Edges of degree in minimally restricted -edge connected graphs

For a connected graph G = ( V , E ) , an edge set S ⊂ E is a restricted edge cut if G − S is disconnected and there is no isolated vertex in G − S . The cardinality of a minimum restricted edge cut of G is the restricted edge connectivity of G , denoted by λ ′ ( G ) . A graph G is called minimally restricted k -edge connected if λ ′ ( G ) = k and λ ′ ( G − e ) < k for each edge e ∈ E . A graph G is λ ′ -optimal if λ ′ ( G ) = ξ ( G ) , where ξ ( G ) is the minimum edge degree of G . In this paper, we prove that every minimally restricted k -edge connected graph has at least three edges of degree k , and if furthermore λ ′ ( G ) ≠ 4 , then there are at least four. Examples show that the lower bounds are best possible.