Forbidding Hamilton cycles in uniform hypergraphs

TL;DR

This paper establishes a new lower bound for the minimum degree needed to guarantee Hamilton cycles in uniform hypergraphs, disproving a longstanding conjecture for certain parameters.

Contribution

It provides a novel lower bound for degree thresholds in hypergraphs and generalizes existing constructions, challenging previous conjectures about perfect matchings.

Findings

New lower bound for degree thresholds in hypergraphs

Disproves a well-known conjecture for certain parameters

Generalizes existing constructions for Hamilton cycles

Abstract

For $1\le d\le \ell< k$, we give a new lower bound for the minimum $d$-degree threshold that guarantees a Hamilton $\ell$-cycle in $k$-uniform hypergraphs. When $k\ge 4$ and $d< \ell=k-1$, this bound is larger than the conjectured minimum $d$-degree threshold for perfect matchings and thus disproves a well-known conjecture of R\"odl and Ruci\'nski. Our (simple) construction generalizes a construction of Katona and Kierstead and the space barrier for Hamilton cycles.