Burnside chromatic polynomials of group-invariant graphs

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 Zaslavsk