Also available at http://amc.imfm.si
ISSN 1855-3966 (printed ed.), ISSN 1855-3974 (electronic ed.)
ARS MATHEMATICA CONTEMPORANEA 1 (2008) 185–205
Cayley Graphs and Symmetric 4-Polytopes
Barry Monson ∗
University of New Brunswick, Fredericton, New Brunswick, Canada E3B 5A3
Asia Ivić Weiss ∗
York University, Toronto, Ontario, Canada M3J 1P3
Received 23 July 2008, accepted 16 November 2008, published online 27 December 2008
Abstract
Previously we have investigated the medial layer graph G for a finite, self-dual, regular or
chiral abstract 4-polytope P . Here we study the Cayley graph C on a natural group generated
by polarities of P , show that C covers G in a readily computable way and construct C as a
voltage graph over G. We then examine such symmetric graphs for several interesting families
of polytopes of type {p, q, p}, p = 3, 4, 5.
Keywords: Abstract regular or chiral polytopes, symmetric graphs.
Math. Subj. Class.: 05C25, 51M20
1 Introduction
In [7] Coxeter studied the graph on the edges and polygonal faces of each of the two self-
dual, regular convex 4-polytopes, then with the second author of this paper, widened the
investigation to include certain families of twisted honeycombs [9, 29]. Much later, in [23]
and [18], we revisited these objects in the even wider setting of abstract polytopes.
To be specific, we focussed there on finite, self-dual, abstract regular (or chiral) 4-poly-
topes P . In fact, self-duality served mainly to help guarantee vertex-transitivity in the associ-
ated graph G. Here we take a closer look at the full group D of dualities and automorphisms
for P . We are particularly interested in the Cayley graph C on the subgroup of D generated
by a natural set of polarities (dualities of period 2); see Definition 3 and the remarks after it.
In Proposition 11 we show that C is symmetric, indeed transitive on t-arcs for some t ≥ 1.
Proposition 12 then describes how C covers the edge-face graph G mentioned earlier; also, in
many cases C can be naturally constructed as a voltage graph over G (Proposition 14). Hav-
ing established this foundation, we then take a detailed look at several interesting classes of
polytopes. Before proceeding, however, we want to thank Marston Conder, Tomo Pisanski,
Egon Schulte and the referees for several helpful comments.
∗Supported by the NSERC of Canada.
E-mail addresses: bmonson@unb.ca (Barry Monson), weiss@mathstat.yorku.ca (Asia Ivić Weiss)
Copyright c© 2008 DMFA – založništvo
186 Ars Mathematica Contemporanea 1 (2008) 185–205
2 Polytopes and graphs
Rather than repeat in detail what can be found elsewhere, we refer the reader to [23] and [18]
for notation and general background, and to [2] and [17] for a deeper discussion of symmetric
graphs and polytopes. Here let us simply recall that an abstract n-polytope P is a partially
ordered set having some of the key combinatorial properties of the face lattice of a convex
n-polytope; in general, however, P need not be a lattice, need not be finite, need not have any
familiar geometric realization.
Beginning with any abstract 4-polytope P , one can construct a simple, bipartite graph
G = G(P), whose nodes are all 1-faces (edges) and 2-faces (polygons) in P , where two
such nodes are joined by a branch when incident as faces of P . (We employ the lesser-used
terms ‘node’ and ‘branch’ since we occasionally discuss the edges and vertices (0-faces) of
P itself.) Following [23, 18] we say that G is the medial layer graph of P . Viewing P as a
ranked, partially ordered set, then G is quite literally the graph on the elements of rank 1 or 2
in a Hasse diagram for P .
Usually P will be equivelar, say with Schläfli type {p, q, r}. This encodes a sort of local
symmetry in which every 2-face of P is a p-gon, with r such p-gons around every edge;
furthermore, in each facet (3-face) of P there are q edges (and hence q polygons) around
each vertex. Consequently, the graph G contains cycles of length 2q and has bipartition
classes consisting, respectively, of p-valent and r-valent nodes.
In fact, we shall enforce equivelarity by demanding that the group A = A(P) of order
preserving bijections (automorphisms) on P be quite large. To frame our discussion we fix a
base flag Φ = {F0, F1, F2, F3} (suppressing the improper faces F−1, F4). For 0 ≤ j ≤ 3,
there is a unique j-adjacent flag Φj which differs from Φ in just the proper face Fj .
Now recall from [17] that P is regular if A is transitive on flags. For 0 ≤ j ≤ 3, let
ρj be the unique automorphism mapping Φ to Φj . Then A is generated by the involutions
ρj , which we usefully think of as ‘reflections’ (see [17, 2B]); moreover, P is equivelar of
type {p, q, r}, where p, q, r are the periods of the ‘rotations’ σ1 := ρ0ρ1, σ2 := ρ1ρ2 and
σ3 := ρ2ρ3, respectively. The rotation group
A+ := 〈σ1, σ2, σ3〉 (1)
has index at most 2 in A. If this index equals 2 we say that P is directly regular. Otherwise,
A+ = A, and we say that P is non-orientably regular (since the order complex for P must
be non-orientable; see [17, 2C] and [26, p. 501]).
Relaxing symmetry a little, we say that P is chiral if A has just two flag orbits, with
adjacent flags always in different orbits [26]. Here the automorphism group is also generated
by ‘rotations’ behaving much like the σj in the directly regular case. In this case, too, we
have A = A+; and again P is equivelar of type {p, q, r}.
If P is regular or chiral, the generating rotations σi satisfy the standard relations
σp1 = σ
q
2 = σ
r
3 = (σ1σ2)
2 = (σ2σ3)2 = (σ1σ2σ3)2 = 1 , (2)
(and usually other independent relations). Moreover, apart from one special case, we have
the following intersection condition on special subgroups:
〈σ1〉 ∩ 〈σ2〉 = {1}
〈σ2〉 ∩ 〈σ3〉 = {1}
〈σ1, σ2〉 ∩ 〈σ2, σ3〉 = 〈σ2〉 .
(3)
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 187
However, if P is non-orientably regular, the very last intersection sometimes becomes
〈σ1, σ2〉 ∩ 〈σ2, σ3〉 = 〈ρ1, ρ2〉.
(For regular polytopes, the subgroup 〈σ2〉 always has index 2 in 〈ρ1, ρ2〉.) One particularly
nice instance of this is the 11-cell described in Example 17 below.
Remarks. When P is a regular 4-polytope, so that A = 〈ρ0, ρ1, ρ2, ρ3〉, the equations in (3)
are related to the perhaps more familiar result that
〈ρi : i ∈ I〉 ∩ 〈ρi : i ∈ J〉 = 〈ρi : i ∈ I ∩ J〉 , (4)
for any subsets I, J ⊆ {0, 1, 2, 3}. This in turn recalls a well-known fact concerning para-
bolic subgroups of general Coxeter groups [15, Theorem 5.5(c)]. Now in either regular or
chiral cases, it is possible to reconstruct P in a natural way as a coset geometry over A ([17,
2E] and [26, §5]). Conditions (3) or (4) translate in this construction into essential properties
of P; for example, in a 4-polytope every 2-face will be incident with exactly two 3-faces.
Lemma 1. σ22 = σ
−1
1 σ3σ1σ
−1
3 ∈ 〈σ1, σ3〉.
Proof. From (2) we have
σ22 = σ
−1
1 σ1σ
2
2
= σ−11 σ1σ2(σ1σ2σ3)
2σ2
= σ−11 (σ1σ2)
2σ3σ1(σ2σ3σ2)
= σ−11 σ3σ1σ
−1
3 .

Corollary 2. If q is odd, then σ2 = (σ3σ−11 σ
−1
3 σ1)
q−1
2 ∈ 〈σ1, σ3〉.
Clearly the graph G inherits considerable symmetry from the polytope P . If P is regular
or chiral, then A acts transitively on each of the two sets of 1-arcs in G, namely those with
initial node a 1-face or 2-face of P .
Of course, no automorphism of P maps a 1-face to a 2-face, so thatA cannot be transitive
on the set of all nodes of G. As a remedy, we may also suppose P to be self-dual. In this
case, A = A(P) has index 2 in the group D = D(P) of all dualities and automorphisms.
Furthermore, the graph G is symmetric, that is, Aut(G) is transitive on the set of all 1-arcs [2,
p. 118]. We refer to [23, 18] for a discussion of such graphs in the case that p = r = 3.
Elaborating on [14, p. 131], we say that P is properly self-dual if there exists a duality
preserving each flag orbit (under the natural action of A). This is forced when P is regular
but can fail in chiral cases. Indeed, an improperly self-dual chiral polytope admits dualities
which swap the two flag orbits [14, Lemma 3.1]. Whenever P is self-dual and regular or
properly self-dual and chiral, there must exist a polarity δ which flips the base flag, that is,
transposes F0, F3 and F1, F2. This implies that
δ2 = 1
δ−1σ1δ = σ−13
δ−1σ2δ = σ−12
(5)
and that p = r.
Conventions. We henceforth assume that the polytope P is self-dual and regular or properly
self-dual and chiral, with Schläfli type {p, q, p}, where p < ∞. (We allow q = ∞.) For
convenience below, we take ‘self-dual’ to mean ‘properly self-dual’ in chiral cases.
188 Ars Mathematica Contemporanea 1 (2008) 185–205
3 A group generated by polarities
Now let us consider the group D+ := 〈σ1, σ2, σ3, δ〉 of all rotations and dualities, which by
(5) splits as
D+ ' A+ o C2 , (6)
with C2 = 〈δ〉. Also, we define, in symmetrical fashion,
δj = σ
1−j
1 δσ
j−1
1 , for 1 ≤ j ≤ p,
so that δ1 = δ. Now we may define the group of most interest to us here:
Definition 3. Let G = 〈δ1, . . . , δp〉 be the subgroup of D+ defined by the above set of
distinguished polarities.
Observe that these generating polarities are distinct: if say δ1 = δj , then
F2δ = F2σ
1−j
1 δσ
j−1
1 ,
so that F1 = F1σ
j−1
1 , which forces j ≡ 1 mod p.
Remarks. By working within the groupD+ (rather than all ofD), we can readily tackle both
regular and chiral polytopes in one go. Our choice of the polarities δj is natural for the study
of local properties of the medial layer graph G, since the δj’s map a base vertex of G to its
neighbours. Looking ahead to Proposition 9, we note as well that the group G can be as large
as D+, so it is hard to see how another set of generators could be more versatile. See also the
comments after Definition 10.
Anticipating the subgroups described in (7) and (8) below, we now have the following
arrangement of groups:
D
index 2
||
||
||
|| index≤ 2
II
II
II
II
I
A
index≤ 2 AA
AA
AA
AA
D+
index 2vv
vv
vv
vv
v
GG
GG
GG
GG
G
A+
GG
GG
GG
GG
G G








xx
xx
xx
xx
x
A+2
FF
FF
FF
FF
F G ∩A2
G ∩A+2
Figure 1: Duality and automorphism groups for self-dual regular or chiral polytopes.
Our ultimate goal is to understand a Cayley graph on the group G (see Definition 10); but
first we must examine G more carefully.
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 189
Lemma 4. Taking subscripts modulo p, we have
(a) δj+k = σ−k1 δjσ
k
1 ;
(b) δjδk = σ
1−j
1 σ
k−j
3 σ
k−1
1 ; in particular, δ1δ2 = σ3σ1 and δpδ1 = σ1σ3 ;
(c) σ22 = δ2δ3δ2δ1 ;
(d) σ3δjσ−13 = δ1δ2δj+1δ2δ1 ;
(e) (σ2σ3)−1δjσ2σ3 = δ3−j ;
(f) σ−12 δjσ2 = δ1δ2δ4−jδ2δ1 and σ2δjσ
−1
2 = δ2δ3δ4−jδ3δ2.
Proof. Part (a) is routine. For (b) note that
δjδk = σ
1−j
1 δσ
j−1
1 σ
1−k
1 δσ
k−1
1 = σ
1−j
1 σ
k−j
3 σ
k−1
1 .
Part (c) follows from (b) and Lemma 1. From (b) we also have σ3 = δ1δ2σ−11 , so that
σ3δjσ
−1
3 = δ1δ2σ
−1
1 δjσ1δ2δ1 = δ1δ2δj+1δ2δ1 .
For part (e) we use (2), which gives σ−11 = σ2σ3σ1σ2σ3, so that (σ2σ3)σ
j
1 = σ
−j
1 (σ2σ3),
for 1 ≤ j ≤ p. Thus
(σ2σ3)−1δj(σ2σ3) = σ2σ3σ
1−j
1 δσ
j−1
1 σ2σ3
= σj−11 σ2σ3δσ2σ3δδσ
1−j
1
= σj−11 σ2σ3σ
−1
2 σ
−1
1 δσ
1−j
1
= σj−11 σ2σ3σ1σ2δσ
1−j
1
= σj−11 σ
−1
1 σ
−1
3 δσ
1−j
1
= σj−21 δσ
2−j
1
= δ3−j .
Finally, part (f) follows at once from (c), (d) and (e). 
Remarks. When p = 2, the polytope P = {2, q, 2} is uniquely specified by its Schläfli
symbol. Both ρ0 and ρ3 act trivially on the graph G, which is then a 2q-cycle.
However, as we observed in [23, §2],D does act faithfully on G when p (= r) ≥ 3. In this
case, we can regard D as a subgroup of Aut(G). A fragment of a polytope of type {3, q, 3} is
displayed in Figure 2. The nodes of the graph G are indicated by black and white discs; and
δ1, δ2, δ3 swap the (black and white) nodes incident with the nearby branches.
It is useful to recall here that, in chiral and directly regular cases, the stabilizer in A+ of
the node v2 = F2 (the base 2-face of P) is the subgroup
A+2 := 〈σ1, σ2σ3〉 , (7)
which is dihedral of order 2p. In the non-orientably regular case, A = A+ and the stabilizer
of F2 is now
A2 := 〈ρ0, ρ1, ρ3〉 , (8)
which has order 4p. Note that the index [A2 : A+2 ] = 2. In Figure 2, σ1 acts on F2 (here
a triangle) like a ‘rotation’ (v1 → v3 → w2 → v1), whereas σ2σ3 acts like a ‘reflection’
(v1 ↔ v3 , w2 ↔ w2).
190 Ars Mathematica Contemporanea 1 (2008) 185–205
v2
δ 2δ 1
δ
3
F
0
v1
2w
F
3
0v 4
v
v3 F
0
F
1
 = v
1 F2
 = v
2 F3, , ,
v5
Base  Flag
Figure 2: A fragment of a polytope of type {3, q, 3}.
Corollary 5. G is a normal subgroup of D+.
Proof. This follows at once from Lemma 4 (a),(d),(f). 
We can now enumerate the cosets of G in D+, though perhaps with some redundancy.
Indeed, we define cosets
aj := Gσ
j−1
1 ; bj := ajσ2 , for 1 ≤ j ≤ p .
(Recall that we assume p <∞.)
Lemma 6. Again taking subscripts modulo p, we have
ajσ1 = aj+1 ajσ2 = bj ajσ−12 = bj ajσ3 = aj−1 ajδ = aj
bjσ1 = bj−1 bjσ2 = aj bjσ−12 = aj bjσ3 = bj+1 bjδ = bj
Moreover, all cosets of G in D+ occur among the aj , bj , 1 ≤ j ≤ p.
Proof. These identitites follow easily from Lemma 4 and Corollary 5. For example,
bjσ2 = Gσ
j−1
1 σ
2
2 = σ
j−1
1 σ
2
2G = σ
j−1
1 G = Gσ
j−1
1 = aj .
Since a1 = G, every coset of G in D+ is counted among the aj , bj . 
Proposition 7. G acts transitively on the nodes of G.
Proof. D+ certainly acts transitively on nodes of G. But from Lemma 6 we find that every
coset of G in D+ has a representative in A+2 (see (7)). Indeed, σ
j−1
1 ∈ A
+
2 , so we need only
observe that Gσj−21 σ2σ3 = aj−1σ2σ3 = bj−1σ3 = bj . It follows that G acts transitively on
nodes of G. 
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 191
Keeping in place a fixed regular or chiral, but self-dual polytope P with Schläfli symbol
{p, q, p}, we now introduce the universal polytope U of type {p,∞, p}. Certainly, U is di-
rectly regular and self-dual; and its automorphism group A(U) is the Coxeter group [p,∞, p]
with diagram
• p • ∞ • p • .
(In fact, we could replace the ‘∞’ by any even multiple of q.) For ease of notation, we let
Ā, D̄+ and so on denote objects attached to U . In particular,
D̄+ = 〈σ̄1, σ̄2, σ̄3, δ̄〉 .
We note that defining relations for D̄+ are obtained by setting p = r, q = ∞ in (2) and (5)
(and replacing σj by σ̄j , δ by δ̄).
Clearly the mapping σ̄j 7→ σj , δ̄ 7→ δ induces an epimorphism g : D̄+ → D+. Thus
K̄ := ker(g) is generated, as a normal subgroup of D̄+, by any relations R which we must
adjoin to (2) in order to establish a presentation for D+.
Even if the cosets in Lemma 6 are redundant, the action described there does suggest that
we also investigate the dihedral group
I2(p) = 〈α0, α1 | α20 = α21 = (α0α1)p = 1〉 ,
of order 2p. It is easy to check that the mapping
σ̄1 7→ α0α1, σ̄2 7→ α0, σ̄3 7→ α1α0, δ̄ 7→ 1 (9)
induces another epimorphism f : D̄+ → I2(p). Clearly, Ḡ ⊆ ker(f). But Ḡ has index at
most 2p in D̄+, by Lemma 6. Thus Ḡ = ker(f), and the index equals 2p in the universal
setting. The situation is summarized in Figure 3.
D+ D̄+
f //
g
oo I2(p)
G K̄Ḡ
f //
g
oo f(K̄)
Ḡ
f //
g
aaCCCCCCCCC
{1}
Figure 3: The index of G in D+.
Returning to the polytope P , we can now discuss a useful covering parameter . We let k =
k(P) denote the order of the subgroup G ∩A+2 (see Figure 1).
Lemma 8. Let f(K̄) be the normal subgroup of I2(p) generated by the images f(R) of the
‘additional’ relations required in a presentation of D+. Then
I2(p)/f(K̄) ' D+/G ' A+2 /(G ∩A
+
2 ) .
Furthermore, k = |f(K̄)| and the index of G in D+ equals 2p/k. Also, in non-orientably
regular cases, |G ∩A2| = 2k.
192 Ars Mathematica Contemporanea 1 (2008) 185–205
Proof. Since K̄Ḡ / D̄+, we immediately get I2(p)/f(K̄) ' D+/G. We easily check that
GA+2 = D
+, so D+/G ' A+2 /(G ∩ A
+
2 ) follows from the second isomorphism theorem.
But A+2 ' I2(p), so that k = |G ∩A
+
2 | = |f(K̄)|.
When P is non-orientably regular, we have A = A+, so that D = D+ = GA+2 = GA2.
Thus, G ∩A2 also has index 2p/k in A2. Since |A2| = 4p, we have |G ∩A2| = 2k. 
Now suppose, for example, that q is odd, so that f(K̄) contains f(σ̄q2) = α
q
0 = α0. Thus
f(K̄) contains the dihedral subgroup generated by the conjugacy class of the reflection α0 in
I2(p). Hence f(K̄) is the full group I2(p) if p is odd as well, but could have index 2 when p
is even. We have verified
Proposition 9. Suppose P is self-dual and regular or chiral of type {p, q, p}, with q odd. If
p is odd, then G = D+. If p is even, then G has index at most 2 in D+.
At last we are ready for the following
Definition 10. Let C = C(P) be the Cayley graph constructed on the groupG, with specified
(involutory) generators δ1, . . . , δp.
Remarks. Recall from [2, ch. 16] that C has node set G, where µ, ν ∈ G are adjacent if and
only if µν−1 = δj for some j ∈ {1, . . . , p}. Since all δ2j = 1, this unambiguously defines C
as a simple p-valent graph.
Since µτ(ντ)−1 = µν−1, for all τ ∈ G, the right regular action of G on itself serves
to embed G as a subgroup of Aut(C). Similarly, if ϕ ∈ Aut(G) globally fixes the set
{δ1, . . . , δp}, then ϕ induces a automorphism of C which stabilizes the vertex 1.
Such considerations also point out the unsuitability of other polarities, such as the σ1−j2 δ
σj−12 , 1 ≤ j ≤ q, as possible generators. For one thing, the corresponding Cayley graph will
have no obvious local symmetry, because of Lemma 4(f).
Let us look more closely at our graph C. It is clear on geometrical grounds, and fol-
lows easily from Lemma 4(a), (e), that the dihedral group A+2 = 〈σ1, σ2σ3〉 normalizes
{δ1, . . . , δp}, and, in fact, acts transitively and faithfully on that set, just as it would act on
the vertices of a p-gon. This gives
Proposition 11. The Cayley graph C is a connected, symmetric, p-valent graph. The node
stabilizer in Aut(C) contains a dihedral subgroup of order 2p.
Remarks. From Proposition 9 we note that A+2 ⊂ G is possible. Thus we cannot generally
hope that Aut(C) contains a subgroup isomorphic to GoA+2 .
We also note that in non-orientably regular cases, it can never happen that the full F2-
stabilizer A2 = 〈ρ0, ρ1, ρ3〉 normalizes {δ1, . . . , δp}. For example, suppose ρ0δ1ρ0 = δj .
Then
F1 = F2ρ0δ1ρ0 = F2δj = F1σ
j−1
1 ,
so that j = 1. But then F0 = F3ρ0δ1 = F3δ1ρ0 = F0ρ0, a contradiction.
Proposition 12. There is a surjective graph homomorphism
h : C → G ,
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 193
which is injective on any neighbourhood (of nodes distant at most 1 from a given node).
Moreover, if P is chiral or directly regular, then C is a k-fold covering of G, where k =
|G ∩A+2 | = |f(K̄)|. If P is non-orientably regular, then C is a 2k-fold covering of G.
Proof. We let h : µ → (F2)µ, for all µ ∈ G. Note that µ is a typical node of C; and h is
surjective by Proposition 7. Now suppose that µ, ν ∈ G are adjacent nodes with µν−1 = δj .
Then
F2µ = F2δjν = F1σ
j−1
1 ν
is incident with F2ν, since F1σ
j−1
1 is a 1-face of F2. Thus h is a graph homomorphism. The
p neighbours of µ in C are the nodes δ1µ, . . . , δpµ. Their images under h are distinct since µ
acts bijectively on P .
Finally, we note that a typical node F2µ of G is covered by λ ∈ G if and only if λµ−1 ∈
StabG(F2) = G ∩ StabA+(F2). But in chiral or directly regular cases, the latter group is just
G ∩A+2 and has order k. The non-orientably regular case also follows from Lemma 8. 
At least in chiral and directly regular cases, we can exploit the foregoing proof to describe
C as a derived voltage graph over the base graph G, with voltage group S := StabG(F2) =
G ∩ A+2 (see [12, Ch. 2] or [2, Ch. 19]). Recall that we must first assign to each 1-arc [u, v]
in G a voltage ϕ[u, v] ∈ S, satisfying ϕ[v, u] = (ϕ[u, v])−1.
Lemma 13. Suppose αδi = βδj , for α, β ∈ S and 1 ≤ i, j ≤ p. Then i = j and α = β.
Proof. Since F2σ1 = F2, we have F2ασ1−i1 δσ
i−1
1 = F2βσ
1−j
1 δσ
j−1
1 , so F1σ
i−1
1 = F1σ
j−1
1 .
Thus i = j. 
By Proposition 7, the nodes of G correspond to a (right) transversal {µ1, . . . , µe} for
the subgroup S in G. Thus, for a typical 1-arc [u, v] in G we may take u = F2µx and
v = F2µy = F2δjµx for uniquely determined x, y ∈ {1, . . . , e} and j ∈ {1, . . . , p}. We
define
ϕ[u, v] := δjµxµ−1y .
Note that ϕ[u, v] ∈ S, and
µyµ
−1
x = (ϕ[u, v])
−1δjϕ[u, v](ϕ[u, v])−1 = δi(ϕ[u, v])−1 ,
say, since S normalizes {δ1, . . . , δp}. (See the remarks after Proposition 11.)
Since δiµyµ−1x ∈ S, we conclude from Lemma 13 that i = j and ϕ[v, u] = (ϕ[u, v])−1.
Proposition 14. Suppose P is chiral or directly regular. Then the Cayley graph C is isomor-
phic to the derived voltage graph Gϕ.
Proof. Recall ([2, Ch. 19]) that the nodes of Gϕ are all pairs (γ, u), where u is a node of G
and γ ∈ S, with (γ, u) ∼ (λ, v) if and only if u ∼ v in G and λ = γ · ϕ[u, v]. But each node
of C can be uniquely written as γµx, where 1 ≤ x ≤ e and γ ∈ S. We may therefore define
a map
F : C → Gϕ
γµx 7→ (γ, F2µx)
Clearly F is a bijection. Consider two adjacent nodes in C, say µ = γµx, ν = λµy , where
γ, λ ∈ S and µν−1 = δj . Again since S normalizes {δ1, . . . , δp}, we have γ−1δjγ = δi and
µy = λ−1δjγµx = (λ−1γ)δiµx .
194 Ars Mathematica Contemporanea 1 (2008) 185–205
Thus F2µy = F2δiµx, so F2µy and F2µx are adjacent in G and
γ · ϕ[F2µx, F2µy] = γδiµxµ−1y = λ .
Thus F preserves adjacency (as does its inverse). 
4 Finite polytopes of type {3,q,3}
Taking p = 3 in the more general discussion of the previous section, we find that both C and
G are symmetric trivalent graphs. Much more is known about this very interesting family
of graphs, both in general ([1, 2, 3, 5, 6]) and in the specific case of medial layer graphs for
regular or chiral polytopes ([23, 18]). Although we could generalize a little, we shall assume
that P is finite, so that the graphs C and G are finite, too.
From Propositions 9, 11 and 12, we immediately get
Proposition 15. Suppose P is finite, self-dual and regular or chiral of type {3, q, 3}, with q
odd. Then D+ = G = 〈δ1, δ2, δ3〉, whose generators satisfy at least the relations implicit in
the diagram
δ
δ δ
q
32
1
bb
b
in which b is the order of the (left) Petrie motion σ1σ3 for P , and q is the order of σ2 (and of
δ1δ2δ3δ2).
In all cases, the Cayley graph C is t-transitive, for some t ≥ 2 (and t ≤ 5).
If P is directly regular (chiral), then C is a 6-fold cover of the 3-transitive (2-transitive)
graph G. If P is non-orientably regular, then C is a 12-fold cover of the 3-transitive graph G.
Proof. From [2, Ch. 18] we recall that a finite, symmetric trivalent graph like C is t-transitive
for some t satisfying 1 ≤ t ≤ 5, and that the node stabilizer has order 3 · 2t−1, which is at
least 2p = 6 for C (Proposition 11). Thus t ≥ 2.
By [23, Thms. 2 and 5], G is 3-transitive (resp. 2-transitive) when P is regular (resp.
chiral). 
Remark. It is very unlikely that C could have transitivity t = 4 or 5, but we have no proof of
this.
Example 16. The regular 4-simplex P = {3, 3, 3}.
The automorphism group for this familiar self-dual, regular convex polytope is, of course,
the Coxeter group [3, 3, 3], so that A ' S5 and A+ ' A5. Recall that the longest element in
A is τ := σ1σ3σ1σ−12 σ1 ∈ A+. Then τδ is a central polarity in D+, so that
D+ ' A5 × C2 .
This group of order 120 is the icosahedral group [3, 5]. With the new generators δ1, δ2, δ3 we
get a Cayley graph C for the group [3, 5], namely the 2-transitive graph 120B in [3]. Evidently
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 195
we can think of δ1, δ2, δ3 as reflections in the sides of a triangular face of the icosahedron
{3, 5}.
The graph C is indeed a 6-fold cover of the medial layer graph G, which in turn is 3-
transitive and is listed as 20B in [3]. We recall that G is the Levi graph for the Desargues
configuration 103.
Example 17. The 11-cell P = {{3, 5}5, {5, 3}5} and its relatives.
This very interesting polytope with 11 facets and (dually) 11 vertices, was independently
discovered by Coxeter [8] and Grünbaum [13]. The enhanced Schläfli symbol indicates that
P is the universal regular 4-polytope with hemi-icosahedral facets and hemi-dodecahedral
vertex-figures. In fact, P is non-orientably regular, with A ' PSL2(11) of order 660. Also,
G = D+ ' PGL2(11).
The Cayley graph C is 2-transitive and has been described as graph C1320.3 in Conder’s
recently expanded census of symmetric trivalent graphs [4]. It is a 12-fold cover of G, which
appears as the 3-transitive graph 110 in Foster’s Census. (Here, and often elsewhere, our
supporting calculations were done using GAP and the subsidiary packages GRAPE and nauty
[10, 28, 16].)
For a somewhat different description of the 11-cell we refer to [21], where the polytope
appears as a singular offshoot from a family of finite, self-dual (directly) regular polytopes
Pπ , all with icosahedral facets and dodecahedral vertex-figures. The parameter π is any
prime in the ring Z[τ ] of algebraic integers in Q(
√
5). In fact, the 11-cell is a natural quotient
of Pπ , when π = −(2 + 5τ). On the other hand, if π = 2, then A ' O(4, 22,−1), the 4-
dimensional orthogonal group overGF (22) with Witt index 1. The polytopeP2 has 68 facets
(and 68 vertices); its medial layer graph G must be C1360.5, the only 3-transitive graph with
1360 nodes listed in [4].
Generally, if π is an odd rational prime ≡ ±2 mod 5, then the number of nodes in C is
about π12.
Remark. Before proceeding, let us comment on the notation used in Example 17. A Petrie
polygon in an abstract n-polytopeQ is defined (inductively) as an edge-path along which any
n − 1 consecutive edges, but no n, belong to a Petrie polygon of a facet of Q; the Petrie
polygon of a polygon (2-polytope) is the polygon itself [17, p. 163].
For example, if Q is a regular polyhedron, say of type {p, q}, then all Petrie polygons
have the same length l, being the period of ρ0ρ1ρ2 in the automorphism group of Q. If
these data determine the combinatorial type of Q, then we may denote Q by {p, q}l. In par-
ticular, the hemi-icosahedron {3, 5}5 has pentagonal Petrie polygons, obtained by antipodal
identification along the decagonal Petrie polygons of the icosahedron {3, 5}.
In many cases, the regular polyhedron Q has a Petrie dual Qπ , which shares the vertices
and edges ofQ but whose faces are the Petrie polygons ofQ. In particular, we have {p, q}πl '
{l, q}p [17, p. 192].
Example 18. The chiral polytopes {3, 5, 3}l, for l = 7, 9.
In [9] we find a discussion of ‘honeycombs’ of type
{3, 5, 3}l ,
for which the group A+ is obtained by adjoining the relation (σ1σ3)l = 1 to the standard
relations listed in (2) (with p = r = 3, q = 5, of course). We recall that l can be interpreted
196 Ars Mathematica Contemporanea 1 (2008) 185–205
as the length of a ‘right-handed’ Petrie polygon; the ‘left-handed’ Petrie polygons have length
l′ equal the period of σ2σ1σ2σ3.
For l ≤ 6 and l = 8 the intersection condition (3) fails and the corresponding group is
non-polytopal. (Even so, the resulting structures still have interesting medial layer graphs.)
For l ≥ 10, it seems, though we have no proof, that A is infinite. Thus, we are left with l = 7
or 9, which somewhat surprisingly give finite, properly self-dual chiral polytopes.
The polytope {3, 5, 3}9 has l′ = 10 and group A+ ' PSL(2, 19), of order 3420. The
medial layer graph G appears as C1140.5 in [4]; and the Cayley graph C for G = D+ '
PGL(2, 19) has 6840 nodes.
The polytope {3, 5, 3}7 has l′ = 29 and group A+ ' PSL(2, 29). However, this poly-
tope admits a central polarity and so D+ ' PSL(2, 29)× C2.
Let us turn now to the case that q is even. We recall from Lemma 8 that the index in D+
of G equals the index in I2(3) of the normal subgroup f(K̄) generated by the images under
(9) of all ‘additional’ relations. Various examples show that all possible quotients do occur:
D+/G could be any one of I2(3), C2 or {1}. Therefore, we shall make do with organizing
various subcases of interest.
Proposition 19. Suppose P is finite, self-dual and directly regular or chiral of type {3, q, 3},
with q even. If |G| = |D+|/6, then G acts regularly on the node set of G and G ' C.
Proof. This follows at once from Proposition 12 and the fact that both G and C have the same
number of nodes. 
Remark. We cannot take P to be non-orientably regular in Proposition 19. Indeed, from
Proposition 12 we see that C is then at least a 2-fold cover of G.
Example 20. The 24-cell and other classical examples.
The medial layer graph for the 24-cell {3, 4, 3} appears as the 3-transitive graph 192A
in [3]. Since A+ ' [3, 4, 3]+ is the rotation subgroup of a Coxeter group, there are no
additional relations to adjoin to those in (2) (with p = r = 3, q = 4). Thus in Lemma 8 we
have f(K̄) = {1}, so k = 1 and G has index 6 in D+. By Proposition 19, G ' C really is a
Cayley graph for the group G of order 192.
Now consider the hemi-24-cell {3, 4, 3}/±1, which is also isomorphic to {3, 4, 3}6,
through imposition of the extra relation (σ1σ3)6 = 1. Consulting (9), we still have f(K̄) =
{1}, so that G ' C is the 3-transitive graph 96 in [3].
The medial layer graph for the spherical polytope {3, 2, 3} is K3,3, which is indeed a
Cayley graph for the group I2(3) ' S3.
Example 21. The universal locally toroidal polytopes
{ {3, 6}s , {6, 3}s } , (10)
for s = (1, 1), (2, 0), (3, 0) .
We recall from [17, 1D] how the toroidal polyhedron {3, 6}s, with s = (b, c), is obtained
from T = {3, 6}, the regular tiling of the Euclidean plane by equilateral triangles. The
rotation group for T is [3, 6]+ = 〈σ1, σ2〉, whose defining relations are a suitable subset of
(2):
σ31 = σ
6
2 = (σ1σ2)
2 = 1 . (11)
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 197
One obtains the toroid by identifying faces of T through the action of the normal subgroup
of translations in [3, 6]+ generated by xbyc, where x := σ22σ
−1
1 and y := σ
−2
2 σ1 are unit
translations along adjacent edges in the tiling. In brief, we adjoin to (11) the extra relation
(σ22σ
−1
1 )
b(σ−22 σ1)
c = 1 . (12)
The resulting toroidal map is actually a 3-polytope precisely when b2 + bc + c2 ≥ 3 and is
regular if and only if bc(b− c) = 0. Otherwise, the polyhedron is chiral. We denote the dual
of {3, 6}s by {6, 3}s. Clearly this dual is obtained in like manner from the hexagonal tiling
{6, 3}. (Later we will need the remaining family of toroidal polyhedra {4, 4}s, for which
s = (b, c) satisfies b2 + c2 ≥ 2.)
A polytope P is locally toroidal if its facets and vertex-figures are spherical or toroidal,
with at least one kind toroidal. The regular locally toroidal polytopes have not yet been fully
classified; see [17, Chs. 10–12].
Now let us consider the universal regular 4-polytope described by (10). This locally
toroidal 4-polytope is known to be finite only when s = (1, 1), (2, 0) or (3, 0) [17, 11E]. In
each of these cases, the polytope is self-dual. Thus D+ is obtained from (2) and (5), with
p = r = 3, q = 6, by adjoining just the relation (12); duality then forces the defining relation
(σ−22 σ3)
b(σ22σ
−1
3 )
c = 1 for the vertex-figure.
From (12) and (9), we conclude that f(K̄) is generated by
(α20α1α0)
b(α−20 α0α1)
c = (α0α1)c−b ∈ I2(3) . (13)
Thus it is trivial to apply Lemma 8. Drawing on [17, Table 11E1], we can summarize the
results in the following table:
s = (b, c) |D+| k = |f(K̄)| |C| |G|
(1, 1) 108 1 18 18
(2, 0) 240 3 120 40
(3, 0) 2916 1 486 486
For s = (1, 1), the medial layer graph G ' C is listed as 18 in [3] and, in fact, is the Levi
graph for the Pappus configuration 93. The corresponding group G of order 18 is actually
isomorphic to the rotation subgroup [3, 2, 3]+ = 〈η1, η2, η3〉 for {3, 2, 3}, if identify δ1 with
η2, δ2 with η1η2η3 and δ3 with η1η2.
When s = (3, 0), we again have C ' G, listed as the 3-regular graph 486C in [3]. We
note from [30] that the polytope P can also be described as {3, 6, 3}6.
Finally, when s = (2, 0), we find that C is again the graph 120B in [3]; and once more G
is the icosahedral group [3, 5]. However, now C is a 3-fold cover of G, which appears as 40 in
[3].
Example 22. The universal polytope {{3, 2b}6, {2b, 3}6}, b ≥ 2.
It is easy to check that
{6, 3}(b,0) ' {6, 3}2b ,
198 Ars Mathematica Contemporanea 1 (2008) 185–205
indicating that the combinatorial type of this toroidal polyhedron is forced by Petrie polygons
having length 2b. Hence, for the dual of the Petrie dual, namely {3, 2b}6, the crucial extra
defining relation is
(ρ0ρ1ρ2)6 = (σ21σ
2
2)
3 = 1 . (14)
The polyhedron {3, 2b}6 has the same group of order 12b2 as the original toroid (with differ-
ent specified generators, of course).
In [31, §5], the second author of this paper proved that the universal, self-dual regular
polytope
{{3, 2b}6, {2b, 3}6}
has automorphism group of order
|A| =
{
36b4 , b odd,
72b4 , b even.
(We require b ≥ 2 for the original toroidal map to be polyhedral.) From (14) and (9) we
conclude that f(K̄) is generated by (α0α1)6 = 1, so that k = 1 in all cases. Thus G ' C is a
3-transitive trivalent graph with 6b4 (resp. 12b4) nodes for b odd (resp. even).
Example 23. Modular polytopes in the class 〈 {3, 6}(b,c) , {6, 3}(b,c) 〉
We begin with the infinite Coxeter group W = [3, 6, 3] and exploit its action on the
conformal model of hyperbolic space H3. Referring to [27, §10] or [22, §6], we therefore
consider the unimodular group L〈−1〉2 (Z[ω]), consisting of all 2 × 2 matrices of determinant
±1 over the domain of Eisenstein integers Z[ω] (where ω = e2πi/3). Next let H be the
subgroup generated by
S1 =
[
1 1
−1 0
]
, S2 =
[
ω2 0
−ω2 −ω
]
, S3 =
[
ω 0
0 ω2
]
, (15)
noting that det(S1) = det(S3) = 1, det(S2) = −1 and S31 = −I [22, Eq. 15]. We recall
that H has index 4 in L〈−1〉2 (Z[ω]) and that the rotation group
W+ = [3, 6, 3]+ ' H/{±I} .
Of course, [3, 6, 3]+ is infinite, as is the corresponding polytope {3, 6, 3}, which is realized
as a self-dual, regular tessellation of H3. However, if we reduce H modulo m ∈ Z[ω], then
almost always we obtain a finite regular or chiral 4-polytope. More precisely, we first reduce
H to Hm, then examine
X := {x ∈ Z[ω]/(m) : xI ∈ Hm} ,
which is a group of units in Z[ω]/(m) arising from the centre of Hm. For any admissible
subgroup S of such units, with {±1} ⊆ S ⊆ X , we set
HSm := Hm/{sI : s ∈ S} ,
and let σj be the natural image of Sj in this quotient. By [22, Th. 6.1], if the non-zero
modulus m - (1 − ω), then A+ := HSm = 〈σ1, σ2, σ3〉 is the rotation group for a finite
4-polytope PSm. Briefly, PSm is directly regular if m | m̄ and S = S̄, but is chiral otherwise.
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 199
For example, if m = 3 then P{±1}3 is the universal regular polytope
{ {3, 6}(3,0) , {3, 6}(1,1) } ,
which is clearly not self-dual. By [18, Th. 2], the medial layer graph G cannot then be
symmetric. Indeed, G is the Gray graph, the smallest trivalent, semisymmetric graph [18,
§3.4].
In fact PSm is self-dual if and only if gcd(m, 1 − ω) = 1, which we now assume. We
also suppose m - 2, since P{1}2 ' {3, 3, 3} collapses to the regular 4-simplex. Restricting
m = b− cω in this way (where b, c ∈ Z), we have that PSm is a self-dual polytope in the class
〈 {3, 6}(b,c) , {6, 3}(b,c) 〉 .
From [22] we note that
|A+| = |HSm| =
2(mm̄)3
|S|
∏
π|m
(1− (ππ̄)−2) , (16)
where the product is over all non-associated prime divisors π of m in Z[ω]. Thus we can
readily compute the the number of nodes in the medial layer graph G. Indeed, a typical node
in G is the basic 2-face F2, whose stabilizer in A+ is the group of order 6 described in (7).
Thus G has |HSm|/3 nodes.
However, recalling our comments in Example 21, it is very unlikely that PSm will be
universal for its facets and vertex-figures. Thus, in order to understand the corresponding
Cayley graph C, we must be a little more devious.
Proposition 24. For any modulus m = b− cω ∈ Z[ω], with gcd(m, 1− ω) = 1 and m 6∼ 2,
and for any admissible unit group S, let PSm be the self-dual 4-polytope with rotation group
HSm. Let G be its (trivalent) medial layer graph. Then
(a) if m | m̄ and S = S̄, then PSm is regular and G is 3-transitive.
(b) if m - m̄ or S 6= S̄, then PSm is chiral and G is 2-transitive.
Furthermore, the associated (trivalent) Cayley graph C is transitive on 2-arcs and is a k-fold
cover of G, where
k :=
{
3, if s2 = 1 for all s ∈ S;
6, if s2 = −1 for some s ∈ S .
Proof. The transitivity properties of G follow at once from [23, Ths. 2 and 5]. By Proposi-
tion 12 and Lemma 8 it remains only to compute k = |f(K̄)|. (Recall that 6 = k [D+ : G]
and that K̄ has nothing to do with complex conjugation.)
From (13) we have (α1α0)b−c ∈ f(K̄) ⊆ I2(3). Now mm̄ = b2 + bc + c2 ≡ (b − c)2
(mod 3). Since, by hypothesis, m is relatively prime to 3 = −ω2(1− ω)2, we conclude that
b− c is relatively prime to 3, and so α0α1 ∈ f(K̄). Thus k = 3 or 6.
Now suppose s2 = 1 for all s ∈ S. Thus, the determinant map is well-defined on
A+ = HSm, and we have det(σj) = det(δσjδ) for j = 1, 2, 3. With (6) this gives a well-
defined homomorphism
h : D+ → {±1}
αδi 7→ det(α), for α ∈ A+, i = 0, 1 . (17)
200 Ars Mathematica Contemporanea 1 (2008) 185–205
Clearly G ⊆ ker(h), which has index 2 in D+, since det(σ2) = −1. Thus k = 3 in all such
cases.
Otherwise we have s2 = −1 for some s ∈ S. But sI = Si1Si2 · · ·Sil for suitable
ij ∈ {1, 2, 3}. Since det(Sl) = −1 only when l = 2, we have ij = 2 for an odd number
of j’s. But in HSm we have 1 = σi1σi2 · · ·σil . From (9) we therefore conclude that f(K̄)
contains a reflection, so that k = 6.

Remarks. Any rational prime p ≡ 1 (mod 3) factors over Z[ω] as p = ππ̄ = b2 +
bc + c2, where π = b − cω is prime in Z[ω]. Let us take our modulus m to be π, with
S = {±1}. We conclude from Proposition 24(b) that P{±1}π is self-dual and chiral of type
{ {3, 6}(b,c) , {6, 3}(b,c)}. From (16) we have
|A+| = (ππ̄)3(1− (ππ̄)−2) = p(p2 − 1) .
Thus the medial layer graph G is 2-transitive and has p(p2 − 1)/3 nodes. Since k = 3, the
Cayley graph C has p(p2 − 1) nodes and has an automorphism group which is transitive on
2-arcs. For example, when p = 7, we find that G and C are the 2-transitive graphs 112A and
336B , respectively, in [3].
Suppose now that the modulus m is an odd rational prime p. For a moment we put aside
the conformal action of W+ on H3 and turn instead to the standard faithful representation of
W = [3, 6, 3] as a crystallographic linear Coxeter group (see [15, Cor. 5.4] and [19, §4]).
Abusing notation a little, we therefore now suppose W ⊂ GL4(Z), then reduce W modulo p
and so obtain a finite representation Wp in some orthogonal space V over Zp. When p > 3,
the reduced group Wp turns out to be the automorphism group of a finite, self-dual regular
polytope Qp in the class
〈 {3, 6}(p,0) , {6, 3}(p,0) 〉 .
(See [20, Eqs. 22, 10]; this reference and [19] give a detailed description of the construction
and applications to several natural familes of polytopes.)
We can now establish a connection with the earlier family of polytopes PSm. For a suitable
matrix B ∈ GL4(Q(ω)), we may define an epimorphism
H → W+
with A 7→ B(A⊗ Ā)B−1 .
As long as ss̄ = 1 for all s ∈ S, this induces an epimorphism
λ : HSp →W+p ,
which in fact becomes an isomorphism when S is chosen carefully. We refer to [19, 20] for
more notation and, without much explanation, summarize our calculations in
Corollary 25. For the rational prime p > 3, let Qp be the self-dual, directly regular 4-
polytope with automorphism group Wp = [3, 6, 3]p. Then the (trivalent) medial layer graph
G for this polytope is 3-transitive with |Wp|/6 nodes. Furthermore, the associated (trivalent)
Cayley graph C is transitive on 2-arcs and is a k-fold cover of G, as indicated below:
p (mod 12) S ⊂ Z[ω]/(p) HSp 'W+p ' |Wp| k
1 {±1,±s} O1(4, p, 1) p2(p2 − 1)2 6
5 {±1} O(4, p,−1) 2p2(p4 − 1) 3
7 {±1} O(4, p, 1) 2p2(p2 − 1)2 3
11 {±1,±s} O1(4, p,−1) p2(p4 − 1) 6
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 201
Proof. For p ≡ ±1 (mod 12) there exists s ∈ Z[ω] with s2 ≡ −1 , ss̄ ≡ 1 (mod p). By
Proposition 24 we then have k = 6. 
Note that C has p2(p2− )(p2− 1) nodes, where  = +1 for p ≡ 1, 7 (mod 12),  = −1
for p ≡ 5, 11 (mod 12). For example, when p = 5, the graph C with 15600 nodes is actually
2-transitive and is a 3-fold cover of the 3-transitive graph G.
5 Finite polytopes P of type {4,q,4}
Suppose P is a finite, self-dual and regular or chiral polytope of type {4, q, 4}. Then our
observations in Section 3 tell us that the corresponding medial layer graph G and Cayley
graph C are symmetric tetravalent graphs. The general theory of these graphs is somewhat
less developed than in the trivalent case, although considerable information can be found in
[2, Ch. 17] and in [11, 24, 32]. We shall have correspondingly less to say here about such
graphs as arise from polytopes. We summarize what we can easily say in the following
Proposition 26. Suppose P is finite, self-dual and regular or chiral of type {4, q, 4}. The
corresponding Cayley graph C and medial layer graph G are symmetric, tetravalent graphs.
Furthermore, C is a l-fold cover of G, as described in the following Table:
P |G| q odd q even
l = l =
directly regular |D+|/8 8 or 4 8, 4, 2 or 1
chiral |D+|/8 8 or 4 8, 4, 2 or 1
non-orientably regular |D+|/16 16 or 8 16, 8, 4 or 2
In particular, if P is directly regular or chiral and |G| = |D+|/8, then C ' G. This isomor-
phism is impossible when q is odd and in all non-orientably regular cases.
Proof. This follows immediately from Propositions 9, 11 and 12. 
Example 27. The cubic toroids {4, 3, 4}s, where s = (b, 0, 0), (b, b, 0) or (b, b, b), for inte-
gers b ≥ 2.
The cubic toroid Ps := {4, 3, 4}s is a finite, self-dual and directly regular 4-polytope
constructed in much the same way as the toroidal polyhedra described in Example 21. Thus
Ps is a quotient of {4, 3, 4}, the familiar tessellation of Euclidean 3-space by cubes; hence,
we can regard Ps as a tessellation of the 3-torus [17, 6D].
The toroidsPs are parametrized by a type vector s = (bi, 03−i), as indicated above. From
[17, 6D5] we find that the groupA+ is defined by the relations in (2), with p = r = 4, q = 3,
along with a single extra relation
(σ1σ3β)b = 1 ,
where β = σ22 , σ1σ3σ2 or (σ1σ3)
2, for s = (b, 0, 0), (b, b, 0) or (b, b, b), respectively. Noting
that σ32 = 1, it is now easy to use (9) and Lemma 8 to show that the covering multiplicity
k = f(K̄) = 4 in every case. Drawing on [17, table 6D1], we summarize our calculations
here:
202 Ars Mathematica Contemporanea 1 (2008) 185–205
s |A| |G| = |A|/8 k |C|
(b, 0, 0) 48b3 6b3 4 24b3
(b, b, 0) 96b3 12b3 4 48b3
(b, b, b) 192b3 24b3 4 96b3
The smallest toroid in this class is in some sense the most interesting. The polytope
P(2,0,0) is flat, meaning that each of the 8 vertices is incident with each of the 8 facets. Of
course, the 24 edges and 24 polygons comprise the 48 nodes of the medial layer graph G,
whose automorphism group is just the full duality group D of order 768. This graph is listed
as C4[48, 10] in [32] and is 1-transitive.
The Cayley graph C has 192 = 4 · 48 vertices; and Aut(C) has the unexpectedly large
order 98304 = 192 · 4 · 27. From [2, p. 134], we conclude that C is 1-transitive, and that the
stabilizer of a 1-arc has order 27.
Example 28. The universal polytope {{4, 2b}4, {2b, 4}4}, b ≥ 2.
This family of finite, self-dual regular polytopes is also described in [31, §5]. The analysis
is very similar to that in Example 22, so we merely record some results. The toroidal map
{4, 4}(b,b) ' {4, 4}2b
has automorphism group of order 16b2. The dual of the Petrie dual is {4, 2b}4 and the self-
dual, universal regular polytope {{4, 2b}4, {2b, 4}4} has |A| = 64b4. Again we have k = 1
in all cases, so that G ' C is a tetravalent graph on 8b4 nodes, with group transitive on 1-arcs.
Example 29. Modular polytopes in the class 〈 {4, 4}(p,0) , {4, 4}(p,0) 〉, for odd primes p.
The infinite Coxeter groupW = [4, 4, 4] is crystallographic. If, as in the previous section,
we reduce the standard linear representation modulo any odd prime p, we find that A = Wp
is the automorphism group of a finite, self-dual and directly regular polytope Pp in the class
〈 {4, 4}(p,0) , {4, 4}(p,0) 〉 [20, pp. 340-343]. This locally toroidal polytope is universal for
its class when p = 3, but almost surely is not for larger primes [17, 10C]. Consulting [20, Eq.
(6)], we note that
A = Wp =

O1(4, p, 1) , if p ≡ 1 (mod 8)
O1(4, p,−1) , if p ≡ 7 (mod 8)
O(4, p, 1) , if p ≡ 5 (mod 8)
O(4, p,−1) , if p ≡ 3 (mod 8).
(18)
From here on the analysis is very similar to that for Corollary 25. We thus obtain
Corollary 30. For any odd prime p, let Pp be the self-dual, directly regular 4-polytope with
automorphism group Wp = [4, 4, 4]p. Then the symmetric tetravalent Cayley graph C is a
k-fold cover of the symmetric, tetravalent medial layer graph G, where
k :=
{
8 if p ≡ ±1 (mod 8)
4 if p ≡ ±3 (mod 8) .
We conclude this section with a look at
Example 31. The universal, locally projective polytope { {4, 3}3 , {3, 4}3 } [25, Table 1].
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 203
The facets of this self-dual, non-orientably regular polytope are hemicubes; dually its
vertex-figures are hemi-octahedra. Along with the standard relations in (2) and (5), with
p = r = 4, q = 3, we need only enforce (ρ0ρ1ρ2)3 = 1, or equivalently
(σ2σ21)
3 = 1 .
This gives |A| = |A+| = 96. From (9), we conclude that f(K̄) is the normal subgroup
generated by (α0(α0α1)2)3 = α1α0α1 in I2(4), so that k = 4.
Note that G has 12 nodes. From Proposition 12, we find that C has 96 nodes and is a
8-fold cover of G. Both C and G are 1-transitive.
6 Other examples and questions
We end our survey of particular examples with a look at
Example 32. The classical 4-dimensional star-polytopes.
Consulting [17, 7D], we find that just three of the 4-dimensional regular star-polytopes
concern us here: {5, 52 , 5}, {
5
2 , 5,
5
2} and {5, 3,
5
2}.
The first two of these are geometrically self-dual, and, as abstract regular polytopes, are
each isomorphic to the universal polytope
{ {5, 5 | 3 }, {5, 5 | 3 } }
([17, Th. 7D16(b)]). The notation indicates, for instance, that the facet {5, 5 | 3 }, with
Schläfli type {5, 5}, is specified by its having triangular holes. This means that (σ1σ−12 )3 =
1. Of course, the full automorphism group A is the Coxeter group H4 of order 14400, so that
G is a symmetric pentavalent graph with 1440 nodes. By Propositions 9 and 12, we see that
C is a symmetric pentavalent graph on 14400 nodes and is a 10-fold cover of G.
Now although the other polytope {5, 3, 52} is clearly not geometrically self-dual, it is
nevertheless self-dual in the combinatorial sense used throughout this paper. In fact, it is
isomorphic to
{ 5, 3, 5 | 3 } ,
whose deep triangular holes are specified by (σ1σ3σ−12 )
3 = 1 [17, Th. 7D16(c)]. We obtain
new pentavalent graphs G and C, again with 1440 and 14400 nodes, respectively.
We conclude by mentioning some open questions. First of all, suppose G is t-transitive,
while the cover C is s-transitive. Invariably we have observed that s ≤ t; however, we have
no proof that this must be so. Of course, it would be very desirable to find examples with say
s = 4 or 5 in the trivalent case (where t = 2 or 3); but this is most unlikely.
In Proposition 14, we have described C as a derived voltage graph. If also A acts as a
group of automorphisms on the stabilizer S, in a manner compatible with the voltage assign-
ment ϕ, it is at least possible to lift t-transitivity on G to C, so that s ≥ t [2, Prop. 19.4].
However, we have no non-trivial examples of this very special situation. Also, we do not
know if the voltage graph construction in Proposition 14 can be extended in a natural way to
non-orientably regular cases (but see [12, Th. 2.2.2]).
Finally, we reconsider improperly self-dual chiral polytopes P of type {p, q, p}. If q is
even, then P might not admit a polarity (duality of period 2); but when q is odd, P does
admit polarities [14, Th. 3.4]. Nevertheless, we have avoided all such polytopes in order to
simplify the definition of the graph C. Perhaps these complications can be overcome to yield
interesting results in improperly self-dual cases.
204 Ars Mathematica Contemporanea 1 (2008) 185–205
References
[1] N. Biggs, Presentations for Cubic Graphs, in: Michael D. Atkinson (ed.), Computational Group
Theory, Academic Press, London, 1984, 57–63.
[2] N. Biggs, Algebraic Graph Theory, second ed., Cambridge University Press, Cambridge, 1993.
[3] I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star (eds.), The Foster Census: R.M. Fos-
ter’s Census of Connected Symmetric Trivalent Graphs, The Charles Babbage Research Centre,
Winnipeg, 1988.
[4] M. Conder, Trivalent (cubic) symmetric graphs:
http://www.math.auckland.ac.nz/˜conder/symmcubic2048list.txt.
[5] M. Conder and P. Lorimer, Automorphism Groups of Symmetric Graphs of Valency 3, Journal of
Combinatorial Theory, Series B 47 (1989), 60–72.
[6] M. Conder and P. Dobcsányi, Trivalent symmetric graphs on up to 768 vertices, J. Combinatorial
Mathematics and Combinatorial Computing 40 (2002), 41–63.
[7] H.S.M. Coxeter, The edges and faces of a 4-dimensional polytope, Congressus Numerantium 28
(1980), 309–334.
[8] H.S.M. Coxeter, A symmetrical arrangement of eleven hemi-icosahedra. in: M. Rosenfeld and
J. Zaks (eds.), Convexity and Graph Theory, Jerusalem, 1981, North-Holland Math. Stud. 87,
North-Holland, Amsterdam, 1984, 103–114.
[9] H. S. M. Coxeter and A. I. Weiss, Twisted honeycombs {3, 5, 3}t and their groups, Geom. Dedi-
cata 17 (1984), 169–179.
[10] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.3 (2002):
http://www.gap-system.org.
[11] A. Gardiner and C. Praeger, On 4-valent symmetric graphs, Europ. J. Combinatorics 15 (1994),
375–381.
[12] J. L. Gross and T. W. Tucker, Topological Graph Theory, Dover, New York, 2001.
[13] B. Grünbaum, Regularity of graphs, complexes and designs, Problèmes combinatoires et théorie
des graphes, Colloq. Internat. C.N.R.S. No. 260, Orsay, 1977, 191–197.
[14] I. Hubard and A. I. Weiss, Self-duality of Chiral Polytopes, Journal of Combinatorial Theory,
Series A 111 (2005), 128–136.
[15] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Univ. Press, Cambridge,
1990.
[16] B.D. McKay, The nauty package:
http://cs.anu.edu.au/people/bdm/nauty/.
[17] P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Mathematics and its
Applications 92, Cambridge University Press, Cambridge, 2002.
[18] B. Monson, T. Pisanski, E. Schulte and A. I. Weiss, Semisymmetric Graphs from Polytopes,
Journal of Combinatorial Theory, Series A 114 (2007), 421–435.
[19] B. Monson and E. Schulte, Reflection Groups and Polytopes over Finite Fields - I, Advances in
Applied Mathematics 33 (2004), 290–317.
[20] B. Monson and E. Schulte, Reflection Groups and Polytopes over Finite Fields - II, Advances in
Applied Mathematics 38 (2007), 327–356.
[21] B. Monson and E. Schulte, Modular Reduction in Abstract Polytopes, Canadian Mathematical
Bulletin, to appear.
B. Monson, A. Ivić Weiss: Cayley Graphs and Symmetric 4-Polytopes 205
[22] B. Monson and A. I. Weiss, Eisenstein Integers and Related C-groups, Geometriae Dedicata 66
(1997), 99–117.
[23] B. Monson and A. I. Weiss, Medial Layer Graphs of Equivelar 4-polytopes, European Journal of
Combinatorics 28 (2007), 43–60.
[24] P. Potočnik and S. Wilson, Tetravalent edge-transitive graphs of girth at most 4, Journal of Com-
binatorial Theory, Series B 97 (2007), 217–236.
[25] E. Schulte, Amalgamation of Regular Incidence-Polytopes, Proc. London Math. Soc. 56 (1988),
303–328.
[26] E. Schulte and A. I. Weiss, Chiral polytopes, in: P. Gritzmann and B. Sturmfels (eds.), Applied
Geometry and Discrete Mathematics. The Victor Klee Festschrift, DIMACS Series in Discrete
Mathematics and Theoretical Computer Science 4, AMS, Providence, 1991, 493–516.
[27] E. Schulte and A. I. Weiss, Chirality and projective linear groups, Discrete Mathematics 131
(1994), 221–261.
[28] L. H. Soicher, The GRAPE package for GAP:
http://www.maths.qmul.ac.uk/˜leonard/grape/.
[29] A. I. Weiss, On trivalent graphs embedded in twisted honeycombs, in: A. Barlotti, P. V. Ceccherini
and G. Tallini (eds.), Combinatorics ’81 (Rome, 1981), North-Holland Math. Stud. 78, North-
Holland, Amsterdam, 1983, 781–787.
[30] A. I. Weiss, An infinite graph of girth 12, Trans. Amer. Math. Soc. 283 (1984), 575–588.
[31] A. I. Weiss, Some infinite families of finite incidence-polytopes, Journal of Combinatorial The-
ory, Series A 55 (1990), 60–73.
[32] S. Wilson and P. Potočnik, A census of edge-transitive tetravalent graphs:
http://jan.ucc.nau.edu/˜swilson/C4Site/.