Covering digraphs with small indegrees or outdegrees by directed cuts

For nonnegative integers k and l , let D ( k , l ) denote the set of digraphs in which every vertex has indegree at most k or outdegree at most l . In this paper, we first compare three existing upper bounds for the number of directed cuts to cover the arcs of digraphs in D ( k , k ) and prove that these bounds can be improved from seven to six in the case k = 5 , 6 . Further, we give a lower bound for the number of directed cuts to cover the digraphs in D ( k , l ) by constructing a digraph in this class.