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Mathematics of Computation

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Counting Hamiltonian cycles in bipartite graphs
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by Harri Haanpää and Patric R. J. Östergård PDF
Math. Comp. 83 (2014), 979-995 Request permission

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.

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Additional Information
  • Harri Haanpää
  • Affiliation: Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076 Aalto, Finland
  • Patric R. J. Östergård
  • Affiliation: Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076 Aalto, Finland
  • Received by editor(s): November 7, 2011
  • Received by editor(s) in revised form: March 12, 2012, and June 25, 2012
  • Published electronically: July 15, 2013
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 979-995
  • MSC (2010): Primary 05C45, 05C85, 05C30, 05C38, 68R10
  • DOI: https://doi.org/10.1090/S0025-5718-2013-02741-X
  • MathSciNet review: 3143701