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 $\widetilde{T}_H(x,y)$ 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 $\widetilde{T}_H(x,y)$ share many properties, and the permutation Tutte polynomial serves as a tool to study the classical Tutte polynomial. We discuss the analogs of Brylawsi's identities and Conde--Merino--Welsh type inequalities. In particular, we will show that if $H$ does not contain isolated vertices, then $$\widetilde{T}_H(3,0)\widetilde{T}_H(0,3)\geq \widetilde{T}_H(1,1)^2,$$ which gives a short proof to the analogous result of Jackson: $$T_G(3,0)T_G(0,3)\geq T_G(1,1)^2$$ for graphs without loops and bridges. We also improve on the constant $3$ in this statement by showing that one can replace it with $2.9243$.