Conflict-Free Coloring of Intersection Graphs of Geometric Objects

Abstract

In 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-free 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 obtain the following strengthening of the two main results of Even et al.: Any family $\mathcal {F}$ of n discs in the plane can be colored with $O(\log n)$ colors in such a way that for any disc B in the plane, not necessarily from $\mathcal {F}$, the set of discs in $\mathcal {F}$ that intersect B contains a uniquely-colored element. 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.