The Merino--Welsh conjecture is false for matroids

2023/11/03 by Beke, Csongor, Csáji, Gergely Kál, Csikvári, Péter +1
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2311.01932

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))≥ 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)≥ 2T_M(1,1), and T_M(2,0)T_M(0,2)≥ T_M(1,1)². We show a counter-example for these inequalities.

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