Noncrossing Hamiltonian Paths in Geometric Graphs

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 a function h, where h(n) is the largest number k such that when we remove arbitrary set of k edges from a complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that $h(n)=\Omega(\sqrt{n})$. We also determine the function exactly in case when the removed edges form a star or a matching, and give asymptotically tight bounds in case they form a clique.