Orthogonal factorizations of graphs Original Research Article

Let G be a graph with vertex set V(G) and edge set E(G), and let g and 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 graph G is called a (g,f)-graph if g(x)⩽dG(x)⩽f(x) for each x∈V(G). Then a spanning subgraph F of G is said to be a (g,f)-factor of G if F itself is a (g,f)-graph. 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 m edges. If Fi,1⩽i⩽m, has exactly one edge in common with H, we say that F is orthogonal to H. In this paper it is proved that every (mg+k,mf−k)-graph contains a subgraph R such that R has a (g,f)-factorization orthogonal to a given subgraph with k edges, where m and k are positive integers with 1⩽k