# Tilings in randomly perturbed graphs: bridging the gap between Hajnal-Szemer\'edi and Johansson-Kahn-Vu

## TL;DR

This paper determines the number of random edges needed in a dense graph with minimum degree lpha n to almost surely contain a perfect K_r-tiling after random perturbation, bridging the gap between known results for purely random and dense graphs.

## Contribution

It establishes thresholds for the number of random edges required in perturbed graphs to guarantee perfect K_r-tilings, connecting previous results for random and dense graphs.

## Key findings

- Number of random edges needed jumps at regular intervals as lpha increases.
- Results are asymptotically optimal within these intervals.
- Bridges the gap between Johansson-Kahn-Vu and Hajnal-Szemeredi results.

## Abstract

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of $K_r$ that together cover all the vertices in $G$. In this paper we consider perfect $K_r$-tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin where one starts with a dense graph and then adds $m$ random edges to it. Specifically, given any fixed $0< \alpha <1-1/r$ we determine how many random edges one must add to an $n$-vertex graph $G$ of minimum degree $\delta (G) \geq \alpha n$ to ensure that, asymptotically almost surely, the resulting graph contains a perfect $K_r$-tiling. As one increases $\alpha$ we demonstrate that the number of random edges required `jumps' at regular intervals, and within these intervals our result is best-possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu (which resolves the purely random case, i.e., $\alpha =0$) and that of Hajnal and Szemer\'edi (which demonstrates that for $\alpha \geq 1-1/r$ the initial graph already houses the desired perfect $K_r$-tiling).