Contractible edges in non-separating cycles

Abstract

An edge of ak-connected graph is said to bek-contractible if the contraction of the edge results in ak-connected graph. We prove that every triangle-freek-connected graphG has an induced cycleC such that all edges ofC arek-contractible and such thatG−V(C) is (k−3)-connected (k≥4). This result unifies two theorems by Thomassen [5] and Egawa et. al. [3].

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References

  1. [1]

    G. Chartrand, andL. Lesniak:Graphs and Digraphs, Second edition, Wadsworth, Belmont, CA (1986).

    Google Scholar 

  2. [2]

    N. Dean: Distribution of contractible edges ink-connected graphs,preprint.

  3. [3]

    Y. Egawa, H. Enomoto, andA. Saito: Contractible edges in triangle-free graphs,Combinatorica 6 (1986) 269–274.

    Google Scholar 

  4. [4]

    W. Mader: Generalization of critical connectivity of graphs,Discrete Math. 72 (1988) 267–283.

    Google Scholar 

  5. [5]

    C. Thomassen: Nonseparating cycles ink-connected graphs,J. Graph Theory 5 (1981) 351–354.

    Google Scholar 

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Dedicated to Professor Toshiro Tsuzuku on his sixtieth birthday

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Egawa, Y., Saito, A. Contractible edges in non-separating cycles. Combinatorica 11, 389–392 (1991). https://doi.org/10.1007/BF01275673

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AMS subject classification (1991)

  • 05 C 40
  • 05 C 38