Abstract. For positive integers r ą , an r-uniform hypergraph is called an -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 vertices. Such cycles are said to be linear when " 1, and nonlinear when ą 1. We determine the sharp threshold for nonlinear Hamiltonian cycles and show that for all r ą ą 1, the threshold pr , pnq for the appearance of a Hamiltonian -cycle in the random r-uniform hypergraph on n vertices is sharp and is pr , pnq " λpr, qp e n q r´ for an explicitly specified function λ. This resolves several questions raised by Dudek and Frieze in 2011. §1. Introduction A basic problem in probabilistic combinatorics concerns locating the critical density at which a substructure of interest appears inside a random structure (with high probability). In the context of random graph theory, the question of when a random graph contains a Hamiltonian cycle has received considerable attention. Indeed, from the foundational works
Recently, Narayanan and Schacht REF pinpointed the precise value of the sharp threshold for the appearance of j-tight Hamilton cycles in k-uniform hypergraphs, provided that k > j > 1.
Sharp thresholds for nonlinear Hamiltonian cycles in hypergraphs
arXiv: Combinatorics
Mathematics