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Counting the number of Hamiltonian cycles that are contained in a geometricgraph 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 toallow the geometric graphs to be 1-plane, that is, each of its edges is crossedat 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 acomplete geometric graph $K_n\/$? We investigate the problem by taking twodifferent situations of $P\/$, namely, when $P\/$ is in convex position, wheelconfigurations position. For points in general position we prove the lowerbound of $k-1\/$ where $n=2^{k}+h\/$ and $0\leq h <2^{k}\/$. In all of thesituations, we investigate the constructions of the graphs obtained.

Packing 1-Plane Hamiltonian Cycles in Complete Geometric Graphs