Arc-disjoint in- and out-branchings in digraphs of independence number at most 2

TL;DR

This paper proves that digraphs with independence number at most 2 and arc-connectivity at least 2 always contain a pair of arc-disjoint in- and out-branchings, settling a conjecture for this class.

Contribution

It establishes the existence of arc-disjoint in- and out-branchings in digraphs with independence number 2 and arc-connectivity at least 2, confirming a conjecture by Thomassen.

Findings

Every such digraph has a good pair.

Small digraphs (up to 6 vertices) with these properties always have a good pair.

Counterexamples exist for larger independence numbers and specific sizes.

Abstract

We prove 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 (we call such branchings good pair). This is best possible in terms of the arc-connectivity as there are infinitely many strong digraphs with independence number 2 and arbitrarily high minimum in-and out-degrees that have good no pair. The result settles a conjecture by Thomassen for digraphs of independence number 2. We prove that every digraph on at most 6 vertices and arc-connectivity at least 2 has a good pair and give an example of a 2-arc-strong digraph $D$ on 10 vertices with independence number 4 that has no good pair. We also show that there are infinitely many digraphs with independence number 7 and arc-connectivity 2 that have no good pair. Finally we pose a number of open problems.