On Oriented Colourings of Graphs on Surfaces

For an oriented graph $G$, the least number of colours required to oriented colour $G$ is called the oriented chromatic number of $G$ and denoted $\chi_o(G)$.For a non-negative integer $g$ let $\chi_o(g)$ be the least integer such that $\chi_o(G) \leq \chi_o(g)$ for every oriented graph $G$ with Euler genus at most $g$. We will prove that $\chi_o(g)$ is nearly linear in the sense that $\Omega(\frac{g}{\log(g)}) \leq \chi_o(g) \leq O(g \log(g))$. This resolves a question of the author, Bradshaw, and Xu, by improving their bounds of the form $\Omega((\frac{g^2}{\log(g)})^{1/3}) \leq \chi_o(g)$ and $\chi_o(g) \leq O(g^{6400})$.