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

Jakob Hofstad

Abstract

Let $T(G)$ be the size of the largest induced tree of $G$, and $G_{n,p}$ 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 $p > n^{-\frac{e-2}{3e-2} + ε}$, 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.

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

Abstract

Let be the size of the largest induced tree of , and be the binomial random graph. Kamaldinov, Skorkin, and Zhukovskii proved that equals one of two consecutive values with high probability if is constant, and more recently, Oropeza extended this result to include all vanishing such that , where is Euler's constant. We further extend this result to all vanishing such that , and furthermore, we show that, for such that cannot be concentrated at the standard expectation threshold.
Paper Structure (13 sections, 5 theorems, 49 equations)

This paper contains 13 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Key definitions and inequality
  3. First moment calculation
  4. Second moment calculation
  5. A general upper bound for $x_{\ell,m}$
  6. Case 1: $\ell \leq k/2$
  7. Case 2: $\ell > k/2$
  8. Subcase 1: $j < \frac{1}{k^2 p^3}$ and $m \leq 2j$
  9. Subcase 2: $j < \frac{1}{k^2 p^3}$ and $m > 2j$
  10. Subcase 3: $j \geq \frac{1}{k^2 p^3}$
  11. Closing remark
  12. Drift from the expectation threshold for small $p$
  13. Closing remarks

Key Result

Theorem 1

There exists an integer function $k_0 = k_0(n,p)$ such that, for all $p = p(n)$ such that $\frac{1}{n} \ll p \ll \frac{1}{\ln^2 n}$:

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • proof : Proof of (iii) of Theorem \ref{['thm: main']}
  • Lemma 4
  • proof
  • Lemma 5
  • proof