Abstract: Let G be a graph with vertex set V(G) and edge set E(G),and let g and f be two integer-valued functions defined on V(G) such that g(x)≤f(x) for all x∈V(G).It is proved that if G is an (mg+rn,mf-rn)-graph,1≤n＜m,r≥2,and g(x)≥k≥1 for all x∈V(G), then there exists a subgraph G′ of G such that G′ has a (g,f)-factorization (n,r )-orthogonal to any given subgraph H of G with ｜E(H)｜=nk.

Key words: graph, factorization, orthogonal