Worked solution: Gladkov–Pak–Zimin's explicit counterexample disproving the bunkbed conjecture (2024)
So the bunkbed conjecture, believed for nearly forty years, turns out to be false in general — but the counterexample is a delicately engineered, specially constructed graph, not a typical one. In fact every graph shape that had already been checked before 2024 — trees, wheels, complete graphs, complete bipartite graphs, and any graph with enough symmetry between and — still genuinely satisfies the conjecture, and always will, since those earlier proofs remain entirely valid.
The honest summary is not "the bunkbed conjecture is meaningless" but "the bunkbed conjecture is not a theorem about all graphs": it draws a real, if still poorly understood, boundary between graphs simple enough for same-level connectivity to always win and graphs complex enough to admit the strange kind of imbalance this proof engineered.
Despite the general disproof, the bunkbed inequality remains proved for trees (an immediate consequence of the original 1985 argument attributed to Kasteleyn, and made explicit by Linusson 2011 for one transversal vertex), for two transversal vertices (Leander 2018, §6.3), for wheels (Leander 2009), for complete graphs and complete bipartite graphs (van den Berg 2016, 2018; Hutchcroft & Leander 2019; Richterich 2022), for graphs symmetric with respect to a automorphism (Richterich 2022), and — remarkably — in the limit for every graph (Hollom, Nachmias & Klivans 2023; Hollom 2024a). Gladkov, Pak and Zimin's counterexample shows only that no proof strategy relying purely on general combinatorial identities (rather than exploiting the specific structure of these graph classes) can extend to arbitrary graphs and arbitrary transversal sets.
The explicit counterexample uses exactly transversal vertices, and the authors note this is the smallest possible number: the conjecture is already known to hold whenever , so three is genuinely the threshold at which the phenomenon first becomes possible. Section 6 of Gladkov, Pak & Zimin (2024) further extends the disproof to a variant called the complete BBC, where the transversal set is chosen uniformly at random rather than fixed in advance, showing the same qualitative failure persists there too.
Finding the precise boundary between the graph classes and probability regimes where the bunkbed conjecture holds and where it can fail — and understanding what makes the difference — is now an active open question in probabilistic combinatorics, alongside variants of the conjecture discussed in the paper's concluding remarks (different percolation models, the random-cluster/Ising analogue proved by Häggström in the 1990s, and the case of infinite graphs).