Guo Jun LI,Gui Zhen LIU
In this paper, 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 ∈ 6 V(G). Then a (g,f)-factor of G is a spanning subgraph H of G such that g(x) ≤ dH(x) ≤ f(x) for all x ∈ V(G). 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, 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 ≥ 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.