Disproof of a conjecture about independent branchings in k-connected directed graphs

For each k ≥ 3, we construct a finite directed strongly k-connected graph D containing a vertex t with the following property: For any k spanning t-branchings, B₁, …, B_k in D (i. e., each B_i is a spanning tree in D directed toward t), there exists a vertex x ≠ t of D such that the k, x, t-paths in B₁, …, B_k are not pairwise openly disjoint. This disproves a well-known conjecture of Frank. © 1995, John Wiley & Sons, Inc.