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 Ok(n 2 k log 4 k n) edges and exactly n vertices. The number of edges is almost as small as possible, as Ω(n2−2/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.