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

(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.

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