The Orbital Bivariate Chromatic Polynomial

TL;DR

This paper introduces the orbital bivariate chromatic polynomial, a new graph polynomial that generalizes existing polynomials and counts colorings considering symmetries, with applications to various graph families and number theory.

Contribution

It defines and explores properties of the orbital bivariate chromatic polynomial, extending previous polynomials and providing new expansions for multiple graph types.

Findings

Derived expansions for complete, bipartite, path, and cycle graphs.

Revealed connections to Fermat's Little Theorem and Lucas numbers.

Outlined open problems for future research.

Abstract

The orbital bivariate chromatic polynomial, introduced in this article, counts the number of ways to color the vertices of a graph with $\lambda$ colors such that adjacent vertices either receive distinct colors from a set of $\lambda$ colors, or the same color from a distinguished subset of $\lambda-\mu$ colors, up to a group of symmetries. This new graph polynomial simultaneously generalizes the orbital chromatic polynomial due to Cameron and Kayibi (2007) and the bivariate chromatic polynomial due to Dohmen, P\"onitz, and Tittmann (2003). We discuss fundamental properties, and provide expansions of this new polynomial for various families of graphs, including complete graphs, complete bipartite graphs, paths, and cycles. Some of these expansions are even new for the orbital chromatic polynomial. In addition to these results, we rediscover Fermat's Little Theorem and a ``Fermat-like'' congruence for Lucas numbers. Finally, we outline several open problems related to the orbital bivariate chromatic polynomial.