Summary: Sparse universal graphs
For every n, we describe an explicit construction of a graph on n vertices with at most O(n2-
edges, for = 0.133 . . ., that contains every graph on n vertices with maximum degree 3 as a
subgraph. It is easy to see that each such graph must have at least (n
3 ) edges. We also show that
the minimum number of edges of a graph that contains every graph with n edges as a subgraph is
). This improves a result of Babai, Chung, Erd¨os, Graham and Spencer.
For a family H of graphs, a graph G is H-universal if it contains every member of H as a (not
necessarily induced) subgraph. The study of universal graphs for various families H is motivated by
problems in VLSI circuit design. See, e.g., ,  and their references.
In this paper we study the minimum possible number of edges in universal graphs for two families