A Simple Algorithm for Coloring m-Clique Holes

An m-clique hole is a sequence $\phi=(\Phi_1,\Phi_2,\dots,\Phi_m)$ of $m$ distinct cliques such that $|\Phi_i| \leq m$ for all $i=1,2,\ldots,m$, and whose clique graph is a hole on $m$ vertices. That is, $\phi$ is an m-clique hole if for all $i\neq j$, $i,j=1,2,\ldots,m$, $\Phi_i \cap \Phi_{j} \neq \emptyset$ if and only if $(j-1)~\mbox{mod}~m = (j+1)~\mbox{mod}~m = i~\mbox{mod}~m$. This paper derives a sufficient and necessary condition on m-colorability of m-clique holes, and proposes a coloring algorithm that colors m-clique holes with exactly m colors.