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Asymptotic enumeration of linear hypergraphs with given number of vertices and edges
Advances in Applied Mathematics ( IF 1.1 ) Pub Date : 2020-04-01 , DOI: 10.1016/j.aam.2020.102000
Brendan D. McKay , Fang Tian

For $n\geq 3$, let $r=r(n)\geq 3$ be an integer. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear $r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically when the number of edges is $m(n)=o(r^{-3}n^{ \frac32})$. As one application, we find the probability of linearity for the independent-edge model of random $r$-uniform hypergraph when the expected number of edges is $o(r^{-3}n^{ \frac32})$. We also find the probability that a random $r$-uniform linear hypergraph with a given number of edges contains a given subhypergraph.

中文翻译:

具有给定顶点和边数的线性超图的渐近枚举

对于 $n\geq 3$,令 $r=r(n)\geq 3$ 是一个整数。如果每条边都是一组 $r$ 顶点,则超图是 $r$-uniform,如果两条边最多在一个顶点相交,则称该超图是线性的。在本文中,当边数为$m(n)=o(r^{-3}n^{\ frac32})$。作为一种应用,当预期边数为 $o(r^{-3}n^{\frac32})$ 时,我们找到随机 $r$-均匀超图的独立边模型的线性概率。我们还发现了具有给定边数的随机 $r$-uniform 线性超图包含给定子超图的概率。
更新日期:2020-04-01
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