Openly disjoint circuits through a vertex in regular digraphs

In 1985, Thomassen [14] constructed for every positive integer r , finite digraphs D of minimum degree δ ( D ) = r which do not contain a vertex x lying on three openly disjoint circuits, i.e. circuits which have pairwise exactly x in common. In 2005, Seymour [11] posed the question, whether an r -regular digraph contains a vertex x such that there are r openly disjoint circuits through x . This is true for r ≤ 3 , but does not hold for r ≥ 8 . But perhaps, in contrast to the minimum degree, a high regularity degree suffices for the existence of a vertex lying on r openly disjoint circuits also for r ≥ 4 . After a survey of these problems, we will show that every r -regular digraph with r ≥ 7 has a vertex which lies on 4 openly disjoint circuits.