Low dimensional middle cubes are Hamiltonian

  • Kunwarjit S. Bagga
  • Frank W. Owens
Track1: Graph Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 507)


Let Qn be the n-dimensional hypercube. If n=2k+1, then the subgraph Mn of Qn induced by the nodes having exactly k or k + 1 ones is called the middle cube of dimension n. A famous conjecture is that Mn is hamiltonian if n>1.

Clearly, M3 has two oriented hamiltonian cycles. This paper includes the following new results:
  1. 1.

    The number of oriented hamiltonian cycles of Mn is divisible by k!(k + 1)!.

  2. 2.

    M5 has exactly 48 oriented hamiltonian cycles.

  3. 3.

    M7 is hamiltonian.

  4. 4.

    Some interesting new combinatorial identities which are relevant to the structure of Mn.



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  1. [1]
    Dwight Duffus, Bill Sands and Robert E. Woodrow. Lexicographic Matchings Cannot Form Hamiltonian Cycles. Abstracts of Contributed Talks, Eleventh British Combinatorial Conference, University of London, Goldsmiths' College, 13–17 July 1987.Google Scholar
  2. [2]
    Niall Graham. center of Q hamiltonian?. E-mail message from to 00ksbagga@bsuvax1.bitnet sent Friday, 1 December 1989, 13:22:05 MST.Google Scholar
  3. [3]
    Raymond E. Pippert. Personal communication, May 1989.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Kunwarjit S. Bagga
    • 1
  • Frank W. Owens
    • 1
  1. 1.Department of Computer ScienceBall State UniversityMuncie

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