Small degree out-branchings

Using a suitable orientation, we give a short proof of a result of Czumaj and Strothmann [3]: Every 2-edge-connected graph G contains a spanning tree T with the property that dT (v) ≤ dG(v)+3 2 for every vertex v. Trying to find an analogue of this result in the directed case, we prove that every 2-arc-strong digraph D has an out-branching B such that d+B(x) ≤ d+D(x) 2 + 1. As a corollary, every k-arc-strong digraph D has an out-branching B such that d+B(v) ≤ d+ D (v) 2r + r, where r = b log 2kc. We conjecture that in this case d+B(x) ≤ d+D(x) k + 1 would be the right (and best possible) answer. We prove that any degree requirement in out-branchings of acyclic digraphs can be polynomially checked.