3 problems

Let $d=d(n)\in[3,n-1]$, and let $G\sim G(n,d)$ be a random $d$-regular graph. An independent spanning tree (IST) family is a collection of spanning trees whose root-to-vertex paths…

Let $G\sim G(n,p)$ be an Erdős–Rényi random graph, and write $\delta(G)$ for its minimum degree. A spanning tree rooted at $r$ is an independent spanning tree (IST) family member w…

Let $d\geq 1$, let $G$ be a $d$-vertex-connected graph, and let $r$ be a vertex of $G$. A family of spanning trees $T_1,\ldots,T_d$ is said to provide independent paths from $r$ if…