# pursuit-evasion based on Schroeder's upper bound for graphs of genus $g$

I am following Schroeder's work on pursuit-evasion games on graphs (often called "cops and robbers"). In his 2001 publication ("The copnumber of a graph is bounded by $$\lfloor 3/2 {\ \rm genus}(G)+3\rfloor$$". In: Categorical perspectives (Kent, OH, 1998). Trends in Mathematics, pp. 243-263. Birkhäuser, Boston 2001) he derived an upper bound for the cop number $$c(G)$$ that depends on the genus $$g$$ of the surface on which the graph $$G$$ can be embedded: $$c(G)\leq \lfloor 3g/2 +3\rfloor$$. My most recent reference for this result is "Topological directions in Cops and Robbers" from 2018, Anthony Bonato and Bojan Mohar, arXiv:1709.09050v2 .

This gives $$c(G)\leq 4$$ if $$G$$ can be embedded on a torus. Now, I have worked extensively to come up with an example of a graph $$G$$ that actually hits this bound, i.e. I have searched for $$G$$ with $$c(G)=4$$, but with no success. So I am starting to see strong evidence for the conjecture $$c(G)\leq 3$$ if $$G$$ can be embedded on a torus. Question: Is someone aware of a more recent reference for this conjecture? It appears lower than any other bound I have seen in the literature so far (N.B. I would also be interested in references beyond torus embeddings)

I recently proved that it is true, i.e. the cop-number of toroidal graphs is at most 3, see this ArXiv preprint. See also this preprint, where a general bound $$c(G) \leq \frac{4g}{3} + \frac {10}3$$ is proved.
• I am very impressed by your result, just took a first look. Your second reference with the general bound is amazing: extremely close to the original Schroeder $g+3$ conjecture!! Jul 31, 2020 at 6:59