Openly disjoint circuits through a vertex in regular digraphs

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