Permutation Tutte polynomial

Csongor Beke, Gergely Kál Csáji, Péter Csikvári, Sára Pituk

Abstract: The classical Tutte polynomial is a two-variate polynomial TG(x,y) associated to graphs or more generally, matroids. In this paper, we introduce a polynomial T˜H(x,y) associated to a bipartite graph H that we call the permutation Tutte polynomial of the graph H. It turns out that TG(x,y) and T˜H(x,y) share many properties, and the permutation Tutte polynomial serves as a tool to study the classical Tutte polynomial. We discuss the analogues of Brylawsi’s identities and Conde–Merino–Welsh type inequalities. In particular, we will show that if H does not contain isolated vertices, then T˜H(3,0)T˜H(0,3)≥T˜H(1,1)2,which gives a short proof of the analogous result of Jackson: TG(3,0)TG(0,3)≥TG(1,1)2 for graphs without loops and bridges. We also give improvement on the constant 3 in this statement by showing that one can replace it with 2.9243.