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The bunkbed of a graph $G$ is the graph $G\times K_2 $. It has beenconjectured that in the independent bond percolation model, the probability for$\left(u,0\right)$ to be connected with $\left(v,0\right)$ is greater than theprobability for $\left(u,0\right)$ to be connected with $\left(v,1\right)$, forany vertex $u$, $v$ of $G$. In this article, we prove this conjecture for thecomplete graph in the case of the independent bond percolation of parameter$p=1/2$.

A proof of the bunkbed conjecture for the complete graph at $p=\frac{1}{2}$