Bounding the vertex cover number of a hypergraph

Combinatorica volume 14, pages23–34(1994)Cite this article

Abstract

For a hypergraphH, we denote by

  1. (i)

    τ(H) the minimumk such that some set ofk vertices meets all the edges,

  2. (ii)

    ν(H) the maximumk such that somek edges are pairwise disjoint, and

  3. (iii)

    λ(H) the maximumk≥2 such that the incidence matrix ofH has as a submatrix the transpose of the incidence matrix of the complete graphK k .

We show that τ(H) is bounded above by a function of ν(H) and λ(H), and indeed that if λ(H) is bounded by a constant then τ(H) is at most a polynomial function of ν(H).

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. [1]

    D. Bienstock, andN. Dean: Some obstructions to small face covers in planar graphs,J. Combinatorial Theory, (B), to appear.

  2. [2]

    A. Blumer, A. Ehrenfeucht, D. Haussler, andM. K. Warmuth: Learnability and the Vapnik-Čhervonenkis dimension,J. Assoc. Comp. Mach. 36 (1989), 929–965.

    Google Scholar 

  3. [3]

    J. Folkman: Notes on the Ramsey numberN (3,3,3,3),J. Combinatorial Theory, (A),16 (1974), 371–379.

    Google Scholar 

  4. [4]

    D. Haussler, andE. Welzl: ε-nets and simplex range queries,Discrete Comput. Geom. 2 (1987), 127–151.

    Google Scholar 

  5. [5]

    J. Komlós, J. Pach, andG. Woeginger: Almost tight bounds for epsilon-nets,Disc. and Comput. Geom.,7 (1992), 163–173.

    Google Scholar 

  6. [6]

    J. Pach, andJ. Törőcsik: Some geometric applications of Dilworth's Theorem,Proc. 9th ACM Symposium on Comput Geom. 1993.

  7. [7]

    E. Slud: Distribution inequalities for the binomial law,Annals of Probability 5 (1977), 404–412.

    Google Scholar 

  8. [8]

    V. N. Vapnik, andA. Ya. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities,Theory Probab. Appl. 16 (1971), 264–280.

    Google Scholar 

Download references

Author information

Affiliations

  1. Department of Mathematics, Louisiana State University, 70803, Baton Rouge, LA

    Guo-Li Ding

  2. Bellcore, 445 South Street, 07960, Morristown, NJ

    Paul Seymour & Peter Winkler

Authors
  1. Guo-Li Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paul Seymour
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter Winkler
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ding, GL., Seymour, P. & Winkler, P. Bounding the vertex cover number of a hypergraph. Combinatorica 14, 23–34 (1994). https://doi.org/10.1007/BF01305948

Download citation

AMS subject classification codes (1991)

  • 05 C 65
  • 05 C 35