Embedding the Erdős-Rényi 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) contains a tight Hamilton cycle. ∗Project sponsored by the National Security Agency under Grant Number H98230-15-1-0172. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon. †Supported in part by NSF grant CCF 1013110. ‡Supported in part by the Polish NSC grant 2014/15/B/ST1/01688 and NSF grant DMS 1102086. §Supported in part by the Knut and Alice Wallenberg Foundation. ¶Part of research performed during a visit to the Institut Mittag-Leffler (Djursholm, Sweden).