A proof of the Bunkbed conjecture on the complete graph for $p\geqslant1/2$

TL;DR

This paper proves the bunkbed conjecture for complete graphs in the independent bond percolation model when the percolation parameter is at least 1/2, confirming a long-standing hypothesis in this setting.

Contribution

It provides the first proof of the bunkbed conjecture for complete graphs for all percolation probabilities p ≥ 1/2.

Findings

Proves the bunkbed conjecture for complete graphs at p ≥ 1/2

Confirms the conjecture's validity in this specific case

Advances understanding of connectivity probabilities in percolation models

Abstract

The bunkbed of a graph $G$ is the graph $G\times\left\{ 0,1\right\} $. It has been conjectured that in the independent bond percolation model, the probability for $\left(u,0\right)$ to be connected with $\left(v,0\right)$ is greater than the probability for $\left(u,0\right)$ to be connected with $\left(v,1\right)$, for any vertex $u$, $v$ of $G$. In this article, we prove this conjecture for the complete graph in the case of the independent bond percolation of parameter $p\geqslant1/2$.