Embedding the Erd\H{o}s-R\'enyi Hypergraph into the Random Regular Hypergraph and Hamiltonicity

TL;DR

This paper demonstrates how to embed Erdős-Rényi hypergraphs into random regular hypergraphs, enabling the transfer of Hamiltonicity results and establishing conditions for the existence of Hamilton cycles in regular hypergraphs.

Contribution

It extends previous results by coupling Erdős-Rényi and regular hypergraphs, allowing for new Hamiltonicity thresholds in regular hypergraphs based on known Erdős-Rényi thresholds.

Findings

Coupling of hypergraph models with high probability

Conditions for Hamilton cycle existence in regular hypergraphs

Extension of sandwiching results to hypergraphs

Abstract

We establish an inclusion relation between two uniform models of random $k$-graphs (for constant $k \ge 2$) on $n$ labeled vertices: $\mathbb G^{(k)}(n,m)$, the random $k$-graph with $m$ edges, and $\mathbb R^{(k)}(n,d)$, the random $d$-regular $k$-graph. We show that if $n\log n\ll m\ll n^k$ we can choose $d = d(n) \sim {km}/n$ and couple $\mathbb G^{(k)}(n,m)$ and $\mathbb R^{(k)}(n,d)$ so that the latter contains the former with probability tending to one as $n\to\infty$. This extends an earlier result of Kim and Vu about "sandwiching random graphs". In view of known threshold theorems on the existence of different types of Hamilton cycles in $\mathbb G^{(k)}(n,m)$, our result allows us to find conditions under which $\mathbb R^{(k)}(n,d)$ is Hamiltonian. In particular, for $k\ge 3$ we conclude that if $n^{k-2} \ll d \ll n^{k-1}$, then a.a.s. $\mathbb R^{(k)}(n,d)$ contains a tight Hamilton cycle.