The Chromatic Polynomials of Signed Petersen Graphs

Abstract

Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs and that they could be told apart by their chromatic polynomials, by showing that the latter give distinct results when evaluated at 3. He conjectured that the six different signed Petersen graphs also have distinct zero-free chromatic polynomials, and that both types of chromatic polynomials have distinct evaluations at \emph{any} positive integer. We developed and executed a computer program (running in {\tt SAGE}) that efficiently determines the number of proper $k$-colorings for a given signed graph; our computations for the signed Petersen graphs confirm Zaslavsky's conjecture. We also computed the chromatic polynomials of all signed complete graphs with up to five vertices.