Summary: Universality, tolerance, chaos and order
Dedicated to Endre Szemer´edi, for his 70th birthday
January 25, 2010
What is the minimum possible number of edges in a graph that contains a copy of every graph
on n vertices with maximum degree a most k ? This question, as well as several related variants,
received a considerable amount of attention during the last decade. In this short survey we describe
the known results focusing on the main ideas in the proofs, discuss the remaining open problems,
and mention a recent application in the investigation of the complexity of subgraph containment
For a family H of graphs, a graph G is H-universal if it contains a copy of any H H. The
construction of sparse universal graphs for various families arises in the study of VLSI circuit design.
See, for example,  and  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 , , , , and universal graphs
for planar graphs and other related families have been investigated in , , , , , .
Universal graphs for general bounded-degree graphs have also been considered extensively. For
positive integers k > 2 and n, let H(k, n) denote the family of all graphs on n vertices with maximum