Vertex-disjoint cycles in bipartite tournaments

Abstract

Let t1,…,tr∈[4,2q] be any r even integers, where q≥2 and r≥1 are two integers. In this note, we show that every bipartite tournament with minimum outdegree at least qr−1 contains r vertex-disjoint directed cycles of lengths t1′,…,tr′ such that ti′=ti for ti=0(mod4) and ti′∈{ti,ti+2} for ti=2(mod4), where 1≤i≤r. The special case q=2 of the result verifies the bipartite tournament case of a conjecture proposed by Bermond and Thomassen, stating that every digraph with minimum outdegree at least 2r−1 contains at least r vertex-disjoint directed cycles.