Let G be a graph with vertex set V(G) and edge set E(G), and let g, f be two nonnegative integer-valued functions defined on V(G) such that

for every vertex x of V(G). We use

to denote the degree of a vertex x of G. A (g, f)-factor of G is a spanning subgraph F of G such that

for every vertex x of V(F). In particular, G is called a (g, f)-graph if G itself is a (g, f)-factor. A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {

,

, ...,

} be a factorization of G and H be a subgraph of G with mr edges. If

,

, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {

,

, ...,

} of E(H) with

there is a (g, f)-factorization F = {

,

, ...,

} of G such that

,

, then we say that G has (g, f)-factorizations randomly r-orthogonal to H. In this paper it is proved that every (0, mf - (m - 1)r)-graph has (0, f)-factorizations randomly r-orthogonal to any given subgraph with mr edges if

for any

.