Sandwiching dense random regular graphs between binomial random graphs

Abstract Kim and Vu made the following conjecture ( Advances in Mathematics , 2004): if $$d\gg \log n$$ d≫logn then random d -regular graph $${\mathscr {G}}(n,d)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">G(n,d) can asymptotically almost surely be “sandwiched” between {G}}(n,p_1)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">G(n,p1) {G}}(n,p_2)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">G(n,p2) where $$p_1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">p1 $$p_2$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">p2 are both $$(1+o(1))d/n$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">(1+o(1))d/n . They proved this for $$\log n\ll d\geqslant n^{1/3-o(1)}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">logn≪d⩾n1/3-o(1) with a defect sandwiching: contains perfectly, but is not completely contained The embedding {G}}(n,p_1) \subseteq {\mathscr xmlns:mml="http://www.w3.org/1998/Math/MathML">G(n,p1)⊆G(n,d) was improved by Dudek, Frieze, Ruciński Šileikis to $$d=o(n)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">d=o(n) In paper, we prove Kim–Vu’s sandwich conjecture, perfect containment on sides, all $$\min \{d, n-d\}\gg n/\sqrt{\log n}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">min{d,n-d}≫n/logn theorem allows translation of many results from {G}}(n,p)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">G(n,p) such as Hamiltonicity, chromatic number, diameter, etc. It also threshold functions phase transitions bond percolation addition sandwiching regular graphs, our cover graphs whose degrees equal. proofs rely estimates probability small subgraph appearances factor pseudorandom graph, which independent interest.