We show that for any constant Δ ≥ 2, there exists a graph Γ with  O ( n  Δ/2 ) vertices which contains every  n -vertex graph with maximum degree Δ as an induced subgraph. For odd Δ 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  Δ/2 ) 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 Δ, an induced subgraph of Γ isomorphic to  H .
['Noga Alon', 'Rajko Nenadov']
Optimal induced universal graphs for bounded-degree graphs