# Conflict-Free Coloring of Intersection Graphs of Geometric Objects

## TL;DR

This paper investigates conflict-free colorings of intersection graphs of geometric objects, establishing bounds for pseudo-discs and other classes, with implications for frequency assignment in wireless networks.

## Contribution

It proves that intersection graphs of pseudo-discs can be conflict-free colored with O(log n) colors and extends results to other geometric classes, strengthening previous findings.

## Key findings

- Conflict-free coloring of pseudo-discs requires O(log n) colors.
- The bound for pseudo-discs is asymptotically sharp.
- Results have applications in wireless frequency assignment.

## Abstract

In FOCS'2002, Even et al. introduced and studied the notion of conflict-free colorings of geometrically defined hypergraphs. They motivated it by frequency assignment problems in cellular networks. This notion has been extensively studied since then. A conflict-free coloring of a graph is a coloring of its vertices such that the neighborhood (pointed or closed) of each vertex contains a vertex whose color differs from the colors of all other vertices in that neighborhood. In this paper we study conflict-colorings of intersection graphs of geometric objects. We show that any intersection graph of n pseudo-discs in the plane admits a conflict-free coloring with O(\log n) colors, with respect to both closed and pointed neighborhoods. We also show that the latter bound is asymptotically sharp. Using our methods, we also obtain a strengthening of the two main results of Even et al. which we believe is of independent interest. In particular, in view of the original motivation to study such colorings, this strengthening suggests further applications to frequency assignment in wireless networks. Finally, we present bounds on the number of colors needed for conflict-free colorings of other classes of intersection graphs, including intersection graphs of axis-parallel rectangles and of \rho-fat objects in the plane.