Marcelino Ibañez, C. M. and Guadalupe RodríguezMarcelino Ibañez, C. M. and Guadalupe Rodríguez

A note on some A note on some inequalities for the Tutteinequalities for the Tuttepolynomial of a matroidpolynomial of a matroid

Matroids: definitionMatroids: definition

A matroid M = (E, r), where EA matroid M = (E, r), where E is a finite is a finite set and r :set and r :(E)(E)→→ is a rank function is a rank function mappingmapping

)itysubmodular(

)()()()(

and ity)(monotonic )()(

, ,)(0

BrArBArBAr

BrArBA

EAAAr

Matroids:Basic conceptsMatroids:Basic concepts

A matroid M=(E,r) A matroid M=(E,r)

AAE is an independent set iff |A|=r(A).E is an independent set iff |A|=r(A).

BBE is a basis of M if B is a maximal E is a basis of M if B is a maximal independent set.independent set.

C is a circuit if it is a minimal C is a circuit if it is a minimal dependent set.dependent set.

Examples: Examples: Graphic MatroidsGraphic Matroids

For G=(V,E ) a connected graph we define the For G=(V,E ) a connected graph we define the matroid M(G)=(E,r) where the rank function ismatroid M(G)=(E,r) where the rank function is

An independent set is a forest of G, a An independent set is a forest of G, a basis is a spanning tree and a circuit basis is a spanning tree and a circuit is a cycle.is a cycle.

r(A) is the maximum size of a r(A) is the maximum size of a spanning forest contained in (V,A).spanning forest contained in (V,A).

Examples: Examples: Uniform MatroidsUniform Matroids

UUr,nr,n=(E,r) where E is [n] and the rank =(E,r) where E is [n] and the rank function isfunction is

][for ),|,min{|)r( nArAA

An independent set is a set of size at An independent set is a set of size at most r, a basis is a set of size r and a most r, a basis is a set of size r and a circuit is a set of size r+1.circuit is a set of size r+1.

Matroids: DualityMatroids: DualityIf M=(E,r) is a matroid, then If M=(E,r) is a matroid, then M*=(E,r*) M*=(E,r*) where r*(A)=|A|-r(E)+r(E-A) is also a matroid, where r*(A)=|A|-r(E)+r(E-A) is also a matroid, called the dual matroid of M.called the dual matroid of M.

An element e of E with r(e)=0 is called a An element e of E with r(e)=0 is called a loop. Dually, an element with r*(e)=0 is loop. Dually, an element with r*(e)=0 is called a coloop.called a coloop.

The Tutte PolynomialThe Tutte Polynomial

)r(||)r()r( )1()1()(TAA

EA

AEM yxyx,

For a matroid M, the two variable polynomialFor a matroid M, the two variable polynomial

is known as the Tutte polynomialis known as the Tutte polynomial.

TTMM(1,1)=number of bases(1,1)=number of bases.

TTMM(2,2)=2(2,2)=2|E||E|

TTMM(x,y)=T(x,y)=TM*M*(y,x)(y,x)

Not entirely obvious but not difficult to Not entirely obvious but not difficult to prove is thatprove is that

where the coefficients twhere the coefficients ti,ji,j are non- are non-negative integers. negative integers.

The Tutte PolynomialThe Tutte Polynomial

ji

jijiM yxtyxT

,, ,),(

There are numerous identities that hold forthe coefficients tij . We need the following.

The Tutte PolynomialThe Tutte Polynomial

Proposition. If a rank-r matroid M with m elements has neither loops nor coloop, then(i) tij = 0, whenever i > r or j > m − r;(ii) tr0 = 1 and t0,m−r = 1;(iii) trj = 0 for all j > 0 and ti,m−r = 0 for all i > 0.

j/i 0 1 2 3 4 5

0 0 24 50 35 10 1

1 24 106 90 20    

2 80 145 45      

3 120 105 15      

4 120 60        

5 96 24        

6 64 6        

7 35          

8 15          

9 5          

10 1          

Theorem. If a matroid M has neither loops nor coloops, then

Inequalities Tutte PolynomialInequalities Tutte Polynomial

ProofProof

)2,2(2

}2,2{

}4,4max{)}4,0(),0,4(max{)(22

Mm

rmr

rmrMM

T

TT

)2,2()}4,0(),0,4(max{ MMM TTT

Observation 1. For a matroid M = (E, r) with dual M*= (E, r*), the following inequalities are equivalent for any AE. |A| |E| − 2(r(E) − r(A)) (1) |E \ A| 2 r*(E \ A) (2)(r(E)-r(A)) + (|A|-r(A)) m − r. (3)

Inequalities Tutte PolynomialInequalities Tutte Polynomial

Inequalities Tutte PolynomialInequalities Tutte Polynomial

By a classical result of J. Edmonds, the matroids in which all subsets A E satisfy the (equivalent) inequalities above are the matroids that contain two disjoint bases; by duality, these are the matroids M whose ground set is the union of twobases of M*.

Observation 2. If a matroid M contains two disjoint bases, then tij = 0, for all i, j such that i + j > m − r. Dually, if its ground set is the union oftwo bases, then tij = 0, for all i, j such that i + j > r.

Inequalities Tutte PolynomialInequalities Tutte Polynomial

Proof:Every term (x−1)r(E)-r(A)(y −1)|A|-r(A) in TM hasxr(E)-r(A)y|A|-r(A) as its monomial of maximum degree. In M, (r(E)-r(A)) + (|A|-r(A)) m − r.

Theorem. If a matroid M contains two disjoint bases, then TM(0, 2a) TM(a, a) ,for all a 2. Dually, if its ground set is the union of two bases of M, then TM(2a, 0) TM(a, a), for all a 2

Inequalities Tutte PolynomialInequalities Tutte Polynomial

Inequalities Tutte PolynomialInequalities Tutte PolynomialProof. Case M has two disjoint bases. In this situation m− r r and

ji

jijiM

rm atT,

,)2,2(4

Note:Even if it has loops

Inequalities Tutte PolynomialInequalities Tutte PolynomialProof.

ji

jiji

ji

ji

ji

ji

jirm

jiji

rm

at

at

ata

,,

,,

,,

22

22

)2(

Multiplying this inequality by (a/2)m−r we get

Inequalities Tutte PolynomialInequalities Tutte PolynomialProof.

),()2()2,0(,

, aaTataaT Mji

jiji

rmM

Corollary. For a matroid M, we have that max{TM(2a, 0), TM(0, 2a)} TM(a, a),for all a 2 whenever M is one of the following:• an identically self-dual matroid M,• a coloopless paving matroid • the uniform matroid Ur,n for 0 r n.• a rank-r projective geometry over GF(q) or its dual, for r 2.

Inequalities Tutte PolynomialInequalities Tutte Polynomial

Corollary. For a graph G, we have that max{TG(2a, 0), TG(0, 2a)} TG(a, a),a 2 whenever G is one of the following:• complete graph Kn, n 3,

• complete bipartite graph Kn,m

• the wheel graph Wn, for n 2,• the square lattice Ln, for n 2,• an n-cycle n 2,• a tree with n edges, for n 1.

Inequalities Tutte PolynomialInequalities Tutte Polynomial

Corollary. For a graph G, we have that max{TG(2a, 0), TG(0, 2a)} TG(a, a),a 2 whenever G is one of the following: • 4-edge-connected graph,[Tutte,Nash- Williams]• a series-parallel graph,[Nash-Williams]• a cubic graph, [Nash-Williams]• bipartite planar graphs, [Nash-Williams]• bridgeless threshold graph.

Inequalities Tutte PolynomialInequalities Tutte Polynomial

ConjectureConjectureC. M. and D.J.A. Welsh, made the following

Conjecture. Let G be a 2-connected graph with no loops, thenmax{TG(2, 0), TG(0, 2)} TG(1, 1) .

Conjecture. If a cosimple matroid M contains two disjoint bases then TM(0, 2) TM(1, 1) .Dually, if the ground set of a simple M is the union of two bases, then TM(2, 0) TM(1, 1) .

ConjectureConjecture

Conjecture:Uniforme Conjecture:Uniforme matroidsmatroids

rm

rrU

ym

my

r

m

r

mx

r

m

xm

xyxTmr

)1()1(1

)1(1

)1(1

)1(),( 1

,

Conjecture:Uniforme Conjecture:Uniforme matroidsmatroids

rm

U

r

m

r

m

r

m

r

mmT

mr

)1(11

111)0,2(

,

0)0,2()1,1()0,2(,,,

r

mTTT

mrmrmr UUU

Conjecture:Complete GraphsConjecture:Complete GraphsNote: for a graphic matroid M(G), the Note: for a graphic matroid M(G), the evaluations Tevaluations TGG(2, 0) and T(2, 0) and TGG (0, 2) have the (0, 2) have the

interpretation of being the number of acyclic interpretation of being the number of acyclic orientations and the number of totally cyclic orientations and the number of totally cyclic orientations of G, respectively.orientations of G, respectively.An acyclic orientation of a graph G is an An acyclic orientation of a graph G is an orientation where there are notorientation where there are notdirected cycles. A totally cyclic orientation is directed cycles. A totally cyclic orientation is an orientation where every edge is in a an orientation where every edge is in a directed cyclic.directed cyclic.

Lemma. For a 2-connected G and a vertex Lemma. For a 2-connected G and a vertex vv of degree of degree dd, we have that, we have that

)2,0()22()2,0( vGd

G TT

vv

Conjecture:Complete GraphsConjecture:Complete Graphs

Conjecture:Complete GraphsConjecture:Complete GraphsIt is a clasical result due to Cayley that TKn(1,1) = nn−2.The edge-set of K3 is the union of two spanning trees and TK3(x,y) = x2+x+yThus, TK3(2,0)=6 > 3 = TK3(1,1).

For n4, Kn has two disjoint spanning trees. TK4(x,y) =x3+ 3x2+ 2x+ 4xy+ 2y+ 3y2+ y3.Thus, TK4(0,2)=24 > 16 = TK4(1,1).

Conjecture:Complete GraphsConjecture:Complete Graphs

)2,0()2,0()22(

)1,1()22()1,1()1(

)1,1()1(1

1

)1()1,1(

1

1

2

1

nn

nn

n

n

KKn

Kn

K

K

n

nK

TT

TTne

Tnnn

nn

nT

22

532

532)1,1( 2

nn

nW LTn

)2()1()2();( xxxxxW nnn

33)( nnW

)()( nn WW

(Sedláček)

Conjecture:Wheels Conjecture:Wheels