Linear Relations for a Generalized Tutte Polynomial

Abstract

Brylawski proved the coefficients of the Tutte polynomial of a matroid satisfy a set of linear relations. We extend these relations to a generalization of the Tutte polynomial that includes greedoids and antimatroids. This leads to families of new identities for antimatroids, including trees, posets, chordal graphs and finite point sets in R n . It also gives a "new" linear relation for matroids that is implied by Brylawski's identities.

Short proof of a theorem of Brylawski on the coefficients of the Tutte polynomial

“…Remark 2.1. Once one conjectures Theorem 1.2, then it can be proved by the deletioncontraction identities via simple induction on h. The more general Theorem 1.1 can be proved by certain recursions too, as it was shown by Gordon [4], but seems to be considerably more work than the proof presented in this paper.…”

“…where r(S) is the rank of a set S ⊆ E. The Tutte polynomial of a graph G simply corresponds to the graphical matroid M of the graph G. In [4], Gordon extended Brylawski's result for ranked sets: besides r(S) ≤ min(r(E), |S|), he assumed the normalization r(∅) = 0, and was able to extend Brylawski's identities for h = |E|.…”

Irreducibility of the Tutte polynomial of an embedded graph

“…Brylawski [8] established a collection of affine relations satisfied by the coefficients of the Tutte polynomial of a matroid. Much later, in a surprising result, Gordon [23] demonstrated that these affine relations hold much more generally.…”

“…For a delta-matroid with element set E, we have σ(∅) = 0, and for any subset A of E we have σ(A) ⩽ max{|A|, σ(E)}. These conditions are sufficient to apply [23,Theorem 11] to show that all of Brylawski's affine relations hold for T (D), giving the following.…”