A note on acyclic vertex-colorings

We prove that the acyclic chromatic number of a graph with maximum degree $\Delta$ is less than $2.835\Delta^{4/3}+\Delta$. This improves the previous upper bound, which was $50\Delta^{4/3}$. To do so, we draw inspiration from works by Alon, McDiarmid and Reed and by Esperet and Parreau.