Super cyclically edge-connected vertex-transitive graphs of girth at least 5

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 Zhang, Z., Wang, B.: Super cyclically edge-connected transitive graphs. J. Combin. Optim., 22, 549–562 (2011)], it is proved that a connected vertex-transitive graph is super-λ _c if G has minimum degree at least 4 and girth at least 6, and the authors also presented a class of nonsuper-λ _c graphs which have degree 4 and girth 5. In this paper, a characterization of k (k ≥ 4)-regular vertex-transitive nonsuper-λ _c graphs of girth 5 is given. Using this, we classify all k (k ≥ 4)-regular nonsuper-λ _c Cayley graphs of girth 5, and construct the first infinite family of nonsuper-λ _c vertex-transitive non-Cayley graphs.