Directed 3-cycle decompositions of complete directed graphs with quadratic leaves

Let GG be a vertex-disjoint union of directed cycles in the complete directed graph DtDt, let |E(G)||E(G)| be the number of directed edges of GG and suppose G≠C⇀2∪C⇀3 or C⇀5 if t=5t=5, and G≠C⇀3∪C⇀3 if t=6t=6. It is proved in this paper that for each positive integer tt, there exist C⇀3-decompositions for Dt−GDt−G if and only if t(t−1)−|E(G)|≡0(mod3).