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set coloring; perfect graph; NP-completeness
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 $\chi _{\rm s}({\overline G})$.
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