Permutation Tutte polynomial

Abstract

The classical Tutte polynomial is a two-variate polynomial $T_G(x,y)$ associated to graphs or more generally, matroids. In this paper, we introduce a polynomial ${\tilde{T}}_H(x,y)$ associated to a bipartite graph $H$ that we call the permutation Tutte polynomial of the graph $H$. It turns out that $T_G(x,y)$ and ${\tilde{T}}_H(x,y)$ 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 ${\tilde{T}}_H(3,0){\tilde{T}}_H(0,3)\ge {\tilde{T}}_H{(1,1)}^2,$which gives a short proof of the analogous result of Jackson: $T_G(3,0)T_G(0,3)\ge T_G{(1,1)}^2$ 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.