Toroidal Cartesian Products Where One Factor is 3-Connected

Abstract In this paper, we show that if G is 3-connected, then the Cartesian product of graphs "Equation missing" embeds on the torus if and only if G is outer-cylindrical and H is a path on two vertices, $$P_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . As a by-product of our work, we also show that "Equation missing" has genus two.