On Acyclic Colorings of Graphs on Surfaces

A proper k-coloring of a graph is acyclic if every 2-chromatic subgraph is acyclic. Borodin showed that every planar graph has an acyclic 5-coloring. This paper shows that the acyclic chromatic number of the projective plane is at most 7. The acyclic chromatic number of an arbitrary surface with Euler characteristic χ = −γ is at most O(γ). This is nearly tight; for every γ > 0 there are graphs embeddable on surfaces of Euler characteristic−γ whose acyclic chromatic number is at least Ω(γ/(log γ)). Therefore, the conjecture of Borodin that the acyclic chromatic number of any surface but the plane is the same as its chromatic number is false for all surfaces with large γ (and may very well be false for all surfaces).