Super-cyclically edge-connected regular graphs

Jin-Xin Zhou () and Yan-Quan Feng ()
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Jin-Xin Zhou: Beijing Jiaotong University
Yan-Quan Feng: Beijing Jiaotong University

Journal of Combinatorial Optimization, 2013, vol. 26, issue 2, No 12, 393-411

Abstract: 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.

Keywords: Cyclic edge-cut; Cyclic edge-connectivity; Super cyclically edge-connected; Edge-transitive (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-012-9472-0

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