The honeycomb toroidal graph 
on 
vertices for 
, 
, and 
positive integers satisfying
and 
even is defined as the graph
on vertex set 
for 
and 
. Edges are then
defined as follows, with the indices 
and 
taken modulo 
and 
, respectively.
1. For each 
from 0 to 
,
is adjacent to 
and 
.
2. For each even 
from 0 to 
, there is an edge from 
to 
for all odd 
.
3. For each odd 
from 1 to 
, there is an edge from 
to 
for all even 
.
4. If 
is even, there is an edge
from 
to 
for all odd 
.
5. If 
is odd, there is an edge
from 
to 
for all even 
.
Torus graph embeddings are illustrated above for the first few honeycomb toroidal graphs, where a single graph may have multiple parameters and therefore multiple embeddings on the torus.
Honeycomb toroidal graphs are cubic, except some cases with 
which give cycle graphs 
. They are also vertex-transitive,
and Cayley graphs (Alspach and Dean 2009).
Honeycomb toroidal graphs have also been called generalized honeycomb tori and brick products (Alspach and Dean 2009).
Known positive exact graph crossing numbers and selected candidates are summarized below. Square brackets denote candidates derived from computed upper bounds, not proved exact values. One honeycomb parameterization is shown for each graph.
| crossing number | honeycomb toroidal graph parameters |
| 1 | 
for odd  |
| 2 | (1, 12, 5) |
| 3 | (1, 14, 5), (1, 18, 5) |
| 4 | (1, 16, 5), (1, 16, 7), (1, 20, 5), (1, 24, 5) |
| 5 | (1, 30, 5), [(1, 20, 9)] |
| 6 | (1, 36, 5), [(1, 24, 7)] |
| 7 | (1, 42, 5), [(1, 28, 7)] |
| 8 | [(1, 26, 7)] |
| 9 | [(1, 30, 7)] |
| 10 | (1, 36, 7), (1, 40, 7), (1, 40, 17) |
| 11 | (1, 38, 7), (1, 44, 11), (1, 44, 21) |
| 12 | (1, 40, 9), (1, 40, 11), (1, 42, 7), (1, 44, 7), (1, 48, 7), [(1, 36, 11)] |
| 13 | (1, 46, 7), (1, 46, 9), [(1, 38, 15)] |
| 14 | (1, 44, 9), (1, 50, 7), [(1, 42, 13)] |
| 15 | (1, 50, 9), (1, 50, 19), [(1, 42, 9)] |
| 16 | [(1, 48, 9)] |
The toroidal graph drawing above exhibits a honeycomb toroidal graph as a translation quotient of the infinite hexagonal grid.
Writing its parameters as 
, where 
in the definition above, let 
be the residue of 
modulo 
. For primitive translation vectors 
and 
, a basis of the
honeycomb cell point lattice is
 |
The corresponding period vectors may be taken as 
and 
.
Since 
and each primitive cell contains two vertices, the
quotient has 
vertices.
The choice of period vectors is not unique: exchanging them, reversing their signs, or applying an integer change
of basis with a unimodular
matrix leaves the translation point
lattice unchanged. The Möbius-Kantor
graph, for example, has the three honeycomb presentations 
, 
, and 
. A particular torus
graph embedding additionally specifies a displayed fundamental
region, representatives of the quotient vertices,
and the translation crossed by each wrapping graph
edge. An edge with wrapping vector 
continues into the region translated by 
. Parallelogram,
hexagonal, and brick-wall drawings
can therefore depict the same periodic graph embedding.
The following table summarizes some special cases.
-
,
,
,
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
,
,
,
, 
A
,
A
A
A
A
A
, 
A
A
A
A
B
A
A
A
A
A
A
A
A
B
A
A
A
A
A
B
B
A
A
A
A
B
A
,
,
,
,
,
, 
, 
, 
,
,
for 
odd
, 
, 
for 
even
