Optimal induced universal graphs for bounded-degree graphs

We show that for any constant Delta >= 2, there exists a graph F with O(n(Delta/2)) vertices which contains every n-vertex graph with maximum degree Delta as an induced subgraph. For odd Delta this significantly improves the best-known earlier bound of Esperet et al. and is optimal up to a constant factor, as it is known that any such graph must have at least Omega(n(Delta/2)) vertices. Our proof builds on the approach of Alon and Capalbo (SODA 2008) together with several additional ingredients. The construction of F is explicit and is based on an appropriately defined composition of high-girth expander graphs. The proof also provides an efficient deterministic procedure for finding, for any given input graph H on n vertices with maximum degree at most Delta, an induced subgraph of Gamma isomorphic to H.