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
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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 = 1
M 3)
) p(x,z)
per ogni x e P ; = 0
per ogni x , y c ~ ,x
#
y
X_ 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 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,
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
,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,(,
.
,
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
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*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
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]
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, 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
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 ( 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
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
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 a2) 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
