The threshold probability for long cycles

TL;DR

This paper determines the threshold probability for a random subgraph of a graph with minimum degree at least k to almost surely contain a long cycle of length at least k+1, generalizing classical results for complete graphs.

Contribution

It establishes a new threshold for long cycle existence in random subgraphs of graphs with minimum degree k, extending known results from complete graphs to broader classes.

Findings

Threshold probability for long cycles is approximately (log k + log log k)/k.

When G is a complete graph, the result matches the classic Hamilton cycle threshold.

The result applies to graphs with minimum degree at least k, not just complete graphs.

Abstract

For a given graph $G$ of minimum degree at least $k$, let $G_p$ denote the random spanning subgraph of $G$ obtained by retaining each edge independently with probability $p=p(k)$. We prove that if $p \ge \frac{\log k + \log \log k + \omega_k(1)}{k}$, where $\omega_k(1)$ is any function tending to infinity with $k$, then $G_p$ asymptotically almost surely contains a cycle of length at least $k+1$. When we take $G$ to be the complete graph on $k+1$ vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.