A result on r-orthogonal factorizations in digraphs

Let G be a digraph with vertex set V(G) and arc set E(G). Let m,r,k be three positive integers, and let f=(f−,f+) be a pair of nonnegative integer-valued functions defined on V(G) with f(x)≥(k+1)r for all x∈V(G). Let H1,H2,…,Hk be k vertex disjoint mr-subdigraphs of G. In this paper, it is proved that every (0,mf−(m−1)r)-digraph has a (0,f)-factorization r-orthogonal to every Hi (i=1,2,…,k).