Properly Colored Geometric Matchings and 3-Trees Without Crossings on Multicolored Points in the Plane

Let X be a set of multicolored points in the plane such that no three points are collinear and each color appears on at most ⌈|X|/2⌉ points. We show the existence of a non-crossing properly colored geometric perfect matching on X (if |X| is even), and the existence of a non-crossing properly colored geometric spanning tree with maximum degree at most 3 on X. Moreover, we show the existence of a non-crossing properly colored geometric perfect matching in the plane lattice. In order to prove these our results, we propose an useful lemma that gives a good partition of a sequence of multicolored points. Keyword(s): red and blue points, multicolored points, alternating matching, alternating tree, properly colored geometric graph, sequence of points. MSC2010: 52C35, 05C70, 05C05.