Bunkbed conjecture for complete bipartite graphs and related classes of graphs

Abstract

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.