Let $${{\mathcal G}_{n,r,s}}$$ denote a uniformly random r-regular s-uniform hypergraph on the vertex set {1, 2, … , 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 ≥ 5, there is a positive constant ρ(s) such that for any r ≥ 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 that tends to 1, this implies that when r ≤ 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 that tends to 1, for any r ≥ 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 ≥ 2. Previously, this asymptotic distribution was only known in the trivial case of 2-regular graphs, or for cubic graphs.

中文翻译：

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随机规则均匀超图中的生成树

让$${{\mathcal G}_{n,r,s}}$$表示均匀随机r-常规的s- 顶点集 {1, 2, ... ,n}。我们为生成树的存在建立一个阈值结果$${{\mathcal G}_{n,r,s}}$$, 限制为n满足必要的可分条件。具体来说，我们表明当s≥5，有一个正常数ρ(s) 这样对于任何r≥ 2，概率$${{\mathcal G}_{n,r,s}}$$包含一棵生成树，如果r>ρ(s)，否则这个概率趋于零。阈值ρ(s) 呈指数增长s. 作为$${{\mathcal G}_{n,r,s}}$$与趋于 1 的概率有关，这意味着当r≤ρ(s）， 最多r-常规的s-均匀超图是连接的，但没有生成树。什么时候s= 3, 4 我们证明$${{\mathcal G}_{n,r,s}}$$包含概率趋于 1 的生成树，对于任何r≥ 2. 我们的证明还提供了生成树数量的渐近分布$${{\mathcal G}_{n,r,s}}$$对于所有固定整数r,s≥ 2。以前，这种渐近分布仅在 2-正则图或三次图的平凡情况下才知道。