Factors in randomly perturbed hypergraphs

TL;DR

This paper determines the minimal number of random edges needed to ensure the existence of specific subgraph factors in hypergraphs with high minimum degree, settling a known problem and extending understanding to less dense graphs.

Contribution

It provides the first precise thresholds for the emergence of F-factors in randomly perturbed hypergraphs for a broad class of F, including resolving a problem posed by Krivelevich, Kwan, and Sudakov.

Findings

Identifies the optimal number of random edges needed for F-factors in dense hypergraphs.

Establishes thresholds for hypergraphs starting from less dense, nearly empty graphs.

Includes a broad class of F, such as all k-partite k-graphs, K_4^{(3)-}, and the Fano plane.

Abstract

We determine, up to a multiplicative constant, the optimal number of random edges that need to be added to a $k$-graph $H$ with minimum vertex degree $\Omega(n^{k-1})$ to ensure an $F$-factor with high probability, for any $F$ that belongs to a certain class $\mathcal{F}$ of $k$-graphs, which includes, e.g., all $k$-partite $k$-graphs, $K_4^{(3)-}$ and the Fano plane. In particular, taking $F$ to be a single edge, this settles a problem of Krivelevich, Kwan and Sudakov [Combin. Probab. Comput. 25 (2016), 909--927]. We also address the case in which the host graph $H$ is not dense, indicating that starting from certain such $H$ is essentially the same as starting from an empty graph (namely, the purely random model).