  The bunkbed conjecture, which has featured in the folklore of probability
theory since at least 1985, concerns bond percolation on the product graph
$G\Box K_2$. We have two copies $G_0$ and $G_1$ of $G$, and if $x^{(0)}$ and
$x^{(1)}$ are the copies of a vertex $x\in V(G)$ in $G_0$ and $G_1$
respectively, then edge $x^{(0)}x^{(1)}$ is present. The conjecture states
that, for vertices $u,v\in V(G)$, percolation from $u^{(0)}$ to $v^{(0)}$ is at
least as likely as percolation from $u^{(0)}$ to $v^{(1)}$. While the
conjecture is widely expected to be true, having attracted significant
attention, a general proof has not been forthcoming.
  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.
