Bang-Jensen and Yeo's strong arc decomposition conjecture

Let D=(V,A)D=(V,A) be a digraph. It is kk-arc-strong if DAD\setminus A^{\prime} remains strong for every subset AA(D)A^{\prime}\subseteq A(D) of size at most k1k-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.

Sources & referencesView supporting material

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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