Dice Question Streamline Icon: https://streamlinehq.com

Transitive tournaments under the unconditioned digraph model (E7)

Prove that the bunkbed inequality holds for every transitive tournament T_n under the unconditioned directed multigraph model E7, in which two copies of T_n are taken, bidirectional vertical edges are added between corresponding vertices, and a uniformly random subset of all directed edges is retained; that is, establish that BBC(E7, T_n) holds for all n.

Information Square Streamline Icon: https://streamlinehq.com

Background

After showing several generalisations of the bunkbed conjecture to be false, the author isolates a structured family of digraphs—transitive tournaments—as a potentially tractable case in the multigraph setting E7.

This conjecture seeks a positive result in a highly ordered directed setting, contrasting with the negative examples built from graphs with numerous cycles.

References

However, one case that does seem approachable, though was found by the author to be more difficult than expected, is expressed by the following conjecture. Conjecture BBC{E_7,T_n} holds for T_n, the transitive tournament on n vertices.

— The bunkbed conjecture is not robust to generalisation (2406.01790 - Hollom, 3 Jun 2024) in Section 6 (Conclusion)