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Spectral Threshold for Extremal Cyclic Edge-Connectivity

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Abstract

The universal cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges whose removal disconnects G into components where every component contains a cycle. We show that for graphs of minimum degree at least 3 and girth g at least 4, the universal cyclic edge-connectivity is bounded above by \((\Delta -2)g\) where \(\Delta \) is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a d-regular graph of girth \(g\ge 4\) is sufficiently small, then the universal cyclic edge-connectivity is \((d-2)g\), providing a spectral condition for when this upper bound on universal cyclic edge-connectivity is tight.

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Notes

  1. Suppose that S is a minimal cyclic edge cut that is not universal. Then \(G\setminus S\) has a tree component T as well as two components \(C_1\) and \(C_2\) that contain a cycle. Now there is some edge \(e \in S\) which is incident to T. Consider the edge cut \(S' = S\setminus e\). Since e is not incident between \(C_1\) and \(C_2\), \(G \setminus S'\) still has two distinct components, \(C_1'\) and \(C_2'\) which contain \(C_1\) and \(C_2\) as subgraphs, respectively. But then \(S'\) is a cyclic edge cut, a contradiction.

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Acknowledgements

The authors would like to thank Carlos Ortiz-Marrero for helpful discussions, and anonymous referees for thoughtful comments which improved the manuscript.

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Authors and Affiliations

  1. Pacific Northwest National Laboratory, Seattle, WA, 98109, USA

    Sinan G. Aksoy

  2. Mathematics Department, Brigham Young University, Provo, UT, 84602, USA

    Mark Kempton

  3. Pacific Northwest National Laboratory, Richland, WA, 99354, USA

    Stephen J. Young

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  1. Sinan G. Aksoy
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  2. Mark Kempton
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Correspondence to Sinan G. Aksoy.

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This work was supported by the High Performance Data Analytics (HPDA) program at Pacific Northwest National Laboratory. Pacific Northwest National Laboratory is operated by Battelle Memorial Institute under Contract DE-ACO6-76RL01830. PNNL Information Release: PNNL-SA-151831 . The authors declare they have no competing interests.

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This work was supported by the High Performance Data Analytics (HPDA) program at Pacific Northwest National Laboratory. Pacific Northwest National Laboratory is operated by Battelle Memorial Institute under Contract DE-ACO6-76RL01830. PNNL Information Release: PNNL-SA-151831.

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Aksoy, S.G., Kempton, M. & Young, S.J. Spectral Threshold for Extremal Cyclic Edge-Connectivity. Graphs and Combinatorics 37, 2079–2093 (2021). https://doi.org/10.1007/s00373-021-02333-6

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