A Note on Coloring Digraphs of Large Girth

The digirth of a digraph is the length of a shortest directed cycle. The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the smallest size of a partition of the vertex-set into subsets inducing acyclic subgraphs. A conjecture by Harutyunyan and Mohar states that $\vec{\chi}(D) \le \left\lceil\frac{\Delta}{4}\right\rceil+1$ for every digraph $D$ of digirth at least $3$ and maximum degree $\Delta$. The best known partial result by Golowich shows that $\vec{\chi}(D) \le \frac{2}{5}\Delta+O(1)$. In this short note we prove for every $g \ge 2$ that if $D$ is a digraph of digirth at least $2g-1$ and maximum degree $\Delta$, then $\vec{\chi}(D) \le (\frac{1}{3}+\frac{1}{3g}) \Delta + O_g(1)$. This improves the bound of Golowich for digraphs without directed cycles of length at most $10$.