Delaunay Graphs of Point Sets in the Plane With Respect to Axis-Parallel Rectangles

Given a point set P in the plane, the Delaunay graph with respect to axis-parallel rectangles is a graph defined on the vertex set P, whose two points p, q is an element of P are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. (SIAM J Comput 33 (2003) 94-136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of it points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(n log(2) log n/ log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of it points almost surely has an independent set of size at least cn log log n/(log n log log log it). We give two further applications of our methods: (1) We construct two-dimensional n-element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matousek and Privetivy (Combinat Probab Comput 15 (2006) 473-475) and improves a result of Kriz and Nesetril (Order 8 (1991) 41-48). (2) For my positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis-parallel rectangle containing at least d points. all of which have the same color. This solves an old problem from the work of Brass et al. (Research Problem in Discrete Geometry Springer-Verlag, New York, 2005). (c) 2008 Wiley Periodicals, Inc. Random Struct. Alg., 34 11-23, 2009