Cycle decompositions of complete digraphs

Abstract

In this paper, we consider the problem of decomposing the complete directed graph Kn∗into cycles of given lengths. We consider general necessary conditions for a directed cycle decomposition of Kn∗into t cycles of lengths m1, m2, …, mt to exist and provide a powerful construction for creating such decompositions in the case where there is one ‘large’ cycle. Finally, we give a complete solution in the case when there are exactly three cycles of lengths α, β, γ ≠ 2. Somewhat surprisingly, the general necessary conditions turn out not to be sufficient in this case. In particular, when γ = n, α + β > n + 2 and α + β ≡ n (mod 4), Kn∗ is not decomposable.