Loose Hamilton Cycles in Regular Hypergraphs

TL;DR

This paper establishes a probabilistic relationship between two models of random hypergraphs and demonstrates the existence of loose Hamilton cycles in regular hypergraphs under certain degree conditions.

Contribution

It extends the switching technique to $k$-graphs and shows that regular hypergraphs contain loose Hamilton cycles when the degree grows faster than logarithmic but slower than a square root of the number of vertices.

Findings

Coupling of $H(n,m)$ and $H(n,d)$ with high probability

Existence of loose Hamilton cycles in $H(n,d)$ for $d o ext{large}$ but $d = o(n^{1/2})$

Conditions on $d$ for Hamilton cycle presence in regular hypergraphs

Abstract

We establish a relation between two uniform models of random $k$-graphs (for constant $k \ge 3$) on $n$ labeled vertices: $H(n,m)$, the random $k$-graph with exactly $m$ edges, and $H(n,d)$, the random $d$-regular $k$-graph. By extending to $k$-graphs the switching technique of McKay and Wormald, we show that, for some range of $d = d(n)$ and a constant $c > 0$, if $m \sim cnd$, then one can couple $H(n,m)$ and $H(n,d)$ so that the latter contains the former with probability tending to one as $n \to \infty$. In view of known results on the existence of a loose Hamilton cycle in $H(n,m)$, we conclude that $H(n,d)$ contains a loose Hamilton cycle when $\log n = o(d)$ (or just $d \ge C log n$, if $k = 3$) and $d = o(n^{1/2})$.