Mader's subdivision conjecture for acyclic digraphs

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Let FF be an acyclic digraph, and let δ+(D)\delta^+(D) denote the minimum out-degree of a digraph DD. A subdivision of FF is obtained by replacing the arcs of FF by directed paths with the same endpoints and internally vertex-disjoint interiors. Mader's subdivision conjecture. For every acyclic digraph FF there exists a number K=K(F)K=K(F) such that every digraph DD with

δ+(D)≥K\delta^+(D)\ge K

contains a subdivision of FF. This remains an open problem concerning degree conditions that force directed subdivisions.

References

Primary source

Raphael Steiner, “Openly disjoint cycles and directed tree-width of regular digraphs”, arXiv:2604.13700 (2026).

Additional references

6 papers in this index state this conjecture (2006–2026). The statement above is taken from the most recent of them; the others are arXiv:2603.13955, arXiv:2403.11470, arXiv:2008.13224, arXiv:1801.08249, arXiv:math/0605178.

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