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Computational Geometry: Theory and Applications
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Let A and B be two disjoint sets of points in the plane such that no three points of A 驴 B arc 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| 驴 (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 驴 B with |B| = n2/16 + n/2 - 1 such that in K(A, B) every path containing the set A has at least one crossing.