Vertex disjoint 4-cycles in bipartite tournaments

Abstract

Let k≥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≥3. Finally, we show that every bipartite tournament with minimum degree δ=min{δ+,δ−} at least 1.5k−1 contains at least k vertex-disjoint 4-cycles.