2004/01/14 by Christina Goldschmidt, Goldschmidt, Christina, James R. Norris +2 · 1 citation
Economics, Econometrics and Finance · Mathematics · #05C65 (Primary) 60C05 #05C80 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #advanced mathematical theories #math.CO #math.PR #msc:05C65 #msc:05C80 #msc:60C05
paper · pdf · doi:10.48550/arxiv.math/0401143
12 pages, 3 figures. Revised version with minor corrections/clarifications and slightly expanded introduction
openalex publication_date 2004/01/14 · arxiv created 2004/04/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider a random hypergraph on a set of N vertices in which, for k between 1 and N, a Poisson(N betak) 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 tends to infinity, of identifiable vertices and hyperedges were obtained by Darling and Norris. 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 sub-graph with minimal vertex degree 2.