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Super-cyclically edge-connected regular graphs

Author

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  • Jin-Xin Zhou

    (Beijing Jiaotong University)

  • Yan-Quan Feng

    (Beijing Jiaotong University)

Abstract

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge-connected, in short, super-λ c , if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In Z. Zhang, B. Wang (Super cyclically edge-connected transitive graphs, J. Combin. Optim. 22:549–562, 2011), it is proved that a connected edge-transitive graph is super-λ c if either G is cubic with girth at least 7 or G has minimum degree at least 4 and girth at least 6, and the authors also conjectured that a connected graph which is both vertex-transitive and edge-transitive is always super cyclically edge-connected. In this article, for a λ c -optimal but not super-λ c graph G, all possible λ c -superatoms of G which have non-empty intersection with other λ c -superatoms are determined. This is then used to give a complete classification of non-super-λ c edge-transitive k(k≥3)-regular graphs.

Suggested Citation

  • Jin-Xin Zhou & Yan-Quan Feng, 2013. "Super-cyclically edge-connected regular graphs," Journal of Combinatorial Optimization, Springer, vol. 26(2), pages 393-411, August.
    • Handle: RePEc:spr:jcomop:v:26:y:2013:i:2:d:10.1007_s10878-012-9472-0
    DOI: 10.1007/s10878-012-9472-0
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    References listed on IDEAS

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    1. Zhao Zhang & Bing Wang, 2011. "Super cyclically edge connected transitive graphs," Journal of Combinatorial Optimization, Springer, vol. 22(4), pages 549-562, November.
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