Packing 1-plane Hamiltonian cycles in complete geometric graphs

• Main Authors: Trao, Hazim Michman, Ali, Niran Abbas, Chia, Gek L., Kilicman, Adem
• Format: Article
• Language: English
• Published: Faculty of Sciences and Mathematics, University of Nis 2019
• Online Access: http://psasir.upm.edu.my/id/eprint/81605/1/PLANE.pdf; http://psasir.upm.edu.my/id/eprint/81605/; http://journal.pmf.ni.ac.rs/filomat/index.php/filomat/article/view/6771

Counting the number of Hamiltonian cycles that are contained in a geometric graph is #P-complete even if the graph is known to be planar [15]. 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 Kn? 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 = 2k + h and 0 ≤ h < 2k. In all of the situations, we investigate the constructions of the graphs obtained.