Randomly orthogonal factorizations in networks

Let m, r, k be three positive integers. Let G be a graph with vertex set V(G) and edge set E(G), and let f: V(G) → N be a function such that f(x)≥(k+2)r−1 for any x ∈ V(G). Let H1, H2, …, Hk be k vertex disjoint mr-subgraphs of a graph G. In this paper, we prove that every (0,mf−(m−1)r)-graph admits a (0, f)-factorization randomly r-orthogonal to each Hi (i=1,2,…,k).