The Merino--Welsh conjecture is false for matroids

Abstract

The matroidal version of the Merino--Welsh conjecture states that the Tutte polynomial $T_M(x,y)$ of any matroid $M$ without loops and coloops satisfies that $$\max(T_M(2,0),T_M(0,2))\geq T_M(1,1).$$ Equivalently, if the Merino--Welsh conjecture is true for all matroids without loops and coloops, then the following inequalities are also satisfied for all matroids without loops and coloops: $$T_M(2,0)+T_M(0,2)\geq 2T_M(1,1),$$ and $$T_M(2,0)T_M(0,2)\geq T_M(1,1)^2.$$ We show a counter-example for these inequalities.