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 $c : V(G) \rightarrow \mathbb{Z}_k$ such that $c(v) \not= \sigma(e) c(w)$ for every edge $e = vw$. Let $\Sigma(G)$ be the set of all signatures of $G$. We study the chromatic spectrum $\Sigma_{\chi}(G) = \{\chi((G,\sigma))\colon\ \sigma \in \Sigma(G)\}$ of $(G,\sigma)$. Let $M_{\chi}(G) = \max\{\chi((G,\sigma))\colon\ \sigma \in \Sigma(G)\}$, and $m_{\chi}(G) = \min\{\chi((G,\sigma))\colon\ \sigma \in \Sigma(G)\}$. We show that $\Sigma_{\chi}(G) = \{k : m_{\chi}(G) \leq k \leq M_{\chi}(G)\}$. 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.