On powers of tight Hamilton cycles in randomly perturbed hypergraphs

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

Hamilton l-cycles in uniform hypergraphs

AbstractWe say that a k-uniform hypergraph C is an ℓ-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ℓ vertices. We prove that if 1⩽ℓ<k and k−ℓ does not divide k then any k-uniform hypergraph on n vertices with minimum degree at least n⌈kk−ℓ⌉(k−ℓ)+o(n) contains a Hamilton ℓ-cycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an ℓ-cycle for any ℓ with 1⩽ℓ<k