Minimum degree conditions for Hamilton $l$-cycles in $ k $-uniform hypergraphs

TL;DR

The paper determines a sharp Dirac-type threshold for Hamilton $2$-cycles in $5$-graphs, proving $h_{2}^{2}(5)=\frac{91}{216}$ and showing that a matching large-$n$ minimum $2$-degree condition guarantees a Hamilton $2$-cycle; this threshold is asymptotically optimal. The authors extend the framework to Hamilton $\ell$-cycles under $d$-degree conditions for $\ell\le d\le k-1$ and $1\le \ell<k/2$, employing an absorbing method augmented by a new connecting lemma and lattice-based absorbers. The key technical contributions include a robust absorbing path lemma, two variants of a connecting lemma (one via Kruskal–Katona, one via non-linear methods), and a path-cover/reservoir strategy to assemble a Hamilton $\ell$-cycle while controlling leftovers. The results advance Dirac-type thresholds in hypergraphs, yielding exact or tight asymptotic bounds for important $k$-graph Hamiltonicity questions and informing tiling thresholds such as $t(5,2,2)$ that underpin the absorbing framework.

Abstract

We show that for $ η>0 $ and sufficiently large $ n $, every 5-graph on $ n $ vertices with $δ_{2}(H)\ge (91/216+η)\binom{n}{3}$ contains a Hamilton 2-cycle. This minimum 2-degree condition is asymptotically best possible. Moreover, we give some related results on Hamilton $ \ell $-cycles with $ d $-degree for $\ell\le d \le k-1$ and $1\le \ell < k/2$.

Figures (1)

• Figure 1: a $\{v_1,\dots,v_{k-\ell}\}$-absorber, where $S'_{A}= \{w_1,\dots,w_{k-\ell}\}$.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2: large
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• proof
• proof
• Proposition 2.1
• proof