Acyclic Orientations and the Chromatic Polynomial of Signed Graphs

TL;DR

This paper establishes a novel connection between acyclic orientations and the bivariate chromatic polynomial of signed graphs, providing multiple proofs and deepening understanding of graph coloring and orientation enumeration.

Contribution

It introduces a new correspondence linking acyclic orientations to a specific evaluation of the signed graph chromatic polynomial, with three different proof techniques.

Findings

The evaluation |_G(-1,2)| equals the number of acyclic orientations modulo source-sink swaps.

Provides three proofs: toric hyperplane arrangements, deletion-contraction, and direct methods.

Enhances understanding of signed graph colorings and their combinatorial properties.

Abstract

We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $\chi_G(k,l)$ that counts the number of signed colorings using colors $0,\pm1,\dots,\pm k$ along with $l-1$ symmetric colors $0_1,\dots,0_{l-1}$. We show that the evaluation of the bivariate chromatic polynomial $|\chi_G(-1,2)|$ is equal to the number of acyclic orientations of the signed graph modulo the equivalence relation generated by swapping sources and sinks. We present three proofs of this fact, a proof using toric hyperplane arrangements, a proof using deletion-contraction, and a direct proof.