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Ralucca Gera, Naval Postgraduate School, Monterey, CA 93943, USA, e-mail: rgera@nps.edu; Futaba Okamoto, University of Wisconsin - LaCrosse, LaCrosse, WI 54601, USA, e-mail: okamoto.futa@uwlax.edu; Craig Rasmussen (corresponding author), Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943, USA, e-mail: ras@nps.edu; Ping Zhang, Western Michigan University, Kalamazoo, MI 49008, USA, e-mail: ping.zhang@wmich.edu
Abstract: For a nontrivial connected graph $G$, let $c V(G)\rightarrow\mathbb{N}$ 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 $\mathop NC(v)$ is the set of colors of the neighbors of $v$. The coloring $c$ is called a set coloring if $\mathop NC(u)\neq\mathop NC(v)$ 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 $\chi_ s(G)$. We show that the decision variant of determining $\chi_ s(G)$ is NP-complete in the general case, and show that $\chi_ s(G)$ can be efficiently calculated when $G$ is a threshold graph. We study the difference $\chi(G)-\chi_ s(G)$, 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 $\chi_ s(G)$ and $\chi_ s({\overline G})$.
Keywords: set coloring, perfect graph, NP-completeness
Classification (MSC 2010): 05C15, 05C17, 05C35, 05C70
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