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 ) ≥ 2 k − 1 contains at least k vertex-disjoint cycles. In this work we prove that every bipartite tournament with minimum out-degree at least 2 k − 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 ( 3 k − 1 ) / 2 contains k vertex-disjoint cycles of length four. [ABSTRACT FROM AUTHOR]