Clique factors in randomly perturbed graphs: the transition points

TL;DR

This paper determines the minimal edge probability needed for a randomly perturbed graph to contain a $K_r$-factor, revealing a polynomial threshold jump at specific density points and introducing extremal examples with pseudorandom subgraphs.

Contribution

It establishes the minimal probability thresholds for $K_r$-factors in perturbed graphs at critical density points, extending previous results and identifying new extremal constructions.

Findings

Threshold probability exhibits a polynomial jump at certain density points.

Extremal examples with pseudorandom subgraphs show optimality of thresholds.

Results apply to all density parameters $ eq 1 - s/r$ for $s eq r$.

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 $K_r$-copies) with high probability? The order of magnitude of the minimal value of $p$ has been determined whenever $\alpha \neq 1- \frac{s}{r}$ for an integer $s$ (see Han, Morris, and Treglown [RSA, 2021] and Balogh, Treglown, and Wagner [CPC, 2019]). We establish the minimal probability $p_s$ (up to a constant factor) for all values of $\alpha = 1-\frac{s}{r} \leq \frac 12$, and show that the threshold exhibits a polynomial jump at $\alpha = 1-\frac{s}{r}$ compared to the surrounding intervals. An extremal example $G_{\alpha}$ which shows that $p_s$ is optimal up to a constant factor differs from the previous (usually multipartite) examples in containing a pseudorandom induced subgraph.