Gao–Isaev–McKay monotone coupling conjecture for random regular graphs

Let $n$ be a positive integer, and let $G_1\sim G(n,d_1)$ and $G_2\sim G(n,d_2)$ denote random regular graphs. Assume that $d_1\le d_2$, $d_1,d_2\in[1,n-1]$, $(d_1,d_2)\neq(1,2)$, $(d_1,d_2)\neq(n-3,n-2)$, and both $d_1n$ and $d_2n$ are even.

Gao–Isaev–McKay conjecture. There exists a coupling such that

$$\mathbb{P}(G_1\subseteq G_2)=1-o(1).$$

This conjecture asks for monotonicity under graph inclusion as the degree increases. The source records several proved regimes, including slowly growing degrees covered by the paper’s theorem, while the full range remains open.