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

TL;DR

This paper establishes the exact minimum degree conditions needed to ensure the presence of Hamilton $( ext{ell},k- ext{ell})$-cycles in large $k$-uniform hypergraphs for certain parameter ranges.

Contribution

It determines the tight minimum $ ext{ell}$-degree threshold guaranteeing Hamilton $( ext{ell},k- ext{ell})$-cycles in large hypergraphs for $k eq 7$ and $rac{k}{2} eq ext{ell}$.

Findings

Identifies the precise minimum degree condition for Hamilton cycles.

Proves the condition is tight for large hypergraphs.

Extends understanding of cycle existence in hypergraph theory.

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 $(L_0,R_0,L_1,R_1,\ldots,L_{t-1},R_{t-1})$ 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 $i=0,1,\ldots,t-1$. 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 $k\geq 7$, $k/2\leq \ell\leq k-1$ and sufficiently large $n\in k\mathbb{N}$.