On Paths in a Complete Bipartite Geometric Graph
نویسندگان
چکیده
Let A and B be two disjoint sets of points in the plane such that no three points of A ∪ B 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 |B| = n2 16 + n 2 − 1 such that in K(A, B) every path containing the set A has at least one crossing.
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