Sparse universal graphs for bounded‐degree graphs

Let ℋ be a family of graphs. A graph T is ℋ‐universal if it contains a copy of each H ∈ℋ as a subgraph. Let ℋ( k , n ) denote the family of graphs on n vertices with maximum degree at most k . For all positive integers k and n , we construct an ℋ( k , n )‐universal graph T with $O_k(n^{2-{2 \over k}} \log ^{4 \over k} n)$ edges and exactly n vertices. The number of edges is almost as small as possible, as Ω( n 2‐2/ k ) is a lower bound for the number of edges in any such graph. The construction of T is explicit, whereas the proof of universality is probabilistic and is based on a novel graph decomposition result and on the properties of random walks on expanders. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007