 
Summary: Coloring graphs with sparse neighborhoods
Noga Alon
Michael Krivelevich
Benny Sudakov
Abstract
It is shown that the chromatic number of any graph with maximum degree d in which the
number of edges in the induced subgraph on the set of all neighbors of any vertex does not exceed
d2
/f is at most O(d/ log f). This is tight (up to a constant factor) for all admissible values of d
and f.
1 Introduction
The chromatic number (G) of a graph G is the minimum number of colors required to color all
its vertices so that adjacent vertices get distinct colors. It is easy and well known that if d is the
maximum degree of G then (G) d + 1. This upper bound can be improved if the graph has
sparse neighborhoods, namely, if no subgraph on the set of all neighbors of a vertex spans too many
edges. The first instance of a result of this type is Brooks' Theorem [5], which asserts that if no
neighborhood contains d
2 edges (that is, if G contains no copy of the complete graph on d + 1
vertices), then (G) d. Molloy and Reed [13] proved that for every > 0 there is some > 0 such
that if no neighborhood contains more than (1  ) d
