A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivity of two vertices u and in a digraph or graph D is the maximum number of edge-disjoint paths in D, and the edge-connectivity of D is defined as . Clearly, for all pairs u and of vertices in D. Let be the minimum degree of D. We call a graph or digraph D maximally edge-connected when and maximally local-edge-connected whenfor all pairs u and of vertices in D.
In this paper we show that some known sufficient conditions that guarantee equality of and minimum degree for an oriented graph D also guarantee that D is maximally local-edge-connected. In addition, we generalize a result by Fàbrega and Fiol [Bipartite graphs and digraphs with maximum connectivity, Discrete Appl. Math. 69 (1996) 271–279] that each bipartite digraph of diameter at most three is maximally edge-connected.