Concentration of the maximum size of an induced subtree in moderately sparse random graphs

TL;DR

This paper extends the known concentration results of the maximum induced subtree size in random graphs from constant probability to a broader range where the edge probability diminishes with the number of vertices.

Contribution

It improves bounds on the second moment of induced subtrees, showing concentration at two points for sparser random graphs.

Findings

Concentration at two points for $p$ in a wider range

Improved bounds on the second moment of induced subtrees

Extension of previous results to sparser graphs

Abstract

Kamaldinov, Skorkin, and Zhukovskii proved that the maximum size of an induced subtree in the binomial random graph $G(n,p)$ is concentrated at two consecutive points, whenever $p\in(0,1)$ is a constant. Using improved bounds on the second moment of the number of induced subtrees, we show that the same result holds when $n^{-\frac{e-2}{3e-2}+\varepsilon}\leq p=o(1)$.