Optimal induced universal graphs for bounded-degree graphs

We show that for any constant ∆ ≥ 2, there exists a graph Γ with O(n) vertices which contains every n-vertex graph with maximum degree ∆ as an induced subgraph. For odd ∆ 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 Ω(n) vertices. Our proof builds on the approach of Alon and Capalbo (SODA 2008) together with several additional ingredients. The construction of Γ 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 ∆, an induced subgraph of Γ isomorphic to H.