 
Summary: Optimal Universal Graphs with Deterministic Embedding
Noga Alon
Michael Capalbo
September 9, 2007
Abstract
Let H be a finite family of graphs. A graph G 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 admissible k and n, we construct an H(k, n)universal graph G with at most ckn2 2
k
edges, where ck is a constant depending only on k. This is optimal, up to the constant factor ck,
as it is known that ckn22/k
is a lower bound for the number of edges in any such graph. The
construction of G is explicit, and there is an efficient deterministic algorithm for finding a copy of
any given H H(k, n) in G.
1 Introduction
For a family H of graphs, a graph G is Huniversal if, for each H H, the graph G contains a subgraph
isomorphic to H. The construction of sparse universal graphs for various families arises in the study of
VLSI circuit design. See, for example, [9] and [16], for applications motivating the study of universal
graphs with a small number of edges for various families of graphs. There is an extensive literature
on universal graphs. In particular, universal graphs for forests have been studied in [8], [12], [13], [14],
