Set colorings in perfect graphs

Gera, Ralucca ; Okamoto, Futaba ; Rasmussen, Craig ; Zhang, Ping

Mathematica Bohemica, Tome 136 (2011) no. 1, pp. 61-68.

Résumé

For a nontrivial connected graph $G$, let $c\colon V(G)\rightarrow \mathbb {N}$ be a vertex coloring of $G$ where adjacent vertices may be colored the same. For a vertex $v \in V(G)$, the neighborhood color set $\mathop {\rm NC}(v)$ is the set of colors of the neighbors of $v$. The coloring $c$ is called a set coloring if $\mathop {\rm NC}(u)\neq \mathop {\rm NC}(v)$ for every pair $u, v$ of adjacent vertices of $G$. The minimum number of colors required of such a coloring is called the set chromatic number $\chi _{\rm s}(G)$. We show that the decision variant of determining $\chi _{\rm s}(G)$ is NP-complete in the general case, and show that $\chi _{\rm s}(G)$ can be efficiently calculated when $G$ is a threshold graph. We study the difference $\chi (G)-\chi _{\rm s}(G)$, presenting new bounds that are sharp for all graphs $G$ satisfying $\chi (G)=\omega (G)$. We finally present results of the Nordhaus-Gaddum type, giving sharp bounds on the sum and product of $\chi _{\rm s}(G)$ and $\chi _{\rm s}({\overline G})$.

Détail

DOI : 10.21136/MB.2011.141450

Classification : 05C15, 05C17, 05C35, 05C70
Keywords: set coloring; perfect graph; NP-completeness