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Let H be a finite family of graphs. A graph G is H-universal if it contains a copy of each H ∈ H as a subgraph. Let H(k,n) denote the family of graphs on n vertices with maximum degree at most k. For all admissible k and n, we construct an H(k, n)-universal graph G with at most ckn2-2/k edges, where ck is a constant depending only on k. This is optimal, up to the constant factor ck, as it is known that c'kn2-2/k is a lower bound for the number of edges in any such graph. The construction of G is explicit, and there is an efficient deterministic algorithm for finding a copy of any given H ∈ H(k,n) in G.