  Given a family $\mathcal{H}$ of graphs, a graph $G$ is called
$\mathcal{H}$-universal if $G$ contains every graph of $\mathcal{H}$ as a
subgraph. Following the extensive research on universal graphs of small size
for bounded-degree graphs, Alon asked what is the minimum number of edges that
a graph must have to be universal for the class of all $n$-vertex graphs that
are $D$-degenerate. In this paper, we answer this question up to a factor that
is polylogarithmic in $n.$
