Sparse universal graphs for bounded-degree graphs

Let ℋ be a family of graphs. A graph T 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 positive integers k and n, we construct an ℋ(k, n)-universal graph T with O_k(n^2-2/k log^4/k n) edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n ^2-2/k) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders.