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

Abstract

Let $n>k>\ell$ be positive integers. We say a $k$-uniform hypergraph $\mathcal{H}$ contains a Hamilton $(\ell ,k-\ell )$-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 $|L_i|=\ell$, $|R_i|=k-\ell$ 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 $\ell$-degree condition that guarantees the existence of a Hamilton $(\ell ,k-\ell )$-cycle in every $k$-uniform $n$-vertex hypergraph for $k\ge 7$, $k/2\le \ell \le k-1$ and sufficiently large $n\in k\mathbb{N}$.

19 pages. arXiv admin note: substantial text overlap with arXiv:2002.12234 by other authors