Cycle Decompositions of Complete Digraphs

In this paper, we consider the problem of decomposing the complete directed graph K∗ n into cycles of given lengths. We consider general necessary conditions for a directed cycle decomposition of K∗ n 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 6= 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), K∗ n is not decomposable. Mathematics Subject Classifications: 05C51, 05B30, 05C20, 05C38