 
Summary: Sparse Universal Graphs for BoundedDegree Graphs
Noga Alon
Michael Capalbo
March 12, 2006
Abstract
Let H be a family of graphs. A graph T is Huniversal 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 Ok(n2 2
k log
4
k n)
edges and exactly n vertices. The number of edges is almost as small as possible, as (n22/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.
1 Introduction
For a family H of graphs, a graph T is Huniversal if, for each H H, the graph T contains a subgraph
isomorphic to H. Thus, for example, the complete graph Kn is Hnuniversal, where Hn is the family
of all graphs on at most n vertices. The construction of sparse universal graphs for various families
