On Paths in a Complete Bipartite Geometric Graph [chapter]

Atsushi Kaneko, M. Kano
2001 Lecture Notes in Computer Science  
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
more » ... o crossings; and (ii) There exists a configuration of A ∪ B with |B| = n 2 16 + n 2 − 1 such that in K(A, B) every path containing the set A has at least one crossing.
doi:10.1007/3-540-47738-1_17 fatcat:3btm6riw4rd55atqv55erxzdde