About some cyclic properties in digraphs

Abstract In this article we are concerned with digraphs in which any two vertices are on a common cycle. For example, we prove that, in a strong digraph of order n and half degrees at least 2 with at least n 2 − 5 n + 15 arcs, any two vertices are on a common cycle. We also consider related properties and give sufficient conditions on half degrees and the number of arcs to insure these properties. In particular, we show that every digraph of order n with half degrees at least r and with at least n 2 − rn + r 2 arcs is 2-linked.