Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U . We give upper and lower bounds for the size of induced universal graphs for the family of graphs with n vertices of maximum degree D. Our new bounds improve several previous results except for the special cases where D is either near-constant or almost n/2. For constant even D Butler [Graphs and Combinatorics 2009] has shown O ( nD/2 ) and recently Alon and Nenadov [SODA 2017] showed the same bound for constant odd D. For constant D Butler also gave a matching lower bound. For generals graphs, which corresponds to D = n, Alon [Geometric and Functional Analysis, to appear] proved the existence of an induced universal graph with (1 + o(1)) · 2(n−1)/2 vertices, leading to a smaller constant than in the previously best known bound of 16 · 2n/2 by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015]. In this paper we give the following lower and upper bound of ( bn/2c bD/2c ) · n−O(1) and ( bn/2c bD/2c ) · 2 (√ D logD·log(n/D) )