On cyclic edge-connectivity of transitive graphs

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 said to be cyclically separable. For a cyclically separable graph G , the cyclic edge-connectivity c λ ( G ) is the cardinality of a minimum cyclic edge-cut of G . In this paper, we first prove that for any cyclically separable graph G , c λ ( G ) ≤ ζ ( G ) = min { ω ( X ) ∣ X induces a shortest cycle in G } , where ω ( X ) is the number of edges with one end in X and the other end in V ( G ) ∖ X . A cyclically separable graph G with c λ ( G ) = ζ ( G ) is said to be cyclically optimal. The main results in this paper are: any connected k -regular vertex-transitive graph with k ≥ 4 and girth at least 5 is cyclically optimal; any connected edge-transitive graph with minimum degree at least 4 and order at least 6 is cyclically optimal.