  A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a
labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever
$\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz
verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every
positive $r,\Delta,\gamma$, there exists $\beta$ such that if $H$ is an
$n$-vertex $r$-chromatic graph with maximum degree at most $\Delta$ which has
bandwidth at most $\beta n$, then any graph $G$ on $n$ vertices with minimum
degree at least $(1 - 1/r + \gamma)n$ contains a copy of $H$ for large enough
$n$. In this paper, we extend this theorem to dense random graphs. For
bipartite $H$, this answers an open question of B\"{o}ttcher, Kohayakawa, and
Taraz. It appears that for non-bipartite $H$ the direct extension is not
possible, and one needs in addition that some vertices of $H$ have independent
neighborhoods. We also obtain an asymptotically tight bound for the maximum
number of vertex disjoint copies of a fixed $r$-chromatic graph $H_0$ which one
can find in a spanning subgraph of $G(n,p)$ with minimum degree $(1-1/r +
\gamma)np$.
