 
Summary: Acyclic Coloring of Graphs
Noga Alon
Department of Mathematics,Sackler Faculty of Exact Sciences, TelAviv University,
Tel Aviv, Israel and IBM Almaden Research Center, San Jose, CA 95120
Colin McDiarmid
Department of Statistics, University of Oxford, Oxford, England
Bruce Reed
Department of Combinatoricsand Optimization, University of Waterloo, Waterloo,
Canada
ABSTRACT
A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same
color and there is no twocolored cycle in G. The acyclic chromatic number of G , denoted
by A(G),is the least number of colors in an acyclic coloring of G. We show that if G has
maximum degree d, then A(G)= O(d413)as d+m. This settles a problem of Erdos who
conjectured, in 1976, that A(G)= o(d2) as dm. We also show that there are graphs G
with maximum degree d for which A(G)= R(d413/(logd ) l " ) ; and that the edges of any
graph with maximum degree d can be colored by O(d) colors so that no two adjacent edges
have the same color and there is no twocolored cycle. All the proofs rely heavily on
probabilistic arguments.
1. INTRODUCTION
