Conflict-Free Coloring of Intersection Graphs

TL;DR

This paper investigates conflict-free coloring in geometric intersection graphs, revealing complexity results, bounds on the number of colors needed, and proposing algorithms and conjectures for specific graph classes.

Contribution

It provides new bounds, complexity results, and algorithms for conflict-free coloring in geometric intersection graphs, including NP-completeness and tight bounds for certain classes.

Findings

Conflict-free chromatic number can be as high as (rac{\u221a{ }n}) for some geometric graphs.

Deciding one-color conflict-free coloring in unit-disk graphs is NP-complete.

Six colors suffice for unit disk graphs of restricted height, with a conjecture that four are enough.

Abstract

A conflict-free $k$-coloring of a graph $G=(V,E)$ assigns one of $k$ different colors to some of the vertices such that, for every vertex $v$, there is a color that is assigned to exactly one vertex among $v$ and $v$'s neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well studied in graph theory. Here we study the conflict-free coloring of geometric intersection graphs. We demonstrate that the intersection graph of $n$ geometric objects without fatness properties and size restrictions may have conflict-free chromatic number in $\Omega(\log n/\log\log n)$ and in $\Omega(\sqrt{\log n})$ for disks or squares of different sizes; it is known for general graphs that the worst case is in $\Theta(\log^2 n)$. For unit-disk intersection graphs, we prove that it is NP-complete to decide the existence of a conflict-free coloring with one color; we also show that six colors always suffice, using an algorithm that colors unit disk graphs of restricted height with two colors. We conjecture that four colors are sufficient, which we prove for unit squares instead of unit disks. For interval graphs, we establish a tight worst-case bound of two.