On Sullivan's conjecture on cycles in 4-free and 5-free digraphs

For a simple digraph G, let .(G) be the size of the smallest subset X . E(G) such that G.X has no directed cycles, and let .(G) be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that .(G) . 0.3819.(G) if G is a 4-free digraph, and .(G) . 0.2679.(G) if G is a 5-free digraph. These improve the results of Sullivan in 2008.