Loose Hamilton Cycles in Regular Hypergraphs
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We establish a relation between two uniform models of random k-graphs (for constant k ≥ 3) on n labelled 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 the switching technique of McKay and Wormald to k-graphs, 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).