Existence of openly disjoint circuits through a vertex

We deal with conditions for a digraph of minimum degree r which imply the existence of a vertex x contained in r circuits which have pairwise only x in common. In particular, we give some positive answers to a question of P. Seymour, whether an r‐regular digraph has a vertex x which is contained in r circuits pairwise disjoint except for x, and show that the answer, in general, is negative. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 93–105, 2010