The oriented-cycle minimum-out-degree conjecture

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Let CC be an oriented cycle, and let ∣V(C)∣|V(C)| denote its order. Oriented-cycle conjecture. Every digraph DD with δ+(D)≥2k−1\delta^+(D)\geq 2k-1 contains a subdivision of any oriented cycle of order kk. Equivalently, the source conjectures the upper bound mader⁡δ+(C)≤2∣V(C)∣−1\operatorname{mader}_{\delta^+}(C)\leq 2|V(C)|-1 for every oriented cycle CC; the source does not give a resolution.

References

Primary source

Pierre Aboulker, Nathann Cohen, Fréderic Havet, William Lochet, Phablo F. S. Moura and Stéphan Thomassé, “Subdivisions in digraphs of large out-degree or large dichromatic number”, arXiv:1610.00876 (2016).

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