Geodesic
Burnside chromatic polynomials of group-invariant graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 55-76

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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.
Keywords: chromatic polynomial, Burnside ring, gain graph, polynomial function 
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White, Jacob A. Burnside chromatic polynomials of group-invariant graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 55-76. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a3/