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 @d(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 [email protected]?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.