Can a closed surface of genus $\geq$ 2 be embedded in a product of graphs?

Question

Let $S$ be a closed orientable surface of genus $g \geq 2$. Is there an embedding of $S$ into the product of two graphs $G_1$ and $G_2$? I can't think of such an embedding but I don't know any obstructions for this sort of thing.

Answer 1

A genus $g$ surface can be found in the product $G_{g+1} \times G_{g+1}$ where $G_n$ is the 2 vertex graph with $n$ edges in between them. To do this for the genus two case, if $e_1,e_2,e_3$ are the edges, you can remove from $G_3 \times G_3$ the three faces $e_i \times e_i$. You can check that it is a closed surface since every edge is on two faces, and then you can compute the Euler characteristic to see it is genus 2.

Higher genus is similar. As mentioned before, these will never be objective on the level of fundamental groups.