Loose Hamilton Cycles in Regular Hypergraphs

We establish a relation between two uniform models of random k-graphs (for constant k ≥ 3) on n labeled vertices: H(k)(n,m), the random k-graph with exactly m edges, and H(k)(n, d), the random d-regular k-graph. By extending to k-graphs the switching technique of McKay and Wormald, we show that, for some range of d = d(n) and a constant c > 0, if m ∼ cnd, then one can couple H(k)(n,m) and H(k)(n, d) so that the latter contains the former with probability tending to one as n→∞. In view of known results on the existence of a loose Hamilton cycle in H(k)(n,m), we conclude that H(k)(n, d) contains a loose Hamilton cycle when d log n (or just d ≥ C log n, if k = 3) and d = o(n1/2). ∗MSC2010 codes: 05C65, 05C80, 05C45. †Supported in part by Simons Foundation Grant #244712. ‡Supported in part by NSF grant CCF1013110. §Research supported by the Polish NSC grant N201 604940. Part of research performed at Emory University, Atlanta. ¶Research supported by the Polish NSC grant N201 604940. Part of research performed at Adam Mickiewicz University, Poznań.