📝 SummarXiv

Abstract

The orbital chromatic polynomial introduced by Cameron and Kayibi counts the number of proper $\lambda$-colorings of a graph modulo a group of symmetries. As a generalization of this polynomial we consider the orbital bivariate chromatic polynomial, itself specializing to the orbital independence polynomial, and establish expansions for path, stars and cycles of arbitrary size. As side results, we rediscover Fermat's Little Theorem and a "Fermat-like" congruence for Lucas numbers.