Degree choosable signed graphs

TL;DR

This paper extends Brooks' theorem to signed graphs with multiple edges, characterizes degree choosable signed graphs, and explores properties of color critical signed graphs.

Contribution

It generalizes Brooks' theorem for signed graphs with multiple edges and provides a characterization of degree choosable signed graphs.

Findings

Extended Brooks' type result to graphs with multiple edges.

Characterized degree choosable signed graphs.

Established basic properties of color critical signed graphs.

Abstract

A signed graph is a graph in which each edge is labeled with $+1$ or $-1$. A (proper) vertex coloring of a signed graph is a mapping $\f$ that assigns to each vertex $v\in V(G)$ a color $\f(v)\in \mz$ such that every edge $vw$ of $G$ satisfies $\f(v)\not= \sg(vw)\f(w)$, where $\sg(vw)$ is the sign of the edge $vw$. For an integer $h\geq 0$, let $\Ga_{2h}=\{\pm1,\pm2, \ldots, \pm h\}$ and $\Ga_{2h+1}=\Ga_{2h} \cup \{0\}$. Following \cite{MaRS2015}, the signed chromatic number $\scn(G)$ of $G$ is the least integer $k$ such that $G$ admits a vertex coloring $\f$ with ${\rm im}(\f)\subseteq \Ga_k$. As proved in \cite{MaRS2015}, every signed graph $G$ satisfies $\scn(G)\leq \De(G)+1$ and there are three types of signed connected simple graphs for which equality holds. We will extend this Brooks' type result by considering graphs having multiple edges. We will also proof a list version of this result by characterizing degree choosable signed graphs. Furthermore, we will establish some basic facts about color critical signed graphs.