Number of 1-Factorizations of Regular High-Degree Graphs


A 1-factor in an n-vertex graph G is a collection of \(\frac{n}{2}\) vertex-disjoint edges and a 1-factorization of G is a partition of its edges into edge-disjoint 1-factors. Clearly, a 1-factorization of G cannot exist unless n is even and G is regular (that is, all vertices are of the same degree). The problem of finding 1-factorizations in graphs goes back to a paper of Kirkman in 1847 and has been extensively studied since then. Deciding whether a graph has a 1-factorization is usually a very difficult question. For example, it took more than 60 years and an impressive tour de force of Csaba, Kühn, Lo, Osthus and Treglown to prove an old conjecture of Dirac from the 1950s, which says that every d-regular graph on n vertices contains a 1-factorization, provided that n is even and \(d \geqslant 2\left[ {\frac{n}{4}} \right] - 1\). In this paper we address the natural question of estimating F(n, d), the number of 1-factorizations in d-regular graphs on an even number of vertices, provided that \(d \geqslant \left[ {\frac{n}{2}} \right] + \varepsilon n\). Improving upon a recent result of Ferber and Jain, which itself improved upon a result of Cameron from the 1970s, we show that \(F\left( {n,\,d} \right) \geqslant {\left( {\left( {1 + o\left( 1 \right)} \right)\frac{d}{{{e^2}}}} \right)^{nd/2}}\), which is asymptotically best possible.

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


  1. [1]

    N. Alon and S. Friedland: The maximum number of perfect matchings in graphs with a given degree sequence, Electron. J. Combin.15 (2008).

  2. [2]

    N. Alon and J. Spencer: The Probabilistic Method, 3rd ed., John Wiley and Sons (2008).

    Google Scholar 

  3. [3]

    L. M. Bregman: Some properties of non-negative matrices and their permanents, Sov. Mat. Dokl.14 (1973), 945–949.

    MATH  Google Scholar 

  4. [4]

    P. J. Cameron: Parallelisms of Complete Designs, London Math. Soc. Lecture Note Ser. 23., Cambridge Univ. Press, Cambridge (1976).

    Google Scholar 

  5. [5]

    B. Csaba, D. Kühn, A. Lo, D. Osthus and A. Treglown: Proof of the 1-factorization and Hamilton decomposition conjectures, Mem. Amer. Math. Soc.244 (2016), monograph 1154.

    MathSciNet  Article  Google Scholar 

  6. [6]

    D. Dubhashi, D. A. Grable and A. Panconesi: Near-optimal, distributed edge colouring via the nibble method, Theoret. Comput. Sci.203 (1998), 225–252.

    MathSciNet  Article  Google Scholar 

  7. [7]

    G. Egorychev: The solution of the Van der Waerden problem for permanents, Dokl. Akad. Nauk SSSR258 (1981), 1041–1044.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    D. Falikman: A proof of the Van der Waerden problem for permanents of a doubly stochastic matrix, Mat. Zametki29 (1981), 931–938.

    MathSciNet  Google Scholar 

  9. [9]

    A. Ferber and V. Jain: 1-factorizations of pseudorandom graphs, arXiv preprint arXiv:1803.10361 (2018).

    Google Scholar 

  10. [10]

    A. Ferber, M. Krivelevich and B. Sudakov: Counting and packing Hamilton l-cycles in dense hypergraphs, J. Comb.7 (2016), 135–157.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    A. Ferber, E. Long and B. Sudakov: Counting Hamilton decompositions of oriented graphs, Int. Math. Res. Not. IMRN22 (2018), 6908–6933.

    MathSciNet  Article  Google Scholar 

  12. [12]

    D. Gale: A theorem on ows in networks, Pacific J. Math.7 (1957), 1073–1082.

    MathSciNet  Article  Google Scholar 

  13. [13]

    R. Glebov, Z. Luria and B. Sudakov: The number of Hamiltonian decompositions of regular graphs, Israel J. Math.222 (2017), 91–108.

    MathSciNet  Article  Google Scholar 

  14. [14]

    P. Hall: On representatives of subsets, J. Lond. Math. Soc.1 (1935), 26–30.

    Article  Google Scholar 

  15. [15]

    W. Hoeffding: Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc.58 (1963), 13–30.

    MathSciNet  Article  Google Scholar 

  16. [16]

    P. Keevash: Counting designs, eprint arXiv:1504.02909 (2015).

    Google Scholar 

  17. [17]

    T. P. Kirkman: On a problem in combinations, Cambridge and Dublin Math. J.2 (1847), 191–204.

    Google Scholar 

  18. [18]

    M. Kwan: Almost all Steiner triple systems have perfect matchings, eprint arXiv:1611.02246 (2016).

    Google Scholar 

  19. [19]

    N. Linial and Z. Luria: An upper bound on the number of Steiner triple systems, Random Structures Algorithms43 (2013), 399–406.

    MathSciNet  Article  Google Scholar 

  20. [20]

    E. Lucas: Recreations mathematiques, Vol. 2., Gauthier-Villars (1883) 161–197.

    Google Scholar 

  21. [21]

    E. Mendelsohn and A. Rosa: One-factorizations of the complete graph - a survey, J. Graph Theory9 (1985), 43–65.

    MathSciNet  Article  Google Scholar 

  22. [22]

    L. Perkovic and B. Reed: Edge coloring regular graphs of high degree, Discrete Math.165/166 (1997), 567–578.

    MathSciNet  Article  Google Scholar 

  23. [23]

    V. Rödl: On a packing and covering problem, European J. Combin.6 (1985), 69–78.

    MathSciNet  Article  Google Scholar 

  24. [24]

    H. J. Ryser: Combinatorial properties of matrices of zeros and ones, Canad. J. Math.9 (1957), 371–377.

    MathSciNet  Article  Google Scholar 

  25. [25]

    E. Shamir and B. Sudakov: Two-Sided, Unbiased Version of Hall’s Marriage Theorem, Amer. Math. Monthly124 (2017), 79–80.

    MathSciNet  Article  Google Scholar 

  26. [26]

    V. G. Vizing: On an estimate of the chromatic class of a p-graph, Diskret. Analiz.3 (1964), 25–30.

    MathSciNet  Google Scholar 

  27. [27]

    W. D. Wallis: One-factorization of complete graphs, Contemporary Design Theory: A Collection of Surveys (1992) 593–631.

    Google Scholar 

Download references


The first author is grateful to Kyle Luh and Rajko Nenadov for helpful discussions at the first step of this project. We would also like to thank the anonymous referees for a very thorough reading of the manuscript and numerous invaluable comments.

Author information



Corresponding author

Correspondence to Benny Sudakov.

Additional information

Research is partially supported by NSF grant DMS 1954395.

Research supported in part by SNSF grant 200021-175573.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ferber, A., Jain, V. & Sudakov, B. Number of 1-Factorizations of Regular High-Degree Graphs. Combinatorica 40, 315–344 (2020).

Download citation

Mathematics Subject Classification (2010)

  • 05C70
  • 05D40