A Simple Algorithm for Coloring m-Clique Holes

Abstract

An m-clique hole is a sequence $\varphi =({\Phi }_1,{\Phi }_2,\dots ,{\Phi }_m)$ of $m$ distinct cliques such that $|{\Phi }_i|\le m$ for all $i=1,2,\dots ,m$, and whose clique graph is a hole on $m$ vertices. That is, $\varphi$ is an m-clique hole if for all $i\ne j$, $i,j=1,2,\dots ,m$, ${\Phi }_i\cap {\Phi }_j\ne \varnothing$ 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.