Two proofs of the Bermond-Thomassen conjecture for tournaments with bounded minimum in-degree (Q960967)

In 1981, \textit{J.C. Bermond} and \textit{C. Thomassen} [``Cycles in digraphs. A survey'', J. Graph Theory 5, 1--43 (1981; Zbl 0458.05035)] conjectured that for any positive integer \(r\), any digraph of minimum out-degree at least \(2r - 1\) contains at least r vertex-disjoint directed cycles. It can be shown that the conjecture is true for \(r = 2\) and \(3\), and that a \(k\)-strongly-connected tournament of order at least \(5k - 3\) contains \(k\) vertex-disjoint directed cycles. The present note gives two proofs of this conjecture for tournaments with minimum in-degree at least \(2r - 1\).