Guo Jun LI, Gui Zhen LIU
Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valuated functions defined on V(G) such that g(x) ≤f(x) for all x∈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.
