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Acyclic digraphs and the directed bunkbed conjecture

Determine whether the directed-graph bunkbed inequality holds for every finite acyclic directed graph under the conditioned model E6^{T,F}: given a digraph D, a vertex set T whose vertical edges are retained, and an edge set F whose edges appear in both bunks (all other edges appearing in exactly one bunk), decide whether for all vertices u and v in D it is true that Prob(u^(0) → v^(0)) ≥ Prob(u^(0) → v^(1)).

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Background

Within the directed setting, the author disproves the conjectured inequality for general digraphs by constructing counterexamples that contain many directed cycles. This leaves open whether acyclicity restores the inequality.

The paper highlights that the negative examples rely on cycles and explicitly remarks that the acyclic case remains unresolved. In the Conclusion, this is reiterated as a central open direction (also framed as a question covering both the conditioned model E6{T,F} and the unconditioned multigraph model E7).

References

The latter is disproved by \Cref{thm:main}, but the example we construct has many cycles, and so it remains open whether the conjecture holds for acyclic directed graphs.

— The bunkbed conjecture is not robust to generalisation (2406.01790 - Hollom, 3 Jun 2024) in Section 4 (Models), after Model E6^{T,F}