Discrete Geometry on Red and Blue Points in the Plane Lattice

Abstract

We consider some problems on red and blue points in the plane lattice. An L-line segment in the plane lattice consists of a vertical line segment and a horizontal line segment having a common endpoint. There are some results on geometric graphs on a set of red and blue points in the plane. We show that some similar results also hold for a set of red and blue points in the plane lattice using L-line segments instead of line segments. For example, we show that if n red points and n blue points are given in the plane lattice in general position, then there exists a noncrossing geometric perfect matching covering them, each of whose edges is an L-line segment and connects a red point and a blue point.