Vertex-disjoint cycles in regular tournaments

The Bermond–Thomassen conjecture states for r≥1r≥1, any digraph of minimum out-degree at least 2r−12r−1 contains at least rr vertex-disjoint directed cycles. In a recent paper, Bessy, Sereni and the author proved that a regular tournament TT of degree 2r−12r−1 contains at least rr 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 76r−73 vertex-disjoint directed cycles.