Essential edges in Poisson random hypergraphs

Consider a random hypergraph on a set of N vertices in which, for 1 ≤ k ≤ N, a Poisson (Nβκ) number of hyperedges is scattered randomly over all subsets of size k. We collapse the hypergraph by running the following algorithm to exhaustion: Pick a vertex having a 1-edge and remove it; collapse the hyperedges over that vertex onto their remaining vertices; repeat until there are no 1-edges left. We call the vertices removed in this process identifiable. Also any hyperedge all of whose vertices are removed is called identifiable. We say that a hyperedge is essential if its removal prior to collapse would have reduced the number of identifiable vertices. The limiting proportions, as N → ∞, of identifiable vertices and hyperedges were obtained in R. W. R. Darling and J. R. Norris [Structure of large random hypergraphs, Ann Appl Probab, to appear. In this paper, we establish the limiting proportion of essential hyperedges. We also discuss, in the case of a random graph, the relation of essential edges to the 2-core of the graph, the maximal subgraph with minimal vertex degree 2. ©2004 Wiley Periodicals, Inc