Time to Cycle

TL;DR

This paper investigates the expected time until the first cycle appears in a randomly evolving graph, revealing a universal property that extends beyond graphs to matroids.

Contribution

It proves that the expected time for the first cycle in a random graph process is n, and extends this result to all graphs and matroids.

Findings

Expected cycle appearance time is n in the random graph process.

The result generalizes to all graphs and matroids.

A new enumeration approach underpins the proof.

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.