Bounding the Cop Number of a Graph by Its Genus

It is known that the cop number $c(G)$ of a connected graph $G$ can be bounded as a function of the genus of the graph $g(G)$. The best known bound, that $c(G) \leq \left\lfloor \frac{3 g(G)}{2}\right\rfloor + 3$, was given by Schröder, who conjectured that in fact $c(G) \leq g(G) + 3$. We give the first improvement to Schröder's bound, showing that $c(G) \leq \frac{4g(G)}{3} + \frac{10}{3}$.