Covering the vertices of a digraph by cycles of prescribed length

AbstractLet D be a strong digraph with n vertices and at least (n − 1)(n − 2) + 3 arcs. For any integers k,n1,n2,…, nk such that n = n1 + n2 +⋯+ nk and ni⩾3, there exis covering of the vertices of D by disjoint directed cycles of length n1,n2,…,nk except in two cases: 1.Case 1: n = 6; n1 = n2 = 3 and D contains a stable set with 3 vertices.2.Case 2: n = 9; n1 = n2 = n3 = 3 and D contains a stable set with 4 vertices