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The chromatic spectrum of signed graphs. (English) Zbl 1339.05169
Summary: The chromatic number \(\chi((G, \sigma))\) of a signed graph \((G, \sigma)\) is the smallest number \(k\) for which there is a function \(c : V(G) \to \mathbb{Z}_k\) such that \(c(v) \neq \sigma(e) c(w)\) for every edge \(e = v w\). Let \(\varSigma(G)\) be the set of all signatures of \(G\). We study the chromatic spectrum \(\varSigma_\chi(G) = \{\chi((G, \sigma)) : \sigma \in \varSigma(G) \}\) of \((G, \sigma)\). Let \(M_\chi(G) = \max \{\chi((G, \sigma)) : \sigma \in \varSigma(G) \}\), and \(m_\chi(G) = \min \{\chi((G, \sigma)) : \sigma \in \varSigma(G) \}\). We show that \(\varSigma_\chi(G) = \{k : m_\chi(G) \leq k \leq M_\chi(G) \}\). 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 E. Máčajová et al. [Electron. J. Comb. 23, No. 1, Research Paper P1.14, 10 p. (2016; Zbl 1329.05116)].

MSC:
05C22 Signed and weighted graphs
05C15 Coloring of graphs and hypergraphs
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References:
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