Vertex-disjoint cycles in regular tournaments

The Bermond–Thomassen conjecture states for $r\ge 1$, any digraph of minimum out-degree at least $2r?1$ 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 $2r?1$ 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 $\frac{7}{6}r?\frac{7}{3}$ vertex-disjoint directed cycles.