A threshold result for loose Hamiltonicity in random regular uniform hypergraphs

TL;DR

This paper establishes a threshold for the appearance of loose Hamilton cycles in random regular uniform hypergraphs, partially confirming a prior conjecture and analyzing the distribution of such cycles.

Contribution

It provides a threshold result for loose Hamiltonicity in random regular uniform hypergraphs and determines the asymptotic distribution of loose Hamilton cycles.

Findings

Threshold for loose Hamilton cycles established

Asymptotic distribution of cycles determined

Probability of other overlapping cycles tends to zero

Abstract

Let $\mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniform hypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may grow with $n$. An $\ell$-overlapping Hamilton cycle is a Hamilton cycle in which successive edges overlap in precisely $\ell$ vertices, and 1-overlapping Hamilton cycles are called loose Hamilton cycles. When $r,s\geq 3$ are fixed integers, we establish a threshold result for the property of containing a loose Hamilton cycle. This partially verifies a conjecture of Dudek, Frieze, Rucinski and Sileikis (2015). In this setting, we also find the asymptotic distribution of the number of loose Hamilton cycles in $\mathcal{G}(n,r,s)$. Finally we prove that for $\ell = 2,\ldots, s-1$ and for $r$ growing moderately as $n\to\infty$, the probability that $\mathcal{G}(n,r,s)$ has a $\ell$-overlapping Hamilton cycle tends to zero.