Exact minimum co-degree conditions for $\ell$-Hamiltonicity in hypergraphs

TL;DR

This paper establishes exact minimum co-degree thresholds for the existence of Hamilton -cycles in large hypergraphs, advancing conjectures and problems in hypergraph Hamiltonicity.

Contribution

It provides the first exact co-degree conditions for -Hamiltonicity in certain hypergraphs, partially confirming conjectures and solving longstanding problems.

Findings

Proves exact co-degree threshold for -Hamiltonicity in specified hypergraphs.

Shows that a slightly higher co-degree suffices for all -Hamiltonian hypergraphs.

Partially verifies a conjecture of Han and Zhao and addresses a problem of R46dl and Ruci
44ski.

Abstract

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 R\"odl and Ruci\'nski. 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 $\ell$.