Long Cycles have the Edge-Erdős-Pósa Property

Article
  • 6 Downloads

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).

Mathematics Subject Classification (2010)

05C70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
  2. [2]
    E. Birmelé, J. A. Bondy and B. Reed: The Erdős-Pósa property for long circuits, Combinatorica 27 (2007), 135–145.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle 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.MATHGoogle 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.MathSciNetMATHGoogle Scholar
  6. [6]
    R. Diestel: Graph theory, fourth ed., Springer, Heidelberg, 2010.CrossRefMATHGoogle 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.MathSciNetMATHGoogle Scholar
  8. [8]
    S. Fiorini and A. Herinckx: A tighter Erdős–Pósa function for long cycles, J. Graph Theory 77 (2014), 111–116.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    T. Huynh, F. Joos and P. Wollan: A unified Erdős-Pósa theorem for constrained cycles, to appear in Combinatorica.Google Scholar
  10. [10]
    N. Kakimura, K. Kawarabayashi and D. Marx: Packing cycles through prescribed vertices, J. Combin. Theory (Series B) 101 (2011), 378–381.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    C.-H. Liu: Packing and Covering immersions in 4-Edge-Connected Graphs, arXiv:1505.00867 (2015).Google Scholar
  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.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    M. Pontecorvi and P. Wollan: Disjoint cycles intersecting a set of vertices, J. Combin. Theory (Series B) 102 (2012), 1134–1141.MathSciNetCrossRefMATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    J.-F. Raymond and D. Thilikos: Recent techniques and results on the Erdős–Pósa property, to appear in Disc. App. Math.Google Scholar
  17. [17]
    N. Robertson and P. Seymour: Graph minors. V. Excluding a planar graph, J. Combin. Theory (Series B) 41 (1986), 92–114.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    K. F. Roth: On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109.MathSciNetCrossRefMATHGoogle 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.MATHGoogle 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.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    C. Thomassen: On the presence of disjoint subgraphs of a specified type, J. Graph Theory 12 (1988), 101–111.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Optimierung und Operations ResearchUniversität UlmUlmGermany
  2. 2.School of MathematicsUniversity of BirminghamBirminghamUK

Personalised recommendations