Almost-spanning universality in random graphs

Abstract

A graph $G$ is said to be $\mathcal H(n,Δ)$-universal if it contains every graph on $n$ vertices with maximum degree at most $Δ$. It is known that for any $\varepsilon > 0$ and any natural number $Δ$ there exists $c > 0$ such that the random graph $G(n,p)$ is asymptotically almost surely $\mathcal H((1-\varepsilon)n,Δ)$-universal for $p \geq c (\log n/n)^{1/Δ}$. Bypassing this natural boundary, we show that for $Δ\geq 3$ the same conclusion holds when $p = ω\left(n^{-\frac{1}{Δ-1}}\log^5 n\right)$.