Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 6 (2013) 69–81
Complex parameterization of
triangulations on oriented maps
Mathieu Dutour Sikirić ∗
Rudjer Bosković Institute, Bijenička 54, 10000 Zagreb, Croatia
Received 12 October 2011, accepted 9 April 2012, published online 1 June 2012
Abstract
We consider here triangulations of oriented maps having a specified set S of vertices of
degree different from 6 and some other vertices of degree 6. Such map can be described by
specifying the relative positions between elements of S using Eisenstein integers. We first
consider the case of 1 parameter, which corresponds to the Goldberg-Coxeter construc-
tion. Then we develop the general theory, the special case of positive curvature studied by
Thurston and finally extend the theory to quadrangulations and some other cases. In the
last section we expose application of parameterizations to the study of zigzags.
Keywords: Maps, graphs, Groups, parameterizations.
Math. Subj. Class.: 05C10, 57M20
1 Triangulations of oriented maps
By a ({v1, . . . , vm}, k)-map we denote a map on an oriented surface with faces of size k, vi
vertices of degree i and “map” being “sphere” (genus g = 0), “torus” (g = 1) or “oriented
map of genus g”. For example, a ({v5 = 2, v7 = 2, v6}, 3)-torus denotes a triangulation
with 2 vertices of degree 5, 7 and an unspecified number of vertices of degree 6. We will be
mostly concerned with the description of (v, 3)-maps of genus g. Euler relation for them
reads as ∑
j≥3
vj(6− j) = 12(1− g). (1.1)
The dual of ({v5 = 12, v6}, 3)-spheres are 3-regular plane graphs, whose faces are 5- or
6-gonal. Such graphs, named fullerenes, occur in chemistry [13] following the discovery
of Buckminsterfullerene (also called truncated icosahedron, soccer ball) in 1985 [22].
∗The author has been supported by the Croatian Ministry of Science, Education and Sport under contract
098-0982705-2707.
E-mail address: mdsikir@irb.hr (Mathieu Dutour Sikirić)
Copyright c© 2013 DMFA Slovenije
70 Ars Math. Contemp. 6 (2013) 69–81
Formula (1.1) can be interpreted as a Gauss-Bonnet formula and 6− j as the curvature
of a vertex of degree j. A triangulation is said to be of positive curvature if all vertices have
non-negative curvature. This implies that the possible vertex degrees belong to {3, 4, 5, 6}
and the v-vector satisfies 3v3 + 2v4 + v5 = 12 with v6 unspecified. All 19 possibilities for
(v3, v4, v5) are given below:
(0, 0, 12) (0, 1, 10) (0, 2, 8) (0, 3, 6) (0, 4, 4) (0, 5, 2) (0, 6, 0)
(1, 0, 9) (1, 1, 7) (1, 2, 5) (1, 3, 3) (1, 4, 1) (2, 0, 6) (2, 1, 4)
(2, 2, 2) (2, 3, 0) (3, 0, 3) (3, 1, 1) (4, 0, 0)
For the enumeration of 3-regular plane graphs with a specified face vector (pi), i.e. number
pi of faces of size i the program CPF by T. Harmuth [2] is very efficient. Another little
known program by the same author is CGF [19], which can enumerate 3-regular oriented
maps of specified genus and face vector. The corresponding program for 4-regular plane
graphs is ENU by O. Heidemeier [2]. All above mentioned programs are available from [1]
and give by duality triangulations and quadrangulations.
The symmetry groups of fullerenes and other plane graphs of positive curvature were
determined in [13, 5, 12, 7, 8, 6]. For a given group G of symmetry of a map, denote by
Rot(G) the subgroup of index 1 or 2 of G formed by the orientation preserving transfor-
mation. The class of a group G is the set of groups G′ having Rot(G′) = G. In Table 1
we give the possible groups of (v, 3)-spheres of positive curvature by their class Rot(G),
where we used Schoenflies nomenclature for point groups. For any class the number of
vertices of positive curvature is finite and the number of vertices of degree 6 is unspec-
ified. Since there is essentially only one 6-regular plane triangulation, one sees that the
positions of the vertices of positive curvature allow to define the map. We want to encode
the positions by complex Eisenstein numbers z ∈ Z[ω] with ω = eiπ/3.
In Section 2 we describe the simple cases of 1 or 2 parameters. The case of 1 parameter
corresponds to the Goldberg-Coxeter construction [9]. In Section 3 we first explain the
general theory of complex parameterization of (v, 3)-maps on oriented surfaces. Then
we explain Thurston’s theory [28] which gives stronger results for the case of spheres of
positive curvature. Finally we explain the extension to (v, 4)-maps, self-dual spheres and
(v, 6)-spheres.
Applications to zigzags are considered in Section 4. A very basic application of param-
eterization is for generating maps efficiently provided that the number of parameters is not
too high. Another application considered in [11] is for eigenvalue estimation where it was
proved that for any interval [a, b] ⊂ [−3, 3] there is a finite number of graphs of positive
curvature having no eigenvalue in I .
We choose to emphasize Eisenstein parameter description but it is of course possible to
consider descriptions by integral parameters. This is done in [16, 15] for fullerenes and this
allows to write parameterizations for each group, not just rotation subgroup. Another real
parameter descriptions, by so called dihedral angles, is developed in [26], but it is more
suited for describing manifolds than graphs.
2 One and two parameter constructions
2.1 The Goldberg-Coxeter construction
The Goldberg-Coxeter construction takes a (v, 3)-mapM, two integers k, l ≥ 0 and returns
another ({v, v6}, 3)-map GCk,l(M) of the same genus by adding triangles and vertices of
M. Dutour Sikirić: Complex parameterization of triangulations on oriented maps 71
({v3 = 4, v6}, 3)-spheres
class all group # param
D2 D2, D2d, D2h 2
T T , Td 1
({v4 = 6, v6}, 3)-spheres
class all group # param
C1 C1, Cs, Ci 4
C2 C2, C2v , C2h 3
D2 D2, D2d, D2h 2
D3 D3, D3d, D3h 2
D6 D6, D6h 1
O O, Oh 1
({v5 = 12, v6}, 3)-spheres
class all group # param
C1 C1, Cs, Ci 10
C2 C2, C2h, C2v 6
C3 C3, C3h, C3v 4
D2 D2, D2h, D2d 4
D3 D3, D3h, D3d 3
D5 D5, D5h, D5d 2
D6 D6, D6h, D6d 2
T T , Th, Td 2
I I , Ih 1
Table 1: The classes of symmetry groups of ({va, v6}, 3)-sphere.
k=2
l=1
Figure 1: The construction of GC2,1(Octahedron).
degree 6. It works by breaking the triangles of M into smaller triangles and gluing the
pieces together in order to get another triangulation. See an example on Figure 1 and [9]
for more details.
IfM has nT triangles then GCk,l(M) has nT (k2 +kl+ l2) triangles. Since k2 +kl+
l2 = |k + lω|2 it makes sense to associate the Eisenstein integer z = k + lω to the pair
k, l. The parameter symmetry z 7→ zωr does not change the resulting map. All the cases
of 1 parameter in Table 1 are described by the Goldberg-Coxeter construction of one plane
graph. For example, if a ({v3 = 4, v6}, 3)-, ({v4 = 6, v6}, 3)-, ({v5 = 12, v6}, 3)-sphere
is of symmetry (I, Ih), (O,Oh) or (T, Td) then it is of the form GCk,l(Icosahedron),
GCk,l(Octahedron) or GCk,l(Tetrahedron) [14, 9]. Additionally, ({v4 = 6, v6}, 3)-
spheres of symmetry (D6, D6h) are obtained as GCk,l(Prism∗6).
2.2 One case of 2 parameters: ({v5 = 12, v6}, 3)-spheres of symmetry D5
The 5-fold axis of such a sphere has to pass through a vertex of degree 5. There are 5
vertices of degree 5 around it; so, by 5-fold symmetry, 1 complex parameter is needed to
72 Ars Math. Contemp. 6 (2013) 69–81
z1
2z
z1
z’2
Op
1
z1
z2
z’1
z’2
Op
2
Figure 2: The parameterization of fullerenes of symmetry D5, D5d or D5h in term of
(z1, z2) ∈ Z[ω]2 and two parameter operations.
describe them. Around those 5 vertices, there are 5 more vertices, so one more parameter
is needed and then the last vertex is uniquely defined.
If one applies the following operations to parameters
• Operation 1: (z1, z2) 7→ (z1, z1 + z2)
• Operation 2: (z1, z2) 7→ (z1 + ω2z2, z1 − z2)
• Operation 3: (z1, z2) 7→ (z1, z2)ωr
then one obtains the same sphere as a result. The group generated by those operations
is named monodromy group. The number of triangles of S is expressed as q(z1, z2) =
10{z1z1 + z1z2−z1z2ω−ω }. So, for a given pair (z1, z2) a sphere may not exist, for example, if
q(z1, z2) < 0.
2.3 Other two parameters descriptions
Of course what has been done for fullerene of class D5 applies just as well for fullerenes
of class D6 and similar simple description are possible for the remaining 2 parameter cases
of Table 1 (See Figure 3 for two such cases).
For the ({v3 = 4, v6}, 3)-spheres a very explicit two parameter description is given in
Figure 4. Geometrically this corresponds to the fact that any ({v3 = 4, v6}, 3)-sphere is
obtained as the quotient of a ({v6}, 3)-torus by a group of order 2 leaving invariant exactly
4 vertices. Clearly, the monodromy group is PSL(2,Z) and the number of triangles is ex-
pressed as 4ω−ω (z1z2−z1z2). This description was used in [21] to compute the eigenvalues
of dual ({v3 = 4, v6}, 3)-spheres.
M. Dutour Sikirić: Complex parameterization of triangulations on oriented maps 73
120
z
1
z2
({v4 = 6, v6}, 3)-spheres of symmetry D2,
D2h, D2d
z
1
z
2
({v5 = 12, v6}, 3)-spheres of symmetry T ,
Th and Td
Figure 3: Two parameters description of two classes of spheres with positive curvature
represented on the plane.
4 triangles in Z[ω] The triangulation
Figure 4: Description of ({v3 = 4, v6}, 3)-spheres.
74 Ars Math. Contemp. 6 (2013) 69–81
3 Parameterization of maps on oriented surfaces
3.1 The general case
The general Eberhard problem is for a given g ≥ 0 and vector (vi)3≤i≤m,i6=6 satisfying∑m
i=3(6 − i)vi = 12(1 − g) to determine the set P (v, g) of values of v6 for which there
exist a ({v, v6}, 3)-oriented map of genus g. It is proved in [20] that P (v, g) is empty only
in the case g = 1 and v = {v5 = 1, v7 = 1} and the exact determination of P (v, g) is an
active subject of research.
Thus for a given v-vector it is interesting to consider how one can parametrize the
({v, v6}, 3)-oriented maps of genus g. Let us callM such a map, M̃ its universal cover,
Γ its fundamental group and S the set of vertices of degree different from 6. By adding
edges one by one, we can build a triangulation T onM having S as vertex set. Note that
the degree of v in T is a priori not related to the degree of the corresponding vertex inM.
Naturally many triangulations are possible and they are mapped on the universal cover M̃
to Γ-invariant triangulations. We will see below that one can build a parameterization by
Eisenstein integers from a triangulation.
Let us a take a triangulation T and encode it combinatorially. A directed edge is an
edge from a vertex to another vertex. Every edge e is composed of a directed edge −→e and
its reversal r(−→e ). The next operator n maps a directed edge to the next one in clockwise
order around the vertex v in which it is contained. A triangulation T is described by the
operators n and r acting on the set of directed edges DE(T ). In particular the vertices,
edges and faces of T correspond to the orbits of n, r and nr. For a given vertex v of T
we denote by ṽ the corresponding vertex in the corresponding ({v, v6}, 3)-map. Edges and
faces of T will have no direct analogs but homology classes will be mapped to homology
classes.
We associate an Eisenstein integer z−→e ∈ Z[ω] to any directed edge −→e of T . For any
face f = {−→e1 ,−→e2 ,−→e3} we impose the relation
z−→e1 + z−→e2 + z−→e3 = 0.
If we have an edge e = {−→e1 ,−→e2} then we impose the consistency relation
ωα(
−→e1)z−→e1 + ω
α(−→e2)z−→e2 = 0.
For any vertex v containing the directed edges (−→ei )1≤i≤m we have the relation
6− deg(ṽ) ≡
m∑
i=1
α(−→ei )− α(r(−→ei )) (mod 6) (3.1)
with deg(ṽ) the degree of ṽ. We write D(v) = deg(ṽ). If the triangulation T is of genus g
then the first homology group H1(T ) is Z2g . For any cycle C composed of directed edges
{−→e1 , . . . ,−→em} with −→e i+1 = r(nr)±1(−→ei ) we define the cycle sum
I(C,α) =
∑
j
α(−→ej )− α(r(−→ej )). (3.2)
This sum depends on the chosen element of the homology class and defines how the ori-
entation is shifted after one moves along C. If one adds 1 to α(−→e ) and α(r(−→e )) then the
M. Dutour Sikirić: Complex parameterization of triangulations on oriented maps 75
6
4
9
10
3
5 8
6
7
1
12
5 8
12
11
2
A ({v3 = 1, v9 = 1, v6}, 3)-torus
1
2
34
6
5 12
11
10
97
8
A
AB
B
C
v
1
v2
The parameterization
Figure 5: A ({v3 = 1, v9 = 1, v6}, 3)-torus and its parameterization. The identifications
along A, B and C yields the equalities ωz12 + z6 = 0, ωz5 + z8 = 0 and z9 − z11 = 0.
We have deg(ṽ1) = 3 and deg(ṽ2) = 9.
resulting map M(T,D, I, α, z) does not change. The same happens if one adds 1 to α(−→e )
for −→e in a face F of the triangulation. In fact if we choose 2g basic cycles Ci of T then
any two vectors α, α′ satisfying Equation (3.1) and I(C,α) ≡ I(C,α′) (mod 6) differ by
repeated application of above two operations. Equation (3.1) and the cycle sums I allow to
find a corresponding function α if it exists, which is not always the case. Henceforth the
data of T , D and I determine the class of maps that one can obtain. In Figure 5 we give an
example of a parameterization for ({v3 = 1, v9 = 1, v6}, 3)-torus.
Theorem 3.1. The parameter spaces of ({v1, . . . , vm}, 3)-oriented maps of genus g have
dimension
∑
3≤i 6=6 vi−1 + 2g if all faces have size divisible by 6 and
∑
3≤i6=6 vi−2 + 2g
otherwise.
Proof. Let us take such a map and build a triangulation T on it. Let us write M =∑
3≤i 6=6 vi. We then construct a spanning tree of M − 1 edges on the set of vertices of
degree different from 6. Since the map is of genus g we have a basis of 2g cycles of the
group H1(G). We add 2g edges to the spanning tree and the remaining edges define a tree
in the dual map. Once we have defined the position of the M − 1 + 2g edges, we have
defined the triangulation uniquely because all other edges can be assigned iteratively. If one
of the vertices has a degree not divisible by 6 then its position is defined uniquely once all
its neighbors are known and so the dimension decreases by 1 in that case. No other relation
exists since one can perturb the remaining parameter and still obtain some corresponding
maps.
Let us call m(v, g) the dimension in the above theorem.
For a given parameterization (T,D, I) we denote by qT (z) the number of triangles of
the obtained triangulation. The number of triangles contained in a face defined by f =
{−→e1 ,−→e2 ,−→e3} is
qf (z) =
1
ω − ω
(z−→e1z−→e2 − z−→e1z−→e2).
The function qT counting the total number of triangles is thus qT =
∑
qf and it is an
Hermitian form. From this, one can deduce that for a fixed v the number of ({v, v6}, 3)-
oriented maps of genus g with at most n triangles grows like O
(
nm(v,g)
)
. Note that [27]
gives the more precise estimate O
(
n9
)
for the number of fullerenes with exactly n tri-
76 Ars Math. Contemp. 6 (2013) 69–81
angles. But we cannot have such a statement in the general case because for example
({v3 = 4, v6}, 6)-spheres exist only if n is divisible by 4.
Conjecture 3.2. If all vertices have degree divisible by 6 then the signature of qT is
(n+, n−, n0) =
(
g +
∑
i
viFr
(
6− i
6
)
, g,m(v, g)− n+ − n−
)
,
otherwise, the signature is
(n+, n−, n0) =
(
g − 1 +
∑
i
viFr
(
6− i
6
)
,m(v, g)− n+, 0
)
with Fr(x) = x− bxc.
The conjecture is proved in the case g = 0 in [24] and has been checked for the regular
maps of genus at most 15 from [3].
For a given triangulation T of a ({v, v6}, 3)-oriented map, we choose m independent
parameters z1, . . . , zm ∈ Z[ω]. The condition of existence of the map M(T,D, I, α, z)
is that qf (z) > 0 for all face f of T and this defines the realizability space. The limit
realizability space is the same space with Z[ω] replaced by C. The limit realizability space
defines a set S in the cone qT > 0. If one approach through a generic point of the boundary
of S, which is not in the boundary of qT > 0 then one can rearrange the triangulation and
get another triangulation T ′. The vertex degree v in T may change but the degree of ṽ
remain the same. Similarly the cycles C are mapped to cycles C ′ and I(C ′, α′) = I(C,α).
Hence D and I are intrinsic to the class of triangulations obtained by rearrangements.
Moreover, the parameter set (z′i) of (T
′, D, I, α′) can be expressed linearly in term of the
parameter set (zi) of (T,D, I, α).
For two quadruples (T,D, I, α) and (T ′, D, I, α′) of parameter set (zi), (z′i) we say
that they are equivalent if there is a mapping from T to T ′ preserving D and I such that
(z′i) can be expressed linearly from (zi). Such an equivalence preserve edge length, triangle
areas and can be extended to adjacent triangulations of T and correspondingly T ′. Note
that some non-trivial equivalence can have T = T ′; this is the case for the 2 parameter
description considered in Section 2 for which one triangulation suffices. The group of such
transformation is the monodromy group and is a subgroup of GLm(Z[ω]) leaving invariant
the form qT . It may be that any two triples (T,D, I), (T ′, D, I) are related by a sequence
of such transformation but we have no proof of it and we do not know a counter-example.
3.2 Thurston theory for maps of positive curvature
Let us take one triple (v3, v4, v5) among the 19 triples of possible curvature. By Theorem
3.1 the number of complex parameters needed to describe ({v3, v4, v5, v6}, 3)-spheres is
m = m({v3, v4, v5}, 0) = v3 + v4 + v5 − 2
The monodromy group is denoted by M({v3, v4, v5}, 3, 0) and it preserves the form q,
which is of signature (1,m− 1, 0). This class of monodromy groups was defined and enu-
merated in [4, 23, 28] and in particular they are discrete groups. The 19 possible (v3, v4, v5)
cases are part of the 94 cases determined there and the form q is the intersection form on
H1(S2 − V,L) with V a set of m+ 2 points and L a line bundle on S2 − V .
M. Dutour Sikirić: Complex parameterization of triangulations on oriented maps 77
30, D5h 40, D5d 50, D5h
Figure 6: Three fullerenes of symmetry D5 or more.
B
B
A
A
B
B
A
A
Figure 7: Two representations of a ({6}, 3)-torus.
In [28] it is proved that if z ∈ Z[ω]m and q(z) > 0 then there exists f ∈M({v3, v4, v5},
3, 0) such that f(z) is realizable as a ({v3, v4, v5, v6}, 3)-sphere. Thus Hm ∩ Z[ω]m up
to the action of the monodromy group is a parameter space for the ({v3, v4, v5, v6}, 3)-
spheres. As a consequence, the quotient
Hm/(R>0 ×M3({v3, v4, v5}, 3, 0))
is of finite covolume because the number of ({v3, v4, v5, v6}, 3)-spheres is finite for any
fixed number of triangles.
In [28] a characterization of the manifolds admitting a cocompact quotient is given.
None of those corresponding to ({v3, v4, v5, v6}, 3)-spheres are compact. Each of the
direction of non-compacity correspond to a partition of the vertices of non-zero curva-
ture into two sets Si for i = 1, 2 each having viq vertices of degree q and satisfying to
3vi3 + 2v
i
4 + v
i
5 = 6. Geometrically those are nanotubes, that is we have two caps Ci with
(vi3, v
i
4, v
i
5) vertices separated by a number of rings of vertices of degree 6 (see Figure 6).
3.3 Extensions and other cases of parameter descriptions
One extension that can be done relatively simply is to consider vertices of degree 1 or 2.
For example, the ({v2 = 3, v6}, 3)-spheres are obtained by applying the Goldberg-Coxeter
construction to the sphere reduced to a cycle of length 3 [18].
78 Ars Math. Contemp. 6 (2013) 69–81
One example that does not fit exactly into the above scheme is the description of
({v6}, 3)-torus by two parameters. It is done by writing down a parallelogram on the
plane and identifying the sides. Since all vertices are equivalent in the torus, we have to
choose one vertices of degree 6 used in the construction. In Figure 7 we show two equiv-
alent representation of a ({v6}, 3)-torus. One of the representation has one horizontal line
in the fundamental domain; it is actually always possible to have such a representation and
this is the basis of a 3 integral parameters construction [25].
Goldberg-Coxeter construction and complex parameterizations for ({v1, v2, v3}, 6)-
spheres are derived in [6]. The method was to apply the truncation operation to each vertex
in order to get a ({v2, v4, v6}, 3)-sphere for which existing theory could be applied.
Yet another extension is for quadrangulations. The theory extends without difficulty,
we are still dealing with triples (T,D, I) but the ring of Eisenstein integers is replaced by
the ring of Gaussian integers. For example, for ({v3 = 8, v4}, 4)-sphere, the number of
Gaussian integer parameters is 6 and the number for the classes (O,Oh), (D4, D4d, D4h),
(D3, D3d, D3h), (D2, D2d, D2h), (C2, C2h, C2v), (C1, Cs, Ci), respectively is 1, 2, 2, 3,
4, and 6. As a byproduct of this parameterization, we also get a method for parametrizing
self-dual plane graphs with faces of size 3 or 4, see [10] for details.
Finally note that all of the above can be specialized to get parameter description of
families of maps having a specific symmetry group G provided that G contains only ori-
entation preserving mappings. This is because the symmetry conditions are translated into
linear equalities in the parameters.
4 Zigzag
In an oriented map a zigzag is a circuit of edges such that two consecutive share a face and
vertex but three do not share a face.
4.1 The Goldberg-Coxeter case
For a triangulationM we define in [9] a permutation group Mov(M) and two elements
L and R. If gcd(k, l) = 1 then the lengths of zigzags of GCk,l(M) is computed from the
cycle structure of the element L k,l R of Mov(M). This element satisfies the defining
relations L1,0 R = L, L0,1 R = R and{
Lk,l R = L k−ql , l RLq if k − ql ≥ 0,
Lk,l R = RqL k , l−qk R if l − qk ≥ 0.
If gcd(k, l) = m > 1 then every zigzag of GCk/m,l/m(M) corresponds to m zigzags of
GCk,l(M) of length multiplied bym. The same method applies as well for 4-regular plane
graphs and their central circuit, see [9] for an exhaustive description.
4.2 The case of ({v3 = 4, v6}, 3)-spheres
All zigzags of ({v3 = 4, v6}, 3)-spheres are simple and the vector enumerating their
lengths is of the form
(4s1)
m1 , (4s2)
m2 , (4s3)
m3 with si,mi ∈ N and simi =
n
4
.
This was first established in [17] but there is another way to establish it: Any ({v3 =
4, v6}, 3)-sphere is obtained as a quotient of a ({v6}, 3)-torus by a group of order 2 formed
M. Dutour Sikirić: Complex parameterization of triangulations on oriented maps 79
h−2i
h
i
z1
z2
Figure 8: The parameter description of a family of ({v4 = 6, v6}, 3)-sphere with simple
zigzags, z1 = hω and z2 = h− 2k + kω and the case (h, k) = (4, 1).
by inversion. The four vertices of degree 3 come from the four invariant vertices of the
torus. All zigzags of a ({v6}, 3)-torus are partitioned into three parallel classes that cover
the vertex set. All the zigzags in a parallel class are of the same length and when passing
to the quotient the parallel classes are preserved hence the above result is proved.
The same argument applies for ({v2 = 4, v4}, 4)-spheres and their central circuits [8].
4.3 Other classes
For other classes of maps with more parameters the structure is more complicated and it
seems very difficult to obtain simple description of the zigzags of fullerenes. However, for
({v4 = 6, v6}, 3)-spheres we have a simple conjecture for the ones with simple zigzags:
Conjecture 4.1. All ({v4 = 6, v6}, 3)-spheres with only simple zigzags are:
• GCk,k(Octahedron) and
• the family of graphs with parameters (m, k) with n = 4h(2h− 3k) triangles whose
parameter description is given in Figure 8. The vector enumerating zigzag length is
z = (6h− 6k)3h−3k, (6h)h−2k, (12h− 18k)k
They have symmetry Oh if k = 0, D6h if h = 3k and D3d otherwise.
References
[1] G. Brinkmann, O. Delgado-Friedrichs, A. Dress and T. Harmuth, CaGe – a virtual environment
for studying some special classes of large molecules, MATCH-Commun. Math. Co. 36 (1997),
233–237.
[2] G. Brinkmann, T. Harmuth and O. Heidemeier, The construction of cubic and quartic planar
maps with prescribed face degrees, Discrete Appl. Math. 128 (2003), 541–554.
[3] M. Conder and P. Dobcsányi, Determination of all regular maps of small genus, Combin. The-
ory Ser. B 81 (2001), 224–242.
[4] P. Deligne and G.D. Mostow, Monodromy of hypergeometric functions and nonlattice integral
monodromy, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 5–89.
80 Ars Math. Contemp. 6 (2013) 69–81
[5] M. Deza and M. Dutour, Zigzag Structure of Simple Two-faced Polyhedra, Comb. Probab.
Comput. 14 (2005), 31–57.
[6] M. Deza and M. Dutour Sikirić, Zigzag and central circuit structure of ({1, 2, 3}, 6)-spheres,
Taiwan. J. Math., to appear
[7] M. Deza, M. Dutour Sikirić and P. Fowler, The symmetries of cubic polyhedral graphs with
face size no larger than 6, MATCH-Commun. Math. Co. 61 (2009), 589–602.
[8] M. Deza, M. Dutour and M.I. Shtogrin, 4-valent plane graphs with 2-, 3- and 4-gonal faces,
”Advances in Algebra and Related Topics” (in memory of B.H. Neumann; Proceedings of ICM
Satellite Conference on Algebra and Combinatorics, Hong Kong 2002), World Scientific Publ.
Co. (2003), 73–97.
[9] M. Dutour and M. Deza, Goldberg-Coxeter construction for 3- and 4-valent plane graphs, Elec-
tron. J. Combin. 11 (2004), R#20.
[10] M. Dutour Sikirić and M. Deza, 4-regular and self-dual analogs of fullerenes, Mathematics
and Topology of Fullerenes, pp. 103–116 ed. by O. Ori, A. Graovac and F. Cataldo, Carbon
materials, Chemistry and Physics, vol 4, Springer Verlag, 2011.
[11] M. Dutour Sikirić and P. Fowler, Cubic ramapolyhedra with face size no larger than 6, J. Math.
Chem. 49 (2011), 843–858.
[12] P. W. Fowler, J.E. Cremona and J. I. Steer, Systematics of bonding in non-icosahedral carbon
clusters, Theory Chem. Acta 73 (1988), 1–26.
[13] P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995.
[14] M. Goldberg, A class of multisymmetric polyhedra, Tohoku Math. J. 43 (1937), 104–108.
[15] J. E. Graver, Encoding fullerenes and geodesic domes, SIAM J. Discrete Math. 17 (2004), 596–
614.
[16] J. E. Graver, Catalog of all fullerenes with ten or more symmetries, Graphs and discovery,
167–188 DIMACS Series Discrete Mathematics Theoretical Computer Science, 69, American
Mathematical Society, 2005.
[17] B. Grünbaum and T. S. Motzkin, The number of hexagons and the simplicity of geodesics on
certain polyhedra, Canadian J. Math. 15 (1963), 744–751.
[18] B. Grünbaum and J. Zaks, The existence of certain planar maps, Discrete Math. 10 (1974),
93–115.
[19] T. Harmuth, The construction of cubic maps on orientable surfaces, PhD thesis, Bielefeld,
2000.
[20] S. Jendrol’, On face vectors of trivalent maps, Math. Slovaca 36 (1986), 367–386.
[21] P. E. John and H. Sachs, Spectra of toroidal graphs, Discrete Math. 309 (2009), 2663–2681.
[22] H. W. Kroto, J. R. Heath, R. F. Curl, R. E. Smalley, C60: Buckminsterfullerene, Nature 318
(1985), 162–163.
[23] G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Publ. Math.
Inst. Hautes Études Sci. 63 (1986), 91–106.
[24] G. D. Mostow, Braids, hypergeometric functions, and lattices, B. Am. Math. Soc. 16 (1987),
225–246.
[25] S. Negami, Uniqueness and faithfulness of embedding of graphs into surfaces, Doctor thesis,
Tokyo Institute of Technology, 1985.
[26] I. Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. Math. 139
(1994), 553–580.
M. Dutour Sikirić: Complex parameterization of triangulations on oriented maps 81
[27] C. H. Sah, A generalized leapfrog for fullerene structures, Fullerene Sci. Techn. 2 (1994), 445–
458.
[28] W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, The Epstein birthday
schrift, Geom. Topol. Monogr. 1 (1998), 511–549.