On vertices of outdegree k in minimally k-arc-connected digraphs

Abstract

Let $k$ be a positive integer, and $D=(V\left(D\right),E\left(D\right))$ be a minimally $k$-arc-connected simple digraph. Mader conjectured (Combinatorics 2 (1996) 423-449) that there are at least $k+1$ vertices of outdegree $k$ in $D$. In this paper we prove that there are at least four vertices of outdegree $k$ for $k\ge 3$.