The toroidal crossing number 
of a graph 
is the minimum number of crossings with which 
can be drawn on a torus.
A planar graph has toroidal crossing number 0, and a nonplanar graph with toroidal crossing number 0 is called a toroidal graph. A nonplanar graph with toroidal crossing number 0 has graph genus 1 since it can be embedded on a torus (but not in the plane) with no crossings.
A graph having graph crossing number or rectilinear crossing number less than
2 has toroidal crossing number 0. More generally, a graph that becomes planar
after the removal of a single edge (in other words, a graph 
with graph skewness 
)
also has toroidal crossing number 0. However, there exist graphs with 
all of whose edge-removed subgraphs are nonplanar,
so this condition is sufficient but not necessary.
If a graph 
on 
edges has toroidal crossing number 
, then 
(Pach and Tóth 2005), where 
denotes the binomial
coefficient. Furthermore, if 
is a graph on 
vertices with maximum
vertex degree 
which has toroidal crossing number 
, then
 |
(1)
|
where 
is a positive constant (Pach and Tóth 2005).
The toroidal crossing numbers for a complete graph 
for 
,
2, ... are 0, 0, 0, 0, 0, 0, 0, 4, 9, 23, 42, 70, 105, 154, 226, 326, ... (OEIS A014543).
The crossing number of 
on the torus is given by
 |
(2)
|
(Guy and Jenkyns 1969, Ho 2005). The first values for 
, 2, ... are therefore 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6,
8, 10, 12, 14, 16, ... (OEIS A008724).
The crossing number of 
on the torus is given by
![nu_t(K_(4,n))=1/2|_n/4_|[n-2(1+|_n/4_|)]](/images/equations/ToroidalCrossingNumber/NumberedEquation3.svg) |
(3)
|
(Ho 2009). The first values for 
, 2, ... are therefore 0, 0, 0, 0, 2, 4, 6, 8, 12, 16, 20,
24, 30, 36, ... (OEIS A182568). Interestingly,
the same result holds for 
, 
, 
, and 
.
The toroidal crossing numbers for a complete bipartite graph 
are summarized in the following table.
 | 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | |
| 3 | 0 | 0 | 0 | 0 | ||
| 4 | 0 | 2 | 4 | |||
| 5 | 5 | 8 | ||||
| 6 | 12 |
." J. Combin. Th. 6, 235-250, 1969.Guy,
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Graph." J. Combin. Th. 4, 376-390, 1968.Harary, F.
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on the Real Projective Plane." Disc. Math. 304, 23-33, 2005.Ho,
P. T. "The Toroidal Crossing Number of 
." Disc. Math. 309, 3238-3248, 2009.Pach,
J. and Tóth, G. "Thirteen Problems on Crossing Numbers." Geocombin. 9,
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