The Orbital Bivariate Chromatic Polynomial
Klaus Dohmen, Mandy Lange-Geisler
TL;DR
This work introduces the orbital bivariate chromatic polynomial $OP_{\Gamma,G}(\lambda,\mu)$, counting μ-proper λ-colorings of a graph up to the action of a subgroup $G$ of automorphisms via the Burnside average $OP_{\Gamma,G}(\lambda,\mu)=\frac{1}{|G|}\sum_{g\in G} P_{\Gamma/g}(\lambda,\mu)$. It unifies and extends the orbital chromatic polynomial and the bivariate chromatic polynomial, providing structural results, degree bounds, and multiplicativity properties. The authors derive closed-form expansions for $OP$ on several graph families (edgeless graphs, complete graphs, complete bipartite graphs, stars, paths, cycles), some of which yield new results even for the non-orbital cases, and present side observations such as Fermat’s Little Theorem and a Lucas-number congruence. They also discuss open problems and directions toward broader orbital analogues of multi-variable graph polynomials. The work advances both theoretical understanding and calculational tools for coloring problems under symmetry constraints, with implications for symmetry-reduced graph invariants and related combinatorial polynomials.
Abstract
The orbital bivariate chromatic polynomial, introduced in this article, counts the number of ways to color the vertices of a graph with $λ$ colors such that adjacent vertices either receive distinct colors from a set of $λ$ colors, or the same color from a distinguished subset of $λ-μ$ 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, Pönitz, 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.