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

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