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- Title
Arc‐disjoint in‐ and out‐branchings in digraphs of independence number at most 2.
- Authors
Bang‐Jensen, Jørgen; Bessy, Stéphane; Havet, Frédéric; Yeo, Anders
- 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 a 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 no good 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.
- Subjects
LOGICAL prediction; BRANCHING processes
- Publication
Journal of Graph Theory, 2022, Vol 100, Issue 2, p294
- ISSN
0364-9024
- Publication type
Article
- DOI
10.1002/jgt.22779