Vertex-disjoint directed cycles of prescribed length in tournaments with given minimum out-degree and in-degree

In a recent paper, Bessy, Sereni and the author (see [3]) have proved that for r ≥ 1 , a tournament with minimum out-degree and in-degree both greater than or equal to 2 r − 1 contains at least r vertex-disjoint directed triangles. In this paper, we generalize this result; more precisely, we prove that for given integers q ≥ 3 and r ≥ 1 , a tournament with minimum out-degree and in-degree both greater than or equal to ( q − 1 ) r − 1 contains at least r vertex-disjoint directed cycles of length q . We will use an auxiliary result established in [3], concerning a union of sets contained in another union of sets. We finish by giving a lower bound on the maximum number of vertex-disjoint directed cycles of length q when only the minimum out-degree is supposed to be greater than or equal to ( q − 1 ) r − 1 .