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

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