Every planar graph (i.e., graph with graph genus 0) has an embedding on a torus. In contrast, toroidal graphs are embeddable on the torus, but not in the plane, i.e., they have graph genus 1. Equivalently, a toroidal graph is a nonplanar graph with toroidal crossing number 0, i.e., a nonplanar graph that can be embedded on the surface of a torus with no edge crossings.
A graph with graph genus 2 is called double-toroidal (West 2000, p. 266).
Examples of toroidal graphs include the complete graphs 
and 
and complete bipartite
graph 
(West 2000, p. 267). Families of toroidal graphs include the 
-crossed prism graphs
for 
and cycle complements 
for 
(E. Weisstein, May 9, 2023).
When it exists, the dual graph of a toroidal graph (on the torus) is also toroidal. Examples of such pairs
include the complete graph 
and Heawood graph, as well
as the Dyck graph and Shrikhande
graph.
A (topological) obstruction for a surface 
is a graph 
with minimum degree at least three that is not embeddable
on 
but for every edge 
of 
,
(
with edge 
deleted) is embeddable on 
. A minor-order obstruction 
has the additional property that for every edge 
of 
, 
(
with edge 
contracted) is embeddable
on 
(Myrvold and Woodcock 2018). The complete list of forbidden
minors for toroidal embedding of a graph is not known, but thousands of obstructions
are known (Neufeld and Myrvold 1997, Chambers 2002, Woodcock 2007; cf. Mohar and
Skoda 2020). Chambers (2002) found 
topological obstructions and 
minor order obstructions that include those on up to eleven
vertices, the 3-regular ones on up to 24 vertices, the disconnected ones and those
with a cut-vertex. Myrvold and Woodcock (2018) found 
forbidden topological minorsTopological Minor and 
forbidden
minors. In addition, Gagarin et al. (2009) found four forbidden minors
and eleven forbidden graph expansions for toroidal
graphs possessing no 
minor and proved that the lists are sufficient.
The following table summarizes forbidden minor obstructions of several types, including with vertex
connectivity 
(Olds 2019). Here, 
denotes vertex contraction and 
denotes a graph join.
| property | count | forbdden minors | reference |
 -free | 4 |  ,
 ,  ,  | Gagarin et al. (2009) |
 | 3 |  ,  ,  | Olds (2019) |
 | 3 |  ,  ,  | Olds (2019) |
 | 68 | Mohar and Skoda (2014) |
The numbers of toroidal graphs on 
, 2, ... nodes are 0, 0, 0, 0, 1, 14, 222, 5365, ... (OEIS
A319114), and the corresponding numbers of
connected toroidal graphs are 0, 0, 0, 0, 1, 13, 207, 5128, ... (OEIS A319115;
E. Weisstein, Sep. 10, 2018).
A toroidal graph has graph genus 
, so the Poincaré
formula gives the relationship between vertex count 
, edge count 
, and face count 
as
 |
(1)
|
However, a toroidal graph also satisfies
 |
(2)
|
as can be derived by taking the sum over every face of the number of edges in each face. Since there are at least 3 edges in a face, this sum is bounded below by 
. On the other hand, because each edge
bounds exactly two faces, its is also exactly 
(Bartlett 2015). Combining these two formulas gives the inequality
 |
(3)
|
which must hold for a graph to be toroidal (West 2000, p. 268).
If is also true that for a toroidal graph,
 |
(4)
|
where 
is the minimum vertex degree. This can be
derived similarly as above by summing the degree of each vertex over all of the vertices.
This sum must be greater than 
by the definition of minimum
vertex degree, but it is also equal to 
(Bartlett 2015).
Plugging the above two inequalities into the Poincaré formula then gives
 |
(5)
|
so 
for any toroidal graph (Bartlett 2015).
Duke and Haggard (1972; Harary et al. 1973) gave a criterion for the genus of all graphs on 8 and fewer vertices. Define the double-toroidal graphs
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
where 
denotes 
minus the edges of 
.
Then a subgraph 
of 
is toroidal if it contains a Kuratowski
graph (i.e., is nonplanar) but does not contain
any 
for 
.
."
Israel J. Math. 11, 452-455, 1972.Gagarin, A.; Myrvold,
W.; and Chambers, J. "The Obstructions for Toroidal Graphs with no 
's." Disc. Math. 309, 3625-3631, 2009.Harary,
F.; Kainen, P. C.; Schwenk, A. J.; and White, A. T. "A Maximal
Toroidal Graph Which Is Not a Triangulation." Math. Scand. 33,
108-112, 1973.Mohar, B. and Škoda, P. "Excluded Minors for
the Klein Bottle I. Low Connectivity Case." 1 Feb 2020.