
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 vertices and no edges, where the edges of are added one at a time in a uniformly chosen random order . Let be the earliest time at which belongs to a cycle in this evolving random graph. By solving the appropriate graph enumeration problem we show that . 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.
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Taxonomy
TopicsStochastic processes and statistical mechanics · Limits and Structures in Graph Theory · Markov Chains and Monte Carlo Methods
