p e c i f y t h e s i z e o f t h e i n p u t . A s f a r a s t h e s e l e c t u r e s a r e c o n c e r n e d , s i n c e we a r e only concerned

w i t h the gap between polynomial and exponential complexity t h e r e i s no l o s s i n p r e c i s i o n i n making t h i s o u r s t a n d a r d i n t e r p r e t a t i o n o f s i z e when d e a l i n g w i t h graph problems. Complexity theory i s based on a Turing machine model o f computation (with which we assume f a m i l i a r i t y ) .

Most modifications o f such a model do

n o t a f f e c t t h e complexity n o t i o n s considered below, though o f course some c a r e h a s t o be e x e r c i s e d , ( p a r t i c u l a r l y i n problems involving l a r g e integers)

-

s e e Aho Hopcroft and Ullman C741 f o r a d i s c u s s i o n o f t h e

r e l a t i o n s h i p between complexity measures with r e s p e c t t o d i f f e r e n t models. For any f u n c t i o n t , a Turing machine M runs i n time t i f on each i n p u t o f s i z e n t h e machine h a l t s w i t h i n t ( n ) s t e p s .

M runs i n polynomial

time i f M runs i n time p f o r some polynomial p. I f Z i s t h e i n p u t a l p h a b e t o f t h e Turing machine M i n q u e s t i o n and n i s a d e c i s i o n problem

i.e.

a p a r t i t i o n of Z * ( t h e s e t of a l l f i n i t e

sequences from E ) i n t o two s e t s L and E*\L corresponding t o t h e o u t p u t

ACCEPT o r NON ACCEPT,

then we nay L belongs t o t h e c l a s s P is there e x i s t s

a Turing machine M such t h a t f o r each s t r i n g I E E*, I i s accepted by i f and only i f I E L and moreover M runs i n polypomial time.

M

The c l a s s P

i s thus t h e c l a s s of s e t s whose membership i s decided by some Turing machine which runs i n polynomial time. Formally, the c l a s s NP i s t h e c o l l e c t i o n of languages which are accepted by a non-deterministic Turing machine i n polynomial time.

A

rigorous and formal defirritionofanon-deterministic Turing machine and t h e c l a s s NP i s given by Garey and Johnson C791 o r Hopcroft and Ullman C791. f o r our purposes it i s probably more i n s t r u c t i v e t o proceed a s follows.

A language L E C * i s i n t h e c l a s s NP i f f o r each member U E L

there i s an algorithm f o r demonstrating t h a t U E L which runs i n polynomial time on a deterministic machine. Because of t h i s e x i s t e n t i a l q u a n t i f i c a t i o n i n t h e notion of acceptance by a non-deterministic Turing machine (NDTM) there i s no obvious reason

why t h e complement of a s e t i n NP should a l s o be i n NP.

This i s b e s t

i l l u s t r a t e d by example.

Example. -

Let I* be the c o l l e c t i o n of inputs corresponding t o a l l graphs,

l e t L be the subset of C* corresponding t o graphs which have Aamiltonian c i r c u i t s ( i . e . c i r c u i t s which v i s i t each vertex once and only once).

Given

a t y p i c a l member of L we can demonstrate i t s membership o f L i n polynomial time by e x h i b i t i n g any one o f i t s Hamiltonian c i r c u i t s . However given a member o f E*\L there i s no known way o f demonstrating t h a t i s doesn't have a Hamiltonian c i r c u i t i n polynomial time!

Accordingly we d e f i n e co-NP t o be t h e c l a s s o f languages L such t h a t C*\L i s i n NP. C l e a r l y i f a language L e P , then E*\LeP and hence we have

5 4.

NP-complete problems A major t o o l i n complexity c l a s s i f i c a t i o n i s computational r e d u c i b i l i t y .

I n s t u d y i n g NP, t h e most u s e f u l notion has been d e t e r m i n i s t i c polynomial reducibility.

For any problems rl, r E Z * we w r i t e rl = r 2 and say rl i s 2

r e d u c i b l e t o s2 i f t h e r e e x i s t s a Turing machine which runs i n polynomial time and which,given an i n s t a n c e o f nl w i l l convert it i n t o an i n s t a n c e o f r 2 such t h a t t h e 'answer' t o an i n s t a n c e of n1 i s ' y e s ' i f and only i f t h e answer t o t h e i n s t a n c e o f w2 i s "yes".

I f L1 i s reducible t o L2 and

L h a s a polynomial time algorithm then s o does L 2 1Formally t h e r e e x i s t s a polynomial transformation o f a language L cC* t o a language L cC* i f t h e r e i s a f u n c t i o n f : Z T + C; 1- 1 2- 2

such t h a t :

(1)

There i s a polynomial time Turing machine which computes f .

(2)

For a l l

X E

Ei,

X E L i ~f and only i f f ( x ) E L2'

I n t h i s case.we w r i t e L 1 a L2A language L i s c a l l e d NP-complete i f : ( 3 ) LENP. (4)

For any o t h e r L ' ENP, L'

a

L.

Cook i n 1971 proved what i s probably t h e most important theory s o f a r i n complexity theory:

Theorem 1.

There e x i s t NP-complete languages.

Cook's proof was e s s e n t i a l l y c o n s t r u c t i v e .

He e x h i b i t e d a n NP-complete

language d e r i v e d from a d e c i s i o n problem i n Boolean l o g i c , which i s u s u a l l y r e f e r r e d t o a s SATISFIABILITY.

We d e s c r i b e it a s follows.

The SATISFIABILITY problem is t h e problem o f p r o p o s i t i o n a l c a l c u l u s o f determining whether a given l o g i c a l formula i s t r u e f o r a t l e a s t one assignment o f t h e values ' t r u e ' NP-complete

and ' f a l s e '

t o the variables.

The a s s o c i a t e d

language i s then t h e c o l l e c t i o n o f l o g i c a l formulae which a r e

satisfiable. By a s l i g h t ( b u t very common) l o s s of p r e c i s i o n w e s h a l l speak o f a d e c i s i o n problem n b e i n g NP-complete

i f t h e a c c e p t i n g l a n uage (corresponding

t o those i n s t a n c e s o f n f o r which t h e answer i s YES) i s an NP-complete language. I n o t h e r words a problem i s NP-complete i f it is a t l e a s t a s hard ( a l g o r i t h m i c a l l y ) a s any o t h e r problem i n t h e c l a s s NP. Once one knows a s i n g l e NP-complete problem n one can prove another problem n

' i s NP-complete by a ) showing t h a t n * c NP and b) showing t h a t n

i s reducible t o n' i n polynomial time.

Karp (1972) used t h i s technique t o

e x h i b i t a number o f n a t u r a l NP-complete problems and t h i s p r o c e s s h a s been contiued almost 'ad nauseam' s o t h a t now t h e number o f known NP-complete problems must be i n t h e thousands.

55.

Some Examp* W e illustrate the &ove concepts w i t h some examples o f well-known

problems.

GRAPH COLOURAEILITY

( :)

INSTANCE:

Graph G = ( V , E) , i n t e g e r k > 2.

QUESTION:

Is G k-colourable i n t h e sense t h a t w e can a s s i g n c o l o u r s t o V(G) from t h e s e t { l ,

...,k}

such t h a t two v e r t i c e s which

a r e joined by an edge have t h e same colour? T h i s i s m - c o m p l e t e even when k = 3 and t h e graphs a r e r e s t r i c t e d t o being planar.

For k = 2 it i s j u s t t h e problem of recognizing when G i s

b i p a r t i t e and c l e a r l y belongs t o P. (2)

CLIQUE INSTANCE:

Graph G=(V,E), p o s i t i v e i n t e g e r k .

QUESTION:

Does G c o n t a i n a s e t V' of

v e r t i c e s such t h a t

I V' I

2 k

and each p a i r o f v e r t i c e s i s joined by an edge? For any O(

1vlk) -

(3)

fixed k

this i s i n P s i n c e t h e exhaustive s e a r c h i s of complexity

However f o r g e n e r a l k t h e problem i s NP-complete.

LINEAR PROGRAMMING INSTANCE:

I n t e g e r valued n-vectors dl,

QUESTION:

...,dm,

C1,

5

m,

I n Gareyilohnson [79, (p. 288)

be i n P.

5

m ) , integers

b

Is t h e r e a v e c t o r x = (xl, for 1 5 i

a s uncertain.

...,Cn'

(v. : 1 5 i

...,xn)

o f r a t i o n a l s such t h a t

v..X6 d and c.x 2 b?

1 t h e e x a c t s t a t u s o f t h i s problem i s s t a t e d

I t was n o t known t o be NP-complete

nor was it known t o

However s t a n d a r d l i n e a r programming d u a l i t y arguments showed

t h a t it is a l s o i n co-NP and t h i s suggested t h a t it was u n l i k e l y t o be NP-complete.

The problem was s e t t l e d i n (1979) by L.G.

Khachian who, i n

one o f t h e most important thoerems of t h e l a s t decade, h a s shown t h a t LINEAR PROGRAMMING i s i n P.

(4)

TOTAL UNIMODULARITY INSTANCE:

An m x n matrix

QUESTION:

Is

A with e n t r i e s from t h e s e t {o, 1, -1).

A =totally

unimodular, t h a t is, i s t h e r e a square

submatrix of A whose determinant i s = i n

the s e t

Again i n Garey-Johnson C79, p. 2 8 8 1 t h e s t a t u s o f t h i s problem i s open. One of t h e most important a p p l i c a t i o n s o f matroid t h e o r y i s Seymour's theorem

which s e t t l e s t h i s problem by showing i t s membership o f P.

We

devote Lecture 6 t o t h i s problem. Another problem which i s open i n Garey-Johnson and which h a s now been s e t t l e d by t h e matroid arguments o f Lovasz i s t h e SPANNING TREE PARITY problem, s e e Lecture 4 below. Of t h e open problems i n complexity theory a t t h e moment, one o f t h e most i n t r i g u i n g is: (5)

GRAPH ISOMORPHISM INSTANCE:

Two graphs G1 = ( V ,E ) and G2 = (V E 1 . 1 1 2' 2

QUESTION:

Are G and G2 isomorphic? 1 This i s a problem known t o be i n NP b u t n o t known t o be NP-complete o r t o be i n P o r i n co-NP.

16.

P-space We c l o s e t h i s b r i e f i n t r o d u c t i o n t o t h e t h e o r y o f P and NP by

mentioning some problems which appear t o be s i g n i f i c a n t l y h a r d e r though still solvable. A d e c i s i o n problem n can be computed i n polynomial space i f t h e r e

e x i s t s a Turing machine which d e c i d e s

I I

and which f o r any i n p u t x never

1

v i s i t s more than p ( x ( ) squares of working tape, where p i s some polynomial. I t i s obvious t h a t i f rr i s computable i n polynomial time it i s

computable i n polynomial space.

Thus Pc_ P-space.

I t i s not much more

d i f f i c u l t t o prove

For example consider our prototype problem of deciding whether o r not a graph has a hamiltonian c i r c u i t .

I f we a r e uninterested i n t h e

time of t h e computation b u t only i n its space requirements we can j u s t exhaustively t e s t a l l n! possible sequences of possible edge

sets

-

this

w i l l be enormously time consuming but of low polynomial space complexity.

In' 1973 Meyer and Stockmeyer i d e n t i f i e d a P-space complete problem which bears W same r e l a t i o n s h i p t o P-space a s SATISFIABILITY does 'to the c l a s s NP.

This problem i s known a s QUANTIFIED BOOLEAN FORMULAS (QBF) and

i s defined a s follows:

QBF INSTANCE:

A well formed quantified Boolean formula

F = (Q1 x1) (Q2 x2 )

...(Qn xn)E

where E i s a Boolean expression i n the variables xl,

...,xn

and each Q. i s e i t h e r "j" o r "r'. QUESTION:

Is F true?

Note t h a t SATISFIABILITY i s t h e case where each Qi i s "j". Since the appearance of QBF many other P-space complete problems have been discovered by the usual technique of exhibiting membership of P-space and then finding some way of showing r e d u c i b i l i t y t o QBF.

Many

of these examples have been of the following type involving games on graphs:-

VERTEX HEX INSTANCE:

Graph G = ( V , E) and two specified v e r t i c e s s , t .

QUESTION:

Does the player have a winning s t r a t e g y i n the following game on G.

Players 1 and 2 a l t e r n a t e l y choose a vertex

from ~ \ { s , t j ,with those chosen by player 1 being coloured "white", the other being coloured "black".

Play continues

u n t i l a l l such v e r t i c e s have been coloured and player 1 wins the game i f f there i s a path i n G from s t o t whick uses only white

vertices.

Even and Tarjan C761 have proved t h a t t h i s i s a problem which i s complete i n P-space.

For other examples of P-space complete problems and

an e x c e l l e n t accout of the theory b r i e f l y reviewed above we r e f e r t o Garey and Johnson C791.

Matroid Theory

2.

51.

Definitions F i r s t some b a s i c n o t a t i o n .

rank r (hence dimension r (AG(r

-

-

V(r,q) w i l l denote t h e v e c t o r space of

1) over t h e f i e l d GF(q) : PG(r

-

1,q)

w i l l denote t h e corresponding p r o j e c t i v e ( a f f i n e ) space.

1,q)

W e w i l l move f r e e l y between v e c t o r spacesand t h e corresponding p r o j e c t i v e

Thus o u r terminology w i l l vary from say a "subspace" t o t h e

space.

corresponding " f l a t " f o r e s s e n t i a l l y t h e same s e t o f p o i n t s depending on circumstances.

Much o f t h e time we s h a l l be working with t h e f i e l d

o f 2 elements where t h e d i f f e r e n c e between t h e geometry and corresponding v e c t o r space i s minimal. Our graph terminology i s standard, G w i l l denote throughout a graph w i t h v e r t e x s e t V and edge s e t E.

A n edge which j o i n s a v e r t e x t o

i t s e l f is c a l l e d a loop, two edges which have a common p a i r of endpoints w i l l be c a l l e d p a r a l l e l and a graph with no loops o r p a r a l l e l elements i s

c a l l e d simple.

The simple graph on n v e r t i c e s i n which each p a i r of

v e r t i c e s i s joined i s t h e complete graph Kn. edges el,

...,e t

cycle of

G is a s e t of

such t h a t ei and ei+l a r e i n c i d e n t 1 5 i i t and ei i s

n o t i n c i d e n t with e . , j # i

1

el.

A

+

1or i

-

1 except t h a t e t i s i n c i d e n t with

I n o t h e r words a c y c l e i s t h e s e t of edges of a simple closed path.

The d e l e t i o n of an edge e from a graph G g i v e s a graph Ge' vertex s e t .

The c o n t r a c t i o n of e from G gives a graph G

on t h e same "

which i s

o b t a i n e d from G by d e l e t i n g e and then i d e n t i f y i n g i t s two endpoints.

I f G and il a r e two graphs, G i s s a i d t o be c o n t r a c t i b l e t o H o r t o have

ti a s a s u b c o n t r a c t i o n i f we can o b t a i n H from G by an a p p r o p r i a t e

sequence of d e l e t i o n s and c o n t r a c t i o n s of edges.

A c u t s e t of G is a

s e t o f edges whose removal i n c r e a s e s t h e number o f connected components o f G.

A

cccycle i s a minimal c u t s e t .

A b r i d g e o r isthmus i s a c u t s e t of s i z e 1.

X u e f o r X u { e l and X\e to denote x\{e]. w i l l be denoted by

1x1.

f o r example i n Bondy A

3

&

We u s u a l l y w r i t e

The c a r d i n a l i t y of a s e t X

Any o t h e r graph terminology used can be found Murty [76!.

matroid i s a p a i r c o n s i s t i n g of a f i n i t e s e t S and a c o l l e c t i o n

of s u b s e t s o f S which a r e c a l l e d independent s e t s and s a t i s f y t h e

following axioms: (11) $

3

E

;

(12)

I f X i s independent and Y E X then Y i s independent;

(13)

If A

c_ S a l l maximal independent s u b s e t s o f

A

have t h e same

c a r d i n a l i t y which i s c a l l e d t h e rank o f A and i s denoted by

-

We u s u a l l y w r i t e M t o denote t h e matroid, i t s S, r ( S ) .

rank i s

t h e rank o f

A s e t i s dependent i f it i s n o t independent and i s a b a s e i f

it i s a maximal independent s u b s e t o f S.

Matroids Ml = (S1,

i s a 1-1 map 4'

:

independent i n M A

dl)

and M2 = (SZ,

E

a r e isomorphic i f t h e r e

S1 + S2 such t h a t X i s independent i n Ml*

YX i s

2-

s u b s e t F of S i s a f l a t o r

each element y

d 2)

S\F, P (F u Y ) > P (F)

a c l o s e d s e t o r a subspace i f f o r

.

A

hyperplane i s a maximal proper

f l a t , t h a t i s i f a matroid h a s rank r a hyperplane i s any f l a t of rank

r

-

1, a * i s

any f l a t of rank 2.

From analogy with graphs (see Example 3 below) we c a l l an element x o f S a loop of M on S i f r ( { x l ) = 0 and a s e t C i s a c i r c u i t i f it i s

minimal dependent.

Two elements a r e p a r a l l e l i f t h e i r union is a c i r c u i t .

The f l a t s o f a matroid, ordered by i n c l u s i o n , forma geometric lattice.

Every geometric l a t t i c e can be obtained from a matroid i n t h i s

way and t h e r e i s a n a t u r a l one-one correspondence between matroids and geometric l a t t i c e s , i n which t h e loops of t h e matroid a r e i d e n t i f i e d w i t h t h e zero element o f t h e l a t t i c e and s e t s o f mutually p a r a l l e l elements a r e i d e n t i f i e d with t h e

atoms (elements

o f rank one i n t h e

l a t t i c e ) i n e x a c t l y t h e same way a s p r o j e c t i v e spaces a r e obtained from v e c t o r spaces by i d e n t i f y i n g elements which a r e mutually l i n e a r l y dependent.

D u a l i t y and Minors Two c r u c i a l i d e a s i n matroid theory a r e those of d u a l i t y and o f t a k i n g minors. The f i r s t , d u a l i t y , corresponds t o o r t h o g o n a l i t y i n v e c t o r space t h e o r y b u t i s r e a l l y much simpler.

I f M i s a matroid on S i t s

dual

matroid i s t h e matroid on S which has a s i t s bases t h o s e s u b s e t s of S o f t h e form S\B, whre B i s a base of M.

I t i s denoted by M* and c l e a r l y

(M*) * = M*. If ~

E weS l e t M' denote t h e matroid on S\e whose independent s e t s

a r e those independent s e t s o f M which a r e contained i n S\e.

We c a l l t h i s

o p e r a t i o n t h e d e l e t i o n o f e from S, i t corresponds e x a c t l y t o removing an edge from a graph. The dual o p e r a t i o n , c o n t r a c t i n y e from S, g i v e s a matroid M" on S \ e whose independent s e t s a r e a l l s e t s X of S \ e with t h e property t h a t

Xu:

i s independent i n M , except i n t h e t r i v i a l case when e i s a loop of

M i r , which case M

" i s t h e same a s M

'.

Contracting i s t h e matroid o p e r a t i o n corresponding e x a c t l y t o c o n t r a c t i n g an

edge

from a graph and e q u i v a l e n t l y t o p r o j e c t i o n i n a

vector space The key f a c t s about t h e s e o p e r a t i o n s a r e :

(1) The c o n t r a c t i o n and d e l e t i o n o f elements a r e commutative matroid o p e r a t i o n s .

(2) Contraction and d e l e t i o n are dual o p e r a t i o n s i n t h e sense t h a t

F i n a l l y because o f (1) we can d e f i n e a

minor

of M t o be any matroid

o b t a i n a b l e from M by a s e r i e s o f c o n t r a c t i o n s and d e l e t i o n s , and w r i t e M > N t o denote t h a t N i s a minor o f M. For p r o o f s of a l l these statements and more on t h e theory of matroids we r e f e r t o t h e books by Bryant and P e r f e c t C801, Brylawski and Kelly CEO],

82.

Crapo and Rota 1701, Lawler [ 7 6 ] , T u t t e

C701 o r Welsh C76!.

Classes o f matroids I n t h i s s e c t i o n we b r i e f l y d e s c r i b e some d i f f e r e n t c l a s s o f matroids

which commonly a r i s e i n combinatorial a p p l i c a t i o n s . Represen t a b l e matroids L e t V be a v e c t o r space over GF(q), l e t S be any s u b s e t o f v e c t o r s i n V and l e t X c_ S be a member o f of v e c t o r s .

3

i f f X i s a l i n e a r l y independent s e t

Any matroid obtained i n this way i s c a l l e d r e p r e s e n t a b l e .

Not a l l matroids a r e r e p r e s e n t a b l e , f o r example t h e following 9-element matroid i s n o t r e p r e s e n t a b l e over any f i e l d . Example.

L e t S = {1,2,

...,91

and l e t

consist of a l l subsets

of cardinality

5

3 except t h o s e which a r e c o l l i n e a r i n Figure 1.

Those

f a m i l i a r w i t h p r o j e c t i v e geometry w i l l recoqnise Figure 1 a s t h e Pappus c o n f i g u r a t i o n with one l i n e missing.

Since Pappus' theorem must hold i n

any geometrical configuration c o o r d i n a t i s e d over a f i e l d , t h i s matroid cannot be embedded i n a v e c t o r space over any f i e l d .

Figure 1 Graphic matroids L e t G be a graph. X c o n t a i n s no cycle.

C a l l a s u b s e t X o f edges o f G independent i f

This g i v e s a matroid on E(G) c a l l e d t h e

o r polygon matroid of G, and i s denoted by M(G)

, and

CJC&

any matroid o b t a i n a b l e

t h i s way i s c a l l e d graphic. A l l such matroids a r e r e p r e s e n t a b l e over any f i e l d .

we show i n Figure 2 t h e equivalence and hence c y c l e matroid M(K ) 4

A s an example

c o o r d i n i s a t i o n of t h e

(Figure 2a) with a c o n f i g u r a t i o n of p o i n t s and l i n e s

i n 2 dimensional space, Figure 2b.

(a) Figure 2 Cographic Matroids An important c l a s s of dual matroids i s t h a t obtained from the

c l a s s of cycle matroids of graphs. Theorem 1.

Whitney proved:

I f M(G) i s the cycle matroid of a graph G then Mf(G) i s

the matroid on E ( G ) whose c i r c u i t s a r e the minimal c u t s e t s of G. G

i s a planar graph

i f f M*(G)

Moreover

i s a l s o the cycle matroid of some graph.

Any matroid obtainable i n t h i s way i s c a l l e d cographic. Since it i s straightforward t o prove t h a t the dual of a representable matroid i s a l s o representable (indeed over the same f i e l d ) then we know:

(1) Cographic matroids a r e representable over any f i e l d . Transversal Matroids One of the major applications of matroid theory has been i n obtaining new r e s u l t s i n transversal theory. of t h i s area we r e f e r t o Mirsky r711. Let S be a f i n i t e s e t and l e t of subsets of S.

A

partial

For a comprehensive survey

W e give the bare o u t l i n e s here.

A

= (Ai

transversal of

: i E I) be a f i n i t e c o l l e c t i o n

i s a s e t X such t h a t

there i s an i n j e c t i o n $ : X + I such t h a t f o r each X E X ,

and i f i # j => Q(x.1 # @(x.). 7

Theorem 2.

The c o l l e c t i o n o f p a r t i a l t r a n s v e r s a l s o f a family

s e t s form t h e independent s e t s o f a matroid

A

of

n(A).

Any matroid isomorphic to a matroid of t h e form collection

A

M(A)f o r

some

i s c a l l e d a t r a n s v e r s a l matroid.

Paving Matroids and S t e i n e r Systems A

matroid i s uniquely defined by its c o l l e c t i o n o f hyperplanes,

and axioms f o r a matroid i n terms o f i t s hyperplanes a r e t h e following.

8

A collection

o f s u b s e t s of S i s t h e s e t o f hyperplanes of a

matroid on S i f : (a)

No member of$

(b>

I f H1,

p r o p e r l y contains another.

Hz a r e d i s t i n c t members of

#

and x

E

H1 u Hz t h e r e

2 (Hl n H2) u x. 3 Using t h e s e axioms it is easy t o prove t h a t if A = (Ai exists H 3

E

$

such t h a t H

: i E I)

is a

family of s u b s e t s o f S such t h a t : (if

111 2 2 ;

(ii) Each member Ai o f A has c a r d i n a l a t l e a s t d ; (iii) Each d-element s u b s e t of S i s contained i n e x a c t l y one o f t h e

s e t s Ai; Then

A

forms t h e s e t o f hyperplanes o f a matroid; matroids

o b t a i n a b l e i n t h i s way a r e c a l l e d paving matroids. A p a r t i c u l a r l y i n t e r e s t i n g paving matroid i s t h e following. A

S t e i n e r system S ( d , k , n) i s a c o l l e c t i o n o f k-subsets c a l l e d

b l o c k s o f an n-set S such t h a t each d-subset of S i s contained i n a unique block. I t i s easy t o s e e t h a t t h e blocks of any S(d, k , n) form the

hyperplanes of a pavlnq matroid.

53.

S p e c i a l Matroids A few matroids w i l l keep on cropping up throughout t h e s e l e c t u r e s .

The s i m p l e s t type o f matroid i s t h e uniform matroid o f rank k on

n elements.

I t i s defined by t h e p r o p e r t y t h a t it i s on a groundset

of n elements and every k-subset o f t h e groundset i s a base.

We

denote t h i s matroid by U k,n' A uniform matroid o f p a r t i c u l a r importance i s t h e matroid U often called the 4 point line.

2,4

I t i s t h e s m a l l e s t matroid which i s n o t

b i n a r y , t h a t i s r e p r e s e n t a b l e o v e r t h e f i e l d of two elements.

Moreover

we h a w t h e basic theorem of T u t t e [65]. Theorem.

A

matroid i s b i n a r y i f and only i f it h a s no minor isomorphic

'2,4. The s m a l l e s t b i n a r y matroid which is n o t r e p r e s e n t a b l e over t h e r e a l s i s t h e Fano matroid which we denote by F7.

It i s t h e s e w n p o i n t

matroid o f rank 3 corresponding t o t h e p r o j e c t i v e p l a n e PG(2,Z) . I t has a b i n a r y r e p r e s e n t a t i o n by t h e

I t s dual matroid F;,

columns o f t h e m a t r i x below.

sometimes c a l l e d t h e

heptahedron, h a s rank 4 and

h a s t h e matrix r e p r e s e n t a t i o n shown over GF(2).

I n general we s h a l l w r i t e M(G) t o denote the

cycle or polygon

matroid of a graph G and M*(G) t o denote the cocycle o r bond matroid of G. However when the graph G i s not planar so t h a t there i s no possible source of confusion we may write G o r G* t o denote respectively M(G) o r M(G*)

.

I n p a r t i c u l a r K5 w i l l be used t o denote the graph K5 and i t s equivalent geometric form,

the three dimensional Desargues' configuration.

I t i s a useful exercise t o c a r r y out this i d e n t i f i c a t i o n .

An i n s t a n t

c o r o l l a r y i s t h a t , the Desargues configuration has automorphism group isomorphic to S5. Now take the Desargues configuration and j o i n by a l i n e each p a i r of p o i n t s which a r e not c o l l i n e a r i n it.

The r e s u l t i n g configuration of

10 p o i n t s and 15 l i n e s w i l l form the Petersen graph PlO.

Its cocyle

matroid occurs frequently i n l e c t u r e s 7 and 8 and i s denoted by P* 10

'

The b i p a r t i t e graph K i n matroid theory.

i s another graph whose matroids recur 3.3 Again we o f t e n write K;I t o denote M* (K3#3)

.

A l l the matroids we have described above have been representable

over some f i e l d , graphic and cographic matroids a r e i n f a c t representable over every f i e l d .

An example of a matroid which i s representable over

no f i e l d i n t h e following matroid which i s sometimes d e s c r i b e d a s t h e Vamos matroid, V8.

I t h a s e i g h t elements a,b,c,d,

1,2,3,4 and rank 4

and i t s b a s e s a r e a l l 4 - s e t s o f t h e groundset except t h e set defined a s follows:

Then

d

2 of

'lines'

Take

is the s e t a u B , a u y , a u 6 , Buy, Bu6, but e y v 6 .

The reason why it i s n o t r e p r e s e n t a b l e over any f i e l d i s e a s y t o s e e . V

8

can be ' r e p r e s e n t e d ' i n 3 dimensional Euclidean space a s follows.

By p o s t u l a t i n g a

u 6 e t c . t o be members o f

a , b 1 , 2 e t c . a r e coplanar. V

8

a r e demanding t h a t

However i f we have a ' t r u e '

embedding o f

i n Euclidean space we have made s o many s e t s o f p o i n t s coplanar t h a t

we would f o r c e c , d , 3,4 t o be coplanar This i s n o t t h e case.

54.

8 we

-

however s i n c e we demand y u 6 4

$

Hence V8 i s n o t r e p r e s e n t a b l e .

Excluded minor theorems W e have already seen one excluded minor c o n d i t i o n i n

53 where we

noted Tut.tels theorem c h a r a c t e r i s i n g b i n a r y matroids a s t h o s e which have no minor isomorphic t o U 2 , 4 . theorems o f t h i s type.

Much o f matroid theory i s concerned with

The p r o t o t y p e

o f t h e s e theorems i s t h e one

quoted above, T u t t e ' s o t h e r b i g theorems a r e a l s o o f this type and more r e c e n t l y we have had o t h e r excluded minor type c o n d i t i o n s found by Reid [ 7 0 3 , Bixby [79] and notably Seymour [771, [793, r80! and r811

.

We summarise t h e s e r e s u l t s below. (1)

M i s binary i f f M

UZ,4.

(2) M i s r e p r e s e n t a b l e over GF (3) i f f M )' U2 ,5, U3,5, Following Walton and Welsh C801 we d e f i n e Ex(M1,M2,.

F7 o r F;

..,\)

be t h e c l a s s of binary matroids w i t h no minor isomorphic t o Mi,

.

to

1 s i 5 k.

Then we have: (3)

M is regular, M E

(4) For ME

-9 *

.

r e p r e s e n t a b l e over every f i e l d ) i f f

EX(F~,F;).

M i s graphic i f f M E EX(F~,F$,K;,K;,~)

$

any c l a s s o f matroids

o ME

(5)

( i .e

. Clearly

3

7*

.

i s t h e d u a l c l a s s defined by

t h e r e f o r e we can d u a l i s e (4) t o g e t

M i s cographic i f f M

6

.

Ex(F~,F;,K~,K~,~)

We s h a l l make e x t e n s i v e use o f t h e s e c r u c i a l l y important theorems below, p a r t i c u l a r l y i n l e c t u r e s 6 , 7 and 8.

85.

Matroid polyhedra I n s t e a d o f axiornatising a matroid by i t s independent s e t axioms

w e could habe defined it as a p a i r (S,r) where r : 2' f o r a l l A,B

c_ S;

(1)

r(+) = 0

(2)

A z B s

(3)

r(A)

(4)

r{x) 5 1

+

r(A)

5

r(B),

r(B) 2 r ( A u B ) + r ( A n B ) , xcS.

+ ? !+ ,

satisfies:

Here o f course r i s t h e rank function. ( 4 ) and have a function

u

I f we drop t h e c o n s t r a i n t

j u s t ( 1 1 , (2) and ( 3 ) we c a l l

satisfying

(S,p) an i n t e g e r polymatroid, and i f we allow

u

t o t a k e non-integer

values we have j u s t a p o l y m a t r o ~ d . p i s c a l l e d t h e ground s e t rank function.

P i n R;

by t h e 2n

+ n i n e q u a l i t i e s (n

=

IS 1)

I t d e f i n e s a polyhedron

o f t h e form:

we c a l l P t h e independence polytope o f t h e polymatroid. r ( + ) of a vector

_a cRs

The v e c t o r rank

i s then given by

We o f t e n w r i t e IP = (S ,P,u) t o i n d i c a t e t h a t P has ground s e t rank f u n c t i o n LI and independence polytope P. An a l t e r n a t i v e d e f i n i t i o n o f a polymatroid i s a s follows.

Definition 2.

A

polymatroid i s a p a i r (S,P) where S , t h e ground s e t

i s a non empty f i n i t e s e t and P, t h e s e t of independent v e c t o r s i n P i s a

non cnpty compact s u b s e t o f IR;

i n t h e space IRS such t h a t

(a)

every subvector o f an independent v e c t o r i s independent

(b)

f o r every v e c t o r

x of a

5 i n IR;,

every maximal independent subvector

has t h e same modulus,

r ( a ) , the

v e c t o r rank o f

5

inP

.

I t i s n o t t h a t d i f f i c u l t t o show t h e equivalence o f t h e two above

d e f i n i t i o n s , d e t a i l s a r e given i n [welsh 76, Chapter 181. The two c r u c i a l p r o p e r t i e s of polymatroids a r e t h e following r e s u l t s of Edmonds C701.

Theorem 1.

I f P i s t h e independence polytope o f an i n t e g e r polymatroid

a l l t h e v e r t i c e s of P a r e i n t e g e r valued. Proof.

(Sketch)

Consider any v e r t e x v o f P; choose a l i n e a r function

which achieves i t s maximum over P ( s e e 93.1)

a t v.

By t h e greedy algorithm

t h e r e i s an i n t e g r a l s o l u t i o n which i s optimum.

Since v i s

t h e o n l y s o l u t i o n v must be i n t e g r a l . Theorem 2.

I f PI, P2 a r e t h e independence polytopes o f two i n t e g e r

polymatroids then a l l t h e v e r t i c e s o f P nP2 a r e i n t e g r a l . 1

Proof.

Much harder:

see Lawler C761.

3.

51.

-reedy

Matroids and Algorithms

algorithm

I n 1957 R. Rado r e a l i s e d t h a t a w e l l known and simple minded algorithm f o r f i n d i n g a spanning t r e e o f a graph which had a maximum weight over a l l weighted t r e e s coulg b e g e n e r a l i s e d t o g i v e an algorithm which would f i n d a base o f maximum ( o r minimum) weight i n any matroid. Indeed more can be s a i d :

w

:

S + R+ i s a weight function.

w(A) =

Let

3

C w(x) xeA

.su_Dpose t h a t S i s a f i n i t e s e t and Extend w (AES)

:

2

S

+R

+ by l e t t i n g

.

b e any c o l l e c t i o n o f s u b s e t s o f S which s a t i s f i e s t h e c o n d i t i o n s (i) $

s

(ii) A

E

3 BE^

~ B ,E A J

-

Define problem n ( 3,w) t o b e t h e problem o f f i n d i n g a member o f

a

o f maximum weight. For a given such

3

and w l e t X ( 3 ,w) be t h e s u b s e t o f S obtained

by t h e following procedure: (1) S e l e c t element x

E

S such t h a t {XI6

3

and such t h a t w{x) i s

a maximum over a l l such x. (2)

Let X = {XI

(3)

S e l e c t y s S\X such t h a t X v {y]

E

3

and such t h a t o f a l l such

y, w(Xuy) i s a maximum. (4) Let X = X

I f no such y e x i s t s stop.

u {y] and r e t u r n t o (3) .

I t is n o t d i f f i c u l t t o prove

Theorem 1.

If

3

i s t h e c o l l e c t i o n of independent s e t s o f a matroid

t h e n f o r any non negative w, X($,w) if

i s a n optimum

set.

Conversely

,w) is a n optimum s e t f o r a l l p o s s i b l e c h o i c e s o f w 2 0, then

x($

3

is t h e c o l l e c t i o n o f independent s e t s o f a matroid. Proof. -

Straightforward s e e Welsh [76, Chapter 192. The procedure o f g e t t i n g t h e s e t X ( 3 ,w) above h a s been a p t l y

c h r i s t e n e d t h e greedy algorithm, and a n appealing axiomatisation of matroids i s " t h a t they a r e t h e only s t r u c t u r e s f o r which t h e greedy a l g o r i t h m works f o r a l l choices of non-negative weight function".

§ 2.

The union o f matroids L e t M. (1Si5k) be matroids on t h e s e t s Si(lSiSk) which may o r may

n o t be d i s j o i n t . The S = S1u

of t h e M.

denoted by

.l'

...usk

i s t h e matroid on

b!lV...VYI(

i n which a s e t X i s independent i f and only i f

X = X

u...

1

where X. is independent i n MIV

(1)

...V

5

n..

I f ri i s t h e rank f u n c t i o n o f Mi,

then

h a s rank function, defined f o r a l l AZS by r ( A ) = min ( r l X + . XCA

.+rkX +

IA\xI)

-

T h i s c o n s t r u c t i o n i n matroid theory has many a p p l i c a t i o n s , s e e f o r example

t h e a r t i c l e by A. Recski

i n t h i s volume.

I t is a l s o important because it can be proved by perhaps t h e f i r s t

example o f a 'good' b u t n o n - t r i v i a l algorithm i n matroid theory.

More

p r e c i s e l y suppose t h a t i n t h e above s i t u a t i o n we wish t o know whether o r not t h e ground s e t S can be p a r t i t i o n e d i n t o sets 11, 1

5

j

k , I . i s independent i n M.. 3 3

5

...,\

such t h a t f o r

Edmonds r651 h a s produced a n algorithm

which does t h i s i n polynomial time provided t h a t whether o r not'a s e t X

i s independent i n M

j

can b e decided i n polynomial time.

This algorithm

known a s t h e matroid p a r t i t i o n i n g alggrithm, produces c o n s t r u c t i v e p r o o f s o f t h e r e s u l t (1) and h a s a n i n t e r e s t i n g 'dual algorithm' c a l l e d t h e i n t e r s e c t i o n algorithm. Consider f i r s t t h e following well known theorem, s e e Welsh

i76, Chapter 81. Theorem 1. wlth

1x1

If M

1

and M2 a r e matroids on S then t h e r e e x i s t s a set X

2 k and which i s independent i n both M1

and M2 i f and only i f

for a l l AES, rl (A)

+ r2(S\A)

I t is not d i f f i c u l t

theory.

2 k.

t o deduce t h i s r e s u l t from (1) by d u a l i t y

Correspondingly, provided t h e r e i s a f a s t a l g o r i t h m f o r

recognising independence i n both M1 and M2 t h e r e i s a ' f a s t ' ,

t h a t is

p o l y n o d a l , algorithm f o r f i n d i n g a s e t X o f maximum weight w(X) and which i s independent i n both M1

and M2.

This algorithm, known a s t h e

'matroid i n t e r s e c t i o n algorithm' i s w e l l described i n Lawler [ 7 6 !

53.

The Shannon switching game A very nice a p p l i c a t i o n of t h e p a r t i t i o n i n g algorithm 1s t h a t i t

g i v e s a polynomial algorithm f o r t h e following game, commonly c a l l e d t h e Shannon-switching game b u t which we s h a l l c a l l EDGE I n 51.6 we defined t h e game VERTEX

-

-

HEX.

HEX i n which two p l a y e r s

a l t e r n a t e l y colour v e r t i c e s o f a graph i n an attempt t o

join

(or

cut)

two s p e c i f i e d v e r t i c e s .

EDGE

-

HEX i s e x a c t l y t h e same game except t h a t i n s t e a d of

c o l o u r i n g v e r t i c e s t h e p l a y e r s a l t e r n a t e l y colour edges o f G. C a l l a graph G (and a p a i r o f s p e c i f i e d v e r t i c e s ) a j o i n graph i f t h e r e i s a winning s t r a t e g y f o r t h e j o i n player whether o r not h e goes first.

S i m i l a r l y G i s a c u t graph i f t h e c u t p l a y e r can win a g a i n s t a l l

p o s s i b l e s t r a t e g i e s of t h e j o i n p l a y e r , no matter who goes f i r s t .

Finally

G i s a n e u t r a l graph i f t h e p l a y e r who goes f i r s t can win a g a i n s t a l l

p o s s i b l e s t r a t e g i e s of t h e o t h e r p l a y e r .

Mutually e x c l u s i v e p o s s i b i l i t i e s

are i l l u s t r a t e d i n t h e example of Figure 1.

(a)

n e u t r a l graph

(b)

j o i n graph

(c)

c u t graph

Figure 1 Lehman r641 c h a r a c t e r i s e d j o i n graphs i n t h e following theorem: Theorem 1.

The game ( G , u , v ) i s a join game i f and only i f t h e r e e x i s t

two edge d i s j o i n t t r e e s T1, T2 on t h e same s u b s e t o f v e r t i c e s V(T1) = V ( T 2 ) E V ( G ) such t h a t { u , v ) c V ( T i ) .

S i m i l a r 'derived'

theorems c h a r a c t e r i s e c u t and n e u t r a l graphs.

More importantly, i t i s reasonably s t r a i g h t f o r w a r d t o apply t h e p a r t i t i o n i n g algorithm of t h e l a s t s e c t i o n t o give a polynomial algorithm which d e c i d e s whether o r not a graph G i s c u t , n e u t r a l o r j o i n .

In

o t h e r words we have t h e dichotomy when comparing w i t h 51.6. (1) Deciding whether t h e f i r s t player can win VERTEX HEX i s complete

i n P-space b u t t h e same problem f o r EDGE HEX c a n be done i n polynomial time.

54.

A polynomial algorithm f o r k-connectivity

A matroid M on S i s connected i f t h e r e e x i s t s no proper subset A of S w i t h

I t has been known f o r some time t h a t t h e r e e x i s t s polynomial algorithms t o

t e s t whether a matroid i s connected o r n o t ( s e e f o r example Cunningham C731). More r e c e n t l y Cunningham and Edmonds (1979) have announced a polynomial algorithm t o t e s t f o r a given matroid M whether o r n o t it has a k-separation,

where both

that is a s e t A

I A ~ and I S \ A (

c_ S,

such t h a t

a r e n o t l e s s than k .

A s we s h a l l see i n 56.4 i n connection w i t h Seymour's decomposition

theorem f o r unimodular m a t r i c e s we w i l l need t o decide whether o r n o t a given matroid has such a s e p a r a t i o n .

The following method shows t h e

e x i s t e n c e of such a polynomial algorithm provided independence i n t h e matroid can be checked i n polynomial tlme.

By t h e matroid i n t e r s e c t i o n theorem, two matroids M1,

rank f u n c t i o n s r

M2 w i t h

r2 have a common independent s e t of s i z e t i f f V Y c ~ ,

For t h e given matroid M on S consider

Then M. on T = S \ ( X

1

u

X ) h a s rank function ri,

r (U) = r ( U 1

2

U

X2)

r (U) = r ( U u X1) 2

Using ( 2 ) , M1,

-

r (XZ)

-

r (X1)

for U

5 T,

given by

.

M 2 have a common independent s e t o f s i z e t i f and o n l y i f

f o r a l l UET,

T h i s proves

t h e v a l i d i t y o f t h e following algorithm:

Algorithm (1)

Select p a i r s of d i s j o i n t k-sets

(2)

For each p a i r ( X I , i f MI,

(XI,

X2) i n t u r n .

X ) use t h e i n t e r s e c t i o n algorithm t o decide

2

M2 above have a common independent s e t o f s i z e t where t i s

chosen s o t h a t

(3)

I f no such independent s e t e x i s t s then t h e corresponding (X1' X2) induces a k-separation of M, otherwise M h a s no k-separation.

Clearly t h i s i s polynomial s i n c e the i n t e r s e c t i o n algorithm i s polynomial and the number o f ways of picking the p a i r (XI, X2) i s 2k O(n 1 . For more on connectivity s e e 56.2 below and a l s o the a r t i c l e by J.G.

Oxley i n t h i s volume.

"

4. LovaszlsAttack on t h e P a r i t y Problem-

01.

The p a r i t y problem The c l a s s i c a l p a r i t y problem seems t o have o r i g i n a t e d with E.

Lawler i n (1971) when it was defined a s follows. Given a matroid M on S and a p a r t i t i o n i n g o f S (taken t o be a s e t o f even c a r d i n a l i t y 2n) i n t o blocks o f s i z e 2 , say

f i n d a n algorithm f o r deciding whether o r n o t M h a s an independent s e t X o f caqdinal t

e. E X .

with t h e property t h a t e . E X i f and only i f i t s mate

Such a s e t i s c a l l e d an independent p a r i t y s e t . The above i s c a l l e d t h e 2-parity problem to d i s t i n g u i s h it from t h e

k-parity

problem i n which we t a k e IS

b l o c k s Ei,

where IE. 1 = k , 1 5 i

5

I

= kn, and we p a r t i t i o n S i n t o d i s j o i n t

n, and we wish t o decide whether M h a s

an independent s e t X , o f c a r d i n a l i t y t such t h a t i f XnEi # We f i r s t remark t h a t t h e 'k-matroid s p e c i a l case o f t h e Proof. -

L e t Mi,

15 i

0 then

XzEi.

i n t e r s e c t i o n problem' i s a

' k - p a r i t y problem'.

<

k , be matroids on S .

Take S1,

...,Sk

t o be d i s j o i n t

c o p i e s o f S and l e t M I be an isomorphic copy o f M . on t h e s e t Si.

Let

Define t h e k-parity s e t s f o r S' = Slu-..USk 1 a given block i s {ei,.

- , ek. 1

i n t h e obvious way:-

where e 3 i s t h e copy o f e . i

E

S i n the s e t S

j'

I t i s e a s y t o see t h a t t h e r e i s a 1-1 correspondence between k - p a r i t y s e t s

of N and i n t e r s e c t i o n s o f t h e k matroids Ml,

...,%.

0

Hence s i n c e t h e k-matroid i n t e r s e c t i o n problem reduces i n a s p e c i a l c a s e t o d e c i d i n g whether k f a m i l i e s o f s e t s have a common t r a n s v e r s a l , and t h i s

is NP-complete,

we have shown:

For k 2 3, t h e k - p a r i t y problem i s NP-hard.

(1)

However f o r k = 2 t h e s i t u a t i o n i s d i f f e r e n t . F i r s t n o t e t h a t t h e 2-parity problem cannot be 'extremely'

easy

s i n c e it c e r t a i n l y includes 2-matroid i n t e r s e c t i o n , which i s n o t t r i v i a l . I t a l s o i n c l u d e s t h e problem o f f i n d i n g a maximum matching i n a graph. A matching i n a graph G = (V,E)

i s a s u b s e t F E E such t h a t no two

members o f F a r e i n c i d e n t w i t h a c o m n vertex.

Finding a maximum

matching.in a graph i s a h i g h l y n o n - t r i v i a l problem b u t it does have a polynomial algorithm (Edmonds (65)). I t i s n o t d i f f i c u l t t o prove:

(2)

Proof.

MAXIMUM MATCHING

Replace each edge e

v e r t e x between them.

i

a

2-PARITY.

o f G by a p a i r o f edges f .

1

, Z .1 w i t h a

Let G' = (V' , E m )be t h e graph s o obtained.

new If

i s t h e c o l l e c t i o n o f s e t s X 5 E ' such t h a t no two edges o f X a r e i n c i d e n t with t h e same v e r t e x o f G ' ,

u n l e s s it i s one of t h e new v e r t i c e s c r e a t e d

by s u b d i v i s i o n , then i t i s n o t d i f f i c u l t t o v e r i f y t h a t o f independent s e t s of a matroid M on E'. with f i and

f.

3 is

the collection

The 2 - p a r i t y problem f o r M

a s mates s o l v e s t h e maximum matching problem f o r G.

0

We have already met polymatroids i n 52.4.

Here we concentrate

on a s p e c i a l c l a s s of polymatroids, c a l l e d by ~ o v z s zC801, 2-polymatroids. Recall t h a t a polymatroid can be defined a s a p a i r (S,u) where

u

s

: 2

+ R+

satisfies

(1)

u(+)

(2)

AEB*

u(A)

(3)

u (A) +

u(B) 2

= 0,

< u(B),

u (A u B) +

IJ (A

n B)

.

The p a i r (S,u) i s a 2-polymatroid i f i n a d d i t i o n we r e s t r i c t

u

to

t a k i n g only i n t e g e r v a l u e s and t o s a t i s f y t h e c o n s t r a i n t 44)

u ({XI)

= 2

f o r a l l x c S.

Note F i r s t t h a t a n immediate consequence o f ( 3 1 , (4) is:

(5) For a l l X E S , p(X)

5 21x1.

F i r s t some examples : Example 1.

I f M1 and M2 on S a r e matroids with rank functions rl and r

2

t h e n ( S , r + r ) i s a 2-polymatroid. 1 2 Example 2.

I f S i s any c o l l e c t i o n o f l i n e s ( = f l a t s o f rank 2) i n a

matroid and we define p{R1,R2,.

.. ,Lk

= r{R1u..

.uRk}

where r i s t h e rank f u n c t i o n o f t h e matroid then (S,U) i s a 2-polymatroid. Example 3 .

I f G = (V,E) i s a graph with

no loops and f o r XEE we d e f i n e

p(X) t o be t h e number o f v e r t i c e s o f G which a r e i n c i d e n t with X then (E,u) i s a 2-polymatroid.

A

s e t Xc_ S i s a matching i n t h e 2-polymatroid

(S,u) i f

and t h e c a r d i n a l i t y o f a maximum matching we denote by v (S ,u) t h e r e i s no

,or

when

ambiguity by v ( S ) .

I n t u i t i v e l y we may t h i n k o f v(S,u) a s t h e ' c a r d i n a l i t y o f t h e l a r g e s t independent s e t i n t h e polymatroid'. Thus i n t h e examples above we have: Example

1.

v ( S , r + r ) i s t h e maxinlum c a r d i n a l i t y o f a s e t 1 2

i n both M1 and M Example 3.

independent

2-

v (E(G) ,!.I)

i s t h e (maximum) c a r d i n a l i t y o f a s e t o f p a i r w i s e

d i s j o i n t edges, o f t e n denoted by a ( G ) . The r e l a t i o n s h i p with t h e 2-parity problem i s a consequence o f t h e following p r o p o s i t i o n . Proposition.

Finding a maximum 2-parity s e t i n a matroid i s no harder

than f i n d i n g a maximum matching i n a 2-polymatroid. Proof. -

We assume M i s simple on S = S u S where 1 2

are disjoint sets. L e t ti (1si.n) ei,ei.

be t h e l i n e o f t h e matroid M which c o n t a i n s t h e p a i r

Let L = {lll,...,fi

n

1

and l e t (L,u) be t h e 2-polymatroid defined by

where r i s t h e rank function o f M.

It is easy t o v e r i f y t h a t t h i s i n f a c t

does g i v e a 2-polymatroid.

Moreover {El,.

..,!.k 1

i s a matching of s i z e

k i n (L,y) i f and only i f

which i s t h e c a s e i f and only i f { e l l ~ l , e k , . - - , e k , ~ k i~s an independent p a r i t y s e t o f s i z e 2k i n M.

5 3.

0

The ~ a l l a i - ~ o v a ) sIzd e n t i t y Consider Example 3 o f t h e previous s e c t i o n s .

A well known i d e n t i t y

o f graph theory due t o G a l l a i C611 r e l a t e s c r ( G ) , t h e maximum number o f d i s j o i n t edges with B ( G ) , t h e minimum number o f edges needed t o cover a l l t h e v e r t i c e s o f G by

Suppose we denote by P (S) = P (S,u) set X

cS

such t h a t p(X) =

u (S) .

c a r d i n a l i t y of a spanning s e t ' .

t h ~minimum c a r d i n a l i t y o f a

I n o t h e r words, p (S) i s ' t h e minimum Then we can prove t h e following r e s u l t

o f Lovasz CEO]. Theorem 1.

I n any 2-polymatroid (S,u1

,

T h i s obviously g i v e s G a l l a i ' s Theorem C611 a s a s p e c i a l case. When y i s t h e sum of rank functions of two matroids M1 and M2, Example 1 above, it r e l a t e s t h e maximum s i z e o f a common independent i n t h e two matroids with t h e minimum s i z e o f a s e t spanning i n both matroids.

I n o r d e r t o prove Theorem 1 we need t h e following lemma.

lemma.

Let (S,v) be a 2-polymatroid and l e t A be a maximum matching.

5

+

:' 2

then ( S , U

If

Z i s defined by

A

)

i s a matroid.

Proof. -

~ ~ ( = $ 0. 1

so t h a t

u is

uAIx} = "Au

submodular, and c l e a r l y i n c r e a s i n g . [XI)

maximum matching

-

p(A), s o t h a t i f X E A , pA{x) = 0 while s i n c e A i s a

uA { X I 5

Proof of Theorem 1.

Moreover

1 when

XPA.

0

Let (S,p) b e a 2-polymatroid and l e t T b e a s u b s e t

.

of S o f minimum c a r d i n a l i t y such t h a t p (T) = p (S)

L e t A be a maximum matching i n T, t h a t i s A i s a s e t o f maximum c a r d i n a l i t y which i s contained i n T and f o r which

Let X = T \ A ,

so that

Thus using t h e lemma

Hence we have

On t h e o t h e r hand, suppose t h a t A i s a maximum matching i n ( S , p ) and suppose t h a t X i s a b a s i s o f t h e matroid ( S , p A ) .

u (A U X)

= 1.1 (A)

+

pA (x)

= U(A1

+

pA(S\A)

Then

= u(S)

s i n c e a l l t h e elements o f A a r e loops i n ( S , u A )

.

Hence p ( S )

S

IA U X I .

and this with (2) completes t h e proof.

Thus

0

We c l o s e t h i s s e c t i o n by remarking t h a t it i s easy t o f i n d examples t o show t h a t Theorem 1 f a i l s when ( S , p ) i s a polymatroid b u t n o t a 2-polymatroid.

54.

A min-max theorem f o r l i n e a r 2-polymatroids

Suppose t h a t P i s a p r o j e c t i v e space. o f P and l e t S = { A ~ , -.., A

-..,A

Let A ~ ,

be s u b s e t s

1- We can d e f i n e a polymatroid (S,y)

on S by d e f i n i n g

where r is t h e rank f u n c t i o n o f t h e underlying s p a c e P .

Any polymatroid

obtained i n t h i s way we c a l l a l i n e a r polymatroid, and when t h e A. a r e a l l l i n e s i n P we have a l i n e a r 2-polymktroid. Lovdsz [80a? proved t h e following remarkable r e s u l t . Theorem 1. L e t (S,!.I) p r o j e c t i v e space P.

be a l i n e a r 2-polymatroid formed by subspaces o f a Then

where A ranges over subspaces o f P and {S1,S2,-. .,S

k

ranges over a l l

p a r t i t i o n s o f S, and where

The proof of t h i s theorem i s d i f f i c u l t , we can do no more than sketch t h e main i d e a . F i r s t , however, we 'describe an e q u i v a l e n t

formulation.

This was

t h e o r i g i n a l geometrical r e s u l t proved by Lovasz i n (1978). Let subfamily$f

86

be a s e t of subspaces o f a p r o j e c t i v e geametry . A

d(

i s c a l l e d independent i f no member o f

3 intersects

the

f l a t spanned by t h e o t h e r members. Theorem 2.

~ e t &be a s e t of l i n e s i n a p r o j e c t i v e space P.

maximum number v ( b C ) o f independent l i n e s i n

fi

Then t h e

i s equal t o t h e minimum

value o f

where A,A1,.

..,%

each l i n e i n

cl(

a r e f l a t s of P such t h a t A :Ai

(i=1,2,.

..,k)

and

which does n o t i n t e r s e c t A i s contained i n some Ai,

i s t h e rank function of t h e underlying p r o j e c t i v e space P

and r

.

We l e a v e it a s a ( n o t d i f f i c u l t ) e x e r c i s e f o r t h e reader t o show t h e equivalence of Theorems 1 and 2. A s p a r t of t h i s we s t a t e without proof:

(1)

Let

3

and only i f

be a s e t o f l i n e s i n P, then

3

r ( 31

(:

213

I

with e q u a l i t y i f

i s an independent s e t of l i n e s .

Sketch Proof o f Lovasz's Theorem 2 F i r s t we show t h a t i f A,Al,

...,%

3

a r e subspaces such t h a t A F A

A o r i s contained i n some Ai,

Let i n Ai

i s any s e t of independent l i n e s and

gi

and

i

and each l i n e of

3

e i t h e r meets

then

% denote t h e s e t of

l i n e s of

3

which a r e contained

and which meet A r e s p e c t i v e l y . L e t A;

be t h e subspace of P spanned by

3; ,$ \ 3 O.

L e t A' be the subspace s p a n n e d by 2 A;

T2\

Sf = Ti\

be the subspace s p a n n e d by

30a n d i n g e n e r a l

Tv2\...

\

let

SO .

Then

a n d m o r e generally

and m o r e o v e r the spaces A;

are clearly i n d e p e n d e n t and hence so are t h e

spaces A! n A , 0

Thus

5

i

5 k.

But r ( A ! n A) = r (A;)

..

+ r (A)

r(Ai)

-r(A)

r(Ai)

-

A

s a n d u s i n g integrality.

2

- r (A; +-

u A)

r(A;nA) 2

r(A)

+ r(A; n A),

which proves one h a l f o f t h e i d e n t i t y . We now prove t h e converse by induction on v ( ? )

.

Case 1. There e x i s t s a p o i n t p i n t h e p r o j e c t i v e space P such t h a t p i s contained i n t h e span of each c o l l e c t i o n of v ( 2 ) independent l i n e s .

P r o j e c t from p onto a hyperplane o f P not going though p , ( t h a t i s

. This gives a collection 1 3(1( 5 ~ ( 3- )1 f o r suppose t h a t

c o n t r a c t p out o f t h e underlying matroid) o f independent l i n e s .

We a s s e r t

Then t h e o r i g i n a l l i n e s i n

3 must

have formed a s e t of

~ ( 3independent )

lines. ~ u pt e span ( f - l ( T 1 ) ) . Hencep and t h e s e t of l i n e s

yl

a r e i n a space o f rank 2 v ( 3 )

3, a r e contained i n a space of

the lines

rank 2v(

3) -

.

Thus

1.

But it i s impossible t o have v ( 3 ) independent l i n e s i n a subspace of rank 2 v ( F ) Hence

-

1.

~ ( 35 v~( J )) -

1.

Hence by t h e i n d u c t i o n hypothesis t h e r e e x i s t f l a t s A 1 , A i , t h i s hyperplane H such t h a t each l i n e of contained i n some A,'

I'

A ' EA;

zl n o t i n t e r s e c t i n g

...,%

A' i s

, and

Now t a k e A t o be the f l a t spanned by A ' and p and A i t o be t h e f l a t spanned by A ! and p f o r 1 5 i f l a t s i n P.

5 k,

ancl we o b t a i n t h e required s e t of

in

For c l e a r l y t h e r i g h t hand s i d e of (*I replace A ' ,

A! by A and A

i

respectively.

i s i n c r e a s e d by 1 i f we

Moreover i f a l i n e L does n o t

i n t e r s e c t A i t s p r o j e c t i o n L ' does n o t i n t e r s e c t A' and hence L ' i s contained i n some A! which i m p l i e s II i s contained i n A . Case 2. in

For each p o i n t p tlP t h e r e e x i s t s a s e t of

3 whose

V(

.

3) independent

lines

span does n o t i n c l u d e p .

This i s t h e h a r d p a r t of t h e p r o o f , and we can only make a few remarks t o i l l u s t r a t e the d i f f i c u l t y . a)

Any A achieving t h e minimum on t h e r i g h t hand s i d e o f t h e equation must be empty f o r otherwise it can be shown t h a t a p e x i s t s f o r which Case 1 a p p l i e s .

b)

Because of ( a ) we need t o show t h a t t h e r e e x i s t p a i r w i s e d i s j o i n t f l a t s Ale..

.,%

i n P such t h a t each l i n e o f

contained i n one of t h e s e Ai

where r ( A ) = 2V. i

+

i

55.

is

and moreover

1, 1 5 i 5 k.

T h i s i m p l i e s t h a t each c o l l e c t i o n o f

v

3

l i n e s which a r e contained i n Ai

v(3)independent

lines i n

3 has

.

On a polynomial algorithm f o r t h e 2-parity problem Consider now t h e a l g o r i t h m i c problem.

nl

: INSTANCE:

QUESTION:

A 2-polymatroid

(S,u)

and an i n t e g e r t.

Does (S,u) have a matching of c a r d i n a l i t y 2 t ?

We a r e i n t e r e s t e d i n t h e q u e s t i o n whether o r not vl h a s a polynomial algorithm s u b j e c t t o t h e p r o v i s o t h a t we can o b t a i n t h e rank U X o f an

a r b i t r a r y s e t X i n time which i s a polynomial

function of

By t a k i n g t h e s p e c i a l c a s e s of p discussed would be a polyncmial

1x1.

i n 92 any such algorithm

algorithm f o r t h e graph matching problem and

t h e matroid i n t e r s e c t i o n problem, and hence i s u n l i k e l y t o be t r i v i a l . Indeed Garey and Johnson r79, p. 2871 pose a s an open problem t h a t of d e c i d i n g whether o r not t h e following problem r Z i s NP-complete. n2 : INSTANCE:

Graph G = (V,E) and a p a r t i t i o n o f E i n t o d i s j o i n t 2-element s e t s El,.

QUESTION:

..,Em .

Is t h e r e a spanning t r e e T = (V,E1) f o r G such t h a t f o r each Ei,

1 < i

I.

m, e i t h e r E i c E 1 o r E. nE' = $?

T h i s is c l e a r l y a s p e c i a l case of t h e matroid p a r i t y problem i n which t h e underlying matroid M i s graphic. Now consider ~ o v & z ' s Theorem 4.1.

Provided we a r e given a 2 - l i n e a r

polymatroid represented i n t h e p r o j e c t i v e space we can n o n d e t e r m i n i s t i c a l l y either "produce a s e t of k independent l i n e s , t h a t i s a matching of s i z e k"

"produce a s e t A and a p a r t i t i o n {S1,

...,Sk)

o f S such t h a t

and more importantly we can check t h e s e a s s e r t i o n s i n polynomial time. Thus i n t h e terminology of Chapter 1 we know nl i s a member o f (NP) n (co-NP)

, and

hence i n view of o u r e a r l i e r remarks it would give some

hope t h a t t h e r e e x i s t s a polynomial algorithm f o r n 1 '

T h i s Lovdsz has produced.

I t i s an extremely complicated algorithm,

o f high polynomial complexity, and o f course only works f o r l i n e a r polymatroids which a r e represented n o t j u s t r e p r e s e n t a b l e i n some p r o j e c t i v e space. There i s f a i r l y s t r o n g evidence ( s e e 5.3 below) t h a t t h e matchching problem f o r g e n e r a l 2-polymatroids i s n o t s o l v a b l e i n polynomial time, however it i s a l s o c l e a r t h a t t h e r e do e x i s t polynomial algorithms f o r c e r t a i n c l a s s e s o f 2-polymatroids which a r e n o t r e p r e s e n t a b l e

-

for

example any polymatroid (S,u) f o r which p can be w r i t t e n a s t h e sum o f two rank f u n c t i o n s .

Extending t h i s c l a s s i s an i n t e r e s t i n g b u t almost

c e r t a i n l y formidable problem.

56.

Pinning p l a n a r s t r u c t u r e s A s a nice a p p l i c a t i o n o f ~ o v g s z ' stheory we c o n s i d e r a problem

a r i s i n g i n t h e theory o f r i g i d s t r u c t u r e s . Consider a graph G whose v e r t i c e s a r e p o i n t s i n t h e Euclidean plane and whose edges a r e r i g i d b a r s with f l e x i b l e j o i n t s a t t h e v e r t i c e s . Suppose t h a t some of the b a r s a r e pinned down t o t h e p l a n e .

An

i n f i n i t e s i m a l motion i s an assignment o f a v e l o c i t y v ( x ) t o each v e r t e x x such t h a t f o r every edge (x,y) o f G ,

and V(X)

= 0

i f x ispinned.

A structure is

x E V(G)

rigid i f i t s o n l y i n f i n i t e s i m a l motion i s v(x) = O

v

.

Deciding whether o r n o t a s t r u c t u r e i s r i g i d i s a s t r a i g h t f o r w a r d e x e r c i s e i n l i n e a r algebra.

The pinning number r ( G ) o f a s t r u c t u r e i s

t h e minimum number of v e r t i c e s which need t o be pinned i n order t o make G rigid.

F i r s t n o t e , t h a t although we r e p r e s e n t w a s a f u n c t i o n only of G, i n r e a l i t y it i s a f u n c t i o n a l s o of i t s r e p r e s e n t a t i o n i n t h e plane.

An

e a s y i l l u s t r a t i o n of t h i s i s provided by t h e following example. Example.

G ani3 G'

0

G

a r e isomorphic graphs b u t n(G) = 3 whereas n(G1) = 2

Now consider the a c t i o n of pinning down a v e r t e x ; i n general such an a c t i o n w i l l reduce t h e dimension o f t h e vector space o f p o s s i b l e motions by e x a c t l y 2.

Using t h i s b a s i c i d e a i t i s n o t d i f f i c u l t t o s e e ( f o r

a r i g o r o u s account see Lov'asz r801) t h a t f i n d i n g t h e pinning number o f a p l a n a r s t r u c t u r e i s e x a c t l y t h e problem of f i n d i n g a minimum spanning s e t i n a 2-polymatroid. But by t h e ~ o v s s z - ~ a l l ai di e n t i t y t h i s i s e q u i v a l e n t t o f i n d i n g t h e c a r d i n a l i t y of a maximum matching and t h i s can then be done by t h e 2-parity algorithm. I t i s i n t e r e s t i n g t h a t Mansfiela '801 has r e c e n t l y shown

t h a t f i n d i n g t h e pinning number o f a s t r u c t u r e i n 3 o r more dimensions i s an NP-complete problem and s o there w i l l only be a polynomial algorithm i f NP = P.

5.

51.

Oracle Bounds on Algorithms

The concept o f an o r a c l e There a r e f ( n ) non-isomorphic matroids where

so a s p o i n t e d o u t i n Robinson and Welsh [80! t h e r e i s l i t t l e hope Of doing l a r g e - s c a l e computing on matroids; f o r what e v e r p o s s i b l e way of r e p r e s e n t i n g a matroid i s chosen t h e ' d a t a b a s e ' o r ' s i z e of i n p u t ' f o r a matroid problem

on a n n-set w i l l be 0(2") Accordingly

.

t h e complexity o f matroid computations i s o f t e n

measured

i n terms of t h e number o f demands made on v a r i o u s p o s s i b l e

oracles.

For example i n Robinson-Welsh an independence o r a c l e ( 4 - o r a c l e )

i s a f u n c t i o n I which f o r any s e t S and matroid M on S , and any A z S , t e l l s u s whether A i s independent o r n o t i n t h e matroid M. YES

if

A E

NO

if

not.

Formally

d(~)

I(A) =

A property P of matroids i s any c o l l e c t i o n of matroids which i s

c l o s e d under isomorphism, i . e .

i f W c P and N - M ,

then N c P .

If P

denotes

t h e p r o p e r t y P r e s t r i c t e d t o matroids on n-sets then t h e complexity o f P n with r e s p e c t t o t h e

$-oracle

i s defined t o be t h e maximum number of

c a l l s on t h e o r a c l e I n a minimum algorithm t o decide whether o r not a

matroid M on a n n-set has o r h a s n o t t h e property P, where t h e maximum i s taken over a l l p o s s i b l e matroids !b on an n-set.

This concept is made

rigorous i n Robinson and Welsh r801 and can c l e a r l y be d e f i n e d f o r o t h e r matroid o r a c l e s such a s t h e following.

(1) A base o r a c l e ( 6 - o r a c l e ) which t e l l s u s whether o r not a given s e t i s a base i n t h e matroid under c o n s t r u c t i o n . (2)

A circuit (4-oracle)

which t e l l s whether o r n o t a s e t i s a

circuit. (3)

A

rank

( 8 - o r a c l e ) which g i v e s t h e rank o f any s e t .

TWOmatroid o r a c l e s

e x i s t polynomials f

,g

Bl,

e2

a r e polynomially e q u i v a l e n t i f t h e r e

such t h a t f o r any property P o f matroids,

and

f o r a l l n , where C Q (Pn) denotes complexity with r e s p e c t t o t h e 0 - o r a c l e . Proposition.

The

d

and

o r a c l e s a r e polynomially e q u i v a l e n t and a r e not

polynomially e q u i v a l e n t t o t h e Proof.

That

4

and

R

(B and

& oracles.

a r e polynomially r e l a t e d i s n o t d i f f i c u l t t o see.

The non-equivalence of t h e o t h e r o r a c l e s i s achieved by c o n s t r u c t i n g various examples, such a s t a k i n g P t o be t h e property LOOP o f having a loop.

Then C (LOOP ) = C (LOOP ) = n, i n e n

whereas f o r n > 1,

S i m i l a r l y i f FREE i s t h e p r o p e r t y o f having every s e t independent,

92.

Examples and f u r t h e r r e s u l t s It i s c l e a r t h a t t h e greedy algorithm d i s c u s s e d i n 13.1 g i v e s a

very f a s t algorithm with r e s p e c t t o t h e

-oracle.

Similarly a l l the other

examples 'which worked' i n Chapter 3 such a s matroid i n t e r s e c t i o n and having a given c o n n e c t i v i t y a r e polynomial algorithms with r e s p e c t t o t h e

$

-oracle. However, our n e x t

theorem shows t h a t t h e s e examples tend t o be

t h e exception r a t h e r than t h e r u l e . We,show t h a t with r e s p e c t t o t h e above o r a c l e s 'most' p r o p e r t i e s a r e exponential.

More p r e c i s e l y we consider b i n a r y o r a c l e s , i . e .

oracles

which a c c e p t a s i n p u t any s e t and only g i v e YES-NO answers, and prove t h e following theorem. Theorem.

For any b i n a r y o r a c l e ah) S e

lim n-

9 and

any a such t h a t

(n-3/2) logn

ID^(^) l / ~ ( n )= 0

where D (n) denotes t h e s e t of p r o p e r t i e s P

o f n-element matroids which

have C (Pn) 5 a ( n ) , and M(n) denotes t h e number of d i f f e r e n t p r o p e r t i e s

8

of matroids on an n-set. Proof. -

See Robinson and Welsh r 8 0 ? . I n o t h e r words we know t h a t "almost a l l " matroid p r o p e r t i e s a r e

e x p o n e n t i a l with r e s p e c t t o any b i n a r y o r a c l e . Since t h e rank o r a c l e , while n o t binary, i s polynomially e q u i v a l e n t t o the

5 -oracle,

a s i m i l a r r e s u l t holds f o r t h e rank o r a c l e .

Other l e s s n a t u r a l o r a c l e s have been considered by some authors. For example Jensen and Korte C801 consider t h e G I R T S o r a c l e ( defined t o g i v e f o r any matroid M on S and any, T

c_ S,

k

-oracle),

the length of the

s m a l l e s t c i r c u i t i n t h e matroid MIT. I t is easy t o f i n d a polynomial p such t h a t f o r any p r o p e r t y P ,

and a t t h e same time t o e x h i b i t p r o p e r t i e s which a r e e x p o n e n t i a l with

.

respect t o

8

$ -oracle

i s s t r i c t l y s t r o n g e r (with r e s p e c t t o t h e s e measures o f

b u t polynomial w i t h r e s p e c t t o

f

I n o t h e r words t h e

complexity) than t h e independence o r a c l e . However t h e s e a r c h f o r t h e ' b e s t o r a c l e ' f o r matroid problems i s i n r e a l i t y a hopeless q u e s t s i n c e i n p r a c t i c e i n o r d e r t o s e t up t h i s ' b e s t o r a c l e ' we would need t o do a n exponential amount o f work.

53.

The 2-PARITY problem i s e x p o n e n t i a l A s an example of t h e ' o r a c l e ' approach we prove t h e following

p r o p o s i t i o n o f ~ o v g s zr801 and Jensen and Korte C801 which i s of d i r e c t i n t e r e s t t o t h e work o f Chapter 4.

Proposition.

With r e s p e c t t o t h e rank o r a c l e t h e 2-parity problem i s

exponential. Proof.

Consider t h e family o f polymatroids o r more p r e c i s e l y t h e 2 - p l y -

matroids (S,p) defined by

T h e n f o r a l l p o s s i b l e assignments o f t h e rank p on t-element s e t s II i s a polymatroid rank f u n c t i o n f o r t 2 1.

L e t A be any algorithm based on t h e

rank o r a c l e and apply t o the 2-polymatroid

We assert that

1x1

=

t+l.

( s , ~ where )

A must ask t h e o r a c l e f o r p ( X I

0

f o r each X

For suppose n o t , then t h e r e e x i s t s Xl

c_ S ,

Ix1I

5.5

which h a s

= t + l whose rank

uO(X1) i s not probed. Now consider t h e 2-polymatroid

(S,%) where

Since f o r a l l s e t s X f o r which t h e algorithm h a s asked f o r

)J

0 (X) we have

Vo(X) = ?J1(X) t h e algorithm A must g i v e t h e same answer t o t h e i n p u t

polymatroid ( S , ! J ~a s it does t o t h e i n p u t polymatroid (S,pJ.

This is

impossible s i n c e t h e maximum s i z e of a matching v(S,pO) i s t while v(S,!.ll) = t + l . Hence t h e algorithm must ask f o r a l l (

t+l

)

v a l u e s vO(X),

1x1

= t+l-

Thus provided we l e t

we have

(,,Yl)

> ( 2 - ~ ) ~

f o r s u f f i c i e n t l y l a r g e n and t h i s prove t h e p r o p o s i t i o n . Note:

T h i s r e s u l t c o n t r a s t s with ~ o v a ' s z ' sTheorem 4.5.1.

However with

IS1 = 4 and t = 1, t h e polymatroid (S,l.tl) above i s t h e Vamos matroid, o r more p r e c i s e l y i s t h e s e t o f l i n e s o f t h e Vamos matroid ( s e e 12.3) which i s w e l l known n o t t o be r e p r e s e n t a b l e over any f i e l d . For a more complete t r e a t m e n t o f t h e s e t o p i c s we r e f e r t o the r e c e n t papers o f Robinson and Welsh [SO], Jensen and Korte C801 (where very many p r o p e r t i e s a r e shown t o be exponential) and Hausmann and Korte C801.

Seymour's C h a r a c t e r i s a t i o n of Regular l l a t r o i d s

6.

51.

Binary and r e g u l a r matroids I n h i s fundamental papers, T u t t e c581, C591 c h a r a c t e r i s e d matroids

which a r e binary, t h a t i s r e p r e s e n t a b l e over t h e f i e l d GF(2) and r e g u l a r , t h a t i s r e p r e s e n t a b l e over every f i e l d by t h e following theorems. Theorem 1.

A matroid i s b i n a r y i f and only i f i t h a s no minor isomorphic

to Theorem 2.

A

u

2r4

matroid i s r e g u l a r i f and only i f it h a s no minor isomorphic

to U

2,4'

F7 o r F* 7

.

An easy consequence o f Theorem 2 i s t h a t M i s r e g u l a r i f and only i f it i s r e p r e s e n t a b l e over every f i e l d .

Another c h a r a c t e r i s a t i o n of r e g u l a r

matroids i s t h a t they a r e b i n a r y matroids which can a l s o be represented over t h e r e a l s by t h e columns o f a matrix o f t h e form (I,,?.) where Ir i s t h e u n i t r x r matrix and A i s an r x (n-r) t o t a l l y unimodular matrix ( i . e a l l i t s submatrices have determinants i n t h e s e t { 0 , 1 , - 1 ) ) . Well known p r o p e r t i e s o f r e g u l a r matroids a r e : (1) A minor o r d u a l o f a r e g u l a r matroid i s r e g u l a r .

(2) Graphic and cographic matroids a r e r e g u l a r . This l a s t r e s u l t i s e s p e c i a l l y s i g n i f i c a n t i n view of t h e r e c e n t r e s u l t o f P.D.

Seymour [80!

which c h a r a c t e r i s e s r e g u l a r matroids a s

t h o s e which can be b u i l t up by ' s t i c k i n g t o g e t h e r ' g r a p h i c and cographic matroids and coples of a s i n g l e matroid on 1 0 elements (we c a l l it R

10

and

define i t l a t e r ) . Apart from t h e beauty o f Seymour's theorem and t h e f a c t t h a t it e x p l a i n s why many theorems about graphs a r e o f t e n e x t e n d i b l e t o r e g u l a r matroids, the methods he used t o develop t h i s proof, n o t a b l y s p l i t t e r s a r e proving t o be a major t o o l i n o t h e r problems i n matroid theory.

5 2.

Splitters Before we can properly d i s c u s s s p l i t t e r s we need t h e more f a m i l i a r

notion o f a k-separation. A matroid M(S) has a k - s e p a r a t i o n f o r some i n t e g e r k i f t h e r e e x i s t s

A c S , with

I A ~5 r (A)

k and

+

Is\AI

2

k such t h a t

r(S\A) 2 r (S) + (k-1)

.

Thus a matroid h a s a l - s e p a r a t i o n i f and only i f i t i s disconnected o r

i s l-separable. If

8 is a

c l a s s o f matroids which i s closed under t h e t a k i n g o f

minors we s a y t h a t N

E

9

is a splitter for

3 if

every M E

3

with a minor

isomorphic to N e i t h e r has a 1 o r 2-separation o r i s isomorphic t o N. The f i r s t t h i n g t o note i s t h a t c l a s s e s of matroids which have s p l i t t e r s a r e n o t easy t o f i n d .

The e x i s t e n c e o f a s p l i t t e r f o r a c l a s s

u s u a l l y seems to mean a f a i r l y s i g n i f i c a n t s t r u c t u r e c o n s t r a i n t on t h e class. For example we can a s s e r t : (1) The c l a s s o f b i n a r y matroids h a s no s p l i t t e r .

Proof. -

Suppose M i s a s p l i t t e r , then t h e r e e x i s t s a l a r g e b i n a r y p r o j e c t i v e 0

space c o n t a i n i n g

and t h i s i s n o t 2-separable,contradiction.

(2)

Proof.

The c l a s s o f g r a p h i c matroids has no s p l i t t e r .

A s i n (1) with a complete graph taking t h e p l a c e o f t h e p r o j e c t i v e

space. Probably t h e f i r s t s p l i t t e r t o appear i n t h e l i t e r a t u r e (though

it was n o t c a l l e d by t h i s name) i s i m p l i c i t i n t h e proof by Wagner r64! t h a t t h e t r u t h o f Hadwiger's c o n j e c t u r e f o r t h e c a s e n = 5 i s e q u i v a l e n t t o t h e 4-colour theorem.

L e t V8 denote t h e graph o f Figure 1, t h e

6 b i u s ladder.

Figure 1 Then

(3)

M(V8)

i s a s p l i t t e r f o r t h e c l a s s of graphic matroids with no

minor isomorphic t o M(K5)

.

Before s t a t i n g t h e key p r o p o s i t i o n i n t h e s e a r c h f o r s p l i t t e r s we need one more d e f i n i t i o n . If

say tJ c

3

i s a c l a s s of matroids closed under t h e t a k i n g of minors we is compressed i n

3

if:

(a)

N i s a geometry, t h a t i s , has no loop o r p a r a l l e l elements;

(b)

If M E

3

and ~ \ =e N then e i t h e r e i s a loop of M o r e i s a

loop o f H* o r e i s p a r a l l e l t o some o t h e r element of M .

I n o t h e r words a geometry N c

3

i s compressed i n

n o r L - t r i v i a l s i n q l e element e x t e n s i o n o f N i n t h e c l a s s

3

i f t h e r e i s no

3.

Then we can prove t h e following p r o p o s i t i o n which g i v e s u s a s t r a i g h t f o r w a r d method o f d e c i d i n g whether o r n o t N i s a s p l i t t e r f o r a c l a s s o f b ~ n a r ymatroids. Theorem 1.

Suppose t h a t

3

i s a c l a s s o f b i n a r y matroids which i s closed

under t h e t a k i n g o f minors and under isomorphism and N non-null,

E

3

.

I f N is

connected and s a t i s f i e s :

(i) N i s compressed i n

(ii) N* i s compressed i n

3,

3 *,

(iii) N i s n o t isomorphic t o t h e polygon matroid

o f t h e wheel W

n

f o r any n 2 3; t h e n N is a s p l i t t e r f o r

3.

Wn i s t h e graph on n + 1 v e r t i c e s , n o f which form t h e

The S

r i m , with t h e remaining v e r t e x ( t h e c e n t r e ) joined t o each o f t h e r i m vertices. We i l l u s t r a t e t h e use o f Theorem 1 by showing how i t g i v e s us a very important s p l i t t e r . (4)

F;

i s a s p l i t t e r f o r t h e c l a s s o f b i n a r y matroids which have

no minor isomorphic t o F

7-

F* i s c e r t a i n l y non-null and connected. I t i s a geometry, and 7 w i l l t h e r e f o r e be compressed i n Ex (F7) i f we can show t h a t i f M c E x (F7)

Proof. -

and ~ \ e = F* t h e n e i s e i t h e r a loop o f M o r o f M* o r e i s p a r a l l e l to 7 some o t h e r element o f M.

But t h i s i s proved by brute' f o r c e checking a l l

p o s s i b l e b i n a r y s i n g l e element e x t e n s i o n s o f F?. we can show (F$)* = F7 i s compressed i n Ex (F;;

I n e x a c t l y t h e same way and s i n c e F* i s c e r t a i n l y

7

n o t a wheel ( 4 ) follows from Theorem 1. The r e a d e r may w e l l puzzle over t h e s i g n i f i c a n c e o f t h e 'wheel c o n d i t i o n ( i i i ) ' i n Theorem 1.

The explanation i s t h a t without it t h e

p r o p o s i t i o n f a i l s , f o r example i f

8

i s t h e f i n i t e c l a s s o f minors of

t h e n W s a t i s f i e s t h e c o n d i t i o n s (i)and (ii)b u t Wn+l n 2-separable.

Wn+l

i s not

Although most of t h e a p p l i c a t i o n s of s p l i t t e r s t o d a t e have been w i t h i n t h e c l a s s o f b i n a r y matroids we s h o u l d n o t e t h a t Theorem 1 can be extended t o non-binary matroids a s follows.

3 is a and N E 3 If

Theorem 2. isomorphism

c l a s s o f m a t ~ o i d s ,c l o s e d under minors, and under

i s non-null,

connected

and s a t i s f i e s i n a d d i t i o n

t o t h e c o n d i t i o n s ( i ) ,(ii)and (iii)o f Theorem 1 t h e a d d i t i o n a l condition: (iv)

N i s n o t isomorphic t o t h e w h i r l splitter for

The

whirl

make {1,2,

...,n)

n 2 3;

then N i s a

9.

mn i s d e f i n e d a s follows.

i s a graph on 2n edges.

mn f o r

Consider t h e wheel Wn,

I f t h e edges a r e l a b e l l e d i n

the r i m (or outer c i r c u i t ) then

0n

it

such a way a s t o

i s t h e non-binary

matroid whose independent s e t s a r e e x a c t l y t h e independent s e t s of Wn e x c e p t that i n

mn,( l , 2 , ...,n )

i s n o t a c i r c u i t b u t a n independent s e t .

Wheels

and w h i r l s were f i r s t s t u d i e d by T u t t e r6E' who showed t h a t i f Iil on S

is a matroid which i s n o t 2-separable and i s n o t isomorphic t o a wheel o r a w h i r l then f o r some p < S , #\p o r #/p i s not 2-separable.

53.

The decomposition theorem A

very i n t e r e s t i n g r e g u l a r matroid i s t h e following 10-element

matroid, which we c a l l R10 and which f i r s t occurred i n t h e work o f Bixby [77!.

I t 1 s the riatroid consisting of t h e 10 5-vectors over GF(2) each of which

have exactly 3 non-zero e n t r i e s .

The following properties a r e routine t o check.

(1) RLO i s regular, self-dual b u t is not graphic o r cographic and f o r any e€RlO,

R

\e = K3,3, R /e = K* 10 10 3,3

.

More important i s the following property (2) Proof.

R10

i s a s p l i t t e r f o r t h e c l a s s of regular matroids.

Tedious checking of one-element extensions of R10

and then using

Theorem 2.1.

0

Now t h i s means t h a t every regular matroid with an R10 2-separable,

except Rlo

itself.

minor is

Hence we may ask, where a r e highly

connected regular matroids t o be found?

They c e r t a i n l y e x i s t , f o r example

graphic and cographic matroids a r e regular and can be a r b i t r a r i l y highly connected.

Seymour's main theorem e s s e n t i a l l y says t h a t these graphic and

cographic matroids together with R10 matroids

a r e t h e only 4-connected regular

.

Before giving Seymour's r e s u l t we need one l a s t s e t of definitions. I f M ,M a r e binary matroids on S1 and S2 respectively we define 1 2 M1AM2

t o be the matroid on t h e symmetric difference SIAS

i t s s e t of cycles a l l subsets of S1AS2 of t h e form C AC 1

cycle of M ~ . (A

2

which has a s

where Ci i s a

cycle of a matroid i s a union of d i s j o i n t c i r c u i t s . )

(3) When S1 n S2 = Ql and (4)

2

Is1 1 ,Is21

<

A

sA,

we c a l l M A M2 a 1-sum. 1

When SlnS2 = { z ) say and z i s not a loop o r coloop of M1 o r M2 and

I ~ ~ 1 , 1< ~ IS~ 11 AS^^,

we say N1AM2

i s a 2-sum of M1 and M 2 -

(5) When

IS 1 n s21 =

3 and Sl n S2 = C say where C i s a c i r c u i t of

both M1 and M2 and IS1

i s a 3-sum o f M1 and M I n each case we c a l l M

1

I ,IS2 ( 2

< IS1 A S2 ( then we say Ml A M2

-

anrl M2 t h e parts of t h e sum.

Now a 1-sum i s j u s t t h e usual d i r e c t sum, t h e 2-sum and 3-sums a r e t h e matroid o p e r a t i o n s corresponding t o ' s t i c k i n g ' two graphs t o g e t h e r by a n edge o r t r i a n g l e r e s p e c t i v e l y and then d e l e t i n g t h e edge o r t r i a n g l e I n e i t h e r c a s e i f we have a matroid M which i s t h e 2 o r 3

i n question.

sum o f o t h e r matroids, t h e n we have a convenient way o f breaking it up i n t o t h e s e smaller s t r u c t u r e . More p r e c i s e l y , i n t h e language of Brylawski r761 we form t h e g e n e r a l i s e d p a r a l l e l connection o f M1,M2

and t h e n d e l e t e t h e modular f l a t

' a c r o s s which' we a r e j o i n i n g M1 and M2. Seymour's decomposition theorem can now be s t a t e d : Theorem 1.

If F

i s a r e g u l a r matroid it i s t h e 1 , 2

or

3 sum of g r a p h i c

matroids, cographic matroids, and copies o f t h e 1 0 element matroid RlOThe f u l l proof o f t h i s theorem i s d i f f i c u l t and long, (94 pages of typescript:)

However we attempt t o give t h e main i d e a s below.

F i r s t we introduce y e t another very s p e c i a l r e g u l a r matroid which we c a l l R12.

I t i s t h e matroid induced by l i n e a r independence over

m ( 2 ) cn t h e columns o f t h e following matrix.

R12

i s r e g u l a r , i s isomorphic b u t n o t e q u a l t o RT2, and does n o t contain

R10

a s a minor.

I t can be a l t e r n a t i v e l y defined a s t h e m a t r o i d obtained

by t a k i n g t h e 3-sum a c r o s s t h e edges a , b , c o f t h e c y c l e matroid o f K \ e 5 of Figure 2

Figure 2 with a copy o f M*(K

3,3

) i n which a , b , c

a r e any t h r e e elements o f K

3,3

which

form a t r i a d o f edges having a common end p o i n t . Throughout, g r e a t use i s made o f t h e following decbmposition lemmas, which a r e s u r p r i s i n g l y awkward t o prove. (6)

I f M i s b i n a r y t h e following a r e e q u i v a l e n t (a)

M i s 1, o r 2-separable

IxlI (b) (7)

r

o r h a s a 3-separation

(X1.X2) with

1x21 > 4

M i s e x p r e s s i b l e a s a 1 , 2 o r 3 sum o f s m a l l e r matroids.

I f M i s b i n a r y and M i s t h e 3-sum of MI and M t h e n i f M h a s 2 no 2-separation, M

1

and M

2

a r e both isomorphic t o minors o f M.

The key s t e p s i n Seymour's proof can now be s t a t e d .

(8)

Every r e g u l a r matroid which i s 3-connected and which i s n e i t h e r graphic nor cographic has R10 o r R12

a s a minor.

(9) I f K i s r e g u l a r and M > R12 then M h a s a 3 s e p a r a t i o n (X1,X2)

(10)

Every r e g u l a r matroid with a minor isomorphic t o R12 i s e x p r e s s i b l e a s t h e 1.2 o r 3 sum of matroids i n t h e c l a s s Ex(F7,F;.R1*)

.

T h i s i s r e a l l y t h e crux o f t h e whole proof s i n c e it e s s e n t i a l l y s a y s t h a t a r e g u l a r matroid which h a s c o n n e c t i v i t y 2 4 cannot have R12 a s a minor and hence by (8) i s e i t h e r graphic o r cographic o r h a s R1 0 as a minor.

But now, by t h e s p l i t t e r theorem (2) we know t h a t i f it h a s R10

a s a minor it i s 2-separable and we can look a t t h e p a r t s of t h e 2-separation and use i n d u c t i v e arguments.

54.

A

polynomial algorithm t o t e s t whether a m a t r i x i s t o t a l l y unimodular An important and immediate p r a c t i c a l consequence o f Seymour's

decomposition theorem i s t h a t it l e a d s t o an e f f i c i e n t polynomial algorithm f o r t e s t i n g whether o r n o t an matrix A with e n t r i e s from t h e s e t (-1,0,1)

i s t o t a l l y unimodular, t h a t i s i s t h e r e a square submatrix o f A

whose determinant i s not i n t h e s e t I-1,0,1)? T h i s q u e s t i o n was one o f t h e outstanding open problems i n computational complexity, s e e f o r example Garey and Johnson r79,p. 2281. Its importance l i e s i n t h e f a c t t h a t a t t h e h e a r t of

many of

the

i n t e g e r programming problems which can be solved i n polynomial time i s a t o t a l l y unimodular matrix.

For example t h e t r i v i a l observation t h a t i f i n a

l i n e a r programming problem we have a t o t a l l y unimodular matrixmeansin a l l

thc d i v i s i o n s used when using t h e simplex method we s h a l l b e d i v i d i n g by

1 and hence w i l l not be moving o u t s i d e t h e c l a s s o f i n t e g e r s .

Theorem 1. an m -

x

There e x i s t s a polynomial algorithm f o r d e c i d i n g whether o r n o t

n matrix with e n t r i e s from {0,1,-1)

i s t o t a l l y unimodular.

T h i s follows almost immediately from Seymour's decomposition theorem with a r e s u l t o f Cunningham and Edmonds r801 which g i v e s a f a s t ( t h a t i s polynomial) algorithm f o r deciding whether o r n o t a matroid i s k-connected f o r any f i x e d i n t e g e r k.

We have d e s c r i b e d such an a l g o r i t h m i n 53.4.

Roughly speaking t h e algorithm works a s follows: Given a b i n a r y matroid b! t e s t i f it i s 1, 2 o r 3 s e p a r a b l e .

If

not t h e n it i s r e g u l a r i f and o n l y i f e i t h e r ( a ) t4 i s graphic o r (b) M i s cographic o r ( c ) F i s R

10

-

P o s s i b i l i t i e s ( a ) and (b) can t h e n be checked by a n a l g o r i t h m o f T u t t e r603 ( i n c a s e (b) a p p l i e d t o P*) .

P o s s i b l i t y ( c ) i s t r i v i a l t o check.

When M h a s a 1, 2 o r 3 s e p a r a t i o n t h e n we examine t h e p a r t s and r e c u r s i v e l y apply t h e above procedure.

I t is not d i f f i c u l t t o see t h a t

s i n c e t h e i n d i v i d u a l subroutines a r e polynomial t h e whole programme can be completed i n polynomial time.

7.

5 1.

Colouring, Flows and Blockinq Problems

The blocking problem I n t h i s l e c t u r e we r e l a t e t h e fundamental graph colouring problem

w i t h t h e l e s s well known problem o f deciding which graphs support flows t a k i n g values i n v a r i o u s a b e l i a n groups and then r e l a t e both ( v i a matroid theory) w i t h a blocking problem i n p r o j e c t i v e spaces which goes back a t l e a s t a s f a r a s Veblen (1912). A s we s h a l l s e e t h e s p l i t t e r . t h e o r y developed i n t h e l a s t c h a p t e r

e n a b l e s u s t o reduce t h i s l a s t , apparently i n t r a c t i b l e geometry problem t o a c o n j e c t u r e about graphs which on t h e s u r f a c e a t l e a s t o f f e r s much more hope. Consider t h e p r o j e c t i v e space P G ( r , u ) .

For any p o s i t i v e i n t e g e r

t , a t-block i s a s e t X o f p o i n t s i n t h i s space such t h a t X n F # each f l a t F o f rank r

-

t.

0

for

I n p a r t i c u l a r t h e 1-blocks a r e t h e s e t s which

have non-empty i n t e r s e c t i o n with every hyperplane o f PG(r,q)

.

I t is

t h e r e f o r e obvious t h a t i f X i s a t-block and Y 2 X then Y i s a t-block.

X

i s a minimal t-block i f it i s a t-block b u t X\p i s n o t a t-block f o r each

- -

Standard v e c t o r space arguments give: (1) The p r o j e c t i v e space PG(t,q) i s a minimal t-block over t h e

f i e l d GF(q) f o r each i n t e g e r t 2 2 and each prime power q . For example PG(2,Z) i s a 2-block over GF(2) and i s a 1-block over GF(4).

More g e n e r a l l y we have:

t

A t-block over GF(p! i s a 1-block over GF(p )

(2)

f o r each p o s i t i v e

i n t e g e r t.

t The converse i s not always t r u e , t h e r e e x i s t 1-blocks over GF(p ) which when regarded a s geometrical c o n f i g u r a t i o n s a r e n o t r e p r e s e n t a b l e ( t h a t i s c o o r d i n a t i s a b l e ) over GF(p).

However we can show:

I f a 1-block over GF(pt ) i s r e p r e s e n t a b l e over GF(p) then

(3)

it i s a t-block over GF(p). Matroids and t h e Blocking Problem Suppose t h a t M i s a matroid on S with rank f u n c t i o n p . chromatic polynomial P ( M ; X )

Its

d e f i n e d by

i s a w e l l known Tutte-Grothendieck i n v a r i a n t ( s e e t h e L e c t u r e s i n t h i s volume by T.H.

Brylawski)

.

The r e l a t i o n s h i p between blocking and t h e chromatic polynomial i s contained i n t h e following remarkable theorem. Theorem 1:

Suppose t h a t M i s a matroid o f rank r on S and t h a t M i s

embeddable i n V ( r , q ) , then t h e r e e x i s t s an r that F n S = @

-

t subspace F o f V(r,q) such

t i f and only i f P(M;o_ ) > 0.

The f i r s t p o i n t t o n o t i c e about Theorem 1 i s t h a t i t s conclusion does not depend on t h e embedding, b u t s a y s t h a t f o r 9 embedding such a subspace exists.

I n f a c t , t h e f u l l v e r s i o n o f Theorem 1 proved by Crapo and Rota

t

r701 shows t h a t P(M;q ) enumerates t h e c o l l e c t i o n s o f hyperplanes whose

i n t e r s e c t i o n i s a f l a t o f t h e type required. The r e l a t i o n s h i p with blocking i s now obvious. (4)

A matroid M which i s r e p r e s e n t a b l e over GF(q) i s a t-block

t

GF(q) i f and only i f P(M;q ) = 0.

over

The statement ( 4 ) i l l u s t r a t e s p r e c i s e l y t h e s l i g h t abuse o f language i n t h e statement "M i s a t-block".

What we mean i s , any o f t h e

v a r i o u s v e c t o r r e p r e s e n t a t i o n s o f M i n V ( r , a ) i s a t-block.

I n o t h e r words

it i s n o t t h e i r c o o r d i n a t i s a t i o n i n V(r,q) which i s important b u t t h e i r geometrical s t r u c t u r e . Example 1:

v e c t o r s o f V(3,2)

Example 2:

i s t h e Fano matroid

I f F7

c o n s i s t i n g o f t h e 7 non-zero

,

More w e r a l l y t h e p r o j e c t i v e geometry M = PG(r,q) h a s chromatic

polynomial

Example 3:

i s t h e complete graph on n v e r t i c e s then i t s cycle

If K

matroid M(K ) h a s chromatic polynomial

Example 4:

If U

2,n

denotes t h e matroid of rank 2 on n p o i n t s i n which

every 2-set i s independent, i n o t h e r words t h e n p o i n t l i n e , then i t s chromatic polynomial is

Thus by e v a l u a t i n g t h e i r r e s p e c t i v e chromatic polynomials a t a p p r o p r i a t e values we have

(5)

The ( q + l )- p o i n t l i n e ( s e e Example 4 ) i s a 1-block over t h e f i e l d GF(q) .

(6)

The Fano matroid F7 i s a 2-block over GF'(2).

(7)

For any prime power q , t h e c y c l e matroid N(Kq+l) i s a

I-block o v e r G F ( q ) , and t h u s i f q = pt it i s a l s o a t-block

52.

o v e r GF(p).

Colouring graphs The r e l a t i o n s h i p between t h e chromatic polynomial o f a matroid and

graph c o l o u r i n g i s s t r a i g h t f o r w a r d . P (G; h )

If G i s a graph i t s c h r o m a t i c polynomial

i s t h a t f u n c t i o n o f h which when e v a l u a t e d a t

)i =

n f o r any non-

n e g a t i v e l n t e g e r n a i v e s t h e number o f p r o p e r c o l o u r i n g s o f t h e v e r t i c e s o f C uslnq n o r fewer c o l o u r s .

A p r o p e r c o l o u r i n g o f G is a c o l o u r i n g o f t h e

v e r t i c e s i n which no two v e r t i c e s which a r e a d j a c e n t i n G have t h e same colour.

For example i t is e a s y t o check t h a t

I f G i s a connected graph it i s shown i n Welsh C76, Chapter 1 6 1 t h a t (1)

P(G;h) = h P(M(G); h )

and t h u s t h e chromatic number x ( G ) o f t h e graph G, t h e s m a l l e s t i n t e g e r n f o r which G h a s an n c o l o u r i n g i s g i v e n by

x ( G ) = i n f n:P(M(G ) ; n ) > 0. n<-~+

Now e v e r y graph G h a s a c y c l e m a t r o i d M(G) which i s r e p r e s e n t a b l e over every f l e l d .

(2)

Hence:

A graph G i s such t h a t M(G)

i s a t-block o v e r GF(q) i f and

o n l y i f i t h a s a chromatic number x ( G ) > q

t

.

Thus t h e g r a p h i c 1-blocks o v e r GF(2) a r e t h e c y c l e matroids o f non-bipartite graphs.

I n f a c t T u t t e L66aJ proves: (3)

M i s a minlmal 1-block over GF(2) i f f M i s t h e cycle matroid of

an odd c i r c u i t . When we come t o 2-blocks over GF(2) however, t h e s i t u a t i o n

i s much more complicated.

Since K5 i s a minimal graph which

i s not 4-colourable we have: (4)

M(K5)

i s a minimal 2-block.

However, t h e r e a r e many o t h e r s , f o r c l e a r l y any graph G which i s not 4-colourable b u t i s such t h a t G\e i s 4-colourable

f o r every edge e , i s

going t o have a cycle matroid which is a minimal 2-block.

Such graphs a r e

c a l l e d e d g e - c r i t i c a l and a n i n f i n i t e family of them can e a s i l y be constructed s e e f o r example Ore r 6 7 1. However a s we s h a l l see i f we consider o n l y t h o s e graphs G such t h a t G i s not 4-colourable b u t every sub-contraction o f G i s 4-colourable t h e n t h e s i t u a t i o n changes.

83.

Flows t a k i n g values i n an a b e l i a n group I f D i s a d i r e c t e d graph and H i s a f i n i t e a b e l i a n group, an H-flow

on D i s a map 41 : E(G)

X

$(e)

et6+(v)

-

+

H : { o ~ such t h a t f o r each v e r t e x v o f D

Z

~ ( e =) 0 mod H ,

efR-(v)

where 6 + ( v ) denotes t h e s e t of edges d i r e c t e d out o f v and 6-(v) denotes t h e s e t o f edges d i r e c t e d i n t o v. flow i n t h e

In o t h e r words an H-flow on D i s a

usual s e n s e , t h a t i s s a t i s f y ~ n gK i r c h o f f ' s laws a t each v e r t e x ,

w i t h t h e two provisos a ) t h a t a r i t h m e t i c i s i n t h e group H and b) no edge

i s allowed t o have a zero flow. i n the literature.

This i s o f t e n c a l l e d a nowhere zero flow

I f t h e r e e x i s t s some H-flow on D we say t h a t D s u p p o r t s an H-flow. W .T.

accommodate?

T u t t e i n 1954 asked t h e q u e s t i o n : what flows can graphs A s we s h a l l s e e , t h e r e a r e some s u r p r i s i n g and b e a u t i f u l

answers. F i r s t note t h a t i f D can support an H-flow and D' i s any digraph which i s o b t a i n a b l e from D by r e o r i e n t a t i o n of some e d g e s , t h e n D ' can support an H-flow.

I n o t h e r words f o r a given group H , whether o r n o t a

dlgraph D s u p p o r t s an H-flow o n l y depends g r a p h i c a l l y on t h e s t r u c t u r e o f t h e underlying graph G ( D ) o b t a i n e d from D by removing t h e d i r e c t i o n s on t h e edges.

Thus, henceforth we speak o f an undirected graph G supporting

an H-flow t o mean t h a t i n any o r i e n t a t i o n o f G, an H-flow i s p o s s i b l e . Secondly, and t h i s i s a very n i c e a p p l i c a t i o n o f T u t t e - Grothendieck theory we have t h e following theorem: Theorem 1.

The number of H-flows on a graph G depends o n l y on t h e o r d e r o f

t h e group H and i s given by e v a l u a t i n g t h e chromatic polynomial o f t h e cocycle matroid M*(G) a t A = O(H), t h e o r d e r of t h e group H. Proof. -

Straightforward application of contraction-deletion,

see the lectures

by T.H. Brylawski i n t h i s volume. Example.

Consider H-flows on any d i r e c t e d v e r s i o n of K4.

chromatic polynomial ( A - 1 ) (A-2) (A-3)

.

M* (K4) has

Hence t h e number o f Z2 x Z2 flows

on K4 i s 3.2.1 = 6 . Thus w e may s e n s i b l y speak o f a graph having a k-flow t o mean t h a t f o r any o r i e n t a t i o n o f G and any a b e l i a n group H o f o r d e r k t h e r e e x i s t s an H-flow on G.

Moreover, and t h i s i s most c r u c i a l , we can decide whether

o r not a p a r t i c u l a r graph s u p p o r t s a k-flow by c a l c u l a t i n g t h e chromatic polynomial o f i t s cocycle matroid and then e v a l u a t i n g t h i s polynomial a t h = k.

A s a f i r s t consequence o f Theorem 1 we have t h e following r e s u l t :

(2)

Proof.

Any p l a n a r graph G without b r i d g e s h a s a 4-flow.

This statement follows from t h e Four Colour Theorem of Appel and

Haken C761.

To s e e t h i s , we note t h a t i f G i s p l a n a r and b r i d g e l e s s

t h e r e e x i s t s a (dual) graph G* which i s p l a n a r and loop f r e e and M*(G) =M(G*). Hence

and by t h e 4-colour theorem and (6.1) we know

s i n c e G* i s p l a n a r .

0

I n h i s fundamental paper i n 1954, W.T.

T u t t e made two c o n j e c t u r e s .

The f i r s t , t h a t t h e r e e x i s t some i n t e g e r n such t h a t any b r i d g e l e s s graph G h a s an n-flow, was s e t t l e d by Jaeger C761 who showed t h a t any b r i d g e l e s s graph G had an 8-flow.

This very n i c e r e s u l t has j u s t been

improved by Seymour 1801 who has proved Theorem 2.

Every b r i d g e l e s s graph has a 6-flow.

The secondconjecture o f T u t t e C541 i s s t i l l u n s e t t l e d , and i s known a s h i s 5-flow c o n j e c t u r e ; it can be s t a t e d a s follows: T u t t e ' s 5-flow Conjecture:

Every b r i d g e l e s s graph has a 5-flow.

To see t h a t t h i s c o n j e c t u r e , i f t r u e , i s b e s t p o s s i b l e consider t h e P e t e r s e n graph P

10

shown i n Figure 1.

Figure 1 The Petersen graph Plo

I t i s non-trivial

t o v e r i f y t h a t P10 h a s no k-flow f o r k

5

4 and t h a t i t i s

t h e s m a l l e s t graph with t h i s p r o p e r t y .

Alternatively, t h e r e a d e r can (with s u f f i c i e n t P(M* (PIO) ;)I) = ( A - 1 )

(A-2) (A-3) (1-4)

2

()I

p a t i e n c e ) show t h a t

-5A+10) and t h u s

P10 h a s no 4-flow.

Now when every v e r t e x v o f G h a s degree 3, t h a t i s G i s

e, it is

n o t d i f f i c u l t t o s e e t h a t provided G i s b r i d g e l e s s , G h a s a 4-flow i f and only i f t h e edges o f G a r e c o l o u r a b l e i n 3-colours s o t h a t no two i n c i d e n t edges have t h e same colour. (4)

I n o t h e r words:

A c u b i c b r i d g e l e s s graph G has a 4-flow

3-edge colourable

i f and 'only i f it i s

.

Using t h i s , a second c o n j e c t u r e of T u t t e 1693 can be formulated a s : A c u b i c b r i d g e l e s s graph G h a s a 4-flow

T u t t e ' s 4-flow Conjecture.

has no subgraph c o n t r a c t i b l e t o t h e P e t e r s e n graph P

i f it

10-

A s t r o n g e r v e r s i o n of t h i s c o n j e c t u r e i s i m p l i c i t i n T u t t e C661

T u t t e ' s 4-flow Conjecture

-

(Strong Version).

&flow i f i t has no subgraph c o n t r a c t i b l e t o P

A b r i d g e l e s s graph has a

10'

I t seems s u r p r i s i n g l y d i f f i c u l t t o show t h a t t h e s e two v e r s i o n s

of T u t t e ' s 4-flow conjecture a r e e q u i v a l e n t .

94.

Tangential Blocks We r e t u r n now t o t h e o r i g i n a l geometrical problem o f T u t t e E661,

namely f i n d i n g t h e minimal 2-blocks over GF(2). Examples of minimal 2-blocks which we have a l r e a d y found a r e : (a)

t h e 7 p o i n t matroid o f rank 3, F7,

(b)

t h e 1 0 p o i n t matroid o f rank 4, Kg,

(c)

t h e 15 p o i n t matroid o f rank 6 , P ; ~

.

From t h e s e i t i s p o s s i b l e t o c o n s t r u c t o t h e r minimal 2-blocks by " s t i c k i n g them t o g e t h e r " i n a n o n - t r i v i a l

fashion.

T h i s i s because o f t h e observation of Oxley C791 t h a t i f M,N a r e two minimal k-blocks then t h e i r s e r i e s connedtion i s a l s o a minimal k-block. (The s e r i e s connection i s t h e matroid o p e r a t i o n corresponding t o t h e graph o p e r a t i o n of ~ a j g sunion).

Continuing t h i s analogy w i t h graph theory we s e e

t h a t a s mentioned e a r l i e r , any edge c r i t i c a l 5-chromatic graph has a polygon matroid which forms a minimal 2-block. matroids of K5,

However each of t h e

F7 and P* h a s t h e a d d i t i o n a l p r o p e r t y t h a t no minor 10

o f them i s a l s o a 2-block. t a n g e n t i a l 2-block.

A 2-block with this p r o p e r t y i s c a l l e d a

T h i s i s n o t T u t t e ' s o r i g i n a l d e f i n i t i o n b u t can be seen

t o be e q u i v a l e n t t o it

s e e Welsh C791.

I n 1966 T u t t e proved t h a t t h e s e

t h r e e matroids were t h e only t a n g e n t i a l 2-blocks o f rank 56.

In 1976,

Datta proved by a complicated geometrical argument t h a t t h e r e i s no t a n g e n t i a l 2-block of rank 7. T u t t e ' s t a n g e n t i a l 2-block c o n j e c t u r e , o r i g i n a l l y made i n (1966) and s t i l l u n s e t t l e d , can be s t a t e d i n t h e following form: T u t t e ' s t a n g e n t i a l block c o n j e c t u r e : F7, K

5

and P* 10

The only t a n g e n t i a l 2-blocks a r e

-

Seymour's theory o f s p l i t t e r s i s a major s t e p towards proving t h i s conjecture.

F i r s t consider Hadwiger's conjecture which i n i t s f u l l form

asserts Conjecture. contains K

n+l

(Hadwiger)

I f a loopless graph G i s n o t n-colourable i t

a s a subcontraction.

Dirac C521 showed t h a t i t was t r u e f o r n = 3 and Wagner C641 showed t h a t f o r n = 4 i t was e q u i v a l e n t t o t h e 4-colour c o n j e c t u r e . h o l d s f o r n = 4.

Therefore it

Thus we know t h a t t h e r e can be no new t a n g e n t i a l 2-block

which i s t h e c y c l e matroid of a graph.

Seymour [ 8 0 ! u s e s h i s c h a r a c t e r i -

s a t i o n o f r e g u l a r matroids t o prove t h e following s t r i k i n g r e s u l t : Theorem I .

Any new t a n g e n t i a l 2-block must be t h e cocycle matroid o f a

graph. In o t h e r words, Seymour's r e s u l t shows t h a t T u t t e ' s t a n g e n t i a l 2-block c o n j e c t u r e i s e x a c t l y e q u i v a l e n t t o t h e s t r o n g form o f t h e 4-flow conjecture.

Thus he has reduced t h i s seemingly i n t r a c t a b l e geometrical

problem t o t h e conceptually much simpler problem o f c h a r a c t e r i s i n g those graphs which have no 4-flow.

More p r e c i s e l y , t h e r e e x i s t s a t a n g e n t i a l

2-block o t h e r than F ,M(K ) and M*(PlO) i f and only i f t h e r e i s a 7 5 b r i d g e l e s s graph 6 n o t

c o n t a i n i n g a sub-graph c o n t r a c t i b l e t o P10 which

has no 4-flow. Sketch proof of Theorem 1.

i s n o t K5,

F

F i r s t consider a t a n g e n t i a l 2-block

7 o r P ; ~ (we use Kg f o r M*(K5), P;o

f o r W*(PlO).

En. which

I f it were

graphic t h e r e would e x i s t a graph G which was n o t c o n t r a c t i b l e t o K5 b u t which was n o t 4-colourable.

T h i s would c o n t r a d i c t Hadwiger's conjecture

f o r t h e c a s e n = 5, which by Wagner's theorem showing t h e equivalence of Hadwiger's 5-chroaatic c o n j e c t u r e w i t h t h e 4-colour theorem o f Appel and Haken, we know t o be t r u e . graphic i s K g . 2-block a minor.

Mo.

Hence t h e only t a n g e n t i a l 2-block which i s

Now consider t h e e x i s t e n c e o f a non-regular t a n g e n t i a l

Since Mo i s n o t r e g u l a r i t must c o n t a i n e i t h e r F7 o r F i a s But because F7 i s a t a n g e n t i a l 2-block,

minor must be F;.

b y minimality, t h i s

Hence s i n c e F* i s a s p l i t t e r f o r b i n a r y matroids with 7

o r M h a s a 2-separation. no F7 minor we know t h a t e i t h e r M0 = F* 7 0

It is

not d i f f i c u l t t o show t h a t a t a n g e n t i a l 2-block cannot have a 2-separation. Hence, Mo = Fj.

But x(F*) = 2 and hence F; i s n o t a t a n g e n t i a l 2-block. 7

So we have shown t h a t t h e r e a r e no non-regular t a n g e n t i a l 2-blocks.

It

remains t o show t h a t t h e r e e x i s t s no t a n g e n t i a l block which i s r e g u l a r b u t n e i t h e r g r a p h i c nor cographic. Suppose such a matroid M e x i s t s .

Then by t h e decomposition theorem

it must be p o s s i b l e t o e x p r e s s M a s a 1 , 2 o r 3 sum o f graphic o r cographic matroids o r copies o f RlO.

An i n d u c t i o n argument shows t h a t t h i s i s

impossible u n l e s s M i s cographic and hence t h e only t a n g e n t i a l blocks which a r e n o t cographic a r e F, and K

55.

5

-

0

A problem on t h e Desargues Configuration

The c o n j e c t u r e s and problems posed s o f a r seem t o be hard, a t l e a s t i n t h e sense t h a t they have stood t h e t e s t o f time. with

We c l o s e this l e c t u r e

new c o n j e c t u r e s which may be e a s i e r t o s e t t l e i n t h e negative b u t '

which i f t r u e would imply o r f u r t h e r r e l a t e some o f t h e e a r l i e r c o n j e c t u r e s . The f i r s t i s a much s t r o n g e r form of T u t t e ' s 5-flow c o n j e c t u r e

-

I f M i s b i n a r y and has no minor isomorphic t o t h e

Conjecture 1.

3-dimensional Desargues c o n f i g u r a t i o n , then x(M) 5 5. Note:

we have p r e s e n t e d t h i s conjecture i n i t s geometrical form; t h e

r e a d e r w i l l q u i c k l y r e a l i s e t h a t t h e 3-dimensional Desargues c o n f i g u r a t i o n

i s a s a matroid i d e n t i c a l w i t h M ( K

)

5

.

The motivation f o r t h i s c o n j e c t u r e i s a r e c e n t paper by Walton and Welsh [801 i n which it i s shown t h a t i f M s a t i s f i e s t h e conditions of t h e c o n j e c t u r e and a l s o h a s no minor isomorphic t o PG(2,2) then x(M)

5

6 and

t h a t i f T u t t e ' s 5-flow c o n j e c t u r e i s t r u e 6 can be replaced by 5. Other problems of t h i s s o r t a r e contained i n Welsh C801.

Possibly

e a s i e r t o s e t t l e i s t h e weaker form o f Conjecture 1. Conjecture 2. t h a t x ( M ) < t.

I f M<Ex(K5) then t h e r e e x i s t s t , independent of M , such

8.

91.

Flows i n Matroids

The max-flow min-cut theorem Many problems i n d i s c r e t e o p t i m i s a t i o n can be formulated i n terms o f

f i n d i n g maximum flows i n capacity c o n s t r a i n e d networks.

A comprehensive

account of t h e theory and i t s a p p l i c a t i o n s i s given by Ford and Fulkerson (1962).

I n t h i s chapter we extend t h e s e i d e a s t o matroids and g e t some

i n t r i g u i n g r e s u l t s which could be viewed a s b r i d g i n g a gap between t h e applied t h e o r y o f flows and f i n i t e geometry. Consider an undirected graph G and two d i s t i n g u i s h e d v e r t i c e s u,v, t o be c a l l e d t h e source and of t h e edge e

i

sink r e s p e c t i v e l y ;

c

i

5 0 is t h e capacity

and r e p r e s e n t s t h e amount o f flow which it can support.

Finding t h e maximum f e a s i b l e flow from u t o v i n t h e c a p a c i t a t e d graph can be found by a well known polynomial method, known a s t h e 'max flow min c u t ' algorithm. I n o r d e r t o consider t h e problem a s a matroid problem we i n s e r t an a d d i t i o n a l d i s t i n g u i s h e d edge e j o i n i n g t h e v e r t i c e s u,v.

L e t C1,

be t h e c i r c u i t s o f t h e matroid M(G) which c o n t a i n t h e edge e . of t h e maximum flow from u t o v i s t h e maximum value of u

flow o r

The value

+...+ uP s u b j e c t

1

t o t h e c o n d i t i o n s t h a t t h e flow along ei i s n o t more t h a n c

u. r 0 represents the

...,CP

i

and where

c i r c u l a t i o n around t h e c i r c u i t C i'

We can now formulate t h e maximum flow problem f o r matroids a s follows.

L e t e be a d i s t i n g u i s h e d element o f matroid M on S = { e , e l ,

...,e

3

and l e t C1,

...,CP be

t h e c i r c u i t s o f M which c o n t a i n e .

(1C i Cn) be t h e c a p a c i t y o f ei.

Let c . 2 0

Let t h e n x p matrix A = ( a ) be defined ij

as

A f e a s i b l e e-flow

i n M is a vector

u=

(ul,

...,uP)

satisfying

P (1)

X

ai j u j C c

(lSiSn),

j =1

and

u=

(ul,-.-,u

P

)

i s a maximum e-flow i f it maximises

t h e c o n s t r a i n t s (1). Xu. i s t h e n c a l l e d t h e

value o f

E u. s u b j e c t t o

t h e maximum e-flow.

Now l e t C* be any c o c i r c u i t o f M which c o n t a i n s e .

We d e f i n e i t s

c a p a c i t y , C(C*), by

The matroid M has t h e max flow min c u t (MFMC) p r o p e r t y i f f o r any element e which i s not a loop and any s e t o f r e a l c a p a c i t i e s c

i

2 0, t h e maximum

v a l u e o f a n e-flow e q u a l s min C(C*) where t h e minimum i s taken over a l l c o c i r c u i t s C* containing e .

We c a l l t h i s minimum t h e min capacity of e .

It i s e a s y t o prove ( s e e Welsh 1761, chapter 19) t h e following:

(2)

I n any matroid t h e maximum value o f a n e-flow i s l e s s than o r equal t o t h e min c a p a c i t y of e .

For general matroids not much more can be s a i d s i n c e it i s easy t o f i n d examples where t h e maximum e-flow has a value s t r i c t l y l e s s than t h e min c a p a c i t y .

For r e g u l a r matrolds however, we can say more.

I n any r e g u l a r matroid t h e maximum value o f a n e-flow e q u a l s

(3)

t h e min c a p a c i t y o f e . I f M i s a matroid on S u e we say t h a t (M,e) has t h e i n t e g e r max flow

(z+ -

min c u t p r o p e r t y

MFMC) p r o p e r t y i f when t h e c a p a c i t i e s c

i

are

r e s t r i c t e d t o being non-negative i n t e g e r s t h e r e e x i s t s a non-negative i n t e g e r flow (ul,

...,uP)

which e q u a l s t h e min c a p a c i t y o f e .

A matroid M has t h e i n t e g e r

the

-

(z'

MFMC) p r o p e r t y V e .

(4) I f M h a s t h e Z+

-

max flow min c u t p r o p e r t y i f (M,e) h a s

I t i s obvious t h a t

MFMC p r o p e r t y then M has t h e (MFMC) -property.

G a l l a i 1591 and Minty r661 independently proved: (5)

Regular matroids have t h e i n t e g e r max flow min c u t property.

However (5) i s n o t b e s t p o s s i b l e s i n c e it i s easy t o check t h a t F7 a l s o has t h e Z+ - MFMC property.

Seymour 1771 completely wrapped up t h e problem

by proving: Theorem 1.

Let

M be a connected matroid.

Then M has t h e i n t e g e r max

flow min c u t p r o p e r t y i f and o n l y i f M i s b i n a r y and has no minor isomorphic t o F;. The o r i g i n a l proof o f t h i s was very involved, we now show how t h e theory o f s p l i t t e r s s i m p l i f i e s t h e proof enormously. Proof. the

F i r s t we l e t t h e reader check t h a t t h e c l a s s

(z+ -

9 of

matroids with

MFMC) -property i s closed under t h e t a k i n g o f minors- t h i s i s

straightforward. Secondly i t i s easy t o v e r i f y t h a t n e i t h e r U (z+-MFMc)

-property.

Hence

3 must

defined by M i

appayently weaker property.

$ if

nor F* have t h e 7

c o n t a i n only b i n a r y matroids.

T h i r d l y it i s easy t o s e e t h a t M i the class

2t4

3

i f and only i f i t belongs t o

and o n l y i f i t has t h e following

For w

:

s \ + ~ X+

and k E Z + suppose t h a t each c o c i r c u i t D containing

e satisfies

Then t h e r e e x i s t s k c i r c u i t s o f M , each c o n t a i n i n g e , b u t no more than w(p) c o n t a i n i n g any o t h e r element p

E

S.

30 -

We need t o prove Ex(F*) = 7 Suppose M E

(a)

3,

belongs t o (b)

and F;

3, , then F*7 M f o r we know every minor o f k 30. Hence &, C_ EX(F;) -

Suppose M E Ex(F;)

we know M cannot be r e g u l a r .

but M e

But F7

E

zO.

By t h e Gallai-Minty r e s u l t

Hence M > F7 o r F;.

But F7 i s a s p l i t t e r f o r Ex(F;),

M also

But M )' F;,

s o M > F7.

thus e i t h e r M = F7 o r M. i s 2-separable.

30 (simple checking) .

Thus M i s 1 o r 2-separable.

But now

f a i r l y s t r a i g h t f o r w a r d arguments show t h a t i f p a r t s o f a 1 o r 2 s e p a r a t i o n b o t h belong t o to

yo.

\30 t h e n t h e 1 o r

Hence i n d u c t i o n on

IS/

2 sum of t h e p a r t s must a l s o belong

shows t h a t M > Pi, and t h i s c o n t r a d i c t i o n

completes t h e proof. Note. -

0

I t i s i n t e r e s t i n g t h a t t h e above proof u s e s s p l i t t e r s together

w i t h t h e Gallai-Minty theorem t h a t t h e r e s u l t holds f o r r e g u l a r matroids. It i s somewhat s u r p r i s i n g t h a t we cannot g e t a d i r e c t proof by t h e

following argument. belong t o

30; it

The Ford-Fulkerson theorem says t h a t graphic matroids i s easy t o prove t h a t cographic matroids belong t o

it i s r o u t i n e t o check t h a t R belonging t o

gow a s

E

FO. Hence i f

Example.

we could prove t h a t

preserved under 1, 2 and 3-sums w e would have a

proof t h a t r e g u l a r matroids a l s o belonged t o t h a t membership of

%;

FO i s preserved under

Take t h e 3-sum o f K4 and F7.

30 .

However it i s n o t t r u e

3-sums!

Both have t h e Z+-MFMC p r o p e r t y b u t

5 2.

Multicommodity flows I f l n s t e a d o f j u s t one source and one sink we have k p a i r s o f

distinct vertices,

(u v ) 1' 1

i d e s t i n e d f o r t h e sink vi,

,

(15 i 5 k ) , where u i s a source o f commodity i

and c . t h e c a p a c i t y of t h e edge e i s an I j

total amount

upper bound on t h e

o f matroid t h a t t h e edge e

j

we have what i s known a s t h e multicommodity flow problem.

can accommodate

It is an obvious

g e n e r a l i s a t i o n of the problem: how many edge d i s j o i n t p a t h s ul

y( +

vk can be drawn i n a graph G?

vl,

+

...,

The case k = 1 i s of course j u s t t h e

max flow problem. 2-commodity flow L e t P = P be t h e edge s e t s o f a l l minimal p a t h s P1, k u

t o v. (12 i

2

k)

.

i

#

V

I$

IS

1 5i

which j o i n

Let v ( P ) be t h e maximum number o f edge d i s j o i n t member

of P and l e t P* be t h e blocker o f B , t h a t 1s P

X nP

-..,Pt

I

*

= {X : X c _ E ( G )

,

t , and X i s a minimal s e t with t h i s p r o p e r t y ) .

I t i s obvlous t h a t (1) v ( 1 P ) 5 m i n l ~ l : D c l P * .

The max flow min c u t theorem o f Ford and Fulkerson s a y s t h a t e q u a l i t y h o l d s i n (1) when k = 1.

4 i s any and 4 * i s

More g e n e r a l l y , i f

{X

: X,Y

c

r6

X f Y 1,

c l u t t e r , t h a t i s a family o f s e t s i t s blocking c l u t t e r ,

i s s a i d t o be

and t h u s t h e max flow min c u t theorem e s s e n t i a l l y s a y s t h a t when k = 1, P

k

i s a Mengerian c l u t t e r .

For k = 2, however t h i s i s f a l s e . Examplel. L e t G be a s shown,

v

v

2

1

p 2 i s 2.

b u t t h e minimum c a r d i n a l i t y o f a blocker o f Nevertheless we do have t h e following: Theorem 1.

(2-commodity flow theorem).

l e t u1,u2,v1,v2

be v e r t i c e s o f G.

and l e t $ . be a flow from u. t o vi,

L e t G be a n undirected graph and

Let c : E 15 i

5

+

2.

Q+ be a c i p a c i t y f u n c t i o n Then t h e maximum value of

$1 + (b2 such t h a t

e q u a l s t h e minimum c a p a c i t y o f a c u t which d i s c o n n e c t s ul from vl and u2 from v

2 -

T h i s r e s u l t s , o r i g i n a l l y proved by Hu [631 by a very long and involved argument now has a very s h o r t e l e g a n t proof by Seymour C781. We i l l u s t r a t e t h e theorem by considering t h e graph G o f Example 1. 0 Assign c a p a c i t y c ( e ) = 1 t o each edge.

Then Figure 2 shows an

assignment o f (non-integer) f l o w s t o t h e edges o f G, t h e f i r s t component c o n t r i b u t e s t o a $ -flow and t h e second t o a $2-flow. 1

Fiaure 2

Multicommodity flows i n matroids

8 3.

We can now extend t h e i d e a s o f multicommodity flows i n graphs t o matroids e x a c t l y analogously t o t h e way we extended t h e s i n g l e commodity problem. For each source sink p a i r ui,vi

we introduce a new s p e c i a l edge e

and l e t t h e c o l l e c t i o n of s p e c i a l edges { el,...,e e.

6

k

) = F-

i

To each edge

F we a s s i g n a demand d = d ( e . ) which r e p r e s e n t s t h e amount o f i

commodity i we wish t o t r a n s p o r t from u t o v i i' To each edge e E E ( G ) \ F we a s s i g n a c a p a c i t y c ( e ) which i s t h e maximum amount o f flow which t h a t edge can accommodate.

With t h i s i n t e r p r e t a t i o n

we can now d e f i n e an (F,c,d) -flow f o r an a r b i t r a r y matroid M on S a s follows. Let other

words

be t h e c o l l e c t i o n o f c i r c u i t s C such t h a t

4

Then a map JI

IC n FI

= 1, i n

can be regarded a s t h e analogues o f ' p a t h s through F ' . :

&F

+ R+ i s an (F,c,d) -flow i f

P r o p o s i t i o n l . I f M has a p-flow through F with p = (c,d) t h e n f o r each c o c i r c u i t C* o f M

Proof.

L e t 4 be such a flow.

I f C i s any c i r c u i t o f M we know

IC

n C*l # 1

and s o i f C n F = {f}

which means I c n c * n F I

5

Icn

(c*\F)

1.

Now by hypothesis

where i n both cases t h e l e f t hand sums a r e over t h o s e C containing e . Hence

S i m i l a r l y c(c*\F) 2

C

4 (C) Ic n (C*\F) I and t h e r e s u l t follows.

CE &F

Example.

(2.1 r e v i s i t e d ) .

s p e c i a l edges el,e2

G

0

Consider t h e corresponding matroid

j o i n i n g (ul,vl)

and (u2,v2) r e s p e c t i v e l y .

-

insert

We f i n d t h a t

t o g e t h e r with t h e s e edges i s e s s e n t i a l l y t h e graph K4, s e e Figure 1 below. This g i v e s an example i n which IF1 = 2, F = {el,e2) and i n which

t h e converse of Proposktion 1fsilafi

we r e s t r i c t a t t e n t i o n t o t h e i n t e g e r s

though it i s i n f a c t t r u e over t h e r a t i o n a l s and ( a f o r t i o r i ) over t h e r e a l s .

Figure 1

+ +

Accordingly we say M i s F-flowing over 74 OR ) i f f o r a l l p € Z

*

t h e e x i s t e n c e of an F-flow w i t h v a l u e s i n Z+CR+)

+@ +) I

. +

We go f u r t h e r and d e s c r i b e M a s k-flowing o v e r Z+ ( o r o v e r lR ) i f F i s F-flowing o v e r Z+ ( o r over R+) f o r a l l s u b s e t s Fc_S w i t h

IF\

= k.

Obviously we have:

(1) M i s k-flowing o v e r Z +

*

+ .

M i s k-flowing o v e r 3?.

However t h e matroid M(K ) above w i t h k = 2 shows t h a t t h e converse i s

4

not t r u e . The max flow min-cut theorem can be r e s t a t e d , a l b e i t e x o t i c a l l y , a s : (2)

Graphic matroids a r e 1-flowing over

z+.

The 2-commodity flow theorem can b e r e s t a t e d a s

(3)

Graphic matroids a r e 2-flowing over

m+.

Cographic m a t r o i d s Consider t h e matroid form o f t h e max-flow min-cut

theorem.

It

can be s t a t e d a s : "The minimum number of c i r c u i t s C. which c o n t a i n e and a r e o t h e r w i s e p a i r w i s e d i s j o i n t i s e q u a l to min I C * l

-

1 where t h e minimum i s

taken The

over a l l C* c o n t a i n i n g e".

dual form o f t h i s i s n o t only t r u e b u t i s very easy t o prove: it

says

(4)

The maximum number of cocircuit.sC? which a r e pairwise d i s j o i n t e x c e p t f o r a s i n g l e edge e i s equal t o t h e minimum

IC I

s i z e of

- 1 taken over a l l c i r c u i t s C which p a s s through e .

For a proof s e e Welsh C76, Chapter 191. I n o t h e r words, it i s e a s y t o prove (5)

54.

Cographic matroids a r e 1-flowing over X

+.

A summary of r e s u l t s A very r e c e n t paper by Seymour C811 almost completely c h a r a c t e r i s e s

k-flowing matroids f o r each k .

We can do no more t h a n summarize some

o f t h e s e d e l i g h t f u l r e s u l t s i n t h e next s e c t i o n . For example one r e s u l t which completely g e n e r a l i s e s (3.5) i s : (1)

Cographlc matroids a r e --flowing

.

Since being 1-flowing i s e q u i v a l e n t t o having t h e max flow min c u t property, the f a c t t h a t U

284

i s n o t 1-flowing, t o g e t h e r with t h e f a c t t h a t

M b e i n g k-flowing i m p l i e s M i s k'-flowing

(2)

for k '

5

k shows:

I f M i s k-flowing then M i s binary.

The max flow min c u t theorem of 5 1 can be r e s t a t e d a s :

+ . (3) M i s 1-flowing over X l f and only i f M

I'

F;

.

We now i n d i c a t e t h e proof i d e a behind a l l t h e following r e s u l t s by proving:

(4) Proof.

M i s 2-flowing i n Z+ i f and only i f M E Ex (M(K4))

.

F i r s t we use t h e f a c t t h a t Ex(M(K4)) i s t h e c l a s s of matroids which

can b e formed by t a k i n g 1 o r 2 sums o f matroids having 5 3 elements.

It is

easy t o prove t h a t a l l o f t h e s e a r e --flowing

( i . e . k-flowing f o r a l l k)

I t i s s t r a i g h t f o r w a r d t o check t h a t i f M1 and M

so a r e t h e 1 and 2 sums of M1 and M2. Slnce M ( K 4 )

i s not ?-flowing i n ' 8

2

a r e k-flowing then

Hence EX(M(K4)) i s --flowing

i n %+.

( s e e Example 3.1) t h e r e s u l t follows.

T h i s l a s t resultcompletely s e t t l e s t h e f i r s t column o f t h e t a b l e below

1-flowing

Ex (FL;)

2-f lowing

Ex ( M ( K 4 ) )

Ex(AG(3,2) ,S8)

3-flowing

Ex ( M ( K 4 ) )

EX ( F ~ , R ~ ~ , M ( H ~ ) )

4-f lowing

EX (M(K4))

Ex (F7,R10,M(K5))

m-f lowing

EX(M(K4)

Ex (F7,R10,M(K5) )

?

Various p o i n t s about t h e above t a b l e need c l a r i f i c a t i o n . First:

t h e matroid

S

8

i s t h e matroid whose b i n a r y r e p r e s e n t a t i o n

i s t h e s e t o f columns o f t h e matrix A :

In o t h e r words S8 with one element d e l e t e d i s F* 7 Secondly:

.

t h e graph H6 whose c y c l e matroid i s 2-flowing b u t n o t

3-flowing has t h e following form:

.

0

Thirdly:

we n o t i c e t h a t t h e only open q u e s t i o n i n t h e above

c l a s s i f i c a t i o n i s c h a r a c t e r i s i n g those matroids n o t 1-flowing over R + . Three exluded minors f o r t h i s are:Tll

AG(3,2), Tll

and Til, where

i s t h e eleven p o i n t matroid g o t from t h e r e p r e s e n t a t i o n o f R10

i n 56.3

by adding on t h e v e c t o r ( 1 , 1 , 1 , 1 , 1 ) . F i n a l l y we c l o s e with a c u r i o s i t y .

Fulkerson C68, 701 showed ( i n

different terminolod that (5)

M* i s 1-flowing i n R+.

M i s 1-flowing over R+

The proof hinges on a l i n e a r programming argument.

No such argument

seems t o be known f o r t h e statement: (6)

M i s 2-flowing over R+

* M*

i s 2-flowing over R+

.

Nevertheless (6) i s t r u e s i n c e by t h e above theory of Seymour M i s 2-flowing over R+ i f and o n l y i f M c l o s e d under d u a l i t y !

E

Ex(AG(3,2) ,S8) and t h i s i s a s e t

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21

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-2

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CENTRO INTERNAZIONALE MATEMATICO ESTIVO (c.I.M.E.)

VOLTAGE-GRAPHIC

MATROIDS

THOMAS ZASLAVSKY

Voltage-Graphic Matroids

Thomas Zaslavsky The Ohio State University

1. A voltage graph i s a p a i r

(N,E) and a voltage, a mapping voltage

m. The

edge i s traversed:

@ = (T,cp)

cp: E

-+

r

c o n s i s t i n g of a graph

6 where @

=

i s a group called the

voltage on an edge depends on the sense i n which the if for

e

i n t h e opposite d i r e c t i o n i t i s

i n one d i r e c t i o n t h e voltage i s -1 cp(e)

.

cp(e)

, then

The voltage on a c i r c l e i s the

product of the edge voltages taken i n order with consistent direction; i f t h e product equals

1 the c i r c l e i s called balanced.

(while i n general

the s t a r t i n g point and o r i e n t a t i o n of

C

no e f f e c t on whether it i s balanced.)

A subgraph i s balanced i f every

c i r c l e i n it i s balanced. SE E

, let

Assuming N

influence i t s voltage, they have

i s finite, l e t

b(S) = the number of balanced components of

n =

IN\

(N,s)

and, f o r

.

Matroid Theorem. of a matrcrid e

The function

G ( @ ) on the s e t

.

E

rk S = n A set

A

5

-

b(S)

E

.4 has an endpoint i n a balanced component of

Sine with edges i n

A

t o form a balanced c i r c l e .

i s t h e rank function

i s closed i f f every edge but does not corn-

(N,A)

A s e t i s a c i r c u i t i f f it

i s a balanced c i r c l e o r a b i c i r c u l a r graph containing no balanced c i r c l e .

Bicircular graphs Theta graphs

We c a l l

Handcuffs

G(Q) a voltage-graphic matroid.

it i s a subgeometry of the Dowling geometry

When it i s a simple matroid,

.

Q~(@)

EXAMPLES

, the

1)

G(T)

2)

Matroids of signed graphs

3) EC(T)

graphic (polygon) matroid:

, the

C :

Q)

,

= (1) cp

@ = (+I,-1)

1

.

.

even-cycle matroid (M. Doob, ~ u t t e ) :

= (+I,-1)

4) B(T) , the b i c i r c u l a r matroid (~imEes-pereira, lee): 6 e ; o r @ = t h e f r e e abelian group on E

I,)

B(TO)

, r0 =

T with a loop a t every node.

s e t of spanning f o r e s t s of

* 0 ) ED(r) 6

=

, the

T

= e

(3imilarl.y one has

.)

-. , e

EDn(T)

,, E

-1

.

~ ( e =)

.

The l a t t i c e of f l a t s i s the

.

equidirected c i r c l e matroid of a digraph

22, cp(e) = + 1 when

when @ = Z

, rp(e)

= Z

, cp

+

T (~atthews):

i s taken i n the d i r e c t i o n assigned by

-f

t h e equidirected c i r c l e matroid modulo

T

. n

,

7)

8)

-+

-+

~ ( r )the ,

group on N

,

,A

=

@ =

@*A

r

anticoherent cycle matroid of

(~atthews): 6 = the free

~ ( e )= vw if e is directed a graph on n nodes; @

v

+

w

.

is A with each edge replaced

by every possible @-labelled edge.

9)

&,(a) , the Dowling geometry of rank

n of

2. Now let 6 have finite order g

.

(Y

, is

G(B*K;)

.

A proper k-coloring of @

is

a mapping K: N

-+

(0)U ((1,...,P)

X (Y)

such that, for any edge e from v to w (including loops), we have K

(v) f 0 or

where

and

l;l

x@ (kg+l)

K

=

(w) f 0 and also

n2

are the numerical and group parts of

the number of proper b-colorings of @

number which do not take the value 0

K

.

Let b

and let x@(P@;) = the

.

Chromatic Polynomial Theorem. X @ ( H + 1) is a polynomial in p Indeed X,(h) of G(@)

=

A~(~)~(A)

, where

p(h)

.

is the characteristic polynomial

.

b Balanced Chromatic Polynomial Theorem. Xm(bg) is a polynomial in p b Indeed ~ ~ ( 1= ) ZA ~($,A)A~(~), summed over balanced flats A s E

.

b

Fundamental Theorem. Let %(A)

denote the balanced chromatic poly-

nomial of the induced voltage graph on X X (A) = @

L Xstable

N

.

%(A-

Then 1)

.

.

This theorem reduces calculation of X (A) @ x;(h)

, which

, or of

~ ( h ), to that of

is often easy.

EXAMPLES (continued)

1) X;(A)

= Xr(h)

4)

=

)):(A)

T

(-l)n-kfk~k , where

b x@(A)

8)

x~(A)=

=

, smed

ZA 2n-rk A x~,~(A/P) g%(~/g)

9 ) p(~n(@);~)

f = the number of k-tree spanning k

r .

forest; in

3)

.

over flats A

of. G(T)

.

.

= g"((~-l)/~)~

, where

(x),

is the falling factorial.

3. There is a geometric realization when set of all hyperplanes x. = cp(e)x J i

@

c' W

.

n in 1 1 where e 6 E

Let

#[@I

be the

is an edge from

Representation Theorem. The lattice of all intersections of subsets

, ordered by reverse flats of G ( @ ) .

of H[@]

Corollary. U[@]

inclusion, is isomorphic to the lattice of

cuts IR"

into

Ix@(-l)l

regions (n-dimensional

cells).

4. Each

@

has a covering graph

graph. If e goes from v from (g,v) to (gcp(e),w).

to w

w

@ = (@ X N, @ x E)

, the

' x E Let p: 3

-t

covering edge

, an unlabelled

(g,e)

extends

E be the covering projection.

Covering Theorem. A set S closed in G@)

5.

E

is closed in G(@)

iff p-l(~) is

.

The Matroid Theorem does not essentially require a voltage.

need is a specified class of "balanced" circles in

r , such that

All we

if two

circles in a theta graph are balanced, then the third is also. The pair

(rye)

is a biased graph. Although a biased graph cannot be colored in the

usual sense, it has algebraically defined "chromatic polynomials" that satisfy the Fundamental Theorem.

References

T. A. Dowling, "A class of geometric lattices based on finite groups", Combinatorial Theory Ser. B, 14 (1973), 61-86. MR 46 47066. Erratum, ibid. 15 (1973), 211. MR 47 48369.

L. R. Matthews, "Matroids from directed graphs", Discrete Math. 24 (1978), 47-61. T. Zaslavsky, "Biased graphs", manuscript,

1977.

T. Zaslavsky, "Signed graphs", submitted. Proofs of the Matroid, Covering, and Representation Theorems for signed graphs, and examples.

T. Zaslavsky, "Signed graph coloring" and "Chromatic invariants of signed graphs", submitted. Proofs of coloring and enumeration results. T. Zaslavsky, "Bicircular geometry and the lattice of forests of a graph", submitted. Details on the bicircular and forest examples and their geometric realizations. Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, Ohio .?3210 U.S.A.

MATROIDS WHOSE GROUND SETS ARE DOMAINS OF FUNCTIONS

James OXLEY

IAS, Australian NatwnaZ University, Canberra, Australia.

K e v i n PREMDERGAST Hydro Electric Comission, Hobart, Australia. D o n ROW*

University of Tasmania, Hobart, Australia.

ABSTRACT

From an integer valued function matroid

Mf

on the domain of

f.

we obtain, i n a natural way, a

f

We show that the c l a s s

f.1

of matroids

so obtained i s closed under restriction, contraction, duality, truncation and elongat-Lon, but not under direct swn. characterisation of

11

and show that

We give an excluded-minor

flconsists

precisely of those

transversa2 matroids with a presentation i n which t h e s e t s i n the presentation are nested. cmct2y

2"

members of

Finally, we show that on an n-set there are

fl .