Ecken von kleinem grad in kritisch n-fach zusammenhängenden digraphen

Abstract

We prove that every critically n-connected, finite digraph has a vertex of outdegree less than 2 n or a vertex of indegree less than 2 n, but there is not always a vertex of outdegree less than 2 n. This bound 2 n−1 for the minimum of the out- and indegrees is, in general, best possible and was conjectured by Y. O. Hamidoune (“Contribution à l'étude de la connectivité d'un graphe”, Thèse d'Etat, Paris, 1980). An immediate consequence is the main result of the author's previous work ( Arch. Math. 25, 1974, 107–112) that every minimally n-edge-connected, finite digraph has a vertex of indegree and outdegree n.