4 problems

Let $\mathcal{G}(n,d)$ denote the uniform random labelled $d$-regular graph. Let $0\leq d_1\leq d\leq n-1$ be integers, excluding $(d_1,d)=(1,2)$ and $(d_1,d)=(n-3,n-2)$. Gao–Isaev…

For $d_1,d_2\in\mathbb{N}$ with $d_1+d_2\leq n-1$, let $\mathcal{G}(n,d)$ denote the uniform random $d$-regular graph. Let $\mathcal{G}(n,d_1)\oplus\mathcal{G}(n,d_2)$ denote the d…

Let $d_1=d_1(n),d_2=d_2(n)\in\mathbb{N}:=\{0,1,\ldots\}$ satisfy … Let $\mathcal{S}_n(d_1,d_2)$ be the set of pairs of edge-disjoint graphs on $[n]$ whose respective degrees are…

Let $0 \leq \delta \leq c$. Define $\mathbf{Q}=G(n,\kappa_1)$ for the constant kernel $\kappa_1\equiv c$, and let $\mathbf{P}=G(n,\kappa_2)$, where $\kappa_2$ equals $c+\delta$ on…