Bunkbed conjecture for complete bipartite graphs and related classes of graphs

TL;DR

This paper proves the bunkbed conjecture for specific classes of graphs, including complete bipartite, certain complete graphs, and symmetric complete k-partite graphs, advancing understanding of percolation connectivity.

Contribution

The paper provides the first rigorous proofs of the bunkbed conjecture for several important classes of graphs beyond complete graphs.

Findings

Proved the conjecture for complete bipartite graphs.

Extended proof to complete graphs minus a complete subgraph.

Established validity for symmetric complete k-partite 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.