Packing 1-plane Hamiltonian cycles in complete geometric graphs

Abstract

Counting the number of Hamiltonian cycles that are contained in a geometric graph is #Pcomplete even if the graph is known to be planar. A relaxation for problems in plane geometric graphs is to allow the geometric graphs to be 1-plane, that is, each of its edges is crossed at most once. We consider the following question: For any set P of n points in the plane, how many 1-plane Hamiltonian cycles can be packed into a complete geometric graph K n ? We investigate the problem by taking three different situations of P, namely, when P is in convex position and when P is in wheel configurations position. Finally, for points in general position we prove the lower bound of k − 1 where n = 2 k + h and 0 ≤ h < 2 k. In all of the situations, we investigate the constructions of the graphs obtained.