Bunkbed conjecture for complete bipartite graphs and related classes of graphs

Let $G=(V,E)$ be a simple finite graph. The corresponding bunkbed graph $G^{\pm }$ consists of two copies $G^{+}=(V^{+},E^{+}),G^{-}=(V^{-},E^{-})$ of $G$ and additional edges connecting any two vertices $v_{+}\in V_{+},v_{-}\in V_{-}$ that are the copies of a vertex $v\in V$. The bunkbed conjecture states that for independent bond percolation on $G^{\pm }$, for all $v,w\in V$, it is more likely for $v_{-},w_{-}$ to be connected than for $v_{-},w_{+}$ to be connected. While this seems very plausible, so far surprisingly little is known rigorously. Recently the conjecture has been proved for complete graphs. Here we give a proof for complete bipartite graphs, complete graphs minus the edges of a complete subgraph, and symmetric complete $k$-partite graphs.