Subdigraphs with orthogonal factorizations of digraphs

Let G=(V,E)G=(V,E) be a digraph and let gg and ff be two pairs of integer-valued functions defined on VV such that n≤g(x)≤f(x)n≤g(x)≤f(x) for every x∈Vx∈V. Let H1,H2,…,HnH1,H2,…,Hn be arc-disjoint kk-subdigraphs of GG. In this article, we prove that every (mg+k−1,mf−k+1)(mg+k−1,mf−k+1)-digraph GG contains a subdigraph RR such that RR has a (g,f)(g,f)-factorization orthogonal to HiHi (1≤i≤n1≤i≤n), where mm and kk are positive integers with 1≤k≤m1≤k≤m.