The covering threshold of a directed acyclic graph by directed acyclic subgraphs

Let H be a directed acyclic graph (dag) that is not a rooted star. It is known that there are constants c = c(H) and C = C(H) such that the following holds for D_n, the complete directed graph on n vertices. There is a set of at most C log n directed acyclic subgraphs of D_n that covers every H-copy of D_n, while every set of at most c log n directed acyclic subgraphs of D_n does not cover all H-copies. Here this dichotomy is considerably strengthened. Let G(n, p) denote the probability space of all directed graphs with n vertices and with edge probability p. The fractional arboricity of H is (Formula presented) where the maximum is over all non-singleton subgraphs of (Formula presented) then H is totally balanced. Complete graphs, complete multipartite graphs, cycles, trees, and, in fact, almost all graphs, are totally balanced. It is proven that: • Let H be a dag with h vertices and m edges which is not a rooted star. For every a^* > a(H) there exists c^* = c^*(a^*, H) > 0 such a.a.s. G ~G(n, n^−1/a*) has the property that every set X of at most c^* log n directed acyclic subgraphs of G does not cover all H-copies of G. Moreover, there exists s(H) = m/2 + O(m^4/5h^1/5) such that the following stronger assertion holds for any such X: There is an H-copy in G that has no more than s(H) of its edges covered by each element of X. • If H is totally balanced then for every 0 < a^* < a(H), a.a.s. G ~G(n, n^−1/a*) has a single directed acyclic subgraph that covers all its H-copies. As for the first result, note that if h = o(m) then s(H) = (1 + o_m(1))m/2 is about half of the edges of H. In fact, for infinitely many H it holds that s(H) = m/2, optimally. As for the second result, the requirement that H is totally balanced cannot, generally, be relaxed.