 
Summary: Large nearly regular induced subgraphs
Noga Alon
Michael Krivelevich
Benny Sudakov
Abstract
For a real c 1 and an integer n, let f(n, c) denote the maximum integer f so that every graph
on n vertices contains an induced subgraph on at least f vertices in which the maximum degree
is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a
regular induced subgraph on at least f(n, 1) vertices. The problem of estimating f(n, 1) was posed
long time ago by Erdos, Fajtlowicz and Staton. In this note we obtain the following upper and
lower bounds for the asymptotic behavior of f(n, c):
(i) For fixed c > 2.1, n1O(1/c)
f(n, c) O(cn/ log n).
(ii) For fixed c = 1 + with > 0 sufficiently small, f(n, c) n(2
/ ln(1/))
.
(iii) (ln n) f(n, 1) O(n1/2
ln1/4
n).
An analogous problem for not necessarily induced subgraphs is briefly considered as well.
