Counting Hamiltonian cycles in bipartite graphs

Abstract

A method for counting Hamiltonian cycles in bipartite graphs is developed with the main focus on the long-standing open case of the 6-cube. Dynamic programming as well as utilization of the automorphism group of the graph are central ingredients of the method. It is further shown how the number of equivalence classes of Hamiltonian cycles can be obtained via a classification of Hamiltonian cycles with nontrivial automorphisms. It turns out that the 6-cube has 35 838 213 722 570 883 870 720 Hamiltonian cycles and 777 739 016 577 752 714 equivalence classes of Hamiltonian cycles. The old result on the number of knight's tours on a $8\times 8$ chessboard is confirmed.