A (g,f)-factorization 2-orthogonal to any subgraph

Let G be a graph with vertex set V(G) and edge set E(G), and let g(x) and f(x) be two integervalued functions defined on V(G) such as 4≤g(x)≤f(x) for every x∈V(G). Then a (g,f)factor of G is a spanning subgraph F of G such as g(x)≤dF(x)≤f(x) for every x∈V(G). The (g,f)factorization of G is a partition of E(G) into edgedisjoint (g,f)factors. Let F={F1,F2,...,Fm} and H be the factorization and a subgraph of G, respectively. If Fi, 1≤i≤m, has exactly two edges in common with H, then it is said that F is 2orthogonal to H. This paper proves that for any 2msubgraph H of an (mg+m-1,mf-m+1)graph G, there exists a (g,f)factorization 2orthogonal to H.