Let $G_{n,d}$ be $d$-regular random graph. We know that for $d \geq 3$, $G \in G_{n,d}$ a.a.s. has a $1$-factorisation when $n$ is even.

So, the resulting graph that obtained from randomly choosing $d$ disjoint perfect matchings is contiguous with $G_{n,d}$.

If we randomly choosing $d$ disjoint matchings of size $(\frac{1}{2}-o(1))n$ , Is there a characterization of the resulting graph distribution?

In particularly, for fixed vertex $v$, we have $Pr(deg(v)=0)\leq o(1)$. Then a.a.s. $deg(v)=d$. And for a vertex set that has constant size, we have all vertex in this set have degree $d$. Furthermore, does resulting graph a.a.s. have a large induced subgraph with large minimum degree?