Tight Hamilton Cycles in Random Uniform Hypergraphs

TL;DR

This paper establishes precise threshold probabilities for the emergence of tight Hamilton cycles and other cycle types in random uniform hypergraphs, advancing understanding of their phase transitions.

Contribution

It determines the sharp threshold for tight Hamilton cycles in random hypergraphs for all uniformities $k \\ge 4$ and asymptotic threshold for $k=3$, also identifying thresholds for other cycle types.

Findings

Threshold for tight Hamilton cycles is e/n for all k ≥ 4.

Threshold for k=3 is asymptotic at 1/n.

Thresholds for other Hamilton cycle types are also identified.

Abstract

In this paper we show that $e/n$ is the sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs, for all $k\ge 4$. When $k=3$ we show that $1/n$ is an asymptotic threshold. We also determine thresholds for the existence of other types of Hamilton cycles.