Inconsequential results on the Merino-Welsh conjecture for Tutte polynomials

Abstract

The Merino-Welsh conjectures say that subject to conditions, there is an inequality among the Tutte-polynomial evaluations T (M ; 2, 0), T (M ; 0, 2), and T (M ; 1, 1). We present three results on a Merino-Welsh conjecture. These results are "inconsequential" in the sense that although they imply a version of the conjecture for many matroids, they seem to be dead ends.

The Merino--Welsh conjecture for split matroids

“…However, as the rank grows, one has to deal separately with a growing finite number of cases that could yield a potential counterexample to Conjecture 1.2(3). See [12] for a brief overview of this and another "inconsequential" result on the Merino-Welsh conjecture.…”