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
