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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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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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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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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DOI: https://doi.org/10.1007/s00373-021-02333-6