Sharp thresholds for nonlinear Hamiltonian cycles in hypergraphs

Abstract

For positive integers $r > \ell$, an $r$-uniform hypergraph is called an $\ell$-cycle if there exists a cyclic ordering of its vertices such that each of its edges consists of $r$ consecutive vertices, and such that every pair of consecutive edges (in the natural ordering of the edges) intersect in precisely $\ell$ vertices. Such cycles are said to be linear when $\ell = 1$, and nonlinear when $\ell > 1$. We determine the sharp threshold for nonlinear Hamiltonian cycles and show that for all $r > \ell > 1$, the threshold $p^*_{r, \ell} (n)$ for the appearance of a Hamiltonian $\ell$-cycle in the random $r$-uniform hypergraph on $n$ vertices is sharp and is $p^*_{r, \ell} (n) = \lambda(r,\ell) (\frac{\mathrm{e}}{n})^{r - \ell}$ for an explicitly specified function $\lambda$. This resolves several questions raised by Dudek and Frieze in 2011.