The chromatic spectrum of signed graphs

The chromatic number $χ((G,σ))$ of a signed graph $(G,σ)$ is the smallest number $k$ for which there is a function $c : V(G) \rightarrow \mathbb{Z}_k$ such that $c(v) \not= σ(e) c(w)$ for every edge $e = vw$. Let $Σ(G)$ be the set of all signatures of $G$. We study the chromatic spectrum $Σ_χ(G) = \{χ((G,σ))\colon\ σ\in Σ(G)\}$ of $(G,σ)$. Let $M_χ(G) = \max\{χ((G,σ))\colon\ σ\in Σ(G)\}$, and $m_χ(G) = \min\{χ((G,σ))\colon\ σ\in Σ(G)\}$. We show that $Σ_χ(G) = \{k : m_χ(G) \leq k \leq M_χ(G)\}$. 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.

No reviews yet.

No comments yet.