A generalization of Noel-Reed-Wu Theorem to signed graphs

TL;DR

This paper extends the Noel-Reed-Wu Theorem to signed graphs, proving that small signed graphs with at most one more vertex than their zero-free chromatic number are zero-free chromatic-choosable, thus generalizing classical graph coloring results.

Contribution

It generalizes the Noel-Reed-Wu Theorem to signed graphs, establishing conditions for zero-free chromatic-choosability based on the number of vertices.

Findings

Signed graphs with at most $ ext{chi}^*( extSigma)+1$ vertices are zero-free chromatic-choosable.

The result strengthens the classical theorem for unsigned graphs to the context of signed graphs.

Provides a new link between graph size and list coloring properties in signed graph theory.

Abstract

Let $\Sigma$ be a signed graph where two edges joining the same pair of vertices with opposite signs are allowed. The zero-free chromatic number $\chi^*(\Sigma)$ of $\Sigma$ is the minimum even integer $2k$ such that $G$ admits a proper coloring $f\colon\,V(\Sigma)\mapsto \{\pm 1,\pm 2,\ldots,\pm k\}$. The zero-free list chromatic number $\chi^*_l(\Sigma)$ is the list version of zero-free chromatic number. $\Sigma$ is called zero-free chromatic-choosable if $\chi^*_l(\Sigma)=\chi^*(\Sigma)$. We show that if $\Sigma$ has at most $\chi^*(\Sigma)+1$ vertices then $\Sigma$ is zero-free chromatic-choosable. This result strengthens Noel-Reed-Wu Theorem which states that every graph $G$ with at most $2\chi(G)+1$ vertices is chromatic-choosable, where $\chi(G)$ is the chromatic number of $G$.