We study a problem on edge percolation on product graphs GΓK2β. Here
G is any finite graph and K2β consists of two vertices {0,1} connected
by an edge. Every edge in GΓK2β is present with probability p
independent of other edges. The Bunkbed conjecture states that for all G and
p the probability that (u,0) is in the same component as (v,0) is greater
than or equal to the probability that (u,0) is in the same component as
(v,1) for every pair of vertices u,vβG.
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.Comment: 13 pages, improved exposition thanks to anonymous referee. To appear
in CP