The chromatic spectrum of signed graphs

The chromatic number $\chi ((G,\sigma ))$ of a signed graph $(G,\sigma )$ is the smallest number $k$ for which there is a function $c : V(G) \rightarrow $\mathbb{Z}$k$suchthat$c(v) \not= \sigma(e) c(w)$foreveryedge$e = vw$. Let $\mathrm{\Sigma }(G)$ be the set of all signatures of $G$. We study the chromatic spectrum ${\mathrm{\Sigma }}_{\chi }(G)=\{\chi ((G,\sigma ))\text{\hspace{0.17em}⁣}:\sigma \in \mathrm{\Sigma }(G)\}$ of $(G,\sigma )$. Let $M{\chi}(G) = \max{\chi((G,\sigma))\colon\ \sigma \in \Sigma(G)}$,and$m_{\chi}(G) = \min{\chi((G,\sigma))\colon\ \sigma \in \Sigma(G)}$\mathrm{.}Weshowthat$\Sigma_{\chi}(G) = {k : m_{\chi}(G) \leq k \leq M_{\chi}(G)}$. We also prove some basic facts for critical graphs. Analogous results are obtained for a notion of vertex-coloring of signed graphs which was introduced by M'{a}\v{c}ajov'{a}, Raspaud, and \v{S}koviera.