Torus Grid Graph
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)
|
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.
 |
graph |
 |
circulant graph  |
 |
generalized quadrangle  |
 |
quartic vertex-transitive graph Qt65 |
 |
tesseract graph  |
Harary et al. (1973) conjectured that the graph crossing number is given by
 |
(2)
|
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)
|
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).
REFERENCES
Adamsson, J. and Richter, R. B. "Arrangements, Circular Arrangements and the Crossing Number of 
." J. Combin. Theory 90, 21-39, 2004.
Harary, F.; Kainen, P. C.; and Schwenk, A. J. "Toroidal Graphs with Arbitrarily High Crossing Numbers." Nanta Math. 6, 58-67, 1973.
Clancy, K.; Haythorpe, M.; and Newcombe, A. §3.1.1 in "A Survey of Graphs with Known or Bounded Crossing Numbers." 15 Feb 2019.
https://arxiv.org/abs/1901.05155.Lawrencenko, S. and Negami, S. "Constructing the Graphs That Triangulate Both the Torus and the Klein Bottle." J. Combin. Theory Ser. B 77, 211-2218, 1999.
Pach, J. and Tóth, G. "Crossing Number of Toroidal Graphs." In International Symposium on Graph Drawing (Ed. P. Healy and N. S. Nikolov). Berlin, Heidelberg: Springer-Verlag: pp. 334-342, 2005.
Riskin, A. "On the Nonembeddability and Crossing Numbers of Some Toroidal Graphs on the Klein Bottle." Disc. Math. 234, 77-88, 2001.
Salazar, G. and Ugalde, E. "An Improved Bound for the Crossing Number of 
: A Self-Contained Proof Using Mostly Combinatorial Arguments." Graphs Combin. 20, 247-253, 2004.
Stewart, I. Fig. 41 in How to Cut a Cake: And Other Mathematical Conundrums. Oxford, England: Oxford University Press, 2006.
