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Stability of large cuts in random graphs

Ilay Hoshen, Wojciech Samotij, Maksim Zhukovskii

Abstract

We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-Ω(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the same manner by all maximum cuts of $G_{n,p}$. Moreover, the analogous statement remains true when one replaces maximum cuts with nearly-maximum cuts. We then demonstrate how one can use this statement as a tool for showing that certain properties of $G_{n,p}$ that hold in a fixed balanced cut hold simultaneously in all maximum cuts. We provide two example applications of this tool. First, we prove that maximum cuts in $G_{n,p}$ typically partition the neighbourhood of every vertex into nearly equal parts; this resolves a conjecture of DeMarco and Kahn for all but a narrow range of densities $p$. Second, for all edge-critical, nonbipartite, and strictly 2-balanced graphs $H$, we prove a lower bound on the threshold density $p$ above which every largest $H$-free subgraph of $G_{n,p}$ is $(χ(H)-1)$-partite. Our lower bound exactly matches the upper bound on this threshold recently obtained by the first two authors.

Stability of large cuts in random graphs

Abstract

We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If , then, with high probability, there is a set of vertices that is partitioned in the same manner by all maximum cuts of . Moreover, the analogous statement remains true when one replaces maximum cuts with nearly-maximum cuts. We then demonstrate how one can use this statement as a tool for showing that certain properties of that hold in a fixed balanced cut hold simultaneously in all maximum cuts. We provide two example applications of this tool. First, we prove that maximum cuts in typically partition the neighbourhood of every vertex into nearly equal parts; this resolves a conjecture of DeMarco and Kahn for all but a narrow range of densities . Second, for all edge-critical, nonbipartite, and strictly 2-balanced graphs , we prove a lower bound on the threshold density above which every largest -free subgraph of is -partite. Our lower bound exactly matches the upper bound on this threshold recently obtained by the first two authors.
Paper Structure (17 sections, 27 theorems, 161 equations)

This paper contains 17 sections, 27 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Neighbourhoods in maximum cuts
  3. Sharp thresholds for Simonovits's theorem in $G_{n,p}$
  4. Relation to previous works
  5. Rigidity
  6. Dynamics of equivalent pairs
  7. Distribution of edges in $G(n,m)$
  8. Proof of \ref{['thm:rigidity']}
  9. Derivation of \ref{['cor:core']}
  10. Proof of \ref{['thm:main-rigidity-theorem']}
  11. Non-uniqueness of maximum cuts and vertices outside of the core
  12. Neighbourhood in maximum cuts
  13. The threshold for Simonovits's theorem in $G_{n,p}$
  14. Preliminaries
  15. Proof of the lower bound on $\mathbb{E}[Z]$
  16. ...and 2 more sections

Key Result

Theorem 1

The following hold for every integer $r \geqslant 2$, real $\alpha > 0$, and all $p = p(n)$ satisfying $1/n \ll p = 1-\Omega(1)$:

Theorems & Definitions (48)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Sim68
  • Theorem 4: HosSam
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • Lemma 8
  • proof : Proof of \ref{['lemma:eq-d-non-edges']}
  • Lemma 10
  • ...and 38 more