The smallest number of vertices in a 2-arc-strong digraph which has no good pair
Abstract
Bang-Jensen, Bessy, Havet and Yeo showed that every digraph of independence number at most 2 and arc-connectivity at least 2 has an out-branching B+ and an in-branching B- which are arc-disjoint (such two branchings are called a good pair), which settled a conjecture of Thomassen for digraphs of independence number 2. They also proved that every digraph on at most 6 vertices and arc-connectivity at least 2 has a good pair and gave an example of a 2-arc-strong digraph D on 10 vertices with independence number 4 that has no good pair. They asked for the smallest number n of vertices in a 2-arc-strong digraph which has no good pair. In this paper, we prove that every digraph on at most 9 vertices and arc-connectivity at least 2 has a good pair, which solves this problem.