A bivariate chromatic polynomial for signed graphs

TL;DR

This paper extends a bivariate chromatic polynomial to signed graphs, providing formulas, evaluations, and reciprocity theorems that deepen understanding of graph colorings and their combinatorial properties.

Contribution

It introduces a signed graph extension of the bivariate chromatic polynomial, including an inclusion-exclusion formula and reciprocity theorems.

Findings

Derived an inclusion-exclusion formula for signed graphs.

Established special evaluations related to balanced subgraphs.

Proved combinatorial reciprocity theorems for the polynomials.

Abstract

We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial $c_\Gamma(k,l)$ which counts all $(k+l)$-colorings of a graph $\Gamma$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an extension of $c_\Gamma(k,l)$ to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_\Gamma(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynomials to acyclic orientations.