Subgraphs with orthogonal factorizations and algorithms

Let G=(V,E) be a graph and let g and f be two integer-valued functions defined on V such that n@?g(x)@?f(x) for every x@?V. Let H"1,H"2,...,H"n be vertex-disjoint subgraphs of G with |E(H"i)|=k (1@?i@?n). In this paper, we prove that every (mg+k,mf-k)-graph G contains a subgraph R such that R has a (g,f)-factorization orthogonal to H"i (1@?i@?n), where m and k are positive integers with 1@?k