An Improved Bound of Acyclic Vertex-Coloring

The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. We show that for all $\alpha>2^{-1/3}$ there exists an integer $\Delta_{\alpha}$ such that if the maximum degree $\Delta$ of a graph is at least $\Delta_{\alpha}$, then the acyclic chromatic number of the graph is at most $\lceil\alpha {\Delta}^{4/3} \rceil +\Delta+ 1$. The previous best bound, due to Gon\c{c}alves et al (2020), was $(3/2) \Delta^{4/3} + O(\Delta)$.Comment: Lemma 5 is wron