Packing Digraphs With Directed Closed Trails

It has been shown [2] that if n is odd and m1,…,mt are integers with mig3 and ∑i=1t mi=vE(Kn)v then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1,…,mt. This result was later generalized [3] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider the corresponding questions for directed graphs. We show that the complete directed graph ****gif image here**** can be decomposed as an edge-disjoint union of directed closed trails of lengths m1,…,mt whenever mig2 and ****gif image here****, except for the single case when n=6 and all mi=3. We also show that sufficiently dense Eulerian digraphs can be decomposed in a similar manner, and we prove corresponding results for (undirected) complete multigraphs.