RANDOMLY ORTHOGONAL FACTORIZATIONS OF (0,mf − (m− 1)r)-GRAPHS

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 g(x) ≤ f(x) for every vertex x of V (G). We use dG(x) 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 g(x) ≤ dF (x) ≤ f(x) 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 = {F₁, F₂, . . . , Fm} be a factorization of G and H be a subgraph of G with mr edges. If Fi, 1 ≤ i ≤ m, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {A₁,A₂, . . . ,Am} of E(H) with |Ai| = r there is a (g, f)-factorization F = {F₁, F₂, . . . , Fm} of G such that Ai ⊆ E(Fi), 1 ≤ i ≤ m, 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 f(x) ≥ 3r − 1 for any x ∈ V (G).