On point covers of multiple intervals and axis-parallel rectangles


In certain families of hypergraphs the transversal number is bounded by some function of the packing number. In this paper we study hypergraphs related to multiple intervals and axisparallel rectangles, respectively. Essential improvements of former established upper bounds are presented here. We explore the close connection between the two problems at issue.

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


  1. [1]

    G. Ding, P. Seymour, andP. Winkler: Bounding the vertex cover number of a hypergraph,Combinatorica 14 (1994), 23–34.

    Article  Google Scholar 

  2. [2]

    D.G. Fon-Der-Flaass, andA. V. Kostochka: Covering boxes by points,Discr. Math.,120 (1993), 269–275.

    Article  Google Scholar 

  3. [3]

    A. Gyárfás, andJ. Lehel: A Helly-type problem in trees, inCombinatorial Theory and its Applications, Eds.: P. Erdős, A. Rényi and V.T. Sós, North-Holland, Amsterdam, (1970), 571–584.

    Google Scholar 

  4. [4]

    A. Gyárfás, andJ. Lehel: Covering and coloring problems for relatives of intervals,Discr. Math.,55 (1985), 167–180.

    Article  Google Scholar 

  5. [5]

    A. Hajnal, andJ. Surányi: Über die Ausflösung von Graphen in vollständige Teilgraphen,Ann. Univ. Sci. Budapest, (1958), 113–121.

  6. [6]

    T. Kaiser: Transversals ofd-intervals, manuscript, 1995.

  7. [7]

    Gy. Károlyi: On point covers of parallel rectangles,Periodica Math. Hung.,23 (1991), 105–107.

    Google Scholar 

  8. [8]

    A. V. Kostochka: personal communication with A. Gyárfás.

  9. [9]

    J. Pach: A remark on transversel numbers, inThe Mathematics of Paul Erdős, Eds.: R. L. Graham and J. Nešetřil, Springer-Verlag, Berlin, 1996.

    Google Scholar 

  10. [10]

    J. Pach, andJ. Törőcsik: Some geometric applications of Dilworth's theorem, inProc. 9th Ann. Symp. Comp. Geom., (1993), 264–269; see alsoDiscr. Comp. Geom. 12 (1994), 1–7.

  11. [11]

    G. Tardos: Transversals of 2-intervals, a topological approach,Combinatorica 15 (1995), 123–134.

    Google Scholar 

  12. [12]

    G. Wegner: Über eine kombinatorisch-geometrische Frage von Hadwiger und Debrunner,Israel J. Math. 3 (1965), 187–198.

    Google Scholar 

Download references

Author information



Additional information

Supported by the Alexander von Humboldt Foundation and the NSF grant No. STC-91-19999

Supported by the NSF grant No. CCR-92-00788, the (Hungarian) National Scientific Research Fund (OTKA) grant No. F014919. The author was visiting the Computation and Automation Institute, Budapest while part of this research was done.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Károlyi, G., Tardos, G. On point covers of multiple intervals and axis-parallel rectangles. Combinatorica 16, 213–222 (1996). https://doi.org/10.1007/BF01844847

Download citation

Mathematics Subject Classification (1991)

  • 05 C 65
  • 05 C 70
  • 05 C 35