Embedding the Erdos-Renyi Hypergraph into the Random Regular Hypergraph and Hamiltonicity

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≪nk we can choose d = d(n) ~ km/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 some previous results 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 nk−2≪d≪nk−1, then a.a.s. R(k)(n, d) containsa tight Hamilton cycle.