The concept of acyclic coloring was introduced by Grünbaum  and is a generalization of point arboricity.
A proper k-coloring of the vertices of a graph Gis said to be acyclic if G contains no two-colored cycle. The acyclic chromatic number of a graph G, denoted by a(G), is the minimum value of k for which G has an acyclic k-coloring. Let a(n) denote the maximum value of the acyclic chromatic number among all graphs of genus n. In , Grünbaum conjectured that a(0) = 5 and proved that a(0)≤9. The conjecture was proved by Borodin  after the upper bound was improved three times in ,  and . In , we proved that a(1)≤a(0) + 3. The purpose of this paper is to prove the following
Theorem. Any graph of genus n>0 can be acyclically colored with 4n + 4 colors.
It is not known for any n>0 whether a(n)>H(n), the Heawood number .