Hamilton $\ell$-cycles in randomly-perturbed hypergraphs

TL;DR

This paper demonstrates that adding random edges to a dense hypergraph with linear minimum codegree likely creates a Hamilton -cycle, extending known results from graphs and 1-cycles to -cycles in hypergraphs.

Contribution

It establishes a threshold for random perturbation ensuring Hamilton -cycles in hypergraphs, generalizing previous results for 1-cycles and graphs.

Findings

High probability of Hamilton -cycle existence after perturbation

Identifies a specific probability threshold for edge addition

Extends known results from graphs and 1-cycles to -cycles in hypergraphs

Abstract

We prove that for integers $2 \leq \ell < k$ and a small constant $c$, if a $k$-uniform hypergraph with linear minimum codegree is randomly `perturbed' by changing non-edges to edges independently at random with probability $p \geq O(n^{-(k-\ell)-c})$, then with high probability the resulting $k$-uniform hypergraph contains a Hamilton $\ell$-cycle. This complements a recent analogous result for Hamilton $1$-cycles due to Krivelevich, Kwan and Sudakov, and a comparable theorem in the graph case due to Bohman, Frieze and Martin.