Bang-Jensen and Yeo's strong arc decomposition conjecture
Let be a digraph. It is -arc-strong if remains strong for every subset of size at most . A strong arc decomposition of is a partition of into disjoint sets and such that the spanning subdigraphs and are both strong.
Bang-Jensen and Yeo's conjecture. There exists an integer such that every -arc-strong digraph has a strong arc decomposition.
A strong arc decomposition would imply the existence of a good -pair for every choice of roots, so this conjecture would imply Thomassen's conjecture on good vertex pairs. The source gives no resolution, and the conjecture remains open.
References
Primary source
Jiangdong Ai, Yiming Hao, Zhaoxiang Li and Qi Shao, “Arc-disjoint in- and out-branchings in semicomplete split digraphs”, arXiv:2410.12575 (2024).
Additional references
4 papers in this index state this conjecture (2018–2024). The statement above is taken from the most recent of them; the others are arXiv:2302.08283, arXiv:2012.06698, arXiv:1805.01687.
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