Let @math and @math be a set of red points and a set of blue points in the plane, respectively, such that @math is in general position, and let @math be a function. We show that if @math , then there exists a non-crossing geometric spanning tree @math on @math such that @math for every @math and the set of leaves of @math is @math , where every edge of @math is a straight-line segment.
In this paper, we give a short survey on discrete geometry on red and blue points in the plane, most of whose results were obtained in the past decade. We consider balanced subdivision problems, geometric graph problems, graph embedding problems, Gallai-type problems and others. 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. 