Inconsequential results on the Merino-Welsh conjecture for Tutte polynomials

TL;DR

This paper explores the Merino-Welsh conjecture related to Tutte polynomial evaluations, providing results that, despite their limitations, contribute to understanding the conjecture's scope and boundaries.

Contribution

The paper presents three results on the Merino-Welsh conjecture, highlighting their limited impact and suggesting they are dead ends despite implying some cases.

Findings

Results imply a version of the conjecture for many matroids

The results are inconsequential and seem to be dead ends

The work clarifies the boundaries of current approaches

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.