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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 bothasymptotically 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 alsoextend this result to near-regular degree sequences.

Kim--Vu's sandwich conjecture is true for $d \geq \log^4 n$