On Weak Chromatic Polynomials of Mixed Graphs

TL;DR

This paper introduces a reciprocity theorem for weak chromatic polynomials of mixed graphs, linking their evaluations at negative integers to combinatorial interpretations using order polynomials of posets.

Contribution

It establishes a reciprocity theorem for weak chromatic polynomials of mixed graphs, providing new combinatorial insights and interpretations.

Findings

Reciprocity theorem for weak chromatic polynomials

Interpretations of polynomial evaluations at negative integers

Connections between mixed graph colorings and order polynomials

Abstract

A \emph{mixed graph} is a graph with directed edges, called arcs, and undirected edges. A $k$-coloring of the vertices is proper if colors from ${1,2,...,k}$ are assigned to each vertex such that $u$ and $v$ have different colors if $uv$ is an edge, and the color of $u$ is less than or equal to (resp. strictly less than) the color of $v$ if $uv$ is an arc. The weak (resp. strong) chromatic polynomial of a mixed graph counts the number of proper $k$-colorings. Using order polynomials of partially ordered sets, we establish a reciprocity theorem for weak chromatic polynomials giving interpretations of evaluations at negative integers.