The torus grid graph is the graph formed from the graph Cartesian product of the cycle graphs and . is isomorphic to .
can be formed starting with an grid graph and connecting corresponding left/right and top/bottom vertex pairs with edges. While such an embedding has overlapping edges in the plane, it can naturally be placed on the surface of a torus with no edge intersections or overlaps. Torus grid graphs are therefore toroidal graphs. The isomorphic torus grid graphs and are illustrated above.
The torus grid graphs are quartic and Hamiltonian and have vertex count
(1)
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Torus grid graphs are circulant graphs iff and are relatively prime, i.e., . In such cases, is isomorphic to . Special cases are summarized in the following table and illustrated above in attractive (but non-toroidal) embddings.
Harary et al. (1973) conjectured that the graph crossing number is given by
(2)
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for all satisfying (Clancy et al. 2019). The conjecture is now known to hold for (Adamsson and Richter 2004 and earlier work cited therein). An asymptotic lower bound of
(3)
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was given by Salazar and Ugalde (2004). Clancy et al. (2019) summarize additional results and details.
Riskin (2001) showed that the Klein bottle crossing numbers of with for , 4, 5, 6 are 1, 2, 4, and 6, respectively.
The torus grid graph is unit-distance since it is isomorphic to the graph Cartesian product , where is the -prism graph (which is itself unit-distance).