Skip to main content

Packing and Covering Triangles in Planar Graphs

Abstract

Tuza conjectured that if a simple graph G does not contain more than k pairwise edge-disjoint triangles, then there exists a set of at most 2k edges that meets all triangles in G. It has been shown that this conjecture is true for planar graphs and the bound is sharp. In this paper, we characterize the set of extremal planar graphs.

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

References

  1. 1

    Haxell P.E.: Packing and covering triangles in graphs. Discrete Math. 195, 251–254 (1999)

    MATH  Article  MathSciNet  Google Scholar 

  2. 2

    Haxell P.E., Kohayakawa Y.: Packing and covering triangles in tripartite graphs. Graphs Combin. 14, 1–10 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3

    Krivelevich M.: On a conjecture of Tuza about packing and covering of triangles. Discrete Math. 142, 281–286 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4

    Tuza, Zs.: Conjecture, Finite and Infinite Sets (Eger, Hungary 1981). In: Hajnal, A., Lovász, L., Sós, V. T. (eds.) Proc. Colloq. Math. Soc. J. Bolyai, vol. 37, p. 888. North-Holland, Amsterdam (1984)

  5. 5

    Tuza Zs.: A conjecture on triangles of graphs. Graphs Combin. 6, 373–380 (1990)

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Qing Cui.

Additional information

Q. Cui was partially supported by Jiangsu Planned Projects for Postdoctoral Research Funds, P. Haxell was partially supported by NSERC and W. Ma was partially supported by an NSERC Undergraduate Student Research Assistantship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cui, Q., Haxell, P. & Ma, W. Packing and Covering Triangles in Planar Graphs. Graphs and Combinatorics 25, 817–824 (2009). https://doi.org/10.1007/s00373-010-0881-5

Download citation

Keywords

  • Packing and covering
  • Triangle
  • Planar graph

Mathematics Subject Classification (2000)

  • 05C70
  • 05C35