Local-edge-connectivity in digraphs and oriented graphs

A digraph without any cycle of length two is called an oriented graph. The local-edge-connectivity  λ(u,v)λ(u,v) of two vertices u   and vv in a digraph or graph D   is the maximum number of edge-disjoint u–vu–v paths in D, and the edge-connectivity of D   is defined as λ(D)=min{λ(u,v)|u,v∈V(D);u≠v}. Clearly, λ(u,v)⩽min{d+(u),d-(v)}λ(u,v)⩽min{d+(u),d-(v)} for all pairs u   and vv of vertices in D  . Let δ(D)δ(D) be the minimum degree of D. We call a graph or digraph D maximally edge-connected   when λ(D)=δ(D)λ(D)=δ(D) and maximally local-edge-connected whenλ(u,v)=min{d+(u),d-(v)}λ(u,v)=min{d+(u),d-(v)}for all pairs u   and vv of vertices in D.In this paper we show that some known sufficient conditions that guarantee equality of λ(D)λ(D) and minimum degree δ(D)δ(D) 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.