Vertex disjoint 4-cycles in bipartite tournaments

Let $k\ge 2$ be an integer. Bermond and Thomassen conjectured that every digraph with minimum out-degree at least $2k?1$ contains $k$ vertex-disjoint cycles. Recently Bai, Li and Li proved this conjecture for bipartite digraphs. In this paper we prove that every bipartite tournament with minimum out-degree at least $2k?2$, minimum in-degree at least $1$ and partite sets of cardinality at least $2k$ contains $k$ vertex-disjoint 4-cycles whenever $k\ge 3$. Finally, we show that every bipartite tournament with minimum degree $\delta =min\{{\delta }^{+},{\delta }^{?}\}$ at least $1.5k?1$ contains at least $k$ vertex-disjoint 4-cycles.