The Edge-Erdős-Pósa Property

Abstract

Robertson and Seymour proved that the family of all graphs containing a fixed graph H as a minor has the Erdős-Pósa property if and only if H is planar. We show that this is no longer true for the edge version of the Erdős-Pósa property, and indeed even fails when H is an arbitrary subcubic tree of large pathwidth or a long ladder. This answers a question of Raymond, Sau and Thilikos.

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

References

  1. [1]

    E. Birmelé, J. A. Bondy and B. Reed The Erdős-Pósa property for long circuits, Combinatorica 27 (2007), 135–145.

    MathSciNet  Article  Google Scholar 

  2. [2]

    H. Bruhn and M. Heinlein K4-subdivisions have the edge-Erdős-Pósa property, to appear in SIAM J. Discrete Math.

  3. [3]

    H. Bruhn, M. Heinlein and F. Joos Long cycles have the edge-Erdős-Pósa property, Combinatorica 39 (2019), 1–36.

    MathSciNet  Article  Google Scholar 

  4. [4]

    H. Bruhn, F. Joos and O. Schaudt Long cycles through prescribed vertices have the Erdős-Pósa property, J. Graph Theory 87 (2018), 275–284.

    MathSciNet  Article  Google Scholar 

  5. [5]

    P. Erdo˝s and L. Po´sa: On independent circuits contained in a graph, Can. J. Math. 7 (1965), 347–352.

    MathSciNet  Article  Google Scholar 

  6. [6]

    T. Huynh, F. Joos and P. Wollan A unified Erdős-Pósa theorem for constrained cycles, Combinatorica 39 (2019), 91–133.

    MathSciNet  Article  Google Scholar 

  7. [7]

    E. Marshall and D. Wood Circumference and pathwidth of highly connected graphs, J. Graph Theory 79 (2015), 222–232.

    MathSciNet  Article  Google Scholar 

  8. [8]

    F. Mousset, A. Noever, N. Škorić and F. Weissenberger A tight Erdős-Pósa function for long cycles, J. Combin. Theory (Ser. B) 125 (2017), 21–32.

    MathSciNet  Article  Google Scholar 

  9. [9]

    M. Pontecorvi and P. Wollan Disjoint cycles intersecting a set of vertices, J. Combin. Theory (Ser. B) 102 (2012), 1134–1141.

    MathSciNet  Article  Google Scholar 

  10. [10]

    J.-F. Raymond, I. Sau and D. M. Thilikos: An edge variant of the Erdős-Pósa property, Discrete Math. 339 (2016), 2027–2035.

    MathSciNet  Article  Google Scholar 

  11. [11]

    N. Robertson and P. Seymour Graph minors. I. Excluding a forest, J. Combin. Theory (Ser. B) 35 (1983), 39–61.

    MathSciNet  Article  Google Scholar 

  12. [12]

    N. Robertson and P. Seymour Graph minors. V. Excluding a planar graph, J. Combin. Theory (Ser. B) 41 (1986), 92–114.

    MathSciNet  Article  Google Scholar 

  13. [13]

    C. Thomassen: On the presence of disjoint subgraphs of a specified type, J. Graph Theory 12 (1988), 101–111.

    MathSciNet  Article  Google Scholar 

  14. [14]

    A. Ulmer and R. Steck Long ladders do not have the edge-Erdős-Pósa property, arXiv:2003.03236, 2020.

    Google Scholar 

Download references

Acknowledgement

We thank the anonymous referees for a very careful reading of the manuscript and many valuable comments that improved the manuscript.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Henning Bruhn.

Additional information

The research leading to these results was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 321904558.

The research leading to these results was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 339933727.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bruhn, H., Heinlein, M. & Joos, F. The Edge-Erdős-Pósa Property. Combinatorica 41, 147–173 (2021). https://doi.org/10.1007/s00493-020-4071-7

Download citation

Mathematics Subject Classification (2010)

  • 05B40