Skip to main content
SpringerLink
Log in

Enumerating Perfect Matchings in n-Cubes

  • Published:
Order Aims and scope Submit manuscript

Abstract

The perfect matchings in the n-cube have earlier been enumerated for n ≤ 6. A dynamic programming approach is here used to obtain the total number of perfect matchings in the 7-cube, which is 391 689 748 492 473 664 721 077 609 089. The number of equivalence classes of perfect matchings is further shown to be 336 in the 5-cube, 356 788 059 in the 6-cube and 607 158 046 495 120 886 820 621 in the 7-cube. The techniques used can be generalized to arbitrary bipartite and general graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Clark, L.H., George, J.C., Porter, T.D.: On the number of 1-factors in the n-cube. Congr. Numer. 127, 67–69 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Dinitz, J.H., Garnick, D.K., McKay, B.D.: There are 526,915,620 nonisomorphic one-factorizations of K 12. J. Comb. Des. 2, 273–285 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fukuda, K., Matsui, T.: Finding all the perfect matchings in bipartite graphs. Appl. Math. Lett. 7, 15–18 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Godsil, C.D.: Algebraic Combinatorics. Chapman & Hall, New York (1993)

    MATH  Google Scholar 

  5. Graham, N., Harary, F.: The number of perfect matchings in a hypercube. Appl. Math. Lett. 1, 45–48 (1988)

    Article  MathSciNet  Google Scholar 

  6. Haanpää, H., Östergård, P.R.J.: Counting Hamiltonian cycles in bipartite graphs.Math. Comput. (2012, in press)

  7. Hulpke, A., Kaski, P., Östergård P.R.J.: The number of Latin squares of order 11. Math. Comput. 80, 1197–1219 (2011)

    Article  MATH  Google Scholar 

  8. Kaski, P., Östergård, P.R.J.: There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations of K 14. J. Comb. Des. 17, 147–159 (2009)

    Article  MATH  Google Scholar 

  9. Kaski, P., Pottonen, O.: Libexact user’s guide, version 1.0. Technical Report HIIT TR 2008-1, Helsinki Institute for Information Technology HIIT, Helsinki (2008)

  10. Lovász, L., Plummer, M.D.: Matching Theory. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  11. Lundow, P.H.: Computation of matching polynomials and the number of 1-factors in polygraphs. Research Reports No. 12, Department of Mathematics, Umeå University (1996)

  12. McKay, B.D., Wanless, I.M.: On the number of Latin squares. Ann. Comb. 9, 335–344 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Östergård, P.R.J.: Classification of binary constant weight codes. IEEE Trans. Inf. Theory 56, 3779–3785 (2010)

    Article  Google Scholar 

  14. Propp, J.: Enumeration of matchings: problems and progress. In: Billera, L.J., Björner, A., Greene, C., Simion, R.E., Stanley, R.P. (eds.) New Perspectives in Algebraic Combinatorics, pp. 255–291. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  15. Uno, T.: Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs. In: Leong, H.W., Imai, H., Jain, S. (eds.) Algorithms and Computation. LNCS 1350, pp. 92–101. Springer-Verlag, Berlin (1997)

    Google Scholar 

  16. Uno, T.: A fast algorithm for enumerating bipartite perfect matchings. In: Eades, P., Takaoka, T. (eds.) Algorithms and Computation. LNCS 2223, pp. 367–379. Springer-Verlag, Berlin (2001)

    Google Scholar 

  17. Valiant, L.G.: The complexity of computing the permanent. Theor. Comp. Sci. 56, 189–201 (1979)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076, Aalto, Finland

    Patric R. J. Östergård & Ville H. Pettersson

Authors
  1. Patric R. J. Östergård
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ville H. Pettersson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ville H. Pettersson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Östergård, P.R.J., Pettersson, V.H. Enumerating Perfect Matchings in n-Cubes. Order 30, 821–835 (2013). https://doi.org/10.1007/s11083-012-9279-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-012-9279-8

Keywords

Search

Navigation