Modular relations of the Tutte symmetric function

Abstract

For a graph G, its Tutte symmetric function XBG generalizes both the Tutte polynomial TG and the chromatic symmetric function XG. We may also consider XB as a map from the t-extended Hopf algebra G[t] of labelled graphs to symmetric functions. We show that the kernel of XB is generated by vertex-relabellings and a finite set of modular relations, in the same style as a recent analogous result by Penaguiao on the chromatic symmetric function X. In particular, we find one such relation that generalizes the well-known triangular modular relation of Orellana and Scott, and build upon this to give a modular relation of the Tutte symmetric function for any two-edge-connected graph that generalizes the n-cycle relation of Dahlberg and vanWilligenburg. Additionally, we give a structural characterization of all local modular relations of the chromatic and Tutte symmetric functions, and prove that there is no single local modification that preserves either function on simple graphs.