Sufficient Conditions for Maximally Edge-connected and Super-edge-connected Digraphs Depending on the Size

Let D be a finite and simple digraph with vertex set V (D). The minimum degree delta of a digraph D is defined as the minimum value of its out-degrees and its in-degrees. If D is a digraph with minimum degree delta and edge-connectivity lambda, then lambda <= delta. A digraph is maximally edge-connected if lambda = delta. A digraph is called super-edge-connected if every minimum edge-cut consists of edges incident to or from a vertex of minimum degree. In this note we show that a digraph is maximally edge-connected or super-edge-connected if the number of arcs is large enough.