Random Hypergraph Processes with Degree Restrictions

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 i = dn−sbdn/sc.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). ∗Some of this work was performed while the first and second authors were visiting the Isaac Newton Institute for Mathematical Sciences, Cambridge, United Kingdom. †Research supported by an Australian Research Council Postdoctoral Fellowship. Current address: School of Mathematics, UNSW, Sydney NSW Australia 2052 ([email protected]). ‡Research supported by the KBN grant 2 P03A 15 23. §Research supported by the Australian Research Council. Current address: Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON Canada N2L 3G1 ([email protected]).