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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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