\(3\)-Restricted Edge Connectivity of Vertex Transitive Graphs

Abstract

A \(3\)-restricted edge cut is an edge cut that disconnects a graph into at least two components each having order at least \(3\). The cardinality \(\lambda_3\) of minimum \(3\)-restricted edge cuts is called \(3\)-restricted edge connectivity. Let \(G\) be a connected \(k\)-regular graph of girth \(g(G) \geq 4\) and order at least \(6\). Then \(\lambda_3 \leq 3k – 4\). It is proved in this paper that if \(G\) is a vertex transitive graph then either \(\lambda_3 = 3k – 4\) or \(\lambda_3\) is a divisor of \(|G|\) such that \(2k – 2 \leq \lambda_3 \leq 3k – 5\) unless \(k = 3\) and \(g(G) = 4\). If \(k = 3\) and \(g(G) = 4\), then \(\lambda_3 = 4\). The extreme cases where \(\lambda_3 = 2k – 2\) and \(\lambda_3 = 3k – 5\) are also discussed.