Abstract: An acyclic coloring of a graph G is a coloring of its
vertices such that:(i) no two neighbors in G are assigned the same
color and (ii) no bicolored cycle can exist in G. The acyclic chromatic
number of G is the least number of colors necessary to acyclically
color G. Recently it has been proved that any graph of maximum
degree 5 has an acyclic chromatic number at most 8. In this paper
we present another proof for this result.