Subdigraphs with orthogonal factorizations of digraphs (II)

Let G be a digraph with vertex set V ( G ) and arc set E ( G ) and let g = ( g − , g + ) and f = ( f − , f + ) be pairs of positive integer-valued functions defined on V ( G ) with f ( x ) ≥ g ( x ) ≥ r − 1 2 for each x ∈ V ( G ) . Let H 1 , H 2 , … , H r be vertex-disjoint k -subdigraphs of G . In this paper, it is proved that every ( m g + ( k − 1 ) r , m f − ( k − 1 ) r ) -digraph G contains a subdigraph R such that R has a ( g , f ) -factorization orthogonal to every H i ( 1 ≤ i ≤ r ) , where k , m and r be three positive integers with k ≤ m .