Bivariate Chromatic Polynomials of Mixed Graphs

TL;DR

This paper extends the bivariate chromatic polynomial to mixed graphs with directed and undirected edges, providing a decomposition formula and a reciprocity theorem.

Contribution

It introduces a new extension of the bivariate chromatic polynomial to mixed graphs and establishes a decomposition formula and reciprocity theorem.

Findings

Decomposition formula expresses $ ext{chi}_G(x,y)$ as a sum of order polynomials.

A combinatorial reciprocity theorem for $ ext{chi}_G(x,y)$ is proved.

The work generalizes previous polynomial concepts to mixed graphs.

Abstract

The bivariate chromatic polynomial $\chi_G(x,y)$ of a graph $G = (V, E)$, introduced by Dohmen-P\"{o}nitz-Tittmann (2003), counts all $x$-colorings of $G$ such that adjacent vertices get different colors if they are $\le y$. We extend this notion to mixed graphs, which have both directed and undirected edges. Our main result is a decomposition formula which expresses $\chi_G(x,y)$ as a sum of bivariate order polynomials (Beck-Farahmand-Karunaratne-Zuniga Ruiz 2020), and a combinatorial reciprocity theorem for $\chi_G(x,y)$.