The bunkbed conjecture is false

TL;DR

This work settles the bunkbed conjecture (BBC) by constructing an explicit planar counterexample with $|V|=7222$ and $|E|=14442$, derived from Hollom's 3-uniform hypergraph and a tailored hyperedge-simulation gadget. A robust hyperedge lemma, together with a weighted hypergraph percolation framework (WZ model), transfers a hyperedge discrepancy into a bunkbed percolation inequality, yielding $\,\\mathbb{P}_{1/2}^{bb}[u\\leftrightarrow v] < \\mathbb{P}_{1/2}^{bb}[u\\leftrightarrow v']$. The construction also extends to Complete BBC with $|V|=7523$, and the authors provide both a detailed combinatorial proof and a discussion of limited success for computer-assisted searches, arguing that computational methods alone cannot reveal such tiny probability gaps. These results resolve the BBC in full generality (including variants like Weighted, Polynomial, and Complete BBC) and illuminate the behavior of percolation on bunkbed graphs, with implications for related models such as the random-cluster model at $q=2$. The work combines hypergraph percolation techniques with explicit gadgetry to embed hyperedge behavior into graph percolation in a planar setting, marking a definitive advance in percolation theory and combinatorial probability.

Abstract

We give an explicit counterexample to the Bunkbed Conjecture introduced by Kasteleyn in 1985. The counterexample is given by a planar graph on $7222$ vertices, and is built on the recent work of Hollom (2024).

Figures (2)

• Figure 3.1: Hollom's 3-uniform hypergraph $H$.
• Figure 4.1: Graph $G_n$ with $n+1$ vertices.

Theorems & Definitions (18)

• Conjecture 1.1: bunkbed conjecture
• Theorem 1.2
• Lemma 3.1: Hollom H
• Lemma 3.2
• proof
• Lemma 4.1
• Remark 4.2
• Lemma 5.1
• proof
• Lemma 5.2