Arc‐disjoint in‐ and out‐branchings in digraphs of independence number at most 2

We prove that every digraph of independence number at most 2 and arc-connectivity least has an out-branching B + in-branching − which are arc-disjoint (we call such branchings a good pair). This is best possible in terms the as there infinitely many strong digraphs with arbitrarily high minimum in- out-degrees have no pair. The result settles conjecture by Thomassen for 2. on 6 vertices pair give example 2-arc-strong D 10 4 also show 7 Finally we pose open problems.