Orthogonal ( g, f )-factorizations of bipartite graphs

Let G be a bipartite graph with vertex set V ( G ) and edge set E ( G ), and let g and f be two positive integer-valued functions defined on V ( G ) such that g ( x ) ≤ f ( x ) for every vertex x of V ( G ). Then a ( g, f )-factor of G is a spanning subgraph H of G such that g ( x ) ≤ d H ( x ) ≤ f ( x ) for each x ∈ V ( H ). A ( g , f )-factorization of G is a partition of E ( G ) into edge-disjoint ( g , f )-factors. Let F = { F 1 , F 2 , …, F m } and H be a factorization and a subgraph of G , respectively. If F i , 1 ≤ i ≤ m , has exactly one edge in common with H , then it is said that F is orthogonal to H. It is proved that every bipartite ( mg + m − 1 , mf − m + 1)-graph G has a ( g , f )-factorization orthogonal to k vertex disjoint m -subgraphs of G if k 2 ≤ g ( x ) for all x ∈ V ( G ). Furthermore, it is showed that the results in this paper are best possible.