Sharp thresholds for nonlinear Hamiltonian cycles in hypergraphs

Abstract

For positive integers r>ℓ, an r‐uniform hypergraph is called an ℓ‐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 ℓ vertices; such cycles are said to be linear when ℓ=1, and nonlinear when ℓ>1. We determine the sharp threshold for nonlinear Hamiltonian cycles and show that for all r>ℓ>1, the threshold for the appearance of a Hamiltonian ℓ‐cycle in the random r‐uniform hypergraph on n vertices is sharp and given by for an explicitly specified function λ. This resolves several questions raised by Dudek and Frieze in 2011.10