Noncrossing Hamiltonian paths in geometric graphs

Abstract

A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: determine the largest number h ( n ) such that when we remove any set of h ( n ) edges from any complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that h ( n ) ⩾ ( 1 / 2 2 ) n . We also establish several results related to special classes of geometric graphs. Let h 1 ( n ) denote the largest number such that when we remove edges of an arbitrary complete subgraph of size at most h 1 ( n ) from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We prove that 1 2 n < h 1 ( n ) < 3 n . Let h 2 ( n ) denote the largest number such that when we remove an arbitrary star with at most h 2 ( n ) edges from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We show that h 2 ( n ) = ⌈ n / 2 ⌉ - 1 . Further we prove that when we remove any matching from a complete geometric graph the resulting graph will have a noncrossing Hamiltonian path.