Sprinkling for random regular graphs and Kim-Vu's sandwich conjecture

We show that the uniform random $d$-regular graph model on $n$ vertices is, in total variation distance, asymptotically close to the union of $d$ uniformly random edge-disjoint perfect matchings, provided that $n$ is even and $\omega (1)=d\le n^{1\mathrm{/}10}$. This confirms a conjecture by Hollom, Lichev, Mond, Portier, and Wang for this range of parameters. This result has a couple of interesting consequences. First, this shows that random $d$-regular graphs on $n$ vertices exhibit sprinkling properties for some range of parameters, thereby confirming a conjecture by Isaev, McKay, Southwell and Zhukovskii in that range. Second, we give a short alternative proof of the upper side of Kim-Vu’s sandwich conjecture in the sparse regime.