Conflict-Free Coloring of Intersection Graphs of Geometric Objects

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 conflictfree 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 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, the set of discs in F that intersect B contains a uniquely-colored element. Moreover, such a coloring can be computed deterministically in polynomial time. 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 ρ-fat objects in the plane.

Non-Monochromatic and Conflict-Free Colorings on Tree Spaces and Planar Network Spaces

“…(If not, we can use a dummy color, thereby increasing the number of colors to four.) CF-coloring nodes with respect to neighborhoods has also been studied for various types of geometric intersection graphs; see the paper by Keller and Smorodinsky [11] and the references therein. As another example, we can let the hyperedges be induced by all the paths in the graph.…”

Conflict-free chromatic number vs conflict-free chromatic index

“…Such a generalization would not be straightforward, as our proof heavily relies on a specific structure of line graphs, hence this will require developing some new ideas. An especially promising subclass of K 1,r -free graphs is the class of intersection graphs of geometric objects (including, for example, unit disk graphs) -such graphs on n vertices have conflict-free chromatic number not greater than O (ln n), as proved by Keller and Smorodinsky [5], and improving the result to O(ln ∆) seems plausible.…”

Conflict-Free Coloring on Open Neighborhoods

“…In this paper, we look at the conflict-free open neighborhood problem, which is considered as the harder of the open and closed neighborhood variants, see for instance, remarks in [8,9]. It is easy to construct examples of bipartite graphs G, for which χ CF (G) is Θ( √ n).…”

“…Restrictions of the conflict-free coloring problem to special classes of graphs have been studied extensively. Of these, graphs arising out of intersection of geometric objects have attracted special interest, see for instance, [9,11,12]. The problem has also been studied for structural classes of graphs such as bipartite graphs and split graphs [5].…”