Exact minimum co-degree conditions for $\ell$-Hamiltonicity in hypergraphs

Abstract

Suppose $1\le \ell <k$ such that $(k-\ell )\nmid k$. Given an $n$-vertex $k$-uniform hypergraph $\mathcal{H}$, for all $k/2<\ell <3k/4$ and sufficiently large $n\in (k-\ell )\mathbb{N}$, we prove that if $\mathcal{H}$ has minimum co-degree at least $\frac{n}{\lceil \frac{k}{k-\ell }\rceil (k-\ell )}$, then $\mathcal{H}$ contains a Hamilton $\ell$-cycle, which partially verifies a conjecture of Han and Zhao and (partially) resolves a problem of Rödl and Ruciński. Moreover, we show that assuming minimum co-degree $\frac{n}{\lceil \frac{k}{k-\ell }\rceil (k-\ell )}+\frac{k^2}{2}$ is enough for all $\ell$.

25 pages, 2 figures