Regular Hypergraphs: Asymptotic Counting and Loose Hamilton Cycles

We present results from two papers by the authors on analysis of d-regular k-uniform hypergraphs, when k is fixed and the number n of vertices tends to infinity. The first result is approximate enumeration of such hypergraphs, provided d = d(n) = o(nκ), where κ = κ(k) = 1 for all k ≥ 4, while κ(3) = 1/2. The second result is that a random d-regular hypergraph contains as a dense subgraph the uniform random hypergraph (a generalization of the Erdős-Rényi uniform graph), and, in view of known results, contains a loose Hamilton cycle with probability tending to one. 1. Regular k-graphs and k-multigraphs. We consider k-uniform hypergraphs (or k-graphs, for short) on the vertex set V = [n] := {1, . . . , n}, that is, families of k-element subsets of V . A k-graph H is d-regular, if the degree of every vertex v ∈ V , degH(v) := deg(v) := | {e ∈ H : v ∈ e} | equals d. Let H(n, d) be the class of all d-regular k-graphs on [n]. Note that each H ∈ H(n, d) has M := nd/k edges (throughout, we implicitly assume that ∗Department of Mathematics, Western Michigan University, Kalamazoo, MI. †Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA. ‡Department of Discrete Mathematics, Adam Mickiewicz University, Poznań, Poland. §Department of Mathematics, Uppsala University, Sweden, [email protected] ¶Corresponding author.