Polynomials associated with graph coloring and orientations

Abstract

We study colorings and orientations of graphs in two related contexts. Firstly, we generalize Stanley's chromatic symmetric function using the k-balanced colorings of Pretzel to create a new graph invariant. We show that in fact this invariant is a quasisymmetric function which has a positive expansion in the fundamental basis. We also define a graph invariant generalizing the chromatic polynomial for which we prove some theorems analogous to well-known theorems about the chromatic polynomial. Secondly, we examine graphs and graph colorings in the context of the combinatorial Hopf algebras of Aguiar, Bergeron and Sottile. By doing so, we are able to obtain a new formula for the antipode of a Hopf algebra on graphs previously studied by Schmitt. We also obtain new interpretations of evaluations of the Tutte polynomial.