Degree conditions for restricted-edge-connectivity and isoperimetric-edge-connectivity to be optimal

For a connected graphG= (V ,E), an edge set S ⊂ E is a k-restricted-edge-cut, ifG−S is disconnected and every component of G− S has at least k vertices. The k-restricted-edge-connectivity of G, denoted by k(G), is defined as the cardinality of a minimum k-restricted-edge-cut. The k-isoperimetric-edge-connectivity is defined as k(G)=min{|[U,U ]| : U ⊂ V, |U | k, |U | k}, where [U,U ] is the set of edges with one end in U and the other end in U =V \U . In this note, we give some degree conditions for a graph to have optimal k and/or k . © 2006 Elsevier B.V. All rights reserved.