Loose Hamilton Cycles in Random Uniform Hypergraphs

TL;DR

This paper establishes the threshold probability for the existence of loose Hamilton cycles in random uniform hypergraphs, showing that above a certain probability, such cycles almost surely exist as the number of vertices grows large.

Contribution

It proves the asymptotically optimal threshold for loose Hamilton cycles in random hypergraphs, extending previous results to a broader range of parameters.

Findings

Threshold for loose Hamilton cycles established

Probability tends to 1 when p n^{k-1}/log n grows large

Result is asymptotically best possible

Abstract

In the random hypergraph $H_{n,p;k}$ each possible $k$-tuple appears independently with probability $p$. A loose Hamilton cycle is a cycle in which every pair of adjacent edges intersects in a single vertex. We prove that if $p n^{k-1}/\log n$ tends to infinity with $n$ then $$\lim_{\substack{n\to \infty 2(k-1) |n}}\Pr(H_{n,p;k}\ contains\ a\ loose\ Hamilton\ cycle)=1.$$ This is asymptotically best possible.