Time to Cycle

Abstract

Consider the random process that starts with $n$ vertices and no edges, where the edges of $K_n$ are added one at a time in a uniformly chosen random order $e_1, e_2,\ldots, e_{\binom{n}{2}}$. Let $T$ be the earliest time at which $e_1$ belongs to a cycle in this evolving random graph. By solving the appropriate graph enumeration problem we show that $\mathbb{E}[T]=n$. This fact turns out to be an instance of a much more general phenomenon and we are able to extend this theorem to all graphs and even to every matroid.