Induced-Universal Graphs for Graphs with Bounded Maximum Degree

For a family F of graphs, a graph U is induced-universal for F if every graph in F is an induced subgraph of U. We give a construction for an induced-universal graph for the family of graphs on n vertices with degree at most r, which has Cn (left perpendicular(r+1)/2right perpendicular) vertices and Dn2(left perpendicular(r+1)/2right perpendicular) edges, where C and D are constants depending only on r. This construction is nearly optimal when r is even in that such an induced-universal graph must have at least cn(r/2) vertices for some c depending only on r. Our construction is explicit in that no probabilistic tools are needed to show that the graph exists or that a given graph is induced-universal. The construction also extends to multigraphs and directed graphs with bounded degree.