Number of Crossings in Hamiltonian Cycles on Given <i>n</i> Points

Let P be a set of n points in general position on the plane. We consider the maximum number of crossings of edges in geometric Hamiltonian cycles on P with straight-line edges. Álvarez-Rebollar et al. proved that for any P, there is a Hamiltonian cycle with at least n2/12-O(n) crossings. In this paper, we give a lower bound 4crP(Kn)/(n-1)(n-2) in terms of the number crP(Kn) of crossings of edges in the complete graph Kn on P. We also give lower bounds of the maximum numbers of mutual crossings, and total number of both self-crossings and mutual crossings for k edge-disjoint Hamiltonian cycles on P.