Powers of tight Hamilton cycles in randomly perturbed hypergraphs

TL;DR

This paper proves that a union of a dense hypergraph and a sparse random hypergraph almost surely contains the r-th power of a tight Hamilton cycle, extending previous results for the case r=1 to general r.

Contribution

It generalizes the existence of Hamilton cycles in hypergraphs to the r-th power in the union of dense and random hypergraphs for all r ≥ 1.

Findings

Union of dense hypergraph and sparse random hypergraph contains the r-th power of a tight Hamilton cycle

Extends previous results from r=1 to general r in hypergraph Hamiltonicity

High probability results for the existence of complex Hamiltonian structures

Abstract

For $k\ge 2$ and $r\ge 1$ such that $k+r\ge 4$, we prove that, for any $\alpha>0$, there exists $\epsilon>0$ such that the union of an $n$-vertex $k$-graph with minimum codegree $\left(1-\binom{k+r-2}{k-1}^{-1}+\alpha\right)n$ and a binomial random $k$-graph $\mathbb{G}^{(k)}(n,p)$ with $p\ge n^{-\binom{k+r-2}{k-1}^{-1}-\epsilon}$ on the same vertex set contains the $r^{\text{th}}$ power of a tight Hamilton cycle with high probability. This result for $r=1$ was first proved by McDowell and Mycroft.