Vertex-disjoint cycles in regular tournaments

The Bermond–Thomassen conjecture states for r ≥ 1 , any digraph of minimum out-degree at least 2 r − 1 contains at least r vertex-disjoint directed cycles. In a recent paper, Bessy, Sereni and the author proved that a regular tournament T of degree 2 r − 1 contains at least r vertex-disjoint directed cycles, which shows that the above conjecture is true for regular tournaments. In this paper, we improve this result by proving that such a tournament contains at least 7 6 r − 7 3 vertex-disjoint directed cycles.