Ramsey-type results for geometric graphs

Abstract

Given a geometric graph, i.e., a collection of segments (edges) between n points in the plane, does it contain a non-crossing configuration of a certain type? It is widely conjectured that all such problems are NP-hard. This has been verified in many special cases, including the existence of a non-crossing spanning tree or k disjoint segments. Here we show that these problems become computationally simpler if we are allowed to choose where to find a non-crossing spanning tree (or k disjoint edges): in the graph or in its complement. We prove that for any 2-coloring of the (2n) segments determined by n points in the plane, at least one of the color classes contains a non-crossing spanning tree, and it can be found in O(nlog log n + O(1)) time. Under the same assumptions, we also prove that there exist [n+1/3] pairwise disjoint segments of the same color, and they can be found with the same efficiency. The non-algorithmic parts of the above theorems were conjectured by Bialostocki and Dierker. Improving an earlier result of Larman et al., we construct a family of m segments in the plane, which has no more than mlog 4/log 27 members that are either pairwise disjoint or pairwise crossing. Finally, we discuss some related problems and generalizations.