Spanning trees in random regular uniform hypergraphs.

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 $\\rho(s)$ such that for any $r\\geq 2$, the probability that $\\mathcal{G}_{n,r,s}$ contains a spanning tree tends to 1 if $r \u003e \\rho(s)$, and otherwise this probability tends to zero. The threshold value $\\rho(s)$ grows exponentially with $s$. As $\\mathcal{G}_{n,r,s}$ is connected with probability which tends to 1, this implies that when $r \\leq \\rho(s)$, 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 $r,s\\geq 2$. This asymptotic distribution was previously only known for cubic graphs.