Exact supported co-degree bounds for Hamilton cycles

TL;DR

This paper establishes exact bounds on supported co-degree conditions that guarantee the existence of Hamilton cycles in large hypergraphs, resolving conjectures and improving previous asymptotic results.

Contribution

It provides the first exact supported co-degree bounds for Hamilton cycles in hypergraphs, fully resolving a conjecture for the case  = k-1 and improving prior asymptotic bounds.

Findings

Bounds are tight for infinitely many (k, ) pairs.

The results are essentially optimal, off by at most 1 for others.

The proof introduces a novel blow-up tiling framework avoiding regularity lemmas.

Abstract

For any $k\ge 3$ and $\ell \in [k-1]$ such that $(k,\ell) \ne (3,1)$, we show that any sufficiently large $k$-graph $G$ must contain a Hamilton $\ell$-cycle provided that it has no isolated vertices and every set of $k-1$ vertices contained in an edge is contained in at least $\left(1 - \frac{1}{\lfloor{\frac{k}{k-\ell}\rfloor}(k-\ell)}\right)n - (k - 3)$ edges. We also show that this bound is tight for infinitely many values of $k$ and $\ell$ and is off by at most $1$ for all others, and is hence essentially optimal. This improves an asymptotic version of this result due to Mycroft and Z\'arate-Guer\'en, and the case $\ell = k-1$ completely resolves a conjecture of Illingworth, Lang, M\"uyesser, Parczyk and Sgueglia. These results support the utility of $\textit{minimum}$ $\textit{supported}$ $\textit{co-degree}$ conditions in a $k$-graph, a recently introduced variant of the standard notion of minimum co-degree applicable to $k$-graphs with non-trivial strong independent sets. Our proof techniques involve a novel blow-up tiling framework introduced by Lang, avoiding traditional approaches using the regularity and blow-up lemmas.