Burnside Chromatic Polynomials of Group-Invariant Graphs

Abstract We introduce the Burnside chromatic polynomial of a graph that is invariant under a group action. This is a generalization of the Q-chromatic function Zaslavsky introduced for gain graphs. Given a group 𝕲 acting on a graph G and a 𝕲-set X, a proper X-coloring is a function with no monochromatic edge orbit. The set of proper colorings is a 𝕲-set which induces a polynomial function from the Burnside ring of 𝕲 to itself. In this paper, we study many properties of the Burnside chromatic polynomial, answering some questions of Zaslavsky.