Embedding the Erdős-Rényi hypergraph into the random regular hypergraph and Hamiltonicity

Abstract We establish an inclusion relation between two uniform models of random k -graphs (for constant k ≥ 2 ) on n labeled vertices: G ( k ) ( n , m ) , the random k -graph with m edges, and R ( k ) ( n , d ) , the random d -regular k -graph. We show that if n log ⁡ n ≪ m ≪ n k we can choose d = d ( n ) ∼ k m / n and couple G ( k ) ( n , m ) and R ( k ) ( n , d ) so that the latter contains the former with probability tending to one as n → ∞ . This extends an earlier result of Kim and Vu about “sandwiching random graphs”. In view of known threshold theorems on the existence of different types of Hamilton cycles in G ( k ) ( n , m ) , our result allows us to find conditions under which R ( k ) ( n , d ) is Hamiltonian. In particular, for k ≥ 3 we conclude that if n k − 2 ≪ d ≪ n k − 1 , then a.a.s. R ( k ) ( n , d ) contains a tight Hamilton cycle.