Clique-coloring of $K_{3,3}$-minor free graphs

TL;DR

This paper extends known clique-coloring results from planar graphs to the broader class of $K_{3,3}$-minor free graphs, showing similar coloring bounds.

Contribution

It generalizes clique-coloring bounds from planar graphs to $K_{3,3}$-minor free graphs, broadening the applicability of these coloring results.

Findings

Every $K_{3,3}$-minor free graph is 3-clique colorable.

The results extend clique-coloring bounds from planar graphs to more general graph classes.

Provides new insights into clique-coloring properties of minor-closed graph families.

Abstract

A clique-coloring of a given graph $G$ is a coloring of the vertices of $G$ such that no maximal clique of size at least two is monocolored. The clique-chromatic number of $G$ is the least number of colors for which $G$ admits a clique-coloring. It has been proved that every planar graph is $3$-clique colorable and every claw-free planar graph, different from an odd cycle, is $2$-clique colorable. In this paper, we generalize these results to $K_{3,3}$-minor free ($K_{3,3}$-subdivision free) graphs.