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≥1r≥1, a tournament with minimum out-degree and in-degree both greater than or equal to 2r−12r−1 contains at least rr vertex-disjoint directed triangles. In this paper, we generalize this result; more precisely, we prove that for given integers q≥3q≥3 and r≥1r≥1, a tournament with minimum out-degree and in-degree both greater than or equal to (q−1)r−1(q−1)r−1 contains at least rr vertex-disjoint directed cycles of length qq. 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 qq when only the minimum out-degree is supposed to be greater than or equal to (q−1)r−1(q−1)r−1.