Subgraphs with orthogonal factorizations and algorithms

Let G = ( V , E ) be a graph and let g and f be two integer-valued functions defined on V such that n ≤ g ( x ) ≤ f ( x ) for every x ∈ V . Let H 1 , H 2 , … , H n be vertex-disjoint subgraphs of G with | E ( H i ) | = k ( 1 ≤ i ≤ n ). In this paper, we prove that every ( m g + k , m f − k ) -graph G contains a subgraph R such that R has a ( g , f ) -factorization orthogonal to H i ( 1 ≤ i ≤ n ), where m and k are positive integers with 1 ≤ k < m . We also prove that there are polynomial-time algorithms for finding the desired subgraph R .