On Vertices of outdegree n in minimally n ‐connected digraphs

Let | D | and |D| + n denote the number of vertices of D and the number of vertices of outdegree n in the digraph D , respectively. It is proved that every minimally n ‐connected, finite digraph D has |D| + n  ≥  n  + 1 and that for n  ≥ 2, there is a c n  > 0 such that $|D|^+_n > C_n \sqrt{|D|}$ for all minimally n ‐connected, finite digraphs D . Furthermore, case n  = 2 of the following conjecture is settled which says that every minimally n ‐connected, finite digraph has a vertex of indegree and outdegree equal to n . © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 129–144, 2002