Packing 1-Plane Hamiltonian Cycles in Complete Geometric Graphs

Counting the number of Hamiltonian cycles that are contained in a geometric graph is {\bf \#P}-complete even if the graph is known to be planar \cite{lot:refer}. 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 two different situations of $P\/$, namely, when $P\/$ is in convex position, wheel configurations position. For points in general position we prove the lower bound of $k-1\/$ where $n=2^{k}+h\/$ and $0\leq h <2^{k}\/$. In all of the situations, we investigate the constructions of the graphs obtained.