Loose Hamilton Cycles in Random Uniform Hypergraphs

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 $$\underset{\begin{array}{c}n\to \infty 2(k-1)|n\end{array}}{\lim }\mathrm{Pr}(H_{n,p;k}containsalooseHamiltoncycle)=1.$$ This is asymptotically best possible.

14 pages