On powers of tight Hamilton cycles in randomly perturbed hypergraphs

TL;DR

This paper proves that adding a sparse random hypergraph to a dense hypergraph guarantees the presence of the r-th power of a tight Hamilton cycle, establishing an optimal probabilistic threshold.

Contribution

It determines the optimal edge probability threshold for the appearance of the r-th power of a tight Hamilton cycle in perturbed hypergraphs, answering a previously open question.

Findings

The union contains the r-th power of a tight Hamilton cycle with high probability.

The threshold for p is optimal up to a small epsilon.

Construction shows the bound on epsilon cannot be improved.

Abstract

For integers $k \geq 3$ and $r\geq 2$, we show that for every $\alpha> 0$, there exists $\varepsilon > 0$ such that the union of $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\alpha n$ and a binomial random $k$-uniform hypergraph $G^{(k)}(n,p)$ with $p\geq n^{-{\binom{k+r-2}{k-1}}^{-1}-\varepsilon}$ on the same vertex set contains the $r^{th}$ power of a tight Hamilton cycle with high probability. Moreover, a construction shows that one cannot take $\varepsilon > C\alpha$, where $C=C(k,r)$ is a constant. Thus the bound on $p$ is optimal up to the value of $\varepsilon$ and this answers a question of Bedenknecht, Han, Kohayakawa, and Mota.