A generalization of orthogonal factorizations in graphs

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 E V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) less than or equal to d(H) (x) less than or equal to f (x) for all x is an element of V(G). A (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} be a factorization of G, and H be a subgraph of G with MT edges. If F-i, 1 less than or equal to i less than or equal to m, has exactly r edges in common with H. then F is said to be r-orthogonal to H. In this paper it is proved that every (mg + kr, mf - kr)-graph where m, k and r are positive integers with k < m and g greater than or equal to r, contains a subgraph R such that R has a (g, f)-factorization which is r-orthogonal to a given subgraph H with kr edges.