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.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (10901015, 11171020), the Fundamental Research Funds for the Central Universities (2011JBM127, 2011JBZ012), and the Subsidy for Outstanding People of Beijing (2011D005022000005).
The authors are indebted to the anonymous referee for many valuable comments and constructive suggestions.
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Zhou, JX., Feng, YQ. Super-cyclically edge-connected regular graphs. J Comb Optim 26, 393–411 (2013). https://doi.org/10.1007/s10878-012-9472-0
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DOI: https://doi.org/10.1007/s10878-012-9472-0