The complete picture for clique factors in randomly perturbed graphs

TL;DR

This paper determines the threshold probability for the appearance of a $K_r$-factor in randomly perturbed graphs with high minimum degree, completing the understanding across all minimum degree regimes.

Contribution

It establishes the threshold probability $p_s$ for $K_r$-factors in the case of high minimum degree $rac{1}{2}<eta<1$, extending previous results to this regime.

Findings

Determined the threshold $p_s$ for $eta > 1/2$.

Proved a fractional stability version of a tiling theorem.

Completed the characterization of $K_r$-factor thresholds in randomly perturbed graphs.

Abstract

A randomly perturbed graph $G^p = G_\alpha \cup G_{n,p}$ is obtained by taking a deterministic $n$-vertex graph $G_\alpha = (V, E)$ with minimum degree $\delta(G)\geq \alpha n$ 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 $\alpha$) 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 $\alpha \neq 1- \frac{s}{r}$ 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 $\alpha = 1-\frac{s}{r}\leq \frac 12$. Here, we complete the picture by establishing $p_s$ in the remaining case $\alpha > \frac12$. 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.