Almost spanning universality in random graphs

Abstract

A graph $G$ is called universal for a family of graphs $\mathcal{F}$ if it contains every element $F \in \mathcal{F}$ as a subgraph. We prove for $\Delta\ge 3$ and $\varepsilon>0$ that $G(n,p)$ is a.a.s.~universal for the family of all graphs on $(1-\varepsilon)n$ vertices with maximum degree $\Delta$ provided that $p=\omega(n^{-1/(\Delta-1)})$. This improves on previously known results by Conlon, Ferber, Nenadov, and Škorić~[{\em Almost-spanning universality in random graphs}, Random Structures \& Algorithms \textbf{50} (2017), no. 3, 380--393] and is asymptotically optimal for $\Delta=3$.