Abstract
We establish a relation between two uniform models of randomk-graphs (for constantk⩾ 3) onnlabelled vertices: ℍ(k)(n,m), the randomk-graph with exactlymedges, and ℍ(k)(n,d), the randomd-regulark-graph. By extending the switching technique of McKay and Wormald tok-graphs, we show that, for some range ofd = d(n)and a constantc> 0, ifm~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 asn→ ∞. 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 whend≫ logn(or justd⩾Clogn, ifk= 3) andd=o(n1/2).
Highlights
A k-uniform hypergraph on a vertex set V = {1, . . . , n} is a family of k-element subsets of V
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, 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 → ∞
We study the behavior of random k-graphs as n → ∞
Summary
The threshold for existence of a loose Hamilton cycle in H(k)(n, p) was determined by Frieze [12] (for k = 3) as well as Dudek and Frieze [9] (for k ≥ 4) under a divisibility condition 2(k − 1)|n, which was relaxed to (k − 1)|n by Dudek, Frieze, Loh and Speiss [10] We formulate these results for the model H(k)(n, m), such a possibility being provided to us by the asymptotic equivalence of models H(k)(n, p) and H(k)(n, m) (see, e.g., Corollary 1.16 in [13]). H(k)(n, d) contains a loose Hamilton cycle a.a.s
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