Subdigraphs with orthogonal factorizations of digraphs (II)

Let GG be a digraph with vertex set V(G)V(G) and arc set E(G)E(G) and let g=(g−,g+)g=(g−,g+) and f=(f−,f+)f=(f−,f+) be pairs of positive integer-valued functions defined on V(G)V(G) with f(x)≥g(x)≥r−12 for each x∈V(G)x∈V(G). Let H1,H2,…,HrH1,H2,…,Hr be vertex-disjoint kk-subdigraphs of GG. In this paper, it is proved that every (mg+(k−1)r,mf−(k−1)r)(mg+(k−1)r,mf−(k−1)r)-digraph GG contains a subdigraph RR such that RR has a (g,f)(g,f)-factorization orthogonal to every Hi(1≤i≤r), where k,mk,m and rr be three positive integers with k≤mk≤m.