Coloring Kk-free intersection graphs of geometric objects in the plane

• Authors: Jacob Fox;János Pach
• Venue: European Journal of Combinatorics
• Year: 2012

The intersection graph of a collection C of sets is the graph on the vertex set C, in which C"1,C"2@?C are joined by an edge if and only if C"1@?C"20@?. Erdos conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant. Here we show that it is bounded by a polylogarithmic function of n, which is the first nontrivial bound for this problem. More generally, we prove that for any t and k, the chromatic number of every K"k-free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most (c"tlognlogk)^c^l^o^g^k, where c is an absolute constant and c"t only depends on t. We establish analogous results for intersection graphs of convex sets, x-monotone curves, semialgebraic sets of constant description complexity, and sets that can be obtained as the union of a bounded number of sets homeomorphic to a disk. Using a mix of results on partially ordered sets and planar separators, for large k we improve the best known upper bound on the number of edges of a k-quasi-planar topological graph with n vertices, that is, a graph drawn in the plane with curvilinear edges, no k of which are pairwise crossing. As another application, we show that for every @e0 and for every positive integer t, there exist @d0 and a positive integer n"0 such that every topological graph with n=n"0 vertices, at least n^1^+^@e edges, and no pair of edges intersecting in more than t points, has at least n^@d pairwise intersecting edges.