Bang-Jensen and Yeo's strong arc decomposition conjecture

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Let D=(V,A)D=(V,A) be a digraph. It is kk-arc-strong if D∖A′D\setminus A^{\prime} remains strong for every subset A′⊆A(D)A^{\prime}\subseteq A(D) of size at most k−1k-1. A strong arc decomposition of DD is a partition of AA into disjoint sets A1A_1 and A2A_2 such that the spanning subdigraphs D1=(V,A1)D_1=(V,A_1) and D2=(V,A2)D_2=(V,A_2) are both strong.

Bang-Jensen and Yeo's conjecture. There exists an integer KK such that every KK-arc-strong digraph has a strong arc decomposition.

A strong arc decomposition would imply the existence of a good (u,v)(u,v)-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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