Numbers of edges in supermagic graphs

For a connected graph the restricted edge-connectivity λ2(G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in GS. A graph G is said to be λ2-optimal if λ2(G) = ξ(G), where ξ(G) is the minimum edge-degree in G defined as ξ(G) = min{d(u) + d(v) - 2:uv e E(G)}, d(u) denoting the degree of a vertex u. A. Hellwig and L. Volkmann [Sufficient conditions for λ2-optimality in graphs of diameter 2, Discrete Math 283 (2004), 113120] gave a sufficient condition for λ2-optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g - 1, g being the girth of the graph, and show that a graph G with diameter at most g - 2 is λ2-optimal. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 7386, 2006