Long Cycles have the Edge-Erdős-Pósa Property

Abstract

We prove that the set of long cycles has the edge-Erdős-Pósa property: for every fixed integer ≥ 3 and every k ∈ ℕ, every graph G either contains k edge-disjoint cycles of length at least (long cycles) or an edge set X of size O(k2 logk+kℓ) such that GX does not contain any long cycle. This answers a question of Birmelé, Bondy, and Reed (Combinatorica 27 (2007), 135-145).

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References

  1. [1]

    ISGCI, http://www.graphclasses.org/smallgraphs.html.

  2. [2]

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

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    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  MATH  Google Scholar 

  4. [4]

    D. Conlon and J. Fox: Graph removal lemmas, Surveys in combinatorics 2013, London Math. Soc. Lecture Note Ser., vol. 409, Cambridge Univ. Press, Cambridge, 2013, 1–49.

    MATH  Google Scholar 

  5. [5]

    I. J. Dejter and V. Neumann-Lara: Unboundedness for generalized odd cyclic transversality, Colloq. Math. Soc. János Bolyai 52 (1987), 195–203.

    MathSciNet  MATH  Google Scholar 

  6. [6]

    R. Diestel: Graph theory, fourth ed., Springer, Heidelberg, 2010.

    Book  MATH  Google Scholar 

  7. [7]

    P. Erdős and L. Pósa: On the maximal number of disjoint circuits of a graph, Publ. Math. Debrecen 9 (1962), 3–12.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    S. Fiorini and A. Herinckx: A tighter Erdős–Pósa function for long cycles, J. Graph Theory 77 (2014), 111–116.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    T. Huynh, F. Joos and P. Wollan: A unified Erdős-Pósa theorem for constrained cycles, to appear in Combinatorica.

  10. [10]

    N. Kakimura, K. Kawarabayashi and D. Marx: Packing cycles through prescribed vertices, J. Combin. Theory (Series B) 101 (2011), 378–381.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    K. Kawarabayashi and Y. Kobayashi: Edge-disjoint odd cycles in 4-edge-connected graphs, J. Combin. Theory (Series B) 119 (2016), 12–27.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    C.-H. Liu: Packing and Covering immersions in 4-Edge-Connected Graphs, arXiv:1505.00867 (2015).

  13. [13]

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

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

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

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    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  MATH  Google Scholar 

  16. [16]

    J.-F. Raymond and D. Thilikos: Recent techniques and results on the Erdős–Pósa property, to appear in Disc. App. Math.

  17. [17]

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

    MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    K. F. Roth: On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    I. Z. Ruzsa and E. Szemerédi: Triple systems with no six points carrying three triangles, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, vol. 18, North-Holland, Amsterdam-New York, 1978, pp. 939–945.

    MATH  Google Scholar 

  20. [20]

    M. Simonovits: A new proof and generalizations of a theorem of Erdős and Pósa on graphs without k+1 independent circuits, Acta Math. Acad. Sci. Hungar. 18 (1967), 191–206.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

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

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Henning Bruhn.

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The research was also supported by the EPSRC, grant no. EP/M009408/1.

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Bruhn, H., Heinlein, M. & Joos, F. Long Cycles have the Edge-Erdős-Pósa Property. Combinatorica 39, 1–36 (2019). https://doi.org/10.1007/s00493-017-3669-x

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Mathematics Subject Classification (2010)

  • 05C70