Clique-relaxed graph coloring

Abstract

We define a generalization of the chromatic number of a graph $G$ called the $k$-clique-relaxed chromatic number, denoted ${\chi }^{\left(k\right)}\left(G\right)$. We prove bounds on ${\chi }^{\left(k\right)}\left(G\right)$ for all graphs $G$, including corollaries for outerplanar and planar graphs. We also define the $k$-clique-relaxed game chromatic number, ${\chi }_g^{\left(k\right)}\left(G\right)$, of a graph $G$. We prove ${\chi }_g^{\left(2\right)}\left(G\right)\le 4$ for all outerplanar graphs $G$, and give an example of an outerplanar graph $H$ with ${\chi }_g^{\left(2\right)}\left(H\right)\ge 3$. Finally, we prove that if $H$ is a member of a particular subclass of outerplanar graphs, then ${\chi }_g^{\left(2\right)}\left(H\right)\le 3$.