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

Question

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)

Answer 1

What you conjecture has been conjectured (more or less explicitly) a few times before. In the paper by Bonato and Mohar that you reference, it is dubbed the Andreae-Schroeder conjecture.

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.