The chromatic spectrum of signed graphs

The chromatic number χ((G,σ))χ((G,σ)) of a signed graph (G,σ)(G,σ) is the smallest number kk for which there is a function c:V(G)→Zkc:V(G)→Zk such that c(v)≠σ(e)c(w)c(v)≠σ(e)c(w) for every edge e=vwe=vw. Let Σ(G)Σ(G) be the set of all signatures of GG. We study the chromatic spectrum Σχ(G)={χ((G,σ)):σ∈Σ(G)} of (G,σ)(G,σ). Let Mχ(G)=max{χ((G,σ)):σ∈Σ(G)}, and mχ(G)=min{χ((G,σ)):σ∈Σ(G)}. We show that Σχ(G)={k:mχ(G)≤k≤Mχ(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 Máčajová, Raspaud, and Škoviera in Máčajová et al. (2016).