Loose Hamilton Cycles in Random 3-Uniform Hypergraphs

TL;DR

This paper proves that in a random 3-uniform hypergraph, a loose Hamilton cycle appears with high probability when the edge probability exceeds a certain threshold proportional to log n/n^2.

Contribution

It establishes a sharp threshold for the existence of loose Hamilton cycles in random 3-uniform hypergraphs, advancing understanding of their probabilistic properties.

Findings

Loose Hamilton cycles appear with high probability above the threshold p>K log n/n^2.

The threshold for cycle existence is explicitly characterized.

The result confirms a conjecture about cycle thresholds in random hypergraphs.

Abstract

In the random hypergraph H=H(n,p;3) each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges {x_i,y_i,x_{i+1}\} for i=1,2,...,n/2. We prove that there exists an absolute constant K>0 such that if p>K\log n/n^2 then lim_{n->oo 4 |n}}Pr(H(n,p;3) contains a loose Hamilton cycle)=1.