Connecting Colored Point Sets # Oswin Aichholzer + Franz Aurenhammer # Thomas Hackl Clemens Huemer Summary: Connecting Colored Point Sets # Oswin Aichholzer + Franz Aurenhammer # Thomas Hackl § Clemens Huemer ¶ Abstract We study the following Ramsey­type problem. Let S = B # R be a two­colored set of n points in the plane. We show how to construct, in O(n log n) time, a crossing­free spanning tree T (B) for B, and a crossing­free spanning tree T (R) for R, such that both the number of crossings between T (B) and T (R) and the diameters of T (B) and T (R) are kept small. The algorithm is conceptually simple and is implementable without using any non­trivial data structure. This improves over a previous method in Tokunaga [16] that is less e#cient in implementation and does not guarantee a diameter bound. Implicit to our approach is a new proof for the result in [16] on the minimum number of crossings between T (B) and T (R). 1 Introduction Let S be a set of n points in general position in the plane. Consider an arbitrary two­coloring of S, that is, S is the disjoint union of B and R such that each point in B is colored blue and each point in R is colored red. This two­coloring induces an edge coloring of the complete geometric straight­ line graph K(S) spanned by S, in the following way: Edges that connect two points of the same color are given this color, and all other edges of K(S) are given a third color, say green. Finding monochromatic subgraphs of K(S) with special properties is a topic of classical Ramsey theory. For example, there always exists a crossing­free perfect matching that uses only green edges, provided |R| = |B|; see [14]. Or, for every (not necessarily induced) two­coloring of the edges of K(S) there Collections: Computer Technologies and Information Sciences