Geodesic
Decompositions of a complete multidigraph into almost arbitrary paths
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 2, pp. 357-372

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For n ≥ 4, the complete n-vertex multidigraph with arc multiplicity λ is proved to have a decomposition into directed paths of arbitrarily prescribed lengths ≤ n - 1 and different from n - 2, unless n = 5, λ = 1, and all lengths are to be n - 1 = 4. For λ = 1, a more general decomposition exists; namely, up to five paths of length n - 2 can also be prescribed.
Keywords: complete digraph, multidigraph, tour girth, arbitrary path decomposition 
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     title = {Decompositions of a complete multidigraph into almost arbitrary paths},
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Meszka, Mariusz; Skupień, Zdzisław. Decompositions of a complete multidigraph into almost arbitrary paths. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 2, pp. 357-372. http://geodesic.mathdoc.fr/item/DMGT_2012_32_2_a13/