Let G=(V,E) be a finite connected graph and let T⊂ V. The bunkbed graph G_T has vertex set V×0,1, two horizontal copies of every edge of G, and a vertical edge between (u,0) and (u,1) for each u∈ T. Let p∈(0,1). Bunkbed conjecture. For Bernoulli percolation with parameter p on G_T, forall o,v∈ V:quad mathbbmP_p((o,0)leftrightarrow(v,0))≥mathbbmP_p((o,0)leftrightarrow(v,1)). This is a natural extension of coordinatewise connectivity monotonicity and is attributed in the source to P. W. Kasteleyn (1985). It remains open, with only a few partial results known.

The bunkbed conjecture

Let $G=(V,E)$ be a finite connected graph and let $T\subset V$ . The bunkbed graph $G_T$ has vertex set $V\times\{0,1\}$ , two horizontal copies of every edge of $G$ , and a vertical edge between $(u,0)$ and $(u,1)$ for each $u\in T$ . Let $p\in(0,1)$ . Bunkbed conjecture. For Bernoulli percolation with parameter $p$ on $G_T$ ,

\forall o,v\in V:\quad \mathbbm{P}_p((o,0)\leftrightarrow(v,0))\geq\mathbbm{P}_p((o,0)\leftrightarrow(v,1)).

This is a natural extension of coordinatewise connectivity monotonicity and is attributed in the source to P. W. Kasteleyn (1985). It remains open, with only a few partial results known.

Sources & referencesView supporting material

Primary source

Philipp König and Thomas Richthammer, “Monotonicity properties for Bernoulli percolation on layered graphs – a Markov chain approach”, arXiv:2207.13173 (2022).

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Progress summary

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Solved

A rigorous counterexample, later published, shows that the bunkbed conjecture is false in its full generality.

The conjecture, attributed to Pieter Kasteleyn in 1985, asserts that same-layer connectivity is always at least as likely as cross-layer connectivity in a two-layer random graph. This assertion has now been disproved.

Known results

de Buyer proved the complete-graph case at $p=\frac{1}{2}$ and then for $p\geq\frac{1}{2}$ .
van Hintum and Lammers proved the complete-graph case for all $p\in[0,1]$ .
A 2023 result established the conjecture for every fixed graph sufficiently close to $p=1$ .
Earlier work covered forests, wheels, outerplanar and cactus graphs, complete bipartite graphs, and symmetric cases.

October 2024 counterexample; June 2025 publication

Aleksei Gladkov, Igor Pak, and Aleksandr Zimin constructed a non-computational counterexample: a connected planar graph with $7{,}222$ vertices, $14{,}442$ edges, and three transversal vertices, violating the inequality at $p=\frac{1}{2}$ . The result was subsequently published in PNAS in June 2025. Lawrence Hollom’s hypergraph counterexample supplied the key construction.

Current status (as of August 2026): The original graph conjecture is resolved negatively by a published explicit counterexample; restricted variants and special cases remain active questions.

Sources

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