Embedding large graphs into a random graph

Abstract

In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $Δ\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree $Δ$. We show that a random graph $G_{n,p}$ with high probability contains a copy of $H$, provided that $p\gg (n^{-1}\log^{1/Δ}n)^{2/(Δ+1)}$. Our assumption on $p$ is optimal up to the $polylog$ factor. We note that this $polylog$ term matches the conjectured threshold for the spanning case.

Incorporated referee comments. To appear in Bulletin of the London Mathematical Society