Concentration of the largest induced tree size of $G_{n,p}$ around the standard expectation threshold

TL;DR

This paper investigates the concentration of the largest induced tree size in random graphs $G_{n,p}$, extending known thresholds and showing non-concentration at certain ranges of $p$, thus deepening understanding of graph structure thresholds.

Contribution

It extends the threshold range for concentration of the largest induced tree size in $G_{n,p}$ and identifies ranges where concentration does not occur.

Findings

Concentration occurs for $p o 0$ with $p o 0$ faster than $n^{-1/2} ext{log}^{3/2} n$.

Concentration fails when $p$ is between $n^{-1}$ and $n^{-1/2}$.

The results generalize previous thresholds for the concentration of the largest induced tree.

Abstract

Let $T(G)$ be the size of the largest induced tree of $G$, and let $G_{n,p}$ be the binomial random graph. Kamaldinov, Skorkin, and Zhukovskii proved that $T(G_{n,p})$ equals one of two consecutive values with high probability if $p$ is constant, and more recently, Oropeza extended this result to include all vanishing $p$ such that $p > n^{-\frac{e-2}{3e-2} + \epsilon}$, where $e$ is Euler's constant. We further extend this result to all vanishing $p$ such that $p \gg n^{-1/2} \ln^{3/2} n$, and furthermore, we show that, for $p$ such that $n^{-1} \ll p \ll n^{-1/2}, \ T(G_{n,p})$ cannot be concentrated at the standard expectation threshold.