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± consists of two copies G+ = (V +, E+), G− = (V −, E−) of G and additional edges connecting any two vertices v+ ∈ V+, v− ∈ V− that are the copies of a vertex v ∈ V . The bunkbed conjecture states that for independent bond percolation on G±, for all v, w ∈ V , it is more likely for v−, w− to be connected than for v−, w+ to be connected. While recently a counterexample for the bunkbed conjecture was found, it should still hold for many interesting classes of graphs, and here we give a proof for complete bipartite graphs, complete graphs minus the edges of a complete subgraph, and symmetric complete k-partite graphs.