On the probability that a random subgraph contains a circuit

Abstract

Let $μ> 2$ and $ε> 0$. We show that, if $G$ is a sufficiently large simple graph of average degree at least $μ$, and $H$ is a random spanning subgraph of $G$ formed by including each edge independently with probability $p \ge \tfrac{1}{μ-1} + ε$, then $H$ contains a cycle with probability at least $1 - ε$.