Small degree out-branchings

Abstract

Using a suitable orientation, we give a short proof of a strengthening of a result of Czumaj and Strothmann: Every 2-edge-connected graph G contains a spanning tree T with the property that dT(v) ≤ dT(v)+3/2 for every vertex v. As 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 dB+(x) ≤ dD+(x)/2 + 1. A corollary of this is that every k-arc-strong digraph D has an out-branching B such that dB+(v) ≤ dD+(v)/2r + r, where r = ⌊log2k⌋. We conjecture that in this case dB+(x) ≤ dD+(x)/k + 1 would be the right (and best possible) answer. If true, this would again imply a strengthening of a result from [4] concerning spanning trees with small degrees in k-connected graphs when k ≥ 2. We prove that for acyclic digraphs the existence of an out-branching satisfying prescribed bounds on the out-degrees of each vertex can be checked in polynomial time. A corollary of this is that the existence of arc-disjoint branchings Fs+, Ft-, where the first is an out-branching rooted at s and the second an in-branching rooted at t, can be checked in polynomial time for the class of acyclic digraphs. © 2003 Wiley Periodicals, Inc.