Subgraphs with (g,f)-factorization orthogonal to star

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 that 0≤g(x)f(x) for every x∈V(G). Then a (g,f) factor of G is a spanning subgraph H of G such that g(x)≤d H(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 n}and H be the factorization and a subgraph of G, respectively. If F i,1≤i≤n,has exactly one edge in common with H,then it is said that F is orthogonal to H. In this paper, it is proved that for any k star H of an (mg+k,mf-k) graph G,1≤km, there exists a subgraph R with a (g,f) factorization orthogonal to H.