Coloring graphs with sparse neighborhoods Michael Krivelevich 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 Collections: Mathematics