New article, co-authored by Gergely Csáji in the journal Advances in Mathematics

The Merino–Welsh conjecture is false for matroids

 

Advances in Mathematics – Volume 446, June 2024 – Available online 25 April 2024

 Abstract: The matroidal version of the Merino–Welsh conjecture states that the Tutte polynomial TM(x,y) of any matroid M without loops and coloops satisfies that

max⁡(TM(2,0),TM(0,2))⩾TM(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: TM(2,0)+TM(0,2)⩾2TM(1,1), andTM(2,0)TM(0,2)⩾TM(1,1)2. We show a counter-example for these inequalities.

MSC: primary 05C30, secondary 05C31, 05C70

 

Keywords: Tutte polynomial, Merino–Welsh conjecture

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