Packing Digraphs with Directed Closed Trails

Abstract

It has been shown [2] that if n is odd and m₁,…,m_t are integers with m_i[ges ]3 and [sum ]_i=1^tm_i=|E(K_n)| then K_n can be decomposed as an edge-disjoint union of closed trails of lengths m₁,…,m_t. This result was later generalized [3] to all sufficiently dense Eulerian graphs G in place of K_n. In this article we consider the corresponding questions for directed graphs. We show that the complete directed graph <?TeX \displaystyle{\mathop{K}^{\raise-2pt\hbox{$\scriptstyle\leftrightarrow$}}}_{n}?> can be decomposed as an edge-disjoint union of directed closed trails of lengths m₁,…,m_t whenever m_i[ges ]2 and <?TeX \sum_{i=1}^t m_{i}=\vert E({\leftrightarrow}_{n})\vert ?>, except for the single case when n=6 and all m_i=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.