Abstract
We propose a novel exact algorithm for generating connected Erdos-Renyi random graphs $G(n,p)$. The method couples the graph exploration process to an inhomogeneous Poisson random walk, which yields an exact sampler that runs in $O(n)$ time in the sparse regime $p=c/n$. We also show how the method extends to the $G(n,M)$ model via an additional acceptance-rejection step.