Minimum codegree threshold for Hamilton l-cycles in k-uniform hypergraphs

TL;DR

This paper establishes the exact minimum codegree threshold for the existence of Hamilton -cycles in large k-uniform hypergraphs, improving previous asymptotic results and confirming the threshold's optimality.

Contribution

It proves the precise minimum codegree condition needed for Hamilton -cycles in large hypergraphs, advancing the understanding of hypergraph Hamiltonicity.

Findings

The minimum codegree threshold is (n/(k-)) for Hamilton -cycles.

The result is optimal and improves previous asymptotic bounds.

The paper confirms the threshold's sharpness for large hypergraphs.

Abstract

For $1\le \ell<k/2$, we show that for sufficiently large $n$, every $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $\frac n{2 (k-\ell)} $ contains a Hamilton $\ell$-cycle. This codegree condition is best possible and improves on work of H\`an and Schacht who proved an asymptotic result.