Toroidal Cartesian Products Where One Factor is 3-Connected

In this paper, we show that if $G$ is $3$-connected, then the Cartesian product of graphs $G\mathrm{\square }H$ embeds on the torus if and only if $G$ is outer-cylindrical and $H$ is a path on two vertices, $P_2$. As a by-product of our work, we also show that $K_4\mathrm{\square }{P}_3$ has genus two.