The honeycomb toroidal graph on vertices for , , and positive integers satisfying and is even is defined as the graph on vertex set for and . Edges are then defined as follows, where
and
adjacency are 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 .
Alspach, B. and Dean, M. "Honeycomb Toroidal Graphs Are Cayley Graphs." Inf. Proc. Lett.109, 705-708, 2009.Altshuler,
A. "Hamiltonian Circuits in Some Maps on the Torus." Disc. Math.1,
299-314, 1972.Marušič, D. and Pisanski, T. "Symmetries
of Hexagonal Molecular Graphs on the Torus." Croatica Chem., 73,
969-981, 2000.
on 
vertices for 
, 
, and 
positive integers satisfying 
and 
is even is defined as the graph on vertex set 
for 
and 
. Edges are then defined as follows, where 
and 
adjacency are taken modulo 
and 
, respectively.
from 0 to 
,
is adjacent to 
and 
.
from 0 to 
, there is an edge from 
to 
for all odd 
.
from 1 to 
, there is an edge from 
to 
for all even 
.
is even, there is an edge from 
to 
for all odd 
.
is odd, there is an edge from 
to 
for all even 
.
which give cycle graphs 
. They are also vertex-transitive,
and a Cayley graphs (Alspach and Dean 2009).
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for 
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