Packing Digraphs With Directed Closed Trails

It has been shown [Balister, 2001] that if n is odd and m1, . . . , mt are integers with mi ≥ 3 and ∑t i=1 mi = |E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1, . . . , mt. This result was later generalized [Balister, to appear] 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 compete directed graph ↔ Kn can be decomposed as an edge-disjoint union of directed closed trails of lengths m1, . . . , mt whenever mi ≥ 2 and ∑ mi = |E( ↔ Kn)|, 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.