Vertices of degree k in a minimally k-edge-connected digraph

AbstractLet k be a positive integer and D=(V,E) be a minimally k-edge-connected simple digraph. For a vertex x∈V(D), its outdegree δ+(x) (indegree δ−(x)) is the number of edges leaving (entering) x. Let u+(D) (resp. u±(D) and u−(D)) denote the number of vertices x in D such that δ+(x)=k<δ−(x) (resp. δ+(x)=δ−(x)=k and δ+(x)>k=δ−(x)). In this paper we prove thatu+(D)+2u±(D)+u−(D)⩾2k+2,which was conjectured by Mader (Combinatorics 2 (1996) 423–449). We also present a lower bound on u+(D)+u±(D)+u−(D) when |D|⩾4k−1