Orthogonal factorizations of digraphs

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) such that g(-)(x) <= f(-)(x) and g(+)(x) <= f(+)(x) for each x is an element of V(G). A (g, f)-factor of G is a spanning subdigraph H of G such that g(-)(x) <= id(H)(x) <= f(-)(x) and g(+)(x) <= od(H)(x) <= f(+)(x) for each x is an element of V(H); a (g, f)-factorization of G is a partition of E(G) into arc-disjoint (g, f)-factors. Let F = {F(1), F(2), ... , F(m)} and H be a factorization and a subdigraph of G, respectively. F is called k-orthogonal to H if each F(i), 1 <= i <= m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m-1, mf-m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k <= min{g(-)(x), g(+)(x)} for any x is an element of V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 <= g(x) <= f(x) for any x is an element of V(G). The results in this paper are in some sense best possible.