The chromatic spectrum of signed graphs

Abstract

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) → Zk such that c(v) ≠ 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 Σχ (G) = {χ((G, σ )): σ ∈ Σ(G)} of (G, σ ). LetMχ (G) = max{χ((G, σ )): σ ∈ Σ(G)}, andmχ (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). © 2016 Elsevier B.V. All rights reserved.

DOI: 10.1016/j.disc.2016.05.013

Cite this paper

@article{Kang2016TheCS, title={The chromatic spectrum of signed graphs}, author={Ying-li Kang and Eckhard Steffen}, journal={Discrete Mathematics}, year={2016}, volume={339}, pages={2660-2663} }