Regular hypergraphs: asymptotic counting and loose Hamilton cycles

Abstract

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(n k), where k = k (k) = 1 for all k ≥ 4, while k(3) = 1/2. The second result is that a random d-regular hypergraph contains as a dense sub-graph 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.KeywordsRandom GraphHamilton CycleAsymptotic EnumerationRandom Regular GraphProper EdgeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.