Vertex-disjoint cycles in tournaments

Abstract: The Bermond-Thomassen conjecture states that, for any positive integer $r$, a digraph of minimum out-degree at least $2r-1$ contains at least $r$ vertex-disjoint directed cycles. Bessy, Sereni and Lichiardopol proved that a regular tournament $T$ of minimum degree $2r-1$ contains at least $r$ vertex-disjoint directed cycles, which shows that the above conjecture is true for tournaments. After that, Lichiardopol improved this result by showing that a $2r-1$-regular tournament contains at least $\frac{7}{6}r-\frac{7}{3}$ vertex-disjoint directed cycles. In this paper, we will extend the result to tournaments with minimum out-degree at least $2r-1$ by proving a more general result.