The largest hole in sparse random graphs

TL;DR

This paper determines the size of the largest induced cycle in sparse random graphs, resolving a long-standing open problem by providing precise asymptotic results.

Contribution

It establishes the asymptotic size of the largest induced cycle in sparse Erdős–Rényi graphs, settling a major open question in the field.

Findings

Largest induced cycle size is approximately (2±ε) * (n/d) * log d.

Results hold for a wide range of average degrees d(n).

Provides a precise asymptotic characterization in sparse regimes.

Abstract

We show that for any $d=d(n)$ with $d_0(\epsilon) \le d =o(n)$, with high probability, the size of a largest induced cycle in the random graph $G(n,d/n)$ is $(2\pm \epsilon)\frac{n}{d}\log d$. This settles a long-standing open problem in random graph theory.