On Percolation and the Bunkbed Conjecture

We study a problem on percolation on product graphs G×K2. Here G is any finite graph and K2 consists of two vertices {0, 1} connected by an edge. In edge percolation every edge in G × K2 is present with probability p. In [3] Olle Häggström stated a conjecture (which he claimed to be folklore) that for all G and p the probability that (u, 0) is in the same component as (v, 0) is greater than the probability that (u, 0) is in the same component as (v, 1) for every pair of vertices u, v ∈ G. We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs G, in particular outerplanar graphs.