The bunkbed conjecture on the complete graph

TL;DR

This paper proves the bunkbed conjecture for the complete graph with a fixed percolation probability, confirming the intuitive idea that vertices are more likely to connect within the same layer than across layers.

Contribution

It provides a proof of the bunkbed conjecture specifically for the complete graph under constant percolation conditions, a case that was previously unresolved.

Findings

Confirmed the bunkbed conjecture for the complete graph

Established the conjecture for fixed percolation parameters

Enhanced understanding of connectivity probabilities in layered graphs

Abstract

The bunkbed conjecture was first posed by Kasteleyn. If $G=(V,E)$ is a finite graph and $H$ some subset of $V$, then the bunkbed of the pair $(G,H)$ is the graph $G\times\{1,2\}$ plus $|H|$ extra edges to connect for every $v\in H$ the vertices $(v,1)$ and $(v,2)$. The conjecture asserts that $(v,1)$ is more likely to connect with $(w,1)$ than with $(w,2)$ in the independent bond percolation model for any $v,w\in V$. This is intuitive because $(v,1)$ is in some sense closer to $(w,1)$ than it is to $(w,2)$. The conjecture has however resisted several attempts of proof. This paper settles the conjecture in the case of a constant percolation parameter and $G$ the complete graph.