The smallest number of vertices in a 2-arc-strong digraph which has no good pair

TL;DR

This paper determines that all 2-arc-strong digraphs with up to 9 vertices necessarily contain a pair of arc-disjoint out- and in-branchings, resolving a question about the minimal size of such digraphs without good pairs.

Contribution

It proves that the smallest 2-arc-strong digraph with no good pair has at least 10 vertices, settling a previously open problem.

Findings

All 2-arc-strong digraphs on up to 9 vertices have a good pair.

The smallest example without a good pair has at least 10 vertices.

This result confirms the minimal size for such digraphs.

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 {\it 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.