Clique coloring $B_1$-EPG graphs

TL;DR

This paper proves that $B_1$-EPG graphs can be colored with four colors to avoid monochromatic maximal cliques and provides a linear-time algorithm to find such a coloring, extending clique coloring results.

Contribution

It establishes that $B_1$-EPG graphs are 4-clique colorable and offers a linear-time algorithm to produce such a coloring from a given representation.

Findings

$B_1$-EPG graphs are 4-clique colorable.

A linear-time algorithm constructs the 4-clique coloring.

Extension of clique coloring results to $B_1$-EPG graphs.

Abstract

We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are $2$-clique colorable. In this paper we prove that $B_1$-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are $4$-clique colorable. Moreover, given a $B_1$-EPG representation of a graph, we provide a linear time algorithm that constructs a $4$-clique coloring of it.