A Simple Algorithm for Coloring m-Clique Holes

TL;DR

This paper establishes a precise condition for coloring m-clique holes with m colors and introduces a simple algorithm to achieve this coloring efficiently.

Contribution

It provides a necessary and sufficient condition for m-colorability of m-clique holes and proposes a straightforward coloring algorithm that uses exactly m colors.

Findings

The algorithm guarantees m-coloring for all m-clique holes.

A characterization of m-clique holes' colorability is provided.

The method simplifies coloring complex clique structures.

Abstract

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.