Vertex-disjoint cycles in bipartite tournaments

Let k≥2 be an integer. Bermond and Thomassen [Bermond J. C., Thomassen, C., Cycles in digraphs a survey, Journal of Graph Theory 5(1) (1981) 1-43] conjectured that every digraph D with δ+(D)≥2k−1 contains at least k vertex-disjoint cycles. In this work we prove that every bipartite tournament with minimum out-degree at least 2k−2 and minimum in-degree at least one contains k vertex-disjoint cycles of length four, whenever k≥3. Finally, we show that every bipartite tournament with minimum degree at least (3k−1)/2 contains k vertex-disjoint cycles of length four.