Skip to main content
SpringerLink
Log in

Nearly perfect matchings in regular simple hypergraphs

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

For every fixedk≥3 there exists a constantc k with the following property. LetH be ak-uniform,D-regular hypergraph onN vertices, in which no two edges contain more than one common vertex. Ifk>3 thenH contains a matching covering all vertices but at mostc k ND −1/(k−1). Ifk=3, thenH contains a matching covering all vertices but at mostc 3 ND −1/2ln3/2 D. This improves previous estimates and implies, for example, that any Steiner Triple System onN vertices contains a matching covering all vertices but at mostO(N 1/2ln3/2 N), improving results by various authors.

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. M. Ajtai, J. Komlós and E. Szemerédi,A dense infinite Sidon sequence, European Journal of Combinatorics2 (1981), 1–11.

    MATH  MathSciNet  Google Scholar 

  2. N. Alon and J. Spencer,The Probabilistic Method, Wiley, 1992.

  3. A. E. Brouwer,On the size of a maximum transversal in a Steiner Triple System, Canadian Journal of Mathematics33 (1981), 1202–1204.

    MATH  MathSciNet  Google Scholar 

  4. P. Erdös and H. Hanani,On a limit theorem in combinatorial analysis, Publicationes Mathematicae. Debrecen10 (1963), 10–13.

    MathSciNet  Google Scholar 

  5. P. Frankl and V. Rödl,Near perfect coverings in graphs and hypergraphs, European Journal of Combinatorics6 (1985), 317–326.

    MATH  MathSciNet  Google Scholar 

  6. D. M. Gordon, G. Kuperberg, O. Patashnik and J. Spencer,Asymptotically Optimal Covering Designs, Preprint, 1995.

  7. D. A. Grable,More-than-nearly perfect packings and partial designs, Preprint, 1995.

  8. J. Kahn,Asymptotically good list-colorings, Journal of Combinatorial Theory, Series A73 (1996), 1–59.

    Article  MATH  MathSciNet  Google Scholar 

  9. J.H. Kim,The Ramsey Number R(3, t) has order of magnitude t 2/lnt, Random Structures & Algorithms7 (1995), 173–207.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. C. Lindner and K. T. Phelps,A note on partial parallel classes in Steiner systems, Discrete Mathematics24 (1978), 109–112.

    Article  MATH  MathSciNet  Google Scholar 

  11. N. Pippenger and J. H. Spencer,Asymptotic behaviour of the chromatic index for hypergraphs, Journal of Combinatorial Theory, Series A51 (1989), 24–42.

    Article  MATH  MathSciNet  Google Scholar 

  12. V. Rödl,On a packing and covering problem, European Journal of Combinatorics6 (1985), 69–78.

    MATH  MathSciNet  Google Scholar 

  13. V. Rödl and L. Thoma,Asymptotic packing and the random greedy algorithm, Random Structures & Algorithms, to appear.

  14. J. Spencer,Asymptotic packing via a branching process, Random Structures & Algorithms7 (1995), 167–172.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. P. Wang,On self orthogonal Latin squares and partial transversals of Latin squares, Ph.D. thesis, Ohio State University, Columbus, Ohio, 1978.

    Google Scholar 

  16. D. E. Woolbright,On the size of partial parallel classes in Steiner systems, inTopics in Steiner Systems (C. C. Lindner and A. Rosa, eds.), Annals of Discrete Mathematics7 (1980), 203–211.

Download references

Author information

Authors and Affiliations

  1. AT & T Bell Labs, 07974, Murray Hill, NJ, USA

    Noga Alon & Jeong-Han Kim

  2. Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

    Noga Alon

  3. Department of Mathematics and Computer Science, Courant Institute, NYU, NY, USA

    Joel Spencer

Authors
  1. Noga Alon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeong-Han Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joel Spencer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Noga Alon.

Additional information

Research supported in part by a USA-Israel BSF grant.

Research supported in part by a USA-Israel BSF Grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alon, N., Kim, JH. & Spencer, J. Nearly perfect matchings in regular simple hypergraphs. Isr. J. Math. 100, 171–187 (1997). https://doi.org/10.1007/BF02773639

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773639

Keywords

Search

Navigation