Non-separating even cycles in highly connected graphs

Abstract

We prove that every k-connected graph (k ≥ 5) has an even cycle C such that G-V (C) is still (k-4)-connected. The connectivity bound is best possible.

In addition, we prove that every k-connected triangle-free graph (k ≥ 5) has an even cycle C such that G-V (C) is still (k-3)-connected. The same conclusion also holds for any k-connected graph (k ≥ 5) that does not contain a K 4 , i.e., K 4 minus one edge.

The first theorem is an analogue of the well-known result (without the parity condition) by Thomassen [9]. It is also a counterpart of the conjecture made by Thomassen [10] which says that there exists a function f(k) such that every f(k)-connected non-bipartite graph has an odd cycle C such that G-V (C) is still k-connected.

The second theorem is an analogue of the results (without the parity condition) by Egawa [2] and Kawarabayashi [3], respectively.

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

References

  1. [1]

    G. Chen, R. Gould and X. Yu: Graph Connectivity after path removal, Combinatorica23(2) (2003), 185–203.

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    Y. Egawa: Cycles in k-connected graphs whose deletion results in a (k-2)-connected graph, J. Combin. Theory Ser. B42 (1987), 371–377.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    K. Kawarabayashi: Contractible edges and triangles in k-connected graphs, J. Combin. Theory Ser. B85 (2002), 207–221.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    K. Kawarabayashi, O. Lee and X. Yu: Non-separating paths in 4-connected graphs, Ann. Comb.9(1) (2005), 47–56.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    K. Kawarabayashi, O. Lee, B. Reed and P. Wollan: A weaker version of Lovász’ path removable conjecture, J. Combin. Theory Ser. B98 (2008), 972–979.

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    M. Kriesell: Induced paths in 5-connected graphs, J. Graph Theory36 (2001), 52–58.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    M. Kriesell: Removable paths conjectures, http://www.fmf.uni-lj.si/~mohar/Problems/P0504Kriesell1.pdf.

  8. [8]

    L. Lovász: Problems in graph theory, in: Recent Advances in Graph Theory (ed. M. Fielder), Acadamia Prague, 1975.

  9. [9]

    C. Thomassen: Non-separating cycles in k-connected graphs, J. Graph Theory5 (1981), 351–354.

    MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    C. Thomassen: The Erdős-Pósa property for odd cycles in graphs of large connectivity, Combinatorica21(2) (2001), 321–333.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    W. Tutte: How to draw a graph, Proc. London Math. Soc.13 (1963), 743–767.

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Shinya Fujita.

Additional information

This work is supported by the JSPS Research Fellowships for Young Scientists.

Research partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by C&C Foundation, by Inamori Foundation and by Kayamori Foundation.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fujita, S., Kawarabayashi, Ki. Non-separating even cycles in highly connected graphs. Combinatorica 30, 565–580 (2010). https://doi.org/10.1007/s00493-010-2482-6

Download citation

Mathematics Subject Classification (2000)

  • 05C38
  • 05C40