The Merino--Welsh conjecture is false for matroids

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...

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Published in:arXiv.org 2024-02
Main Authors: Beke, Csongor, Csáji, Gergely Kál, Csikvári, Péter, Pituk, Sára
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Inequalities
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Csáji, Gergely Kál
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description 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.
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Polynomials
title The Merino--Welsh conjecture is false for matroids
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