3 research outputs found
Essential edges in Poisson random hypergraphs
Consider a random hypergraph on a set of N vertices in which, for k between 1
and N, a Poisson(N beta_k) 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.Comment: 12 pages, 3 figures. Revised version with minor
corrections/clarifications and slightly expanded introductio