Random Hypergraph Processes with Degree Restrictions

Abstract.

A d-process for s-uniform hypergraphs starts with an empty hypergraph on n vertices, and adds one s-tuple at each time step, chosen uniformly at random from those s-tuples which are not already present as a hyperedge and which consist entirely of vertices with degree less than d. We prove that for d≥2 and s≥3, with probability which tends to 1 as n tends to infinity, the final hypergraph is saturated; that is, it has n−i vertices of degree d and i vertices of degree d−1, where This generalises the result for s=2 obtained by the second and third authors. In addition, when s≥3, we prove asymptotic equivalence of this process and the more relaxed process, in which the chosen s-tuple may already be a hyperedge (and which therefore may form multiple hyperedges).