A note on some inequalities for the Tutte polynomial of a matroid

Abstract

We prove that if a matroid M contains two disjoint bases (or, dually, if its ground set is the union of two bases), then TM(a,a) max{TM(2a,0),TM(0,2a)} for a 2. This resembles the conjecture that appears in C. Merino and D.J.A. Welsh, Forests, colourings and acyclic orientations of the square lattice, Annals of Combinatorics 3 (1999) pp. 417‐429: If G is a 2-connected graph with no loops, then TG(1,1) max{TG(2,0),T(0,2)}. We conjecture that TM(1,1) max{TM(2,0),TM(0,2)} for matroids which contains two disjoint bases or its ground set is the union of two bases. We also prove the latter for some families of graphs and matroids.