Embedding partial bipartite directed cycle systems

A 2k-dicycle is a 2k-cycle of a directed bipartite graph and a 2k-dicycle system of order (m,n) is a triple (X, Y,D), where D is a collection of edge disjoint 2k-dicycles which partitions the edge set of the complete directed bipartite graph Dm,n with parts X and Y . A partial 2k-dicycle system of order (s, t) is a triple (S, T, P ), where P is a collection of edge disjoint 2k-dicycles of Ds,t which does not necessarily partition the edge set of Ds,t. The partial 2k-dicycle system (S, T, P ) of order (s, t) is said to be embedded in the 2k-dicycle system (X, Y,D) of order (m,n) provided that S ⊆ X, T ⊆ Y , and P ⊆ D. A partial 2k-dicycle system of order (s, t) can always be embedded in a 2k-dicycle system of order (ks, kt).