On Paths in a Complete Bipartite Geometric Graph

Abstract

Let A and B be two disjoint sets of points in the plane such that no three points of A ∪ U are collinear, and let n be the number of points in A. A geometric complete bipartite graph K(A, B) is a complete bipartite graph with partite sets A and B which is drawn in the plane such that each edge of K(A, B) is a straight-line segment. We prove that (i) If |B (n + 1)(2n - 4)+1, then the geometric complete bipartite graph K(A, B) contains a path that passes through all the points in A and has no crossings; and (ii) There exists a configuration of A ∪ B with $\left| B \right| = \tfrac{{n^2 }} {{16}} + \tfrac{n} {2} - 1$ such that in K(A, B) every path containing the set A has at least one crossing.