On the probability that a random subgraph contains a circuit

TL;DR

This paper proves that in large graphs with high average degree, a random spanning subgraph formed with a certain probability almost certainly contains a cycle, highlighting probabilistic thresholds for cycle existence.

Contribution

It establishes a probabilistic threshold for the appearance of cycles in random spanning subgraphs of high-degree graphs, extending understanding of random subgraph properties.

Findings

High probability of cycle presence in random subgraphs with edge probability above threshold

Threshold for edge inclusion probability ensuring cycle existence

Results applicable to large graphs with average degree exceeding a fixed constant

Abstract

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