Vertex disjoint 4-cycles in bipartite tournaments

Let k≥2 k ≥ 2 be an integer. Bermond and Thomassen conjectured that every digraph with minimum out-degree at least 2k−1 2 k − 1 contains k 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 2 k − 2 , minimum in-degree at least 1 1 and partite sets of cardinality at least 2k 2 k contains k k vertex-disjoint 4-cycles whenever k≥3 k ≥ 3 . Finally, we show that every bipartite tournament with minimum degree δ=min{δ + ,δ − } δ = min { δ + , δ − } at least 1.5k−1 1 . 5 k − 1 contains at least k k vertex-disjoint 4-cycles.