In this paper we consider the problem of embedding almost-spanning, bounded degree graphs in a random graph. In particular, let $\Delta\geq 5$, $\varepsilon > 0$ and let $H$ be a graph on $(1-\varepsilon)n$ vertices and with maximum degree $\Delta$. We show that a random graph $G_{ n,p}$ with high probability contains a copy of $H$, provided that $p\gg (n^{ -1}\log^{ 1/\Delta}n)^{ 2/(\Delta+1)}$. 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.
Categories: math.CO