A toroidal graph is a graph with graph genus 1. Equivalently, it is a nonplanar graph that admits a torus graph embedding but no planar graph embedding, and hence has toroidal crossing number 0. Every planar graph also has a torus graph embedding. Some authors therefore use "toroidal graph" inclusively for any graph embeddable on a torus, including planar graphs (Gross and Tucker 1987).
A singlecross graph is necessarily toroidal, since its single crossing can be removed by adding a handle. The converse is false:
for example, the complete graphs 
and 
are toroidal but have graph
crossing numbers 3 and 9, respectively, and so are not singlecross.
A nonplanar graph with a projective plane graph embedding is toroidal iff that embedding
has face-width 2 or 3. More generally, its graph
genus is 
,
where 
is the face-width and 
is the floor function
(Fiedler et al. 1995).
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 vertex degree
at least three that is not embeddable on 
, although 
is embeddable on 
for every edge 
. A minor-order obstruction additionally requires 
to be embeddable on 
for every graph edge 
, where 
is obtained by edge
contraction (Myrvold and Woodcock 2018). The complete list of forbidden
minors for a torus graph embedding 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
minors and 
forbidden minors. In addition, Gagarin et al.
(2009) found four forbidden minors and eleven
forbidden graph expansions for toroidal graphs
possessing no 
graph 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 graph
face count 
as
 |
(1)
|
However, a toroidal graph also satisfies
 |
(2)
|
because the sum of the numbers of edge occurrences around all graph faces is at least 
, while each graph edge contributes
two occurrences and makes the same sum equal to 
(Bartlett 2015). Combining these formulas gives
 |
(3)
|
which must hold for a graph to be toroidal (West 2000, p. 268).
It is also true that for a toroidal graph,
 |
(4)
|
where 
is the minimum vertex degree. Indeed, the
sum of the vertex degrees is 
and is at least 
(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 graph genus of all graphs on 8 and fewer vertices. Define the double-toroidal graphs
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
where 
is obtained from 
by edge deletion of 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.Fiedler,
J. R.; Huneke, J. P.; Richter, R. B.; and Robertson, N. "Computing
the Orientable Genus of Projective Graphs." J. Graph Th. 20, 297-308,
1995. 
's."
Disc. Math. 309, 3625-3631, 2009.Gross, J. L. and
Tucker, T. W.