Covering the edges of digraphs in D(3,3)D(3,3) and D(4,4)D(4,4) with directed cuts

For nonnegative integers kk and ll, let D(k,l)D(k,l) denote the family of digraphs in which every vertex has either indegree at most kk or outdegree at most ll. In this paper we prove that the edges of every digraph in D(3,3)D(3,3) and D(4,4)D(4,4) can be covered by at most five directed cuts and present an example in D(3,3)D(3,3) showing that this result is best possible.