Sufficient conditions for graphs to be maximally 4-restricted edge connected

For a subset S of edges in a connected graph G, the set S is a k-restricted edge cut if G - S is disconnected and every component of G - S has at least k vertices. The k-restricted edge connectivity of G, denoted by lambda(k)(G), is defined as the cardinality of a minimum k-restricted edge cut. A connected graph G is said to be lambda(k)-connected if G has k-restricted edge cut. Let xi(k)(G) = min{|[X, X]| : |X| = k, G[X] is connected}, where X = V(G)\X. A graph G is said to be maximally k-restricted edge connected if lambda(k)(G) = xi(k)(G). In this paper we show that if G is a lambda(4)-connected graph with lambda(4)(G) <= xi(4)(G) and the girth satisfies g(G) >= 8, and there do not exist six vertices u(1), u(2) , u(3), v(1), v(2) and v(3) in G such that the distance d(u(i), v(j) ) >= 3, (1 <= i, j <= 3), then G is maximally 4-restricted edge connected.