The chromatic spectrum of signed graphs

Abstract: 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.