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.
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© 2001 Springer-Verlag Berlin Heidelberg
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Kaneko, A., Kano, M. (2001). On Paths in a Complete Bipartite Geometric Graph. In: Akiyama, J., Kano, M., Urabe, M. (eds) Discrete and Computational Geometry. JCDCG 2000. Lecture Notes in Computer Science, vol 2098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47738-1_17
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DOI: https://doi.org/10.1007/3-540-47738-1_17
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