The Merino–Welsh conjecture is false for matroids

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 thatmax⁡(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.