Poisson随机超图的本质边
Essential edges in Poisson random hypergraphs
摘要：
考虑一个N点集上的随机超图，其中k在1和N之间，Poisson（N beta_k）数的超边随机散布在所有大小为k的子集上。我们通过运行以下算法来将超图折叠到耗尽：选择一个具有1-边的顶点并将其删除;将该顶点上的超边折叠到其剩余顶点上;重复直到没有1-边。我们把这个过程中去掉的顶点称为“可识别的”。所有顶点都被移除的超边也被称为“可识别的”。我们说一个超边是“必要的”，如果它的删除之前，崩溃将减少可识别的顶点的数量。当N趋于无穷大时，可识别顶点和超边的极限比例由Darling和Norris得到。本文建立了本质超边的极限比例。在随机图的情况下，我们还讨论了本质边与图的2-核（最小顶点度为2的最大子图）的关系。
英文摘要：
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.
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