The bunkbed conjecture is not robust to generalisation

Abstract

The bunkbed conjecture, which has featured in the folklore of probability theory since at least 1985, concerns bond percolation on the product graph $G\square K_2$. We have two copies $G_0$ and $G_1$ of $G$, and if $x^{\left(0\right)}$ and $x^{\left(1\right)}$ are the copies of a vertex $x\in V\left(G\right)$ in $G_0$ and $G_1$ respectively, then edge $x^{\left(0\right)}{x}^{\left(1\right)}$ is present. The conjecture states that, for vertices $u,v\in V\left(G\right)$, percolation from $u^{\left(0\right)}$ to $v^{\left(0\right)}$ is at least as likely as percolation from $u^{\left(0\right)}$ to $v^{\left(1\right)}$.

In this paper we consider three natural generalisations of the bunkbed conjecture; to site percolation, to hypergraphs, and to directed graphs. Our main aim is to show that all these generalisations are false, and to this end we construct a sequence of counterexamples to these statements. However, we also consider under what extra conditions these generalisations might hold, and give some classes of graph for which the bunkbed conjecture for site percolation does hold.