A bound on 4-restricted edge connectivity of graphs (Q2384424)

An edge cut of a connected graph \(G\) is \(4\)-restricted if it disconnects \(G\) with each component having order at least four. The cardinality of the minimum \(4\)-restricted edge cut of \(G\) is called the \(4\)-restricted edge connectivity of \(G\) and is denoted by \(\lambda_4(G)\). Let \(\xi_4(G)=\min\{\partial (F): F\) is a connected vertex-induced subgraph of order four of \(G\}\), where \(\partial (F)\) denotes the number of edges of graph \(G\) with exactly one endpoint in \(F\). This paper shows that for a connected graph \(G\) with order at least \(11\), if \(G\) contains \(4\)-restricted edge cuts, then \(\lambda_4(G)\leq\xi_4(G)\). Moreover, if \(G\) is a \(k\)-regular vertex-transitive graph of girth at least five, then \(\lambda_4(G)=\xi_4(G)\) when \(k\geq 4\).