For 3-connected G, G □ H embeds on the torus iff G is outer-cylindrical and H = P₂; additionally K₄ □ P₃ has genus two.
Paul Bonnington, Nathaniel Dean, Nora Hartsfield, and Katherine Scott,Obstruc- tion sets for outer-cylindrical graphs, J
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Toroidal Cartesian Products Where One Factor is 3-Connected
For 3-connected G, G □ H embeds on the torus iff G is outer-cylindrical and H = P₂; additionally K₄ □ P₃ has genus two.