On powers of tight Hamilton cycles in randomly perturbed hypergraphs

Abstract

We show that for $k \geq 3$, $r\geq 2$ and $\alpha> 0$, there exists $\varepsilon > 0$ such that if $p=p(n)\geq n^{-{\binom{k+r-2}{k-1}}^{-1}-\varepsilon}$ and $H$ is a $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\alpha n$, then asymptotically almost surely the union $H\cup G^{(k)}(n,p)$ contains the $r^{th}$ power of a tight Hamilton cycle. The bound on $p$ is optimal up to the value of $\varepsilon$ and this answers a question of Bedenknecht, Han, Kohayakawa and Mota