A (g, f) - Factorization (m, r) - Orthogonal to an Arbitrary Graph

Let G be a graph with its vertex set V (G) and frontier set E (G), and let g (x) and f (x) be non-negative integer-valued functions defined on V (G) such that g (x) < f (x) for each x∈V (G), hence a (g, f) - factor of G is a spanning subgraph F of G if g (x)≤ dF (x)≤ f (x) for each x∈V (G). The (g, f) - factorization of G is a partition of E (G) into edge-disjoint (g, f) -factors. Let F = {F1, F2, L, Fm} and H be a factorization and a subgraph of G respectively. Then F is (m, r) -orthogonal to H, providing that Fi has r edges in common with H, where 1≤ i≤ m. It is proved in this paper through Theorem 1 that, suppose G is an ( mg + ( m −1) r , mf - m +1) graph and H is a subgraph of G with mr edges, then there exists a (g, f)-factorization (m, r)-orthogonal to H.