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 sqrt(2)) sqrt(n). We also establish several results related to special classes of geometric graphs. Let h1 (n) denote the largest number such that when we remove edges of an arbitrary complete subgraph of size at most h1 (n) from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We prove that frac(1, sqrt(2)) sqrt(n) < 3 sqrt(n). Let h2 (n) denote the largest number such that when we remove an arbitrary star with at most h2 (n) edges from a complete geometric graph on n vertices the resulting graph still has a noncrossing Hamiltonian path. We show that h2 (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. © 2006 Elsevier B.V. All rights reserved.