Acyclic Coloring of Graphs

A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two-colored 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 d-m. We also show that there are graphs G with maximum degree d for which A ( G ) = R(d413/(log d ) 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 two-colored cycle. All the proofs rely heavily on probabilistic arguments.