Note on induced paths in sparse random graphs

TL;DR

This paper proves that sparse random graphs typically contain long induced paths, improving previous bounds and answering a longstanding open question in graph theory.

Contribution

It establishes a new lower bound on the length of induced paths in $G(n,d/n)$ for large average degree, extending prior work and generalizing recent results on induced matchings.

Findings

High probability of long induced paths in $G(n,d/n)$ for large $d$

Improved lower bounds over previous results

Generalization of induced matching results to induced paths

Abstract

We show that for $d\ge d_0(\epsilon)$, with high probability, the random graph $G(n,d/n)$ contains an induced path of length $(3/2-\epsilon)\frac{n}{d}\log d$. This improves a result obtained independently by Luczak and Suen in the early 90s, and answers a question of Fernandez de la Vega. Along the way, we generalize a recent result of Cooley, Dragani\'c, Kang and Sudakov who studied the analogous problem for induced matchings.