Conflict-Free Coloring: Graphs of Bounded Clique Width and Intersection Graphs

TL;DR

This paper studies conflict-free coloring in various graph classes, analyzing how clique-width and intersection properties influence the minimum number of colors needed, providing bounds and complexity results.

Contribution

It establishes that clique-width and the minimum number of colors do not bound each other and solves an open problem for interval graphs.

Findings

Clique-width and coloring number are independent parameters.

Determined complexity of conflict-free coloring for multiple graph classes.

Resolved an open problem regarding interval graphs.

Abstract

A conflict-free coloring of a graph $G$ is a (partial) coloring of its vertices such that every vertex $u$ has a neighbor whose assigned color is unique in the neighborhood of $u$. There are two variants of this coloring, one defined using the open neighborhood and one using the closed neighborhood. For both variants, we study the problem of deciding whether the conflict-free coloring of a given graph $G$ is at most a given number $k$. In this work, we investigate the relation of clique-width and minimum number of colors needed (for both variants) and show that these parameters do not bound one another. Moreover, we consider specific graph classes, particularly graphs of bounded clique-width and types of intersection graphs, such as distance hereditary graphs, interval graphs and unit square and disk graphs. We also consider Kneser graphs and split graphs. We give (often tight) upper and lower bounds and determine the complexity of the decision problem on these graph classes, which improve some of the results from the literature. Particularly, we settle the number of colors needed for an interval graph to be conflict-free colored under the open neighborhood model, which was posed as an open problem.