 
Summary: UNIVERSALITY AND TOLERANCE
(Extended Abstract)
Noga Alon
Michael Capalbo
Yoshiharu Kohayakawa
Vojtech Ršodl§
Andrzej RuciŽnski¶
Endre SzemerŽedi
Abstract
For any positive integers r and n, let H(r, n) denote
the family of graphs on n vertices with maximum degree
r, and let H(r, n, n) denote the family of bipartite graphs
H on 2n vertices with n vertices in each vertex class, and
with maximum degree r. On one hand, we note that any
H(r, n)universal graph must have (n22/r
) edges. On
the other hand, for any n n0(r), we explicitly construct
H(r, n)universal graphs G and on n and 2n vertices,
and with O(n2( 1
r log r )
