Minimum vertex degree conditions for loose Hamilton cycles in $3$-uniform hypergraphs

TL;DR

This paper establishes near-optimal minimum vertex degree conditions in 3-uniform hypergraphs that guarantee the existence of loose Hamilton cycles, advancing understanding of cycle conditions in hypergraph theory.

Contribution

It proves an asymptotically best possible minimum vertex degree threshold for loose Hamilton cycles in 3-uniform hypergraphs.

Findings

Minimum degree threshold of (7/16 + o(1)) * binomial(n, 2) guarantees loose Hamilton cycles.

The bound is asymptotically tight, matching known lower bounds.

Provides new insights into cycle existence conditions in hypergraphs.

Abstract

We investigate minimum vertex degree conditions for $3$-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex. We prove that every $3$-uniform $n$-vertex ($n$ even) hypergraph $\mathcal{H}$ with minimum vertex degree $\delta_1(\mathcal{H})\geq \left(\frac7{16}+o(1)\right)\binom{n}{2}$ contains a loose Hamilton cycle. This bound is asymptotically best possible.