Subdigraphs with orthogonal factorizations of digraphs

Let G = ( V , E ) be a digraph and let g and f be two pairs of 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 arc-disjoint k -subdigraphs of G . In this article, we prove that every ( m g + k − 1 , m f − k + 1 ) -digraph G contains a subdigraph 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 .