Geodesic
Decomposition of complete bipartite digraphs and even complete bipartite multigraphs into closed trails
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 2, pp. 241-249 Cet article a éte moissonné depuis la source Library of Science

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It has been shown [3] that any bipartite graph K_a,b, where a, b are even integers, can be decomposed into closed trails with prescribed even lengths. In this article, we consider the corresponding question for directed bipartite graphs. We show that a complete directed bipartite graph ^↔ K_a,b is decomposable into directed closed trails of even lengths greater than 2, whenever these lengths sum up to the size of the digraph. We use this result to prove that complete bipartite multigraphs can be decomposed in a similar manner.
Keywords: trail, decomposition 
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Cichacz, Sylwia. Decomposition of complete bipartite digraphs and even complete bipartite multigraphs into closed trails. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 2, pp. 241-249. http://geodesic.mathdoc.fr/item/DMGT_2007_27_2_a2/

[1] P.N. Balister, Packing Circuits into Kₙ, Combin. Probab. Comput. 10 (2001) 463-499, doi: 10.1017/S0963548301004771.

[2] P.N. Balister, Packing digraphs with directed closed trails, Combin. Probab. Comput. 12 (2003) 1-15, doi: 10.1017/S0963548302005461.

[3] M. Hornák and M. Woźniak, Decomposition of complete bipartite even graphs into closed trails, Czechoslovak Math. J. 128 (2003) 127-134.