Exact minimum co-degree conditions for $ll$-Hamiltonicity in hypergraphs

Suppose $1\le \ell <k$ such that $(k-\ell)\nmid k$. Given an $n$-vertex $k$-uniform hypergraph $\mathcal H$, for all $k/2<\ell< 3k/4$ and sufficiently large $n\in (k-\ell)\mathbb N$, we prove that if $\mathcal H$ has minimum co-degree at least $\frac{n}{\lceil \frac{k}{k-\ell}\rceil (k-\ell)}$, then $\mathcal H$ contains a Hamilton $\ell$-cycle, which partially verifies a conjecture of Han and Zhao and (partially) resolves a problem of Rödl and Ruciński. Moreover, we show that assuming minimum co-degree $\frac{n}{\lceil \frac{k}{k-\ell}\rceil (k-\ell)}+\frac{k^2}2$ is enough for all $\ell$.

💡 Research Summary

The paper addresses the long‑standing problem of determining exact minimum co‑degree thresholds that guarantee the existence of Hamilton ℓ‑cycles in k‑uniform hypergraphs. A Hamilton ℓ‑cycle is a spanning ℓ‑overlapping cycle: the vertices can be ordered cyclically so that each edge consists of k consecutive vertices and any two consecutive edges intersect in exactly ℓ vertices. The necessary divisibility condition (k − ℓ) | n is assumed throughout.

The authors focus on the regime where (k − ℓ) does not divide k, a case in which the co‑degree threshold is expected to be substantially lower than the “tight” case (k − ℓ) | k. They define s := ⌈k/(k − ℓ)⌉ and prove two main theorems.

Theorem 1.3 (Exact result for ℓ < 3k/4).
For integers k ≥ 3 and ℓ satisfying k/2 < ℓ < 3k/4 with (k − ℓ)∤k, and for sufficiently large n that is a multiple of (k − ℓ), any n‑vertex k‑uniform hypergraph H with minimum co‑degree \