Discrete Mathematics 124 (1994) 179-191

North-Holland

179

Regular maps from Cayley graphs II Antibalanced Cayley maps

Jozef h-S Department of Mathematics, Slovak Technical University, 813 68 Bratislava, Slovak Republic

Martin Skoviera Department of Computer Science, Comenius University. 842 15 Bratislava, Slovak Republic

Received 2 June 1990

Abstract

A Cayley map is a Cayley graph embedded in some orientable surface so that the local rotations at

every vertex are identical. In this series we consider two types of such maps: the balanced and

antibalanced Cayley maps. Part I was devoted to giving conditions under which a balanced Cayley

map is regular or reflexible, and to determining the corresponding map automorphism groups. In

Part II, similar questions will be pursued for antibalanced maps. Results and their proofs are based

on the theory of groups with sign structure.

1. Introduction

This paper is the second (and last) part of a series devoted to embeddings of Cayley

graphs possessing certain local symmetry. These embeddings, known as Cayley maps,

are characterized by the property that the local rotations in the whole graph are all

identical. More formally, let C(G, Q) be a Cayley graph of a finite group G with

a unit-free generating set 52 satisfying 52 - 1 = Sz. The Cayley map CM(G, 0, p) is a 2-cell

embedding of C(G, 52) in some orientable surface such that the local rotations induce

a unique cyclic permutation p of s2.

One of the basic problems about Cayley maps consists in determining when

the local symmetry encoded by p extends to a global symmetry, i.e. when the

automorphism group of a Cayley map acts transitively on the set of directed edges.

(Maps admitting this type of global symmetry are known as regular maps.) Both

the previous and the present part of our series have been motivated by that

problem.

Correspondence to: Jozef &iii, Dept. of Algebra & Number Theory, Comenius University, Mlynski dolina, 842 15 Bratislava, Slovak Republic.

0012-365X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved

SSDI 0012-365X(92)00060-X

180 J. .&ii, M. .?koviera

In [l] we were dealing with Cayley maps fulfilling the identity

for every XEQ. Observe that this identity forces any generator to be placed in

p ‘opposite’ to its inverse. For obvious reasons, such Cayley maps were termed

balanced in [l]. As shown therein, the regularity of a balanced Cayley map

CM(G, s2, p) is equivalent to the existence of an automorphism of the group G which

extends p. Adopting a more geometrical interpretation, balanced Cayley maps encode central

symmetry of the permutation p. From this point of view, it is natural to consider axial symmetry in place of the central one. The axial symmetry of p corresponds to the

identity

p(x_‘)=(p-l(x))-’

for every x~s2. Cayley maps having this property are called antibalanced. The main goal of this article is to establish conditions under which an antibalanced

Cayley map is regular and/or reflexible (roughly speaking, isomorphic to its mirror

image). Moreover, in case of regularity we describe the corresponding automorphism

groups.

It turns out that dealing with antibalanced Cayley maps requires to develop

a special algebraic machinery of groups with sign structure and their antiautomor- phisms. In particular, both regularity and reflexibility can be characterized in these

terms. Here we summarize, without proof, just some basic facts about such groups and

antiautomorphisms, deferring the details to a separate paper [2].

2. Preliminaries

This section provides a very brief review of some necessary concepts concerning

maps (i.e. graphs embedded in surfaces) and map automorphisms. A more detailed

discussion can be found in Section 2 of Part I [l].

We shall confine ourselves to maps on orientable surfaces. Thus, a map can be

defined as a pair (K, P) where K is a connected graph and P is a certain permutation

called rotation. In more detail, if D(K) is the set of arcs (=directed edges) of K then

P is a permutation of D(K) which, for every vertex UE K, cyclically permutes the arcs

emanating from 0.

Let T be a permutation of D(K) sending any arc to its reverse. An automorphism of

the map M=(K,P) is a permutation A of D(K) which commutes with both P and T, i.e. AP= PA and AT= TA. Note that every map automorphism is necessarily an

automorphism of the graph K. The set of all map automorphisms constitutes a group

denoted by Aut M. The map M is said to be regular if Aut M acts transitively on D(K) (equivalently, if 1 Aut MI = 1 D(K)I).

Cayley graphs 181

A rejection of the map M is a permutation B of D(K) such that BP=P-‘B and

BT= TB. A map admitting a reflection is called rejexible. Again, every reflection is

a graph automorphism. The set of all reflections together with the set of all automor-

phisms of M forms a group in which Aut M has index two or one. This group will be

denoted by E Aut M and referred to as the extended automorphism group of M. If M is

regular and reflexible then, obviously, 1 E Aut M ( = 2 ID(K

We shall be mainly interested in maps whose underlying graph is a Cayley graph.

Let G be a finite group and 52 a unit-free generating set of G such that 52- ’ =sZ. The

Cayley graph C(G, 52) is a graph with vertex set G and arc set G x 52, the incidence

being given by the condition that an arc (g, X)EG x Sz has g as its initial vertex and gx

as terminal vertex. Thus, T(g, x) = (gx, x-l). Note that if Q containsj involutions then

the valency of each vertex in C(G, Q) is 2 1 SZI -j.

Let C(G, Q) be a Cayley graph and let p be a cyclic permutation of 52. The map

(C(G, Sz), P) where P(g, x)=(g, p(x)) for any gEG and XEQ will be called Cayley map

and denoted by CM(G, 52, p). It is known that Cayley maps are vertex-transitive,

although in general they need not be regular.

Finally, recall that a Cayley map CM(G, Q, p) is said to be balanced if, for every x~sZ,

P(x-‘)=(p(x))_‘.

Analogously, it will be called antibalanced if

p(x_‘)=(p-l(x))-’

for every xEs2.

3. Groups with sign structure and antiautomorphisms

A group with sign structure is a group G together with a homomorphism

G--P{ + 1, - 1). The image of an element geG under this homomorphism will be

denoted by 191 and will be referred to as the sign of g.

Let G+ be the subgroup of G consisting of all elements with sign + 1. Clearly, the

index of G+ in G is 1 or 2. If G+ = G then the corresponding sign structure will be

called trivial. Each group obviously admits the trivial sign structure. On the other

hand, a nontrivial sign structure of a finite group can only exist when the group has

even order and is not simple (for G+ is a normal subgroup of G).

In this paper we shall mainly deal with sign structures induced by a set of generators.

Let 52 be a unit-free generating set of a group G, satisfying 52-l =Q. Define G+ to be

the set of all elements of G expressible as a product of even number of generators in Q.

Since G+ is obviously a subgroup of index 1 or 2 in G, it defines a sign structure on G.

This kind of sign structure plays the key role in the subsequent theory of antibalanced

regular Cayley maps.

The concept of a group with sign structure makes it possible to introduce certain

bijective mappings of the group onto itself, called antiautomorphisms. For a group

182 J. hiA, M. Skoviera

G with sign structure, a bijection cp: G+G will be said to be an antiautomorphism of

G provided

(1) for any XEG, (q(x) I= 1x1 (i.e. cp preserves signs), and

(2) for any x,yG cp(x~)=cp(4d”~(~).

To give an example, consider the mapping Pi: G+G given by x+gxg-IX1 where geG

is an arbitrary fixed element. It is easily verified that conditions (1) and (2) above are

satisfied for pLe. Thus, pe is an antiautomorphism of G; we shall refer to it as the inner

antiautomorphism corresponding to the element geG.

In what follows, we list some properties of antiautomorphisms which will be needed

for dealing with antibalanced maps. For a detailed exposition of the theory of

antiautomorphisms the reader is referred to [2]. Throughout, let G be a group with

sign structure.

Proposition 3.1. Let G have a nontrivial sign structure. Then any sign-preserving

involutory automorphism of G is an antiautomorphism of G. In particular, the identity mapping id is an antiautomorphism. Conversely, if cp is both an automorphism and an antiautomorphism of G then (p2 = id.

Proposition 3.2. If cp is an antiautomorphism of G then its restriction (plG+ is an automorphism of G+. Consequently, rp(e)=e where e is the unit element of G.

Proposition 3.3. Let cp be an antiautomorphism of G and let k be an integer. Then cpk is again an antiautomorphism of G. Moreover, for any x, LEG it holds that

cpk(xy)=cpk(x)cpk9y) (1) and

cp”(x-‘)=(cpk’X’(x))-1. (2)

An alternative approach to groups with nontrivial sign structure consists in viewing

such a group G as an extension of G + by E2. The language of group extensions turns

out to be a convenient tool in our further exposition. We therefore present a simple

construction which provides a description of all extensions of a given group by Z2

(briefly, Z2-extensions).

Let H be a group with unit element e. For every hEH let vh denote the inner

automorphism of H corresponding to h, i.e. v,,(x) = hxh- ’ for XEH. Further, let 8 be an

automorphism of H and t an element of H such that

e(t) = t (3) and

tP=v,. (4)

In order to define a Z,-extension of H, consider the set H(8; t) of all pairs gi where gEH and Ibiza = (0, l}, endowed with a binary operation defined as follows:

gi.hj=gOi(h)tij(i+j), g,hEH, i,jEZ2. (5)

Cayley graphs 183

Formula (5) requires a further explanation. Therein as well as in all similar cases,

elements of B2 used as exponents are to be interpreted as integers. Moreover, if an

exponent includes an operation in the ring Z, then the operation is to be carried out

first, and next the result is to be considered integer. For example, ei+j is either equal to

8 or to id, the identity automorphism, depending on whether i#j or i=j, respectively.

Analogously, tii= t or tii=e according as i=j= 1 or not.

It is easy to see that operation (5) provides H(8; t) with a group structure. The pair

e0 serves as the unit element of H(8; t). The inverse elements are given by

(sj)-l=~-i(g-l)t-i j (6)

for every geZ-Z and iEB,. Here y-‘(iEi2,) stands for (y-l)’ whenever y is an element of

H or an automorphism of H.

In [2] it is proved that every Z,-extension of If can be described in the following way.

Theorem 3.4. Let G be a Z,-extension of a group H. Then there exists an automorphism B of H and an element tgH with the property g(t)= t and g’=v,, such that G is

isomorphic to H(g; t).

In a similar spirit, antiautomorphisms can be understood as extensions of automor-

phisms [2], as supported by Proposition 3.2.

Theorem 3.5. Let G= H(tI; t) be a Z2-extension of H. Let $ be an automorphism of

H and let ZEH. Assume that the following two conditions are satisfied:

+e(zt) = zt (7)

and

(+e)z = v,[. (8)

Then the mapping

q(hi)=$(h)z’i, hiEH(g; t)

de$nes an antiautomorphism of H(8; t). Conversely, every antiautomorphism of H(B; t)

has this form.

Formally, (7) and (8) resemble the conditions (3) and (4). Indeed, the existence of an

antiautomorphism with a given restriction to H is equivalent to the existence of

certain h,-extension of H. The interested reader is referred to [Z] for further details.

4. Antibalanced Cayley maps and their regu!arity

The balanced Cayley maps CM(G, Q, p) considered in the first part of this series [l]

were the maps satisfying the identity p(x-‘)=(p(x))-’ for every XEL?. As already

184 J. &GA. M. ~koviera

indicated, here we shall be interested in maps CM(G,Q,p) where

P(x-‘)=(P-‘(x))-l,

i.e. in antibalanced Cayley maps.

(9)

Observe first that, unlike the balanced case, if CM(G, 52, p) is an antibalanced map

then Q may contain both involutory and noninvolutory generators. However, if XEQ

is an involution and y =pk(x), then y-’ = P-~(X), which can be easily proved by

induction from (9). Thus, the number of involutions in Q cannot exceed 2. A more

detailed analysis shows that if Q is involution-free then [521= 2d for some d, and there

exist y,z~Q such that p(y-I)= y, p(z)=z-l, and pd(yml)=z. If s2 contains exactly one

involution x then 1 L? 1 is odd, say ( Sz I = 2d + 1, and there is a generator ycQ such that

p(y- ‘) = y and pd(y)=x. Finally, in case Sz contains two involutory generators x and

y then I&?1 =2d again, and pd(x)= y.

To start with, we show that groups with sign structure are an appropriate tool for

describing regularity of antibalanced Cayley maps.

Proposition 4.1. Let M = CM(G, Qp) be an antibalanced regular Cayley map of

valency r. Then either r 62 or the sign structure on G induced by Q is nontrivial; in

this case !SCG~.

Proof. Assume that the sign structure on G induced by 52 is trivial, i.e. Gf = G. Thus,

any fixed y~sZ can be expressed as a product of an even number of generators in 8, say

y=x1x2 “‘X2”, Xi~s2. Let A be the automorphism of M which sends the arc (e, y) to

(e,p(y)). For the elements x1,x2, . . . ,x2,,, y~s2 introduced above, we now prove (by

induction on k) the relation

/t(Xi+ ..‘xk,y)= i

(P(x,)P-'(x,)...P-'(~~),P(Y)) for k even,

(p(x,)p-'(x,)'.'p(xk),p-'(Y)) for k odd,

where 0 < k < 2n (if k = 0, the products are considered to be equal to e E G). Since for

k=O we have A(e, y)=(e,p(y)), we may pass directly to the induction step. Suppose

that our relation holds for an even k <2n (the case k odd is analogous). Let

y = #(xk;li) and xk + r = p’(y). Then

A(xl ..‘XkXk+i,y)=/t(X1 “‘xkxk+l,pi(xk;ll))=APi T(x, “‘xk,&+l)

=AP’TPj(xl . ..Xk.y)=PiTPiz‘i(Xl . ..Xk.y)

=P’TP’(P(x1)...P-‘(Xk)rP(Y))

=PiT(p(x,)...P-‘(xk),P(xk+,))

=pi(P(xl)~~~P-‘(xk)P(xk+l)~P(xk+l)-l)

=pi(P(xl)~~~P-‘(xk)P(xk+l)~~-‘(xk;’l))

=(P(Xl)“‘P(Xk+l)~P-l(Y))>

which completes the induction step.

Cayley graphs 185

In this way, setting k=2n we have just proved that

A(y~Y)=~(xI)p-l(xZ)~~~p(xZn-i)p-l(xzn)~P(y)).

But, on the other hand, if y =pm(y-I) then

A (y, y) = AP(y, y - ’ ) = AP” T(e, y) = Pm TA(e, y) = Pm T(e, p(y))

=P”(P(Y)>P-‘(y-‘))=(p(y)K’(Y)).

Comparing the second coordinates in both the expressions for A(y,y), we obtain

p(y) = p-‘(y), which is possible only if the valency of the map M does not exceed 2. The

fact that Qc G- if G+ #G is a direct consequence of the definition of a sign structure

on G induced by Q. 0

Corollary 4.2. Let M =CM(G, Qp) be an antibalanced regular Cayley map. Then

either the group G has even order and the underlying Cayley graph C(G, 52) is bipartite,

or M is an odd cycle.

Proof. Obvious and therefore omitted. 0

In [l] .we proved that balanced regular Cayley maps can be characterized by means

of certain group automorphisms. Now, we shall show that an analogous characteriza-

tion exists also in the antibalanced case, this time by means of antiautomorphisms.

Theorem 4.3. Let G be aJinite group and Sz a generating set of G. Assume that Sz induces

a nontrivial sign structure on G. If there exists an antiautomorphism cp of G whose

restriction p = cp ) l2 is a cyclic permutation of elements in 52, then M = CM(G, 52, p) is an

antibalanced regular Cayley map. Conversely, if CM(G, 0, p) is antibalanced and regular

then such an antiautomorphism of G exists.

Proof. Let cp be an antiautomorphism of the group G for which p = cp ( D is a cyclic

permutation of 52. Observe first that QEG-. Now, define for every bEG and k,

0 <k < ) Ql, a mapping Ab,k on the set of arcs of the map M = CM(G, 52, p), p = cp 1 C! as

follows:

4 & 4 = (b cp“(4, @ “(4). 1 7

To show that A,,, k (which is obviously a bijection) is an automorphism of the map M,

it is sufficient to verify that it commutes with both P and T:

Ab,kP(a>x)=Ab,k(a&‘(x))=(b4’k(a)&””’k(~(x)))

=(b qk(a), cP(q’a’k(x)))= P(b cpk(a), cPlalk(x))= PAb,k(% X),

A~,~T(a,x)=A,,k(ax,x-‘)=(b~k(Ux),~”X’k(x-l))

=(bcpk(a)cp’“~k(x),(p-~s~k(x-1))=(bcpk(a)(p~a~k(x),(cp~a~k(x))-1)

= T(b qk(U), f~9”‘~(X))= Ti&,k(a X) 5 7

186 .I. &rciA, M. .?kouiera

the properties of antiautomorphisms (Section 3) were used throughout, Since

A,,+(e, x) =(b, q”(x)), we have 1 GI. 1521 automorphisms of M, whence M is a regular

Cayley map. Moreover, the inclusion RGG- together with the fact that cp is an

antiautomorphism of G imply that M is antibalanced.

To prove the converse, let M =CM(G, Q,p) be an antibalanced regular map,

whereby G is endowed with the sign structure induced by Q. Let A be the automor-

phism of M which sends the arc (e, x) to (e,p(x)). Define cp : G+G by putting

p(a) =q A(a, x) where 4 is the projection onto the first coordinate. Clearly, cp is

a well-defined bijection on the group G. Before proving that cp is an antiautomorphism

of G, we show by induction that A(a, z) =(~(a), p’“’ (z)). This is obvious for a = e. The

induction step with respect to the length of the word a in generators from Q is carried

out as follows (where z=pk(y- ‘) for a suitable k):

A(uy,z)=A(uy,pk(y-‘))=APk(uy,y-‘)=APkT(u,y)=~kTA(u,y)

= pk T(cp(u), p’“‘(y))= Pk(cp(u) P’“‘(Y), (P’“’ (y))_ I)

=pk(cp(~)P’~‘(Y),P-‘~‘(Y-l))=pk(cp(~)P’a’(Y), P’““(Y-9

=(cp(tW(Y)> P’““(Z)).

At the same time, we have proved the identity tp(uy)=cp(u)p’“‘(y) for every UEG and

HER; in particular, 9 =p on 0. Now, it is not difficult to see that cp is the required

antiautomorphism. 0

The antiautomorphism of G appearing in the formulation of Theorem 4.3 will be

referred to as the rotary untiuutomorphism associated with the regular map M.

5. Reflexibility

Now, we turn our attention to conditions under which an antibalanced regular

Cayley map is reflexible. As we saw in [ 11, the reflexibility of a regular balanced Cayley

map depends on the existence of an involutory automorphism r of the group which,

together with the corresponding rotary automorphism p, satisfies the identity

pz=Tp - ‘. Surprisingly enough, in the antibalanced case, reflexibility is completely

independent of the rotary antiautomorphism.

Theorem 5.1. Let M = CM(G, 52, p) be an antibalanced regular Cuyley map. Then M is

rejlexible fund only if there is an untiautomorphism o of the group G such that, for every

x1,x2, . . . EQ,

o(x1 x1 *..)=x;‘x;‘... . (10)

Proof. Let M be reflexible. Then there exists a reflection B such that B(e, x) = (e, x-r)

for a fixed XEQ. Obviously, B(u, z) =(a(~), z’) where IS is a certain bijection G-+G. We

Cayley graphs 187

want to show that r~ is the required antiautomorphism; so far we know that a(e)=e.

A straightforward computation shows that B(e,z)=(e,z-‘) for every 2~52. By

induction on the length of the word a in generators from Q, we show that

B(a,z)=(cr(a),z-‘) and, at the same time, we prove the identity a(az=a(a)z-’

for every UEG and ZEQ. The following computation realizes the induction step

(y,z&, z=pk(y_‘)):

B(uy,z)=BP~(uy,y-~)=P-~B(uy,y-~)=P-~BT(a,y)=P-~TB(u,y)

=(by the induction hypothesis)=P-kT(o(u),y-‘)=P-k(a(u)y-’,y)

=(o(u)y_‘,p-k(y))=(o(u)y_‘,z_‘).

Consequently, o(ay) = a(u) y - I, which immediately implies that o is an involutory

automorphism of G satisfying (12). By Proposition 3.1, (T is the required

antiautomorphism.

For the converse, assume that (T is an antiautomorphism of G such that

a(xy . ..)=x-‘y-l . . . for everyxy,y, . ..EsZ. Then, a(ux)=o(u)x-’ for every uEG and

XEQ. Define on the set of arcs of M, a mapping B as follows: B(u, x) = (a(u), x- ‘). B is

obviously a bijection. It is, in fact, a reflection of’the map M, because

~P(~,x)=W4P(x))=(~(4~(x))-‘)=(W,P-’(x-’))

=P-‘(a(u),x-‘)=P-‘B(u,x) and

BT(a,x)= B(ux,x-‘)=(o(ux),x)=(o(u)x-‘,x)

= T(a(a),x-‘)= TB(u,x).

The proof is complete. Cl

The antiautomorphism c appearing in the statement of Theorem 5.1 will be called

the reflective untiuutomorphism of G associated with M. As an immediate consequence of Theorem 5.1 we obtain the following result.

Corollary 5.2. Let M = CM(G, Q,p) be an antibalanced regular Cuyley map. If the group G is Abeliun then the map M is reflexible.

It turns out that the last result can still be strengthened. However, for the proof we

need much stronger means, namely the theory of Z,-extensions of groups.

Theorem 5.3. Let M = CM(G, Q, p) be a regular antibalanced Cayley map such that the group G+ is Abeliun. Then M is reflexible.

Proof. In view of Corollary 5.2, it is sufficient to consider the case that G+ # G. Thus,

we may assume that G= G+ (0; t) is a Z,-extension introduced in Section 3. Let

Q = {xl, yl, . . . >. By Theorem 5.1 it is sufficient to specify an antiautomorphism r~ of

G such that o(xl,yl, . ..)=(xl)-‘(yl)-’ ... for anyxl,yl, . ..EQ Since G+ is Abelian,

188 J. .%rriii. M. ikoviercr

by(4)wehave82=id,whence(xl)-‘=B(x-’)t-’1.Now,defineamapping~:G-*Gby

setting a(gi)=@g-‘)t-‘i. Obviously, 0 is an involution satisfying a(xl)=(xl)-’ for

every xl EQ. It is easy to see that to finish the proof we only have to verify that

a(gi . hj) = cr(gi) o(hj). (11)

Using the fact that G+ is Abelian, 82 =id and e(t)=t we obtain

a(gi)a(hj)=8(g-‘)t-ii~8(h-1)t-jj=8(g-’)t-i8i(8(h-1)t-j)tij(i+j)

=e(g-l)ei+l(h-l)t-it-jtij(i+j).

On the other hand,

a(gi.hj)=o(g~‘(h)t’j(i+j))=B(t-‘jB’(h-’)g-’)t-(‘+j)(i+j)

=e(g-l),i+l(h-l)t-ijt-(i+i)(i+j).

Thus, (11) reduces to verification of the identity

t-it-jtij,__-ijt-(i+j)

But this is easily done. 0

6. Automorphism groups

In [l] we showed that the automorphism group of a regular balanced Cayley map

M = CM(G, 52,~) is simply the split extension of G by the rotary automorphism of

G associated with M. Recall that if H is an arbitrary group and c( an automorphism of

H, the split extension H(a) ofH by c( is a group whose elements are pairs g/I where gEH and /I is a power of a; the multiplication in H(u) is given by

g P . hy = dW) BY.

A similar concept of extension can be introduced replacing the automorphism by an

antiautomorphism. Let G be a group with sign structure and let K be an antiautomor-

phism of G. The extension G[K] of G by the antiautomorphism K is a group whose

elements are the pairs g2 where gEG and A is a power of rc. This time, however, the

multiplication law is

Such a multiplication coincides with the one in a split extension whenever K is both an

antiautomorphism and an automorphism. It should be pointed out that, unlike the

split extension, G now need not be a normal subgroup of G[rc].

Note that G[rc] itself can be naturally endowed with a sign structure by setting

(G[K])+ =G+ [rc].

Now we can pass to the description of automorphism groups of regular anti-

balanced Cayley maps.

Cayley graphs 189

Theorem 6.1. Let M = CM(G, Q, p) be an antibalanced regular Cayley map. Then the

automorphism group of M is isomorphic to G[q] where q is the rotary antiautomor-

phism of G associated with M.

Proof. We have already shown in the proof of Theorem 5.1 that the mappings

Ab,k: (a, x)-(bqk(a), @lk (x)) constitute the group Aut M. The composition of two

such map automorphisms Ab,k and A,,, yields

A,,,A,,k(a,~)=A,,,(b~~(a),cp’“‘~(x))

= (ccp”(bqk (a)), cp Ib”“‘4”((p14k(X)))

where d = c’p” (b) and m = 1 b In + k. By the definition of G [q] it is now easy to see that

the mapping

Aut M+G[q],

Ab,k-‘(b, Vk)

is a group isomorphism Aut M z G[q]. I7

As regards reflexibility we have the following result.

Theorem 6.2. Let M = CM (G, 52, p) b e a regular antibalanced Cayley map, with cp as its

rotary antiautomorphism. If M is reflexible and a is the corresponding reJective

antiautomorphism of G then the extended automorphism group E Aut M of the map M is

isomorphic to the split extension of the group Aut M g G[q] by the (anti)automorphism

13 of G[q] giuen by gq’+a(g)cp-‘.

Proof. Every element of E Aut M is expressible as a product A,, i. Bj= Ce,i,j where

Ae,i is the map automorphism sending (1, x) to (g, q’(x)) and B is the reflection which

maps (1, x) to (1, x- ‘). It follows that Cs,i,j(a, x)=(gpi J(a), (pl”ti rrj(x)) for arbitrary

aeG and x~s2. A straightforward computation shows that

wheref=gcp’aj(h), m=lh(i+(-l)jk, and n=j+l.

Now, consider a mapping 8: G[q]-G[q] given by g$+o(g)q-‘. Obviously, 6 is

an involutory automorphism (thus, at the same time an antiautomorphism) of G[q].

Let H be the (split) extension of G[q] by 8; the elements of H will be denoted by pairs

(g$, @), and the multiplication rule in H is (g#, 0j) (hqk, gt)=(gqi@(hqk), gj+t). Using

the definition of 0 and the multiplication in G[q], we further obtain

190 J. L?irciA, M. ikoviera

Comparing these computations with the composition of automorphism CB,i,j in

E Aut M we conclude that the mapping E Aut M-H, Cg,i,j+(gv’, f3j) is the required

isomorphism. 0

7. Examples of antibalanced regular maps

The theory presented in Section 4 makes it possible to reduce the construction of

a regular antibalanced Cayley map to finding a suitable antiautomorphism of a group

with sign structure. Perhaps the simplest example to start with is a cyclic group of

even order.

Example 7.1. Take the cyclic group Ez, and let s2= { 1,3, . . . . 2n- l}. For the sign

structure induced by 0 it clearly holds that (Z,,)’ = 2H, = {0,2, . . . ,2n - 2). Consider

a mapping cp : Z12n-+Z2n defined as follows: cp(x)=x+ 1-1x1, xeZZn. It is easy to see

that cp is an antiautomorphism of Zzn and the restriction cpls2=p is the cyclic

permutation (1,3, . . . , 2n- 1) of Q. By virtue of Theorem 4.3, the map

Ml = CM&,, R, p) is regular and by Corollary 5.2 it is reflexible. As it can be readily

seen, Ml represents the well-known regular map with the underlying graph

K,,, where each face is bounded by a Hamiltonian cycle [3]. The genus of this map is

(n - l)(n - 2)/2.

If a new vertex is placed into the interior of each face of the map Ml and joined by

a new edge with every vertex on its boundary then an embedding of the graph K,,.,, in

the same surface is obtained. Such an embedding of K,,,,, is known to form a regular

map again [3]. Our next example shows that the latter map is the dual to a regular

antibalanced Cayley map Mz = CM(G, 0, p) where G+ is Abelian. By Theorem 5.3,

M2 is reflexible. Consequently, the above map of K,,,,,, as the dual of Mz, is also

reflexible, which has recently been observed in [4].

Example 7.2. Let G= G+(tI; t) where G+ =Z, x E,, e(x,y)=(x+y, -y) for any

(x, y)~& x Z,, and t=(l, 0). By Theorem 3.4, G is well defined and the multiplication

there is (x,y)i.(u,u)j=((x,~)+8~(u,~)+ijt)(i+j)=(x+U+iu+ij,y+(-l)~u)(i+j),

where (x,y),(u,u)~&,xZ, and i,jEZ,. Put sZ={(O,O)l,(-l,O)l, (0, -1)l). It is not

difficult to verify that 52 generates G and induces there the same sign structure as G+.

Define a mapping cp: G+G by setting

cp((x,y)i)=(-x-y-i,x)i,

where (x, Y)EZ, x H, and iE (0, l}. A straightforward computation shows that cp is an

antiautomorphism of G such that cp 1 Sz = p = ((0,O) 1, ( - 1,0) 1, (0, - 1) 1). The corres-

ponding regular map Mz = CM(G, Q, p) has 2n2 vertices of valency 3, 3n2 edges and 3n faces of length 2n. Its genus is therefore (n - 1) (n - 2)/2. A little more detailed analysis

yields that M2 is indeed the dual of the above regular map of K,,,,,.

Cayley graphs 191

The results of this paper together with the machinery of groups with sign structure

and their antiautomorphisms make it possible to produce several new regular maps.

We shall illustrate this in our last example.

Example 7.3. Let G = G+(8; t) where G+ = H, x Z, x h,, 6’ is the automorphism of G+

which sends the element (a, b, c) to (b, a, c), and t = (0, 0,l). Since 8’ = pt = id and e(t) = t,

the group G is a well-defined Z,-extension of H, x Z, x Z,. In order to construct

an antiautomorphism of G, consider the automorphism $ of G+ which maps the

element (a, b,c) to (b, c, a), and take z=(O, LO). Obviously, $e(z+ t)=z+ t and

($0)’ =pz+t=id. Theorem 3.5 now implies that the formula

q((a,b,c)i)=($(u,b,c)+iz)i=(b,c+i,u)i

defines an antiautomorphism of G whose restriction to G+ is II/. It is easy to

see that if e=(O,O,O) then q3k(el)=(k,k,k)1, (p3k+1(el)=(k,k+l,k)l and

cp 3k+2(el)=(k+1,k+1,k)1. Let SZ={cp’(el); r=O,l,...}. Then 1521=3n and cp is

a cyclic permutation of 0. Moreover, routine calculations show that Sz generates G. By

Theorem 4.3, M3 = CM (G, R, cp IQ) is a regular antibalanced Cayley map; it comprises

2n3 vertices of valency 3n. Since G+ is Abelian, M, is reflexible. A standard computa-

tion yields that each face of M3 is a Zn-gon, which implies that the genus of M3 is

1 + n3(3n - 5)/2. The smallest example in this series for n = 2 yields a regular map of the

graph KS, 8 - 4C4 in the orientable surface of genus 5.

References

[l] J. &ran and M. Skoviera, Regular maps from Cayley graphs I: balanced Cayley maps, Discrete Math.

109 (1992) 265-276. [2] J. Siran and M. Skoviera, Groups with sign structure and antiautomorphisms, Discrete Math. 108

(1992) 189-202.

[3] A.T. White, Symmetrical maps, Congr. Numer. 45 (1984) 257-266.

[4] S. Wilson, Cantankerous maps and rotary embeddings of K,, J. Combin. Theory Ser. B 47 (1989)

262-273.