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

Abstract: The Merino-Welsh conjecture states that for a graph $G$ without loops and bridges we have $\max(T_G(2,0),T_G(0,2))\geq T_G(1,1).$ Later Jackson proved that for any matroid $M$ without loop and coloop we have $T_M(3,0)T_M(0,3)\geq T_M(1,1)<sup>2.$ The value $3$ in this statement was improved to $2.9242$ by Beke, Cs\'aji, Csikv\'ari and Pituk. In this paper, we further improve on this result by showing that $T_M(2.355,0)T_M(0,2.355)\geq T_M(1,1)<sup>2.$ We also prove that the Merino--Welsh conjecture is true for matroids $M$, where all circuits of $M$ and its dual $M<sup>*$ have length between $\ell$ and $(\ell-2)<sup>4$ for some $\ell\geq 6$.