A bound on 4-restricted edge connectivity of graphs

An edge cut of a connected graph is 4-restricted if it disconnects this graph with each component having order at least four. The size of minimum 4-restricted edge cuts of graph G is called its 4-restricted edge connectivity and is denoted by 4(G). Let 4(G)=min{ (F ) : F is a connected vertex-induced subgraph of order four of graph G}, where (F ) denotes the number of edges of graph G with exactly one endpoint in F. For connected graphs that contain 4-restricted edge cuts, 4(G) is proved to be an upper bound on 4(G) if G has order at least 11. If G is a k-regular vertex-transitive graph of girth at least five, then 4(G)= 4(G) when k 4. © 2007 Elsevier B.V. All rights reserved.