The Cyclic Edge-Connectivity of Strongly Regular Graphs

Let G be a connected graph. An edge cut set M of G is a cyclic edge cut set if there are at least two components of \(G-M\) which contain a cycle. The cyclic edge-connectivity of G is the minimum cardinality of a cyclic edge cut set (if exists) of G. In this paper, we show that the cyclic edge-connectivity of a connected strongly regular graph G (not \(K_{3,3}\)) of degree \(k\ge 3\) with girth c is equal to \((k-2)c\), where \(c=3, 4\) or 5. Moreover, if G is not the triangular graph srg-(10, 6, 3, 4), the complete multi-partite graph \(K_{2,2,2,2}\) or the lattice graph srg-(16, 6, 2, 2), then each cyclic edge cut set of size \((k-2)c\) is precisely the set of edges incident with a cycle of length c in G.