Han–Zhao's exact minimum co-degree conjecture for Hamilton ll-cycles

Let $k$ and $k/2<\ell<k$ be integers such that $(k-\ell)\nmid k$, and let $n\in (k-\ell)\mathbb{N}$ be sufficiently large. Define

$$s:=\left\lceil \frac{k}{k-\ell}\right\rceil.$$

For an $n$-vertex $k$-graph ${\mathcal H}=(V,E)$, write $\delta_{k-1}({\mathcal H})$ for its minimum co-degree.

Han–Zhao's conjecture. If

$$\delta_{k-1}({\mathcal H})\geq \frac{n}{s(k-\ell)},$$

then ${\mathcal H}$ contains a Hamilton $\ell$-cycle.

This conjecture proposes the exact minimum co-degree threshold in the remaining range $k/2<\ell<k$ with $(k-\ell)\nmid k$, extending the cases already resolved for several parameter ranges. The divisibility condition on $n$ is necessary for a Hamilton $\ell$-cycle, and the conjectured threshold is expected to be sharp.