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On Cyclic Edge-Connectivity and Super-Cyclic Edge-Connectivity of Double-Orbit Graphs

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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 said to be cyclically separable. For a cyclically separable graph G, the cyclic edge-connectivity \(\lambda _c(G)\) is the cardinality of a minimum cyclic edge-cut of G. Let \(\zeta (G)=\min \{\omega (X) | X \text{ induce } \text{ a } \text{ shortest } \text{ cycle } \text{ in } G\}\), where \(\omega (X)\) is the number of edges with one end in X and the other end in \(V(G)-X\). A cyclically separable graph G with \(\lambda _c(G)=\zeta (G)\) is said to be cyclically optimal. In particular, we call a graph G super cyclically edge-connected if every minimum cyclic edge-cut isolates a shortest cycle of G. In this work, we first discuss the cyclic edge-connectivity of vertex transitive graphs, regular double-orbit graphs, and the double-orbit graphs with two orbits of same size; moreover, we also discuss the super-cyclic edge-connectivity of double-orbit graphs mentioned above.

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Acknowledgments

The authors would like to thank the referees for the valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, Shanxi, China

    Weihua Yang

  2. Department of Mathematics, Guangxi Teachers Education University, Nanning, 530001, Guangxi, China

    Chengfu Qin

  3. Department of Mathematics and Physics, Xiamen University of Technology, Xiamen, 361024, Fujian, China

    Meirun Chen

Authors
  1. Weihua Yang
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  2. Chengfu Qin
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  3. Meirun Chen
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Corresponding author

Correspondence to Weihua Yang.

Additional information

Communicated by Sanming Zhou.

The research is supported by NSFC (Nos. 11301371, 11101345, 11401119), and Natural Sciences Foundation of Shanxi Province (No. 2014021010-2).

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Yang, W., Qin, C. & Chen, M. On Cyclic Edge-Connectivity and Super-Cyclic Edge-Connectivity of Double-Orbit Graphs. Bull. Malays. Math. Sci. Soc. 39 (Suppl 1), 13–27 (2016). https://doi.org/10.1007/s40840-015-0286-y

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  • DOI: https://doi.org/10.1007/s40840-015-0286-y

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