Marcelino Ibañez, C. M. and Guadalupe Rodríguez A note on some inequalities for the Tutte polynomial of a matroid

Matroids: definition A matroid M = (E, r), where E is a finite set and r :  (E) →  is a rank function mapping

Matroids:Basic concepts A matroid M=(E,r) A  E is an independent set iff |A|=r(A). B  E is a basis of M if B is a maximal independent set. C is a circuit if it is a minimal dependent set.

Examples: Graphic Matroids For G=(V,E ) a connected graph we define the matroid M(G)=(E,r) where the rank function is An independent set is a forest of G, a basis is a spanning tree and a circuit is a cycle. r(A) is the maximum size of a spanning forest contained in (V,A).

Examples: Uniform Matroids U r,n =(E,r) where E is [n] and the rank function is An independent set is a set of size at most r, a basis is a set of size r and a circuit is a set of size r+1.

Matroids: Duality If M=(E,r) is a matroid, then M*=(E,r*) where r*(A)=|A|-r(E)+r(E-A) is also a matroid, called the dual matroid of M. An element e of E with r(e)=0 is called a loop. Dually, an element with r*(e)=0 is called a coloop.

The Tutte Polynomial For a matroid M, the two variable polynomial is known as the Tutte polynomial is known as the Tutte polynomial. T M (1,1)=number of bases T M (1,1)=number of bases. T M (2,2)=2 |E| T M (x,y)=T M* (y,x)

Not entirely obvious but not difficult to prove is that where the coefficients t i,j are non-negative integers. The Tutte Polynomial

There are numerous identities that hold for the coefficients t ij. We need the following. The Tutte Polynomial Proposition. If a rank-r matroid M with m elements has neither loops nor coloop, then (i) t ij = 0, whenever i > r or j > m − r; (ii) t r0 = 1 and t 0,m−r = 1; (iii) t rj = 0 for all j > 0 and t i,m−r = 0 for all i > 0.

j/i

Theorem. If a matroid M has neither loops nor coloops, then Inequalities Tutte Polynomial Proof

Observation 1. For a matroid M = (E, r) with dual M*= (E, r*), the following inequalities are equivalent for any A  E. |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 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 two bases of M*.

Observation 2. If a matroid M contains two disjoint bases, then t ij = 0, for all i, j such that i + j > m − r. Dually, if its ground set is the union of two bases, then t ij = 0, for all i, j such that i + j > r. Inequalities Tutte Polynomial Proof: Every term (x−1) r(E)-r(A) (y −1) |A|-r(A) in T M has x r(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 T M (0, 2a)  T M (a, a),for all a  2. Dually, if its ground set is the union of two bases of M, then T M (2a, 0)  T M (a, a), for all a  2 Inequalities Tutte Polynomial

Proof. Case M has two disjoint bases. In this situation m− r  r and Note:Even if it has loops

Inequalities Tutte Polynomial Proof. Multiplying this inequality by (a/2) m−r we get

Inequalities Tutte Polynomial Proof.

Corollary. For a matroid M, we have that max{T M (2a, 0), T M (0, 2a)}  T M (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 U r,n for 0  r  n. a rank-r projective geometry over GF(q) or its dual, for r  2. Inequalities Tutte Polynomial

Corollary. For a graph G, we have that max{T G (2a, 0), T G (0, 2a)}  T G (a, a), a  2 whenever G is one of the following: complete graph K n, n  3, complete bipartite graph K n,m the wheel graph W n, 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 Polynomial

Corollary. For a graph G, we have that max{T G (2a, 0), T G (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 Polynomial

Conjecture C. M. and D.J.A. Welsh, made the following Conjecture. Let G be a 2-connected graph with no loops, then max{T G (2, 0), T G (0, 2)}  T G (1, 1).

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

Conjecture:Uniforme matroids

Conjecture:Complete Graphs Note: for a graphic matroid M(G), the evaluations T G (2, 0) and T G (0, 2) have the interpretation of being the number of acyclic orientations and the number of totally cyclic orientations of G, respectively. An acyclic orientation of a graph G is an orientation where there are not directed cycles. A totally cyclic orientation is an orientation where every edge is in a directed cyclic.

Lemma. For a 2-connected G and a vertex v of degree d, we have that v Conjecture:Complete Graphs

It is a clasical result due to Cayley that T Kn (1,1) = n n−2. The edge-set of K 3 is the union of two spanning trees and T K3 (x,y) = x 2 +x+y Thus, T K3 (2,0)=6 > 3 = T K3 (1,1). For n  4, K n has two disjoint spanning trees. T K4 (x,y) =x 3 + 3x 2 + 2x+ 4xy+ 2y+ 3y 2 + y 3. Thus, T K4 (0,2)=24 > 16 = T K4 (1,1).

Conjecture:Complete Graphs

 (Sedláček) Conjecture:Wheels