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The complete picture for clique factors in randomly perturbed graphs

Sylwia Antoniuk, Nina Kamčev, Christian Reiher, Tadej Petar Tukara

Abstract

A randomly perturbed graph $G^p = G_α\cup G_{n,p}$ is obtained by taking a deterministic $n$-vertex graph $G_α= (V, E)$ with minimum degree $δ(G)\geq αn$ and adding the edges of the binomial random graph $G_{n,p}$ defined on the same vertex set $V$. For which value $p$ (depending on $α$) does the graph $G^p$ contain a $K_r$-factor -- a spanning collection of vertex-disjoint copies of $K_r$ -- with high probability? The order of magnitude of the minimum such $p$ was determined whenever $α\neq 1- \frac{s}{r}$ for an integer $s$ by Balogh, Treglown and Wagner, and by Han, Morris and Treglown. In earlier work, the first three authors determined this threshold probability $p_s$ up to a constant factor for all values of $α= 1-\frac{s}{r}\leq \frac 12$. Here, we complete the picture by establishing $p_s$ in the remaining case $α> \frac12$. A key ingredient in our approach is an extremal result of independent interest: we prove a fractional stability version of a tiling theorem due to Shokoufandeh and Zhao.

The complete picture for clique factors in randomly perturbed graphs

Abstract

A randomly perturbed graph is obtained by taking a deterministic -vertex graph with minimum degree and adding the edges of the binomial random graph defined on the same vertex set . For which value (depending on ) does the graph contain a -factor -- a spanning collection of vertex-disjoint copies of -- with high probability? The order of magnitude of the minimum such was determined whenever for an integer by Balogh, Treglown and Wagner, and by Han, Morris and Treglown. In earlier work, the first three authors determined this threshold probability up to a constant factor for all values of . Here, we complete the picture by establishing in the remaining case . A key ingredient in our approach is an extremal result of independent interest: we prove a fractional stability version of a tiling theorem due to Shokoufandeh and Zhao.
Paper Structure (4 sections, 13 theorems, 9 equations, 1 figure)

This paper contains 4 sections, 13 theorems, 9 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Probabilistic bounds
  4. Regularity

Key Result

Theorem 1.1

For all integers $1 < s \leq r$ and any $\alpha \in \left(1-\frac{s}{r}, 1-\frac{s-1}{r} \right)$, there exists $C$ such that the following holds. For any $n$-vertex graph $G_\alpha$ with minimum degree at least $\alpha n$, w.h.p. $G_\alpha \cup G_{n,p}$ with $p = Cn^{-2/s}$ contains a $K_r$-factor.

Figures (1)

  • Figure 1.1: The extremal construction $G_{\alpha}$ which, crucially, contains an independent set of size $(1-\alpha)n$.

Theorems & Definitions (18)

  • Theorem 1.1: Clique-factors in randomly perturbed graphs btw18, hmt21
  • Theorem 1.2: Clique-factors at the transition points akr24
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof : Proof sketch.
  • Lemma 2.5: akr24, Lemma 4.3
  • ...and 8 more