arXiv 2005.073502020-05math.CO

随机正则均匀超图中的生成树

Spanning trees in random regular uniform hypergraphs

Catherine Greenhill, Mikhail Isaev, Gary Liang

备注：: 40 pages. Fixed some typos
发表信息：: Combinator. Probab. Comp. 31 (2022) 29-53
DOI：: 10.1017/S0963548321000158

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AI总结：

本文研究了随机正则均匀超图中生成树的存在性，建立了阈值结果，并给出了生成树数量的渐近分布，填补了此前仅知于特殊情形的空白。

摘要：

设 $\mathcal{G}_{n,r,s}$ 表示顶点集为 $\{1,2,\ldots, n\}$ 上的均匀随机 $r$-正则 $s$-均匀超图。我们建立了 $\mathcal{G}_{n,r,s}$ 中存在生成树的阈值结果，并将 $n$ 限制为满足必要的可整除性条件。具体地，我们证明当 $s\geq 5$ 时，存在一个正常数 $\rho(s)$，使得对于任意 $r\geq 2$，若 $r > \rho(s)$，则 $\mathcal{G}_{n,r,s}$ 包含生成树的概率趋于 1；否则该概率趋于 0。阈值 $\rho(s)$ 随 $s$ 指数增长。由于 $\mathcal{G}_{n,r,s}$ 以趋于 1 的概率连通，这意味着当 $r \leq \rho(s)$ 时，大多数 $r$-正则 $s$-均匀超图是连通的但没有生成树。当 $s=3,4$ 时，我们证明对于任意 $r\geq 2$，$\mathcal{G}_{n,r,s}$ 包含生成树的概率趋于 1。我们的证明还给出了对于所有固定的整数 $r,s\geq 2$，$\mathcal{G}_{n,r,s}$ 中生成树数量的渐近分布。此前，该渐近分布仅在 2-正则图或三次图的平凡情形下已知。

英文摘要：

Let $\mathcal{G}_{n,r,s}$ denote a uniformly random $r$-regular $s$-uniform hypergraph on the vertex set $\{1,2,\ldots, n\}$. We establish a threshold result for the existence of a spanning tree in $\mathcal{G}_{n,r,s}$, restricting to $n$ satisfying the necessary divisibility conditions. Specifically, we show that when $s\geq 5$, there is a positive constant $ρ(s)$ such that for any $r\geq 2$, the probability that $\mathcal{G}_{n,r,s}$ contains a spanning tree tends to 1 if $r > ρ(s)$, and otherwise this probability tends to zero. The threshold value $ρ(s)$ grows exponentially with $s$. As $\mathcal{G}_{n,r,s}$ is connected with probability which tends to 1, this implies that when $r \leq ρ(s)$, most $r$-regular $s$-uniform hypergraphs are connected but have no spanning tree. When $s=3,4$ we prove that $\mathcal{G}_{n,r,s}$ contains a spanning tree with probability which tends to 1, for any $r\geq 2$. Our proof also provides the asymptotic distribution of the number of spanning trees in $\mathcal{G}_{n,r,s}$ for all fixed integers $r,s\geq 2$. TPreviously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

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