Around the Merino--Welsh conjecture: improving Jackson's inequality

TL;DR

This paper improves bounds related to the Merino-Welsh conjecture for the Tutte polynomial of matroids, advancing from Jackson's inequality and establishing the conjecture for specific classes of matroids.

Contribution

The authors refine Jackson's inequality by lowering the constant from 2.9243 to 2.355 and prove the Merino-Welsh conjecture for matroids with circuits of certain lengths.

Findings

Improved the inequality constant to 2.355

Confirmed the conjecture for matroids with circuits within specific length bounds

Extended the validity of the conjecture to new classes of matroids

Abstract

The Merino-Welsh conjecture states that for a graph $G$ without loops and bridges the Tutte polynomial $T_G(x,y)$ satisfies the inequality $$\max(T_G(2,0),T_G(0,2))\geqslant T_G(1,1).$$ Later Jackson proved that for any matroid $M$ without loops and coloops we have $$T_M(3,0)T_M(0,3)\geqslant T_M(1,1)^2.$$ The value $3$ in this statement was improved to $2.9243$ by Beke, Cs\'aji, Csikv\'ari and Pituk. In this paper, we further improve on this result by showing that $$T_M(2.355,0)T_M(0,2.355)\geqslant T_M(1,1)^2.$$ We also prove that the Merino--Welsh conjecture is true for matroids $M$, where all circuits of $M$ and its dual $M^*$ have length between $\ell$ and $(\ell-2)^2(\ell^2-4\ell+2)$ for some $\ell\geqslant 4$.