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 Dt , let |E(G)| be the number of directed edges of G and suppose G 6= ⇀ C 2 ∪ ⇀ C 3 or ⇀ C 5 if t = 5, and G 6= ⇀ 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 Dt − G if and only if t(t − 1)− |E(G)| ≡ 0 (mod 3). © 2008 Elsevier B.V. All rights reserved.