Minimum degree thresholds for Hamilton $(\mathrm{\ell },k-\mathrm{\ell })$-cycles in $k$-uniform hypergraphs

Let $n>k>\mathrm{\ell }$ be positive integers. We say a $k$-uniform hypergraph $\mathcal{H}$ contains a Hamilton $(\mathrm{\ell },k-\mathrm{\ell })$-cycle if there is a partition $(L_0,R_0,L_1,R_1,\dots ,L_{t-1},R_{t-1})$ of $V(\mathcal{H})$ with $\mathrm{\mid }{L}_i\mathrm{\mid }=\mathrm{\ell }$, $\mathrm{\mid }{R}_i\mathrm{\mid }=k-\mathrm{\ell }$ such that $L_i\cup R_i$ and $R_i\cup L_{i+1}$ (subscripts module $t$) are all edges of $\mathcal{H}$ for $i=0,1,\dots ,t-1$. In the present paper, we determine the tight minimum $\mathrm{\ell }$-degree condition that guarantees the existence of a Hamilton $(\mathrm{\ell },k-\mathrm{\ell })$-cycle in every $k$-uniform $n$-vertex hypergraph for $k\ge 7$, $k\mathrm{/}2\le \mathrm{\ell }\le k-1$ and sufficiently large $n\in k\mathbb{N}$.