Hamiltonian Berge cycles in random hypergraphs

TL;DR

This paper establishes the threshold for the emergence of Hamiltonian Berge cycles in random hypergraphs, showing that such cycles almost surely appear once the minimum degree reaches 2, and confirms related conjectures.

Contribution

It proves an optimal stopping-time result for Hamiltonian Berge cycles in random hypergraphs and determines the threshold probability for Berge Hamiltonicity, resolving a conjecture.

Findings

Hamiltonian Berge cycles appear almost surely when minimum degree reaches 2

Threshold probability for Berge Hamiltonicity in Erdős–Rényi hypergraphs is identified

Almost sure existence of such cycles in 2-out random hypergraphs

Abstract

In this note, we study the emergence of Hamiltonian Berge cycles in random $r$-uniform hypergraphs. For $r\geq 3$, we prove an optimal stopping-time result that if edges are sequently added to an initially empty $r$-graph, then as soon as the minimum degree is at least 2, the hypergraph almost surely has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erd\H{o}s--R\'enyi random $r$-graph, and we also show that the $2$-out random $r$-graph almost surely has such a cycle. We obtain similar results for \textit{weak Berge} cycles as well, thus resolving a conjecture of Poole.