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Stability of the independence number of $G(n, r, 1)$ graphs

M. Koshelev

TL;DR

The paper contributes a rigorous stability theorem for the independence number of Johnson graphs in the case $s=1$ by providing the correct proof of a key lemma that was previously flawed. It analyzes the random subgraph $G_{p}(n,r,1)$ with a focus on the case $p=1/2$ and establishes that, w.h.p., any large independent set must resemble a star, leading to $\,\alpha(G_{1/2}(n,r,1))=\binom{n-2}{r-2}$ for $r\ge4$. The approach combines a careful combinatorial decomposition relative to a maximal star, auxiliary quantities like $X(A)$ and $I(X)$, and probabilistic tools including Chernoff bounds and edge-count estimates to rule out troublesome configurations. By correcting the previous arguments, the work solidifies the understanding of stability for independence numbers in Johnson graphs with $s>0$ and strengthens the probabilistic method in extremal combinatorics.

Abstract

In this paper we obtain the stability theorem for the independence number of $G(n, r, 1)$ graphs. This result was previously stated in the paper of M. Pyaderkin but the proof there was incorrect. We introduce the correct proof of the key lemma and thus finally complete the proof of this theorem.

Stability of the independence number of $G(n, r, 1)$ graphs

TL;DR

The paper contributes a rigorous stability theorem for the independence number of Johnson graphs in the case by providing the correct proof of a key lemma that was previously flawed. It analyzes the random subgraph with a focus on the case and establishes that, w.h.p., any large independent set must resemble a star, leading to for . The approach combines a careful combinatorial decomposition relative to a maximal star, auxiliary quantities like and , and probabilistic tools including Chernoff bounds and edge-count estimates to rule out troublesome configurations. By correcting the previous arguments, the work solidifies the understanding of stability for independence numbers in Johnson graphs with and strengthens the probabilistic method in extremal combinatorics.

Abstract

In this paper we obtain the stability theorem for the independence number of graphs. This result was previously stated in the paper of M. Pyaderkin but the proof there was incorrect. We introduce the correct proof of the key lemma and thus finally complete the proof of this theorem.
Paper Structure (5 sections, 10 theorems, 33 equations)

This paper contains 5 sections, 10 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Proof of Lemma \ref{['fix']}
  3. Proof of Lemma \ref{['Tech']}
  4. Proof of Lemma \ref{['fixBipartite']}
  5. Proof of Lemma \ref{['fixTuran']}

Key Result

Theorem 1

Let $\varepsilon > 0$ be an arbitrary number, and let $n \geq 2r+1$. Then for every $p \geq (1 + \varepsilon)p_0$, where the graph $G_p(n,r,0)$ will with high probability (w.h.p.) have independence number equal to $\binom{n-1}{r-1}$. Moreover the equality $\alpha(G_p(n,r,0)) = \binom{n-1}{r-1}$ w.h.p. does not hold for any $p \leq (1 - \varepsilon)p_0$. We will use the notation w.h.p. hereinafter

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Theorem 3
  • Lemma 2.1
  • Theorem 4
  • Lemma 2.2
  • ...and 1 more