Note on Disjoint Cycles in Multipartite Tournaments

Abstract: In 1981, Bermond and Thomassen conjectured that for any positive integer $k$, every digraph with minimum out-degree at least $2k-1$ admits $k$ vertex-disjoint directed cycles. In this short paper, we verify the Bermond-Thomassen conjecture for triangle-free multipartite tournaments and 3-partite tournaments. Furthermore, we characterize 3-partite tournaments with minimum out-degree at least $2k-1$ ($k\geq 2$) such that in each set of $k$ vertex-disjoint directed cycles, every cycle has the same length.