  Kim and Vu made the following conjecture (\textit{Advances in Mathematics},
2004): if $d\gg \log n$, then the random $d$-regular graph $G(n,d)$ can be
``sandwiched'' between $G(n,p_*)$ and $G(n,p^*)$ where $p_*$ and $p^*$ are both
asymptotically equal to $d/n$.
  This famous conjecture was previously proved for all $d\gg (n\log n)^{3/4}$.
  In this paper, we confirm the conjecture when $d \ge \log^4 n$. We also
extend this result to near-regular degree sequences.
