On the codegree threshold for Hamilton $\mathrm{\ell }$-cycles in $k$-uniform hypergraphs

In this note, we resolve the remaining open case of a conjecture by Han and Zhao concerning the codegree threshold for Hamilton $\mathrm{\ell }$-cycles in $k$-uniform hypergraphs. Specifically, we prove that for integers $k\ge 3$, $3k\mathrm{/}4\le \mathrm{\ell }<k$, with $k\equiv ̸0(\mathrm{m}\mathrm{o}\mathrm{d}k-\mathrm{\ell })$, and for all sufficiently large $n$ divisible by $k-\mathrm{\ell }$, every $n$-vertex $k$-uniform hypergraph $H$ satisfying $${\delta }_{k-1}(H)\ge \frac{n}{(k-\mathrm{\ell })\lceil \frac{k}{k-\mathrm{\ell }}\rceil }$$ contains a Hamilton $\mathrm{\ell }$-cycle. Our proof builds on the framework of Gan, Han and Xu, and refines their argument to obtain, at the exact threshold, the required family of paths.