Graphs, groups and surfaces

GRAPHS, GROUPS AND SURFACES

NORTH-HOLLAND MATHEMATICS STUDIES

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

8

GRAPHS, GROUPS AND SURFACES ArthurT. WHITE Western Michigan University Kalamazoo, Michigan U.S.A.

Completely revised and enlarged edition

1984

NORTH-HOLLAND -AMSTERDAM

0

NEW YORK

OXFORD

Elsevier Science Publishers E.V., 1984

All rights reserved. No part of this publication may be reproduced, stared in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

First edition: First printing 1973 Revised edition: First printing 1984

ISBN: 0 444 87643 x

Publishers:

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 EZ AMSTERDAM THE NETHERLANDS

Soledistributors forthe U.S.A.andCanada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52VANDERBILT AVENUE NEW YORK, N.Y. 10017 U.S.A.

Library of Congress Cataloging in Publlcation Data

White, Arthur T. Graphs, groups, and surfaces. (North-Holland lllathematics studies ; 8)

Bibliography: p. Includes indexes.

1. Graph theory. 2. Groups, Theory of. 3. Combinat o r i a l topolo I. Title. 11. Series. W66.W45 19% 511'. 5 84-18853 ISBN 0-444-87643-X

.

PRINTED IN THE NETHERLANDS

V

FOREWORD TO THE FIRST EDITION

This text is based on a course that I taught at Western Michigan University in 1971. The material is suitable for presentation at the graduate or advanced undergraduate level, and assumes only an introductory knowledge of group theory and of point-set topology.

One of

the features of the subject matter is that many results of the general theory can be readily visualized, or modeled, so that the student is constantly reassured that what he is doing has meaning, is more than a formal manipulation of symbols. The focus of attention is an interaction amoung graphs, groups, and surfaces (see Figure 0-1). After a brief introduction to the theory of graphs, the relationship between graphs and groups will be explored.

For every graph there is an associated group, called the

automorphism group of the graph.

Conversely, for every group presen-

tation, there is an associated graph, called a Cayley color graph of the group. Both associations will be studied, with emphasis on the latter.

An excursion into combinatorial topology will follow;

Euler's generalized polyhedral formula and the classification theorem for closed 2-manifolds will be discussed. Imbedding problems in graph theory will be examined at some length (here a graph and a surface come together, often with the aid of a group); it will be seen that the famous four-color conjecture can be stated in this context. The five-color theorem will be established. The Heawood mapcoloring theorem will then be studied in detail. (This theorem determines a chromatic number for every closed 2-manifold except the sphere

--

a truly astounding result!)

The theory of quotient graphs

and quotient manifolds (and quotient groups) is at the heart of the proof of this theorem (as well as many others) and will be presented; this theory relies upon each of the concepts mentioned in this paragraph and serves as the unifying feature of this text.

Foreword t o t h e F i r s t E d i t i o n

vi

Several peripheral (but s i g n i f i c a n t ) r e s u l t s a r e s t a t e d without proof.

An e f f o r t h a s been made t o p r o v i d e t h o s e p r o o f s of theorems

which a r e most i n d i c a t i v e o f t h e charm and b e a u t y o f t h e s u b j e c t and which i l l u s t r a t e

t h e t e c h n i q u e s employed.

Proofs missing i n the

t e x t c a n be s u p p l i e d by t h e r e a d e r , a s p a r t o f t h e problem s e t s (problems marked

lo*"

a r e d i f f i c u l t ; t h o s e marked

s o l v e d ) , o r c a n be found i n t h e r e f e r e n c e s .

"**'I

a r e a s y e t un-

A bibliography is pro-

v i d e d , f o r f u r t h e r r e a d i n g : items ( c ) o r ( j ) , ( g ) , and ( h ) r e s p e c t i v e l y are p a r t i c u l a r l y s u i t a b l e f o r more e x t e n s i v e t r e a t m e n t s of t h e t h e o r i e s o f g r a p h s , g r o u p s , and s u r f a c e s , which a r e s e e n i n t e r a c t i n g in this text.

A.T.W.

vii

FOREWORD TO THE SECOND EDITION

The field of topological graph theory has greatly expanded in the ten years since the first edition of this book appeared, with many able mathematicians contributing to this expansion. As with the first edition, I have tried to focus on those aspects of the subject germane to the interaction of the structures listed in the title. I apologize in advance, for any omissions of relevant material. The nine chapters of the first edition have been revised and updated. Thirty-two new problems have been distributed among these chapters. Six new chapters have been added, dealing with: voltage graphs (extending the work of Jacques even further), non-orientable imbeddings (emphasizing parallels to the orientable theory and results), block designs associated with graph imbeddings (tying topological graph theory securely to more traditional combinatorics) , hypergraph imbeddings (as generalizing graph imbeddings and as depicting block designs), map automorphism groups (groups acting (primarily) on graphs of groups on surfaces), and change ringing (finding how to ring church bells by picturing the right graph of the right group on the right surface.) Several new problems have been given for each new chapter, so that there are now 181 problems in all; 22 of these have been designated as "difficult" ( * ) and 9 as "unsolved" ( * * ) . Three of the four unsolved problems from the first edition have been solved during the ten years between editions; they are now marked as "difficult." I wish to thank everyone who read the first edition, particularly those who sent me comments and corrections, and especially Jonathan Gross. I also owe thanks (but delegate no responsibility) to many, including the following, for their contributions to my

viii

Foreword t o t h e Second E d i t i o n

u n d e r s t a n d i n g of tlic m a t e r i a l in t h e new c h a p t e r s as i n d i c a t e d :

Thomas T u c k e r ( l o ) , S a u l Stahl (11) a n d ( 1 3 ) , S e t h A l p e r t (12), D e r e k Waller

(13) and ( 1 4 )

,

Norman B i g g s ( 1 4 )

J o a n H u t c h i n s o n , a n d T. J e f f e r s o n Smith (15).

,

a n d Sabra Anderson, Finally,

1 thank

Margo J o h n s o n f o r t y p i n g t h e m a n u s c r i p t , a n d P a u l H i m c l w r i g h t €or drawing t h e f i g u r e s .

A.T.W.

Figure 0-1

This Page Intentionally Left Blank

xi

TABLE OF CONTENTS

Chapter 1: Chapter 2 : 2.1 2.2 2.3 2.4 2.5

Chapter 3: 3.1 3.2 3.3 3.4 3.5 3.6

. . . . . . . . A Brief Introduction to Graph Theory . . . Definition of a graph . . . . . . .. Variations of Graphs . . . . . . . Additional Definitions . . . . . . .. Operations on Graphs . . . . . . . Problems . . . . . . . . . . . The Automorphism Group of a Graph . . . .. Definitions . . . . . . . . . . Operations o n Permutation Groups . . . .. Computing Automorphism Groups of Graphs Graphs with a Given Automorphism Group . .. Problems Other Groups of a Graph . . . . . . . . . . . . . . . . . Historical Setting

Chapter 4: 4.1 4.2 4.3 4.4 4.5 4.6

Chapter 5: 5.1 5.2 5.3 5.4 5.5 5- 6

. . .. .. . . ... .. .

The Cayley Color Graph of a Group Presentation

... . ..

..

.. ..

. . .. . .

Definitions Automorphisms Properties Products Cayley Graphs Problems

.. ... .

... .. .

... .. .

... .. .

... .. .

... .. .

... .. .

An Introduction to Surface Topology

. . . . . . . . . .

. ...

. ...

. ...

. ...

Definitions Surfaces and Other 2-manifolds The Characteristic of a Surface Two Applications Pseudosurfaces Problems

... ...

... .. . . .. .. ..

... .. . . .. .. ..

. . . 6.1 . Answers to Some Imbedding Questions 6 . 2 . Definition of "Imbedding" . . . . . 6.3 . The Genus of a Graph . . . . . . . 6.4 . The Maximum Genus of a Graph . . . . 6.5 . Genus Formulae for Graphs . . . . . . . . 6.6 . Edmonds' Permutation Technique 6.7 . Imbedding Graphs on Pseudosurfaces . . 6.8 . Other Topological Parameters for Graphs 6.9 . Applications . .. .. .. .. .. .. .. .. .. 6.10 . Problems

Chapter 6:

. . .. .. . . ... .. .

Imbedding Problems in Graph Theory

... .. . . .. .. ..

. ... ... .. ..

1

. 1 9 12

15 15 17 18 20 20 21 23

... .. . . .. .. ..

. ... ... ... .

23 26

28 32 35 36 39 39 41 42 47 53 54 57 58 60 61 64 67 70 73 75 79 81

xi i

Table of Contents

. . . . . . . . . 1 . 7 . Imbeddings of Cayley Color Graphs . . . . 2 . Genus Formulae for Groups . . . . . . . . 7 3 . 7 . Related Results . . . . . . . . . . 4 . The Characteristic of a Group . . . . . . . 7 5 . Problems . . . . . . . . . . . . . . 7 Chapter 8: Map-coloring Problems . . . . . . . . . 1 . 8 . Definitions . . . . . . . . . . . . 8 . 2 . The Four-color Conjecture . . . . . . . 8.3 . TheFive-color Theorem . . . . . . . . 8 . 4 . Other Map-coloring Problems; the Heawood Map-coloring Theorem . . . . . . . . 5 . 8 . A Related Problem . . . . . . . . . . 86. . Four-color Theorem for the Torus . . . . 7 . A Nine-color Theorem f o r the Torus and . 8 Klein Bottle . . . . . . . . . . . .. k-degenerate Graphs . . . . . . . . . Coloring Graphs on Pseudosurfaces . . . . 8.10 . The Cochromatic Number of Surfaces . . . . 8.11 . Problems . . . . . . . . . . . . . Chapter 9: Quotient Graphs and Quotient Manifolds (and Quotient Groups!) . . . . . . . . 9 . 1 . The Genus of Kn . . . . . . . . . . Theory of Quotient Graphs and Quotient . TheManifolds. as Applied to Kn . . . . . . . . . . . . 9 . 3 . The Genus of Kn (again) 9 . 4 . Extending the Theory . . . . . . . . . . The General Theory . . . . . . . . . 9 . 6 . Applications to Known Imbeddings . . . . . 9 . 7 . New Applications . Problems . . . . .. .. .. .. .. .. .. .. .. Chapter 10: Voltage Graphs . . . . . . . . . . . 10.1 . Covering Spaces . . . . . . . . . . 10.2 . Voltage Graphs . . . . . . . . . . . 10.3 . Examples . . . . . . . . . . . . . 10.4 . The Heawood Map-coloring Theorem (again) . . 10.5 . Strong Tensor Products . . . . . . . . 10.6 . Problems . . . . . . . . . . . . . Chapter 11: Nonorientable Graph Imbeddings . . . . . . 11.1 . General Theory . . . . . . . . . . . 11.2 . Nonorientable Covering Spaces . . . . . . 11.3 . Nonorientable Voltage Graph Imbeddings . . . 11.4 . Examples . . . . . . . . . . . . . 11.5 . The Heawood Map-coloring Theorem, Nonor ientable Version . . . . . . . . 11.6 . Other Results . . . . . . . . . . . 11.7 . Problems . . . . . . . . . . . . .

Chapter 7:

The Genus of a Group

A

88.

89.

9.2

95.

98.

83 83 88 95

97 98

101 102

102 106

107 111 114 117 118 120 122 123

125 125 127 131 136

142 149 152 154 157

157 160 164 171 173

171

177 177 180 180 182 184 185

187

xiii

Table of Contents

. . . . . . . . . . . . . 12-1. Balanced Incomplete Block Designs 1 2 - 2 . BIBDs and Graph Imbeddings . . . . . . 1 2 - 3 . Examples . . . . . . . . . . . . 1 2 - 4 . Strongly Regular Graphs . . . . . . . 1 2 - 5 . Partially Balanced Incomplete Block Designs . . . . . 1 2 - 6 . PBIBDs and Graph Imbeddings 1 2 - 7 . Examples . . . . . . . . . . . . 12-8. Doubling a PBIBD . . . . . . . . . 1 2 - 9 . Problems . . . . . . . . . . . . Chapter 1 3 : Hypergraph Imbeddings . . . . . . . . 3-1. Hypergraphs . . . . . . . . . . . . . . . . 3-2. Associated Bipartite Graphs 3-3. Imbedding Theory for Hypergraphs . . . . 3-4. The Genus of a Hypergraph . . . . . . 3 - 5 , The Heawood Map-coloring Theorem, for Hypergraphs . . . . . . . . . . . . . . 1 3 - 6 . The Genus of a Block Design 1 3 - 7 . An Example . . . . . . . . . . . . . . . . . . 1 3 - 8 . Nonorientable Analogs 1 3 - 9 . Problems . . . . . . . . . . . . Chapter 1 4 : Map Automorphism Groups . . . . . . . 1 4 - 1 . Map Automorphisms . . . . . . . . . 1 4 - 2 . Symmetrical Maps . . . . . . . . . 1 4 - 3 . Cayley Maps . . . . . . . . . . . 1 4 - 4 . Complete Maps . . . . . . . . . . 14-5. Other Symmetrical Maps . . . . 1 4 - 6 . Self-complementary Graphs . . . . . . 1 4 - 7 . Self-dual Maps . . . . . . . . . . 14-8. Paley Maps . . . . . . . . . . . 14-9. Pr ob 1ems . . . . . . . . . . . . Chapter 15: Change Ringing . . . . . . . . . . 15-1. Definitions . . . . . . . . . . . 1 5 - 2 . Notation . . . . . . . . . . . 15-3. General Theory . . . . . . . . . . 1 5 - 4 . Four-bell Extents (Minimus) .. .. .. .. .. 15-5. Five-bell Extents (Doubles) 1 5 - 6 . A New Composition . . . . . . . . . 15-7. Problems . . . . . . . . . . . . References . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . Index of Symbols . . . . . . . . . . . . . Index of Definitions . . . . . . . . . . . . Chapter 1 2 :

Block Designs

I

.

1

. .. .. ... .. . .. .. ... .. . .. .

. .. .. . .. .

189 189 190 191 192 194 197 198 201 203 205 205 207 207 211 212 213 215 217 217 219 219 225 229 233 236 237 239 243 254 257

... .

257 260 262 266 270 272 276

.

279

.

. . .

303 305 309

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1

CHAPTER r HISTORICAL SETTING

In coloring the regions of a map, one must take care to color differently any two countries sharing a common boundary line, so that the two countries can be distinguished.

One would think that an

economy-minded map-maker would wish to minimize the number of colors to be used for a given map, although there appears to be no historical evidence of any such effort. Nevertheless a conjecture was made, over thirteen decades ago, to the effect that four colorswould always suffice for a map drawn on the sphere, the regions of which were all connected.

The first reported mention of this problem (see [ B C L l ]

and [Ol]) was by Francis Guthrie, through his brother Frederick and Augustus de Morgan, in 1852. Cayley, in 1878 and 1879.

The first written references were by

Incorrect "proofs" of the Four Color Con-

jecture were published soon after by Kempe and Tait. The error in Kempe's "proof" was found by Heawood [ H 3 ] in 1890; this error has reappeared in various guises in subsequent years.

Ore and Stemple [Osl]showedthat any counterexample to the conjecture must involve a map of at least 72 regions. The conjecture continued to provide one of the most famous unsolved problems in mathematics, until Appel and Haken affirmed it in 1976 [ A H l ] . It is an astonishing fact that several related, seemingly much more difficult, map-coloring problems have been completely solved. Chief among these is the Heawood Map-coloring Conjecture, which gives the chromatic number for every closed 2 -manifold other than the spere; we state the orientable case:

where

k

is the genus of the closed orientable

Heawood showed in 1980 [ H 3 ] that

x(sk) 5

f (k) ,

2 -manifold

Sk.

and in 1891 Heffter

[H4] showed the reverse inequality for a possibly infinite set of natural numbers k ; almost eight decades passed before it was shown that x(Sk) 1. f (k) , for all k > 0 . In 1965 this problemwasgiven the place of honor on the dust jacket for Tietze's Famous Problems

2

Chapt. 1

Historical Setting

of

Mathematics [T2]. An outline of the major portion of the solution now follows. The dual of a map drawn on S k is a pseudograph imbedded in

Sk

,

for

and it can be shown (see Section 8-4) that k > 0 ,

provided the complete graph

Heawood established ( * ) for

n = 7

Kn

x(Sk)

2 f (k) ,

has qenus given by

in 1890, and Heffter for

8 5 n 5 12 in 1891; Ringel handled n = 13 in 1952. The first major breakthrough occurred in 1954, when Ringel showed ( * ) for n : 5 (mod 12). During 1961-1965, Ringel treated the residue cases 7 , 10 , and 3 (mod 12) , while independently Gustin settled the cases

3 ,4

,

and

7

.

Gustin's method involved the powerful and

beautiful idea of quotient graph and quotient manifold, and relies can be regarded as a Cayley color graph for upon the fact that Kn a group presentation; thus graph theory, group theory, and surface topology are combining to solve this famous problem of mathematics. case cases

23;

In 1965, Terry, Welch, and Youngs announced their solution to 0. Gustin, Ringel, and Youngs finished the remaining residue

,

except for the isolated values n 1 8 , 2 0 , and their work was announced in 1968 [RYl]. In 1969, Jean Mayer (mod 12)

(a Professor of French Literature) [M4] eliminated the last three obstinate graphs by ad hoc techniques. Much of the work of Ringel, Terry, Welch, and Youngs was made possible by Gustin's theory of quotient graphs andquotientmanifolds; this theory was developed and modified by Youngs, who also introduced the theory of vortices [ Y 3 ] .

The theory is considerably more

general than was needed to prove the Heawood Map-coloring Theorem, and has been unified and developed in more generality by Jacques [JZ], in 1969. Jacques' results will be presented in Chapter 9, together with many applications to other imbedding problems in graph theory.

This will be a focal point of the text, and it

illustrates vividly the fruitful interaction among graphs, groups, and surfaces. We continue this development through the theory of voltage qraphs and by extending to nonorientable imbeddings. We consider the related structures of block designs and hypergraph imbeddings. In map automorphism groups we study groups acting on graphs ofgroups on surfaces. Finally, in studying change ringing, we use graphs of

Chapt. 1

Historical Setting

3

groups on surfaces to compose pieces of music. The conjunctions of graph theory, group theory, and surface topology described above are foreshadowed, in this text, by several pairwise interactions among these three disciplines. The Heawood Map-coloring Theorem is proved by finding, for each suface, a graph of largest chromatic number that can be drawn on that surface. Equivalently (as it turns out) we find, for each complete graph, the surface of smallest genus in which it can be drawn.

The extension of

this latter problem to arbitrary graphs is natural; the solution is particularly elegant for graphs which are the Cayley color graphs of a group. We are led in turn to the problem of finding, for a given group, a surface of minimum genus which represents the group in some way. Dyck [D5] (see also Burnside [B18], Chapters 18 and 19) considered maps, on surfaces, that are transformed into themselves in accordance with the fixed group r , acting transitively on the regions of the map. Any such map gives an upper bound for the parameter y ( T ) discussed in Chapter 7 of this text, as a "dual" formed in terms of Burnside's white regions gives a Cayley color graph for

r . (Cayley [C4] defined his color graphs as complete symmetric digraphs, corresponding to the choice r less the identity element as a generating set for r ; it is sensible to extendhisdefinition to any generating set for the group in question.) Brahana [B15] studied groups represented by regular maps on surfaces; these maps correspond to presentations on two generators, one of which is of order two.

In this context the group acts transitively on the edges

of the map, and again an upper bound for

y(T)

is obta.ined. In

Chapter 7, we regard r as acting transitively on the vertices of the map induced by imbedding a Cayley color graph CA(r) for r in a surface; in Chapter 4 , we show that the automorphism group of CA(T) is isomorphic to r , independent of the generating set

A

selected for r , so that in this sense C A ( r ) provides a "picture" of r . But more: many properties of r , such as commutivity, normality of certain subgroups, the entire multiplication table, can be "seen" from the picture provided by CA(T) . Thus it is natural to seek the simplest surface on which to draw this picture; this is given by the parameter y ( r ) . This point of view may give a surface of lower genus for a given group than the other two approaches listed above; for example, the group r = Z x Z 4 is toroidal for Dyck (or Burnside) and for

4

Historical Scttinq

y ( Z 2 x 2,)

Brahana, yet

=

0

Chapt. 1

.

There is one correspondence depicted in Figure 0-1 which we do riot discuss in this text:

unique group, the groups by

Lk

relation

il(Sk)

Q(Sk)

generators albla;lb;'.

,

to every surface

Sk

there corresponds a

called the fundamental qroup of the surface;

have been coniplrtely determined a1

, bl , .

..

--

they are yivcn

..

, a k , bk arid thc single defining -1 -1 a b a bk = e (see, for example, l S 6 1 . )

Each of the other five correspondences illustrated in Figure 0-1 (where the inner triangle commutes, for proper choice of A ) is germdne, as outlined a b o v e , to the conjunction of qraph theory, group theory, and surface topology described in this introduction and which we now begin to develop.

5

CHAPTER 2 A BRIEF INTRODUCTION TO GRAPH THEORY

In this chapter we introduce basic terminology from the theory of graphs that will be used in this text.

We will give several

binary operations on graphs; these will enable us to construct more complicated graphs, and hence to build up our store of examples of frequently encountered graphs. We emphasize that the material introduced here is primarily for the purpose of later use in this text; for a considerably more thorough introduction to graph theory, see [BCLl] or [H2].

Definition of a Graph

2-1.

A graph

Def. 2-1.

G

consists of a finite non-empty set

V(G)

of

vertices together with a set E(G) of unordered pairs of distinct vertices, called edges. If x = [u , v] E E(G) ,

u , v E V(G) , we say that u and v are adjacent vertices, and that vertex u and edge x are incident with each other, as are v and x . We also say that the edges [ u , v] and [u , w] , w # v , are adjacent. The degree, d(v) , of a vertex v is the number of edges with which v is incident. (Equivalently, d(v) is the number of vertices to which v is adjacent; i.e., for

If the vertices of

G

are labeled,

G

is said to be a

labeled graph. As a matter of notation, we usually write p = Iv(G) I ; q =

size of

~

G

IE(G)

1

is given by

.

The order of q .

G

uv for is given by

[u , v ] ; p . The

Example:

Chapt. 2

A B r i e f I n t r o d u c t i o n t o Graph T h e o r y

6

Let

be d e f i n e d b y :

G

then

G

may be r e p r e s e n t e d by e i t h e r F i g u r e 2 - l a

or 2-lb,

w h e r e t h e l a t t e r r e p r e s e n t a t i o n i s more a c c u r a t e , i n a sense we w i l l d e s c r i b e i n Chapter 6 .

F i g u r e 2-1.

Note : -

A g r a p h may b e more b r i e f l y d e f i n e d as J. f i n i t e o n c -

d i m e n s i o n a l s i m p l i c i a 1 complex.

Proof:

I n summing t h e d e g r e e s , e a c h e d g e i s c o u n t e d

e x a c t l y twice. Cor. 2 - 3 .

I n any g r a p h

t h e number of v e r t i c e s of odd d e g r e e

G ,

i s even.

2-2.

Dcf. 2 - 4 .

A

V a r i a t i o n s of G r a p h s

loop i s a n e d g e of t h e f o r m

vv

.

A multiple

a n e d g e t h a t a p p e a r s more t h a n o n c e i n E ( G ) edye i s a n o r d e r e d p a i r of d i s t i n c t v e r t i c e s . g r a p h allows m u l t i p l e e d g e s . and m u l t i p l e e d g e s . edge d i r e c t e d .

.

edge i s Adirected A multi-

A pseudograph a l l o w s loops

A d i r e c t e d graph

(digraph) has every

The c o r r e s p o n d i n g g r a p h ( w i t h a l l e d g e

d i r e c t i o n s d e l e t e d ) i s c a l l e d t h e underlying graph. graph h a s i n f i n i t e v e r t e x set. infinite -

F o r e x a m p l e , see F i g u r e 2 . 2 .

An

Additional Definitions

Sect. 2 . 3

loop

multiple edge

7

directed edge

a directed pseudograph

Figure 2-2.

The term "graph," unless qualified appropriately, disallows any and all of the above variations.

2-3.

Additional Definitions

Def. 2-5.

A graph H is said to be a subgraph of a graph G if V(H) g V(G) and E(H) c E(G) If V(H) = V(G) , H is called a spanning subgraph. For any $I # S G V(G) I the induced subgraph (S) is the maximal subgraph of G with vertex set S .

Notation:

For v E V(G) , G - v denotes (V(G) - v) For x E E(G) I V ( G - x ) = V(G) , and E(G - x ) = E(G) - x .

.

.

Certain subgraphs are given special names. by a series of definitions. Def. 2-6.

We indicate these

A of a graph G is an alternating sequence of vertices and edges vo , x l , v l I l ~ n - l ,l~v n (or, and ending briefly: vo , v l I l ~ n - l , ~ nbeginning ) with vertices, in which each edge is incident with the two vertices immediately preceding and following it; n is the length of the walk. If vo = vn I the walk is said to be closed; it is said to be open otherwise. The

...

...

8

A Brief Introduction to Graph Theory

Chapt. 2

walk is called a trail if all its edges are distinct, and a path if all the vertices are distinct. A gycle_ is a closed walk with n 3 3 distinct vertices (i.e., but otherwise the vi are distinct). vo = v n '

Two famous problems in graph theory may be described in terms of the above definitions. A graph is said to be eulerian if the graph itself can be expressed as a closed trail. (This corresponds to the "highway inspector" problem; eulerian graphs have been completely and simply characterized: see Harary IH21, p . 6 4 - 6 5 . ) A graph is said to be hamiltonian if it has a spanning cycle. (This corresponds to the "traveling salesman" problem; hamiltonian graphs have not been completely characterized. See Harary, p. 65-69, for some partial results.) Def. 2-7.

A graph G is connected if u , v E V ( G ) implies there exists a path in G joining u to v . A component of G i s a maximal connected subgraph of G .

Def. 2-8.

The distance, d(u,v) , between two vertices u and v of G is the length of a shortest path joining them if such exists; if not, d(u,v) = m .

Thm. 2-9.

A connected graph may be regarded as a (finite) metric

space. Proof:

See Problem 2 - 7 .

#

For a partial converse to the above theorem, see Chartrand and Kay LCKl]. By Theorem 2-9, every connected graph may be regarded as a topological space. (Actually, since the metric induces the discrete topology, we knew this already.) In Chapter 6 we will see that this is true in another sense also: that is every graph may be regarded as a subspace of R 3 , with all edges represented as straight lines. If we consider G as a topological space in this latter sense, then G is connected as a graph if and only if it is connected as a topological space (see Problem 2-8). The term "component" is easily seen to mean the same in both contexts. Furthermore, a graph (as a subspace of R3) is connected if and only if it is path connected; (see Problem 2.9.)

Additional Definitions

Sect. 2.3

Def. 2-10.

Two graphs

(G ;G2 1

,

Note : ___

uv

and

or

onto map is,

G1

G1

=

0 : V(G1) t

E(G1)

G2

are said to be isomorphic

if there exists a one-to-one,

G2) +

9

V(G2)

preserving adjacency; that

if and only if

O(u)B(v)

E

.

E(G2)

Isomorphism is an equivalence relation on the set of a l l graphs.

Notation:

Def. 2-11.

6(G) = minId(v) lv E V(G) 1 A(G) = max{d(v) / v E V(G) 1

If

6(G) = A(G) = r , r

degree Thm. 2-12.

Let

@ :

.

+

G

we say that

(If r = 3

V(G1)

.

.

, G

V(G2)

is regular of

is said to be cubic.)

give

GI

G2

2

then

;

~~

d(@(v)) = d(v)

Cor. 2-13.

Cor. 2-14.

,

for all

v

Proof:

See Problem 2-10.

If

G1

is regular of degree

G2

is regular of degree

V(G1)

E

. #

r

and

G1

G2,

then

r.

Let the vertices of a graph G1 have degrees and the vertices of a graph dl 5 d2 5 < dn ,

. ..

have degrees some

c1 5 c2

1 5 i 5 n ,

5

. ..

- cn

.

G1

and

G2

then

If

di # ci ,

G2 for

are not isomorphic.

The inverse of the above corollary need not be true; see Problem 2-3. Def. 2-15.

The complement G of a graph G has E(G) = {uvlu # v and uv ,d E(G)}

V ( G ) = V(G)

.

2-4.

Operations on Graphs

We now define several binary operations on graphs. follows, we assume that

V(G1)

n V(G2)

= @

.

In what

and

Chapt. 2

A Brief Introduction to Graph Theory

10

Notation:

2.)

2G nG

The joill G

=

=

= G

G U G (n-l)G U G, n F 3.

+

G2

has:

V(C) = V(G1) U V(G2) E(G) = E ( G ~ )u E ( G ~ ) u 3.)

The Cartesian product V(G) E(G)

4.)

= =

{v1v2/vic v ( G ~ ) ,i = 1 , 2 1 . G

=

G

x

G2

has:

V(G1) X V(G2) ~[(ul,u2),(vl,v2)l~ul = v1 and u2v2E E(G2) or u 2 = v2 and ulvl E E ( G 1 ) 1 .

The composition (or lexicoqraphic product) G = G1[G2] has: V(G) E(G)

= =

V(Gl) x V(G2) or { t (ul,u2),(vlfv2) 1 \ u l v l E E(G]) (u, = v1 and u2v2 E E(G2) 1 3 .

For additional products, see Harary and Wilcox [HWI]. We are now in a position to conveniently define several infinite families of graphs. Def.

2-17.

a.) b.) c.)

Pn denotes the p a t h of length n-1 ( i . e . of order n . ) Cn denotes the cycle of length n. Kn denotes the complete graph on n vertices; that is a l l

d.)

(;)

possible edges are present.

Kn denotes the totally disconnected graph on vertices; that is, E(En) = 9. Km,n denotes a complete bipartite graph:

-

KmIn = Km + (Equivalently,

K m,n

En.

is defined by:

n

Sect. 2.4

Operations on Graphs

f.)

K

P1'P2 graph:

I

denotes a complete n-partite

n'

* . -

an iterated join.

In the special case where

- pn ( = m, p1 = p2 = * . - complete n-partite graph:

We introduce for this graph. g.)

Q,

11

Kn (m)

say), we get a regular

as a shorter notation

denotes the n-cube and is defined recursively:

The complete bipartite graphs are a subclass of an extremely

--

important class of graphs

the bipartite graphs.

Def. 2-18.

A bipartite graph G is a graph whose vertex set V(G) can be partitioned into two non-empty subsets V' and V" so that every edge of G has one vertex in V' and the other in V".

Thm. 2-19.

A nontrivial graph G its cycles are even. Proof:

(i).

tite graph that

Let

G,

v1 E V ' ;

is bipartite if and only if all

v1v2..-vnv1

be a cycle in a bipar-

and assume, without loss of generality, then

vn E V",

and

n

must be even.

(ii). We may assume that G is connected, with only even cycles, since the argument in general follows directly from this special case. Consider a fixed vo t V(G). Let vi = {u t V(G) ld(u,vo) = i > , i = 0,1, n. Then n is finite, since G is connected, and

...,

A B r i e f I n t r o d u c t i o n t o Graph Theory

12

Vo

,V1 , . . . ,Vn

p r o v i d e s a p a r t i t i o n of

no t w o v e r t i c e s i n c e n t , or cycle.

are a d j a c e n t , s i n c e

I n f a c t , every edge i n

Letting

, v E Vi+l

,

.

NOW,

contains

G

a r e adja-

V2

Vi

i = 0,l , . . . , n - 1 .

for

Vi

for

uv,

i s of t h e f o r m

G

f o r some

be t h e u n i o n of t h e

V'

be t h e union of t h e

V"

V(G)

would c o n t a i n e i t h e r a 3 - c y c l e o r a 5-

G

u 6 Vi

where

G

V1

A l s o , no two v e r t i c e s i n

no 3 - c y c l e s .

Chapt. 2

i

odd,

and

e v e n , w e see t h a t

i

#

is bipartite.

T h i s c o m p l e t e s o u r b r i e f i n t r o d u c t i o n ; o t h e r t e r m s w i l l b e def i n e d , and t h e o r e m s d e v e l o p e d , a s n e e d e d .

2.5.

Problems

-G

is n o t connected, then

G

i s connected.

2-1.)

Prove t h a t i f

2-2. )

Given a n example t o show t h a t t h e c o n v e r s e n e e d n o t h o l d . A g r a p h i s s a i d t o be p e r f e c t i f no t w o v e r t i c e s h a v e t h e same d e g r e e .

2-3. )

of o r d e r 2-4.)

P r o v e t h a t no g r a p h i s p e r f e c t , e x c e p t

Show t h a t , e v e n t h o u g h 6

G2

and

K2

x

G = K 1' are b o t h r e g u l a r

K3

3, t h e y a r e not i s o m o r p h i c . i s b i p a r t i t e i f and o n l y i f b o t h

and d e g r e e

Prove t h a t and

K3 , 3

GI

x G2

are b i p a r t i t e .

G1

Give a n example t o show t h a t a

similar r e s u l t need n o t h o l d f o r t h e l e x i c o g r a p h i c p r o d u c t .

1-

2-5. )

Show t h a t

2-6.)

Show t h a t

2-7.)

P r o v e Theorem 2-9.

2-8.)

Consider t h e graph

G1

=

-+

G1[c21,

G2 = G1 U G

c2.

as a subspace of

R

3

.

Show t h a t

G

i s c o n n e c t e d as a t o p o l o g i c a l s p a c e i f a n d o n l y i f i t i s conn e c t e d as a g r a p h .

2-9. )

Show t h a t t h e f i r s t o c c u r e n c e of "connected" i n P r o b l e m 2-8 may be r e p l a c e d w i t h " p a t h - c o n n e c t e d . ' '

(Recall t h a t a path-

c o n n e c t e d t o p o l o g i c a l s p a c e m u s t be c o n n e c t e d , b u t t h a t t h e c o n v e r s e does n o t a l w a y s h o l d .

However, a c o n n e c t e d s p a c e

f o r which e v e r y p o i n t h a s a p a t h c o n n e c t e d n e i g h b o r h o o d must be p a t h - c o n n e c t e d . )

Sect. 2.5

2-10.) 2-11.)

Problems

13

Prove Theorem 2-12. Show that the set of all graphs, under the operation of Cartesian product, forms a commutative semigroup with unity

2-12.)

2-13.)

(i.e. a commutative monoid). Let G and H both be hamiltonian. Show that G[H] and G x H are hamiltonian also. Show that Qn is hamiltonian, for n 2 2. The line graph H = L(G) of a graph G is defined by V ( H ) = E(G), E ( H ) = {efle and f are adjacent, e,f E E(G)}. Show that L(Km,n) = K x Kn. (This result appears in Palmer

[P21 . )

This Page Intentionally Left Blank

15

CHAPTER 3 THE AUTOMORPHISM GROUP OF A GRAPH

In this chapter we show that there is associated, with each graph, a group, known as the automorphism group of the graph.

We

introduce various binary operations on permutation groups to aid in computing automorphism groups of graphs. Several powerful results relating graph and group products are stated, sometimes without proof (see [ ~ 2 ] for a further discussion); these results will not be used in the sequel. Indeed, the concept of automorphism group of a graph is, in the main, peripheral to the present course; it is introduced here primarily as one example of an interaction between graphs and groups. (In Section 2 of Chapter 4 we find a more direct bearing on subsequent material.)

3-1.

Def. 3-1.

Definitions

A one-to-one mapping from a finite set onto itself is

called a permutation. A permutation group is a group whose elements are all permutations acting on the same finite set, called the object set. is composition of mappings.) and

A

If

(The group operation X

the permutation group, then

the group, and

1x1

is the object set / A \ is the order of

is the degree.

A permutation P partitions its object set by the k equivalence relation x G y if and only if P (x) = y for some integer k. The equivalence classes are called

the orbits of X I under the action of P. If there is just one orbit in the action of A on X, then A is said to be transitive on X. If / A \ = 1x1, and if A is transitive on X, then A is said to be a regular permutation group. Def. 3-2.

Two permutation groups A and B are said to be isomorphic - (A B ) if there exists a one-to-one onto map

16

0 : A + B such that al , a 2 E A.

Def. 3 - 3 .

Chapt. 3

The Automorphism Group of a Graph

O(alaZ)

=

b(al)u(a,)

,

for all

Two permutation groups A and B (acting on object sets X and Y respectively) are said to be identical (or equivalent) (A 5 B ) if: (i) A B (given by 0 : A B) (ii) there exists a one-to-one, onto map f: X Y such that f(ax) = o(a)f(x), for all x t X and a t A. -f

-f

For a general treatment of permutation groups acting on combinatorial structures, see Biggs and White [BWl]. Here, we consider only the graph automorphism case. Def. 3-4.

Remark.

Def. 3-5.

An automorphism of a graph G is an isomorphism of G with itself. (The set of all automorphisms of G forms acting on the object set a permutation group, G ( G ) , V(G) .) G(G) is called the automorphism group of G . An automorphism of G, which is a permutation of V ( G ) , also induces a permutation of E ( G ) , in the obvious manner. An identity is a graph G having trivial automorphism group: that is, the identity permutation on V(G) is the only automorphism of G.

It is easy to see that the graph pictured in Figure 3-1 is an identity graph. That there is no identity graph of smaller order (other than K1) is established in Problem 3-1.

Figure 3-1

S e c t . 3-2

O p e r a t i o n s on P e r m u t a t i o n Groups

G(G)

Thm. 3 - 6 .

5

11

G(G).

proof:

Let

0:

G(G)

+

G(G)

and

f: V(G)

+

V(G)

b o t h be

i d e n t i t y maps, a n d o b s e r v e t h a t a d j a c e n c y i s p r e s e r v e d i n #

a g r a p h i f and o n l y i f n o n - a d j a c e n c y i s p r e s e r v e d .

3-2.

O p e r a t i o n s on P e r m u t a t i o n Groups

From a t h e o r e m d u e t o C a y l e y , w e r e c a l l t h a t a n y f i n i t e g r o u p

i s a b s t r a c t l y isomorphic ( a s opposed t o n e c e s s a r i l y b e i n g i d e n t i c a l ) with a permutation group; i n f a c t , i f t h e group then

is isomorphic t o a subgroup of

G

Sn.

h a s order

G

n

,

In this light, the

o p e r a t i o n s soon t o be d e f i n e d c o u l d be regarded as a p p l y i n g t o g r o u p s i n g e n e r a l ; however,

t h e d e f i n i t i o n s w i l l be g i v e n i n t e r m s

of a c t i o n upon a s p e c i f i e d o b j e c t set. Let X

and

A

Y

and

b e p e r m u t a t i o n g r o u p s a c t i n g o n o b j e c t sets

B

respectively.

W e d e f i n e t h r e e b i n a r y o p e r a t i o n s on t h e s e

permutation groups as f o l l o w s : Def. 3-7.

1.)

sum,

The

+

A

B

d i s j o i n t union

,

(or d i r e c t p r o d u c t ) a c t s on t h e X U Y; A + B = { a + b l a E A ,b E B I ,

and

(a+b)

r {

(2) =

az, bz,

2.)

The p r o d u c t , X

A

b) ( x

if z E Y . ( o r C a r t e s i a n p r o d u c t ) a c t s on

x

B = { a x b l a E A , b E B},

X

, yl

and

=

( o r w r e a t h p r o d u c t ) a c t s on a E A and a n y seq u e n c e b l t b 2 r . . . , b d ( w h e r e d = 1x1) i n B , t h e r e i s a u n i q u e p e r m u t a t i o n i n A[B] , w r i t t e n X

Y

as f o l l o w s :

(a;bl , b 2 , ( a ; b l ,b 2 , Note:

,

z E X

(ax , b y ) . The c o m p o s i t i o n , A[B] , (a

3.)

Y;

X

A x B

if

The o r d e r of

A[B]

f o r each

... , b d )

,

.. .

and , b d ) ( x i , y j ) = (axi

is

(A1

(Bid.

,b . y . ) , 1 3

18

The Automorphism Group of

Thm. 3-8.

A

+

B

A

x

Let

Proof:

Graph

a

Chapt. 3

B. O:A + B

-f

A

B

x

be given by

O(a+b) = ( a x b ) . #

A + B

Note that, in general, we cannot claim

~

B.

A

Computing Automorphism Groups of Graphs

3-3.

The following theorems indicate some connections between the graphical operations defined in Section 2-4 and the group operations defined above. The groups Sn , An , Zn , Dn are respectively the symmetric and alternating groups of degree

n

order n , and the dihedral group of order is due to Frucht [ F 5 ] . Thm. 3-9. Thm. 3-10.

If

G

is a connected graph, then

If no component of of

G2

Proof: Thm. 3-11.

Any graph G r = nr = 1 ,

Thm. 3-12.

then

G1

2n.

The first theorem

G(nG)

Sn[G(G)I,

is isomorphic with a component

G(G1 Li G2) = Ci(G1) + G(G2).

See Problem 3-2.

Let G = n 1G 1 u n2G2 U number of components of

Proof:

Note:

,

, the cyclic group of

#

... u G

nrGr , where n i isomorphic to Gi.

is the Then

Apply Theorems 3-9 and 3-10, using induction.

#

may be written as in Theorem 3-11, but if the theorem gives no information.

-

If no component of G1 Of G2 I then G(G1 +

is isomorphic with a component G2)

f

G(G1) + G(G2).

Proof: Apply Theorems 3-6 and 3-10, together with problem 2-6.

#

Sect. 3-3

The following two theorems are due to Sabidussi respectively). Def. 3-13.

19

Computing Automorphism Groups of Graphs

( [ S 2 1 and [ S 2 1

A non-trivial graph G is said to be prime if G = G1 x G2 implies that either G1 or G2 must be trivial (1.e. = K1). If G is not prime, G is composite. Two graphs G1 and G2 are relatively prime if GI = G3 G4 and G2 = G3 x G5 i m p l y G3 = K1.

Thm. 3-14.

b(G1 x

G2)

?

G(G1) x G(G2)

if and only if

G1

and

G2

are relatively prime. Def. 3-15.

Thm. 3-16.

The neighborhood of a vertex

u

N(u)

The closed neighborhood

=

is

“u]

If

G1

{v t V(G) [ uv =

E(G) J

.

is given by:

N(u) U {ul.

is not totally disconnected, then , if and only if:

G(G~[G~]) Z

G(G1) [ f i ( G 2 ) l

(i) if there are two vertices in same neighborhood, then

G2

G1

with the

is connected.

and (ii)

if there are two vertices in G1 with the same closed neighborhood, then G2 is connected.

We are now able to list the automorphism groups for several common families of graphs.

The Automorphism Group of a Graph

20

3-4.

Chapt. 3

Graphs with a Given Automorphism Group

Every finite group is the automorphism group of some

Thm. 3-18.

graph. For a proof of this theorem, due to Frucht [ F 4 ] , see Section

4-2.

Other Groups of a Graph

3-5.

Two rjraphs

Def. 3-19.

and

G

H

(with non-empty edge sets) are

said to he edge-isomorphic if there exists a one-to-one, onto map

$:

E(G)

+

E(EI)

preserving adjacency; that is

share a vertex in G if and only if @(x) , $ ( y ) share a vertex in H. 4 is called an edge-isomorphism.

x , y

If

Thm. 3-20.

G

and

€1

are isomorphic (with non-empty edge sets),

then they are edge-isomorphic. Proof: Def. 3-21.

See Problem 3 - 6 .

An induced edge-isomorphism is an edge isomorphism

E(H) determined by $ ( u v ) = OuOv , where * V ( 1 I ) is an isomorphism. An edge-automorphism of a non-empty graph G is an edge-automorphism of G with itself. (The set of all edge-automorphisms of G forms a permutation group, Gl(G), acting on the object set E(G) . ) G1(G) is called the edqe-automorphism group of G. Similarly, the induced 9 - a u t o m o r p h i s m group of G , G * ( G ) , is the set of all induced edgeautomorphisms of G under composition. $:

E(G)

$J:

V(C)

+

We note that G * ( G ) is a (possibly proper; see Problem 3-8) subgroup of G1(G). However, in almost every case, the three groups introduced in this chapter are isomorphic (see, for example, [ B C L L I ) ; denote the graph obtained by adding one edge to let Klf3

K1,3:

+

x

Sect. 3-6

Thm. 3-22.

Problems

21

# K1 , C ( G ) G*(G) if and only if G contains K2 as a component nor two or more isolated vertices. For

G

neither

Thm. 3-23.

Let (1)

(2)

Thm. 3 - 2 4 .

Let G(G)

E ( G ) # 0; then G1(G) G*(G) if and only if: not both C3 and K are components of G , and 1r3 none of K + x , K 4 - x , K4 is a component lI3 of G .

G

be a connected graph of order G1(G)

G # K1,3 + X

2

G*(G)

r

K4 -

p

2 3; then

if and only if X

r

K4.

Thus it is customary to focus attention on the automorphism group,

G(G). 3-6.

3-1.)

3-2.) 3-3.) 3-4.) 3-5.) 3-6.) 3-7.) 3-8.)

Problems

K1) of order five or less? (Hint: Check to see that every graph in Appendix 1, Harary [H2], with p 5 5, has at least one nontrivial automorphism.) Prove Theorem 3-10. Prove Theorem 3-17. Let G = K find G ( G ) . P,q,r' G(K4) z S4, yet K 4 is the 1-skeleton of the tetrahedron, and the symmetry group of the tetrahedron is A4. Explain! Prove Theorem 3-20. Show that the converse of Theorem 3-20 is false. Show that not all edge-automorphisms are induced. Does there exist

-

an identity graph (other than

This Page Intentionally Left Blank

23

CHAPTER 4 THE CAYLEY COLOR GRAPH OF A GROUP PRESENTATION I n t h i s c h a p t e r w e see t h a t a n y g r o u p may be d e f i n e d i n terms o f g e n e r a t o r s a n d r e l a t i o n s and t h a t c o r r e s p o n d i n g t o s u c h a p r e s e n t a t i o n t h e r e i s a u n i q u e g r a p h , c a l l e d t h e C a y l e y c o l o r g r a p h of t h e presentation.

A "drawing" o f

t h i s graph gives a " p i c t u r e " of t h e

g r o u p , from w h i c h may b e d e t e r m i n e d c e r t a i n p r o p e r t i e s o f t h e g r o u p . W e w i l l e s t a b l i s h some b a s i c r e s u l t s a b o u t C a y l e y c o l o r g r a p h s , i n -

c l u d i n g a r a t h e r n a t u r a l correspondence between d i r e c t p r o d u c t s o f g r o u p s and C a r t e s i a n p r o d u c t s o f a s s o c i a t e d C a y l e y c o l o r g r a p h s .

In

C h a p t e r 7 w e w i l l a s k which g r o u p s h a v e Cayley c o l o r g r a p h s t h a t c a n be r e p r e s e n t e d p r o p e r l y i n t h e p l a n e , and a s s o c i a t e d q u e s t i o n s .

In

C h a p t e r 9 a p p r o p r i a t e a n s w e r s w i l l solve t h e Heawood m a p - c o l o r i n g p r o b l e m , a s w e l l a s many o t h e r s .

The v o l t a g e g r a p h t h e o r y o f Chap-

t e r 1 0 , t h e b l o c k d e s i g n c o n n e c t i o n of C h a p t e r 1 2 , a n d t h e map automorphisms o f C h p a t e r 1 4 w i l l be e s p e c i a l l y n a t u r a l i n t h e cont e x t of Cayley g r a p h s .

And, e a c h c h a n g e - r i n g i n g

graph of Chapter

1 5 w i l l be a C a y l e y c o l o r g r a p h .

4-1. Def.

4-1.

r

Let

b e a g r o u p , w i t h {g1,g2,g3,...)

word

element set of

G.

f i n i t e product

flf2-. .fn

set

tgl , g 2 , g 3

element of

.. . for

,

then

r.

r

I

A

...

W

,

, g1-1

in

a s u b s e t of t h e

g1 , g 2 , g 3

where e a c h

-1 -1 ,g2 ,g3 ,

fi

...1 .

, ...

is a

is i n the If every

c a n b e e x p r e s s e d a s a word i n g 1 ' 4 2 ' 4 3 g1

, g2

I

g3 I . .

.

I

are s a i d t o be g e n e r a t o r s

A r e l a t i o n i s an e q u a l i t y between t w o words i n

91 ' 4 2 I 4 3 Thm. 4 - 2 .

Definitions

I . . .

-

Given a n a r b i t r a r y s e t o f symbols and a n a r b i t r a r i l y p r e s c r i b e d s e t ( p o s s i b l y e m p t y ) of r e l a t i o n s i n t h e s e symb o l s , t h e r e i s a unique ( u p t o isomorphism) group w i t h t h e

24

The C a y l e y C o l o r Graph o f a Group P r e s e n t a t i o n

Chapt. 4

symbols a s g e n e r a t o r s and w i t h s t r u c t u r e d e t e r m i n e d by the prescribed relations.

see [MKSl]. )

(For a proof,

Dcf.

4.3.

~-

i'

If

g1 , g 2

i s g e n e r a t e d by

relation in

#

I

, . ..

yj

and i f e v e r y

c a n b e d e d u c e d from t h e r e l a t i o n s

G

.

R' , . . , P' , Q = Q ' , R

, R

P =

I ' = ( g1 , g 2 , R' }, and t h e r i g h t hand s i d e o f t h e e q u a t i o n i s said t o be a p r e s e n t a t i o n PI

,Q

= Q'

g3 ,

. .. 1

of

I'

P =

.

then we w r i t e

...

=

A p r e s e n t a t i o n i s said t o be a f i n i t e l y gener-

a t e d ( f i n i t e l y r e l a t e d ) i f t h e number of g e n e r a t o r s ( d e fining relations) is finite.

A finite presentation is

boLh f i n i t e l y y e n c r a t e d a n d f i n i t e l y r e l a t e d .

Thm. 4 - 4 .

Every f i n i t e group has a f i n i t e p r e s e n t a t i o n . Proof:

Take

' i t s e l f a s t h e s e t of g e n e r a t o r s , w i t h all

r e l a t i o n s o f t h e form group o p e r a t i o n .

gig,

= qk

,

a s d e t e r m i n e d by t h e

(1.e. t h e m u l t i p l i c a t i o n t a b l e s e r v e s #

as a f i n i t e presentation.) pef.

4-5.

For e v e r y group p r e s e n t a t i o n t h e r e i s a s s o c i a t e d a Cayley c o l o r graph:

. -

t h e group;

t h e v e r t i c e s c o r r e s p o n d t o t h e e l e m e n t s of

next,

imacjine t h e g e n e r a t o r s o f t h e g r o u p t o

be a s s o c i a t e d w i t h d i s t i n c t c o l o r s .

v2

correspond t o group elements

If vertices

q1

and

g2

v1

and

respec-

t i v c l y , then t h e r e i s a d i r e c t e d edge (of t h e color (or l a b e l ) of g e n e r a t o r glh

= g2;

h)

from

see F i g u r e 4 - 1 .

F i g u r e 4-1.

vl

to

v2

i f and o n l y i f

Sect. 4-1

Definitions

Let P be a presentation for the qroup Cayley color graph of P for r by C p ( r ) , by

C?(;'),

where

A

25

; we denote the or (when convenient)

denotes the generating set.

(Since a group

may have more than one generating set, the Cayley color graph depends on A , as well as 1'). Then CA(r ) is a labeled, directed graph, with a color (or label) assigned to each edge.

We

observe that the following correspondences occur: Group element

vertex

generator

a set of directed edges of the same color

inverse of a generator

the same set of edges (now

word

directed against the arrow) walk

multiplication of elements identity word

closed walk

solvability of

(weakly) connected di-graph

Note:

~

Cayley Color Graph ____--

rx = s

succession of walks

A characterization is given in [MKSl] of those graphs

G

which can be oriented and colored so as to form Cayley color graphs. Historical

note:

Max Dehn (in 1911) formulated three fundamental decision problems concerning group presentations. One of these is: "determine in a finite number of steps, f o r two arbitrary words

W

and

W'

in

the generators, whether W = W' or not." Equivalently: "construct the Cayley color graph for a given group presentation." The term "Connected" may have a "stronger" meaning for directed graphs than for graphs in general, since we may be allowed to travel only in the direction of the arrow along a given directed edge. Def. 4-6.

A directed graph

D is said to be strongly connected if, for every pair u , v of distinct vertices, there is a directed path from u to v. D is said to be unilat-

erally connected if, for every pair of distinct vertices, one is joined to the other by a directed path.

D

is

The C a y l e y C o l o r Graph of a Group P r e s e n t a t i o n

26

Chapt.

4

c a l l e d weakly connected i f t h e ( u n d i r e c t e d ) pseudograph underlying

i s connected.

D

F o r e x a m p l e , see F i g u r e 4-2, D'

where

D

is strongly connected,

--

i s u n i l a t e r a l l y connected ( b u t n o t s t r o n g l y c o n n e c t e d ) and

D"

i s weakly connected ( b u t n o t u n i l a t e r a l l y c o n n e c t e d ) .

D:

A

D':

D":

F i g u r e 4-2.

4-2.

Autornorphisrns

W e h a v e p r e v i o u s l y d e f i n e d a n automorphisrn of a g r a p h

V(G)

p e r m u t a t i o n of

preserving adjacency).

G

(as a

An a u t o m o r p h i s m of a

d i r e c t e d g r a p h m u s t p r e s e r v e d i r e c t e d a d j a c e n c y ; a n d a n autornorphisrn

of a C a y l e y c o l o r g r a p h m u s t a l s o p r e s e r v e t h e c o l o r c o r r e s p o n d i n g t o e a c h a d j a c e n c y . We summarize i n :

Def.

4-7.

An a u t o m o r p h i s m mutation in

T

ti

and

of h

of a

Cayley c o l o r graph

V(CA(I'))

in

such t h a t ,

glh = g 2

A,

CA(T)

f o r each

i s a per-

g1 , g 2

i f and o n l y i f

0 (gl)h = 8(G2).

E q u i v a l e n t l y (see P r o b l e m 4 - l ) , C,(i')

B(gh)

i f and o n l y i f : = B(g)h;

i.e.

f o r each

g

8 in

i s a n autornorphism of

r

and g e n e r a t o r

t h e d i a g r a m i n F i g u r e 4-3

I 0 t

% 4-=

0 (g

F i g u r e 4-3

commutes.

h

in

A,

Automorphisms

S e c t . 4-2

21

A s e x p e c t e d , t h e c o l l e c t i o n o f a l l automorphisms of

forms a g r o u p , c a l l e d t h e automorphism g r o u p of n o t e d by Thm. 4 - 8 .

.

G(Cb(I') )

r

and de-

be any Cayley color g r a p h f o r t h e f i n i t e

:

r

t i ( C A ( l ' ) )2

then

sentation selected for Proof:

,

The n e x t r e s u l t i s p e r h a p s n o t e x p e c t e d .

C,(I')

Let

group

CA(r')

CA(T)

Define

r

ii:

( i n d e p e n d e n t of t h e p r e -

G.)

+

G(CA(l'))

rz(g) = 8

by

where

9'

u . v ( C A ( I ' ) ) + V ( C A ( r ) ) i s g i v e n by 0 ( g i ) = g g i . 4' 9 E G(CA(T)). Clearly 0 i s oneF i r s t w e show t h a t e 4 9 t o - o n e and o n t o ( a n d hence p e r m u t e s V(CA(T))). A l s o , 0 (g.h) = g ( g . h ) = ( g g i ) h = tI ( g i ) h , 4 1 9 well-defined.

so t h a t

a

is

preserves products: a(gg*) = 8 * degg * ( g i ) = g g * g . = 0 ( g * g i ) = 8 ( 6 *(gi)) = 99 1 g 9 9 ( 0 0 * ) (gi) ; t h a t i s ( r ( g g * ) = a ( g ) t x ( g * ) . g q I t i s c l e a r t h a t u i s one-to-one, s i n c e

u

Now,

0

f i n e d by

ker

= {el.

N

I t r e m a i n s t o show t h a t 0

t G(CA(r)).

r.

t i t y of

Let

U(e)

Now any

g*

i n the generators for where

h,

e = o

4 ' proof.

=

i.e.

is a generator f o r

so t h a t

is onto.

a

where

r

in

r' ;

is onto.

a g ,

r

Let

e

i s t h e iden-

c a n b e w r i t t e n a s a word a a a m g* = h l h hm , 1 2 and a: = ?- 1. Then

This completes t h e

#

From t h e above t h e o r e m ( a n d i t s p r o o f ) it i s e v i d e n t t h a t any v e r t e x of

CA(r)

can b e l a b e l e d w i t h t h e i d e n t i t y

e

of

r ,

and

t h a t o n c e t h i s h a s been done ( f o r f i x e d a s s i g n m e n t o f c o l o r s t o t h e generators) a l l o t h e r vertex labelings a r e determined; t h a t i s ,

C,(r)

is vertex transitive.

p e r m u t a t i o n g r o u p , on

We 3-18:

Moreover,

V (C, (I') )

G(CA(T))

is a regular

.

a r e now a b l e t o p r o v i d e a p r o o f of F r u c h t ' s Theorem, Theorem Every f i n i t e g r o u p i s t h e automorphism g r o u p of some g r a p h .

The C a y l e y C o l o r Graph of a Group P r e s e n t a t i o n

28

Proof:

Let

set for 4-8,

1'

.

Form t h e C a y l e y c o l o r g r a p h

w e know t h a t t o a graph

Cn(l')

Replace each edge

,u i j ,u i j

, v 3. .

1'

G(CA ( 1 ' ) )

.

(gi

A

let

,9 . 1 , 2k

-

=

, A 2 , . .. ,

{A,

uij (u. . 11

1'.

An}.

gj - gidk , by a p a t h : w e a t t a c h a new p a t h

5

1 ( 2 k ) (1 5 k

n) ;

see F i g u r e 4 - 4 ,

I n t h i s way t h e " n o n - g r a p h i c a l "

of d i r e c t i o n and l a b e l , p r e s e n t i n G.

: by Theorem

I t only remains t o c o n v e r t

where

3

A t vertex

P. . (P. . ' ) of l e n g t h 11 1 3 f o r t h e case k = 2 . i n t o t h e graph

be a g e n e r a t i n q

C,(l')

h a v i n g t h e same a u t o m o r p h i s m g r o u p ,

G

T h i s i s done as f o l l o w s : vi

A

b e a f i n i t e g r o u p , and l e t

I'

4

Chapt.

,

C,(I')

It is clear that

features

a r e incorporated

G(G)

G(CA(I'))

#

1'.

?

V.

u..

17

u

v

i j

j

F i g u r e 4-4

4-3

Properties

I t i s c l e a r t h a t e v e r y Cayley c o l o r g r a p h i s b o t h r e g u l a r and

c o n n e c t e d ( a s a g r a p h ) ; t h e c o n v e r s e i s not t r u e

(see P r o b l e m 4 - 1 0 ) .

F o r t w o c h a r a c t e r i z a t i o n s of g r a p h s w h i c h may b e r e g a r d e d as C a y l e y c o l o r g r a p h s , see [ J 2 ] a n d [MKSl]. W e may s t u d y a d d i t i o n a l p r o p e r t i e s f o r

r

( a p a r t from t h e

m u l t i p l i c a t i o n t a b l e so c o n v e n i e n t l y s u m m a r i z e d i n

cA(T)

Cn(l'))

from

as follows:

Thm. 4 - 9 .

r

i s c o m m u t a t i v e i f a n d o n l y i f , f o r e v~. e r y p a i r of g e n e and

h. 3 '

t h e walk

rators

hi

Proof:

See Problem 4-2.

-1 -1 h.h.hi h 1 3 j

i s closed. #

S e c t . 4-3

Def.

4-10.

Properties

29

r

An e l e m e n t o f a g e n e r a t i n g s e t f o r a g r o u p

is said

t o b e r e d u n d a n t i f i t c a n be w r i t t e n a s a word i n t h e A generating set i s said t o be

remaining g e n e r a t o r s .

minimal i f it c o n t a i n s no redundant g e n e r a t o r s . Example:

{x2

, x3 i

i s a minimal g e n e r a t i n g s e t f o r

Thm. 4-11.

r

Let

2 = (x1x6 = e), 6 i s a g e n e r a t i n g set w i t h fewer elements.

{x }

even though

be a f i n i t e ( i n f i n i t e ) group.

i s redundant colored

h

A generator

h

i f and o n l y i f t h e d e l e t i o n o f a l l e d g e s in

l e a v e s a s t r o n g l y (weakly) con-

Ca(T)

nected d i r e c t e d graph. Proof: Thm.

4-12.

If

#

S e e P r o b l e m 4-3.

i s n o t r e d u n d a n t , t h e r e m o v a l of a l l e d g e s col-

H

ored

h

leaves a c o l l e c t i o n of isomorphic d i s j o i n t

r

subgraphs, each r e p r e s e n t i n g t h e subgroup of r a t e d by t h e g e n e r a t i n g s e t o f

T

Let

#

b e a f i n i t e g r o u p w i t h minimal g e n e r a t i n g s e t

{hl , h 2 ,

... , h r l

,

R

and

group w i t h g e n e r a t i n g set C1

gene-

h.

See Problem 4 - 4 .

Proof: Thm. 4-13.

minus

r'

, C 2 , . . . , Ck ,

graph chl(r) edges c o l o r e d

a ( n e c e s s a r i l y p r o p e r ) sub{h2

, h3 , . . .

hr].

Let

b e t h e weak c o m p o n e n t s o f t h e d i r e c t e d

CA(T)

o b t a i n e d from hl.

R

Then

is

by d e l e t i n g t h e

r

normal i n

i f and

o n l y if t h e d e l e t e d d i r e c t e d e d g e s f r o m a n y g i v e n component

Ci

Proof:

( i ) Assume t h e c o n d i t i o n h o l d s .

a l l l e a d t o a s i n g l e o t h e r component

be t h e component c o n t a i n i n g

r

f

T.

W e m u s t show t h a t

...

bl b2 al a 2 b . = t 1.

b . = +1

bm a m r If

and

where

hl

v

ai

occurs i n

t i m e s with

corresponding t o

r

w -v

ending i n

components,

,

e rgr

-1

C1

Let

g E C1

let E C1.

exactly

b . = -1 1

l e a d s from

e

cl+w-v'

,

j'

= R

and

We write

r =

r

and

i s a g e n e r a t o r of

r

C

w

t i m e s with

t h e n t h e walk

(in

C1)

The w a l k

through

The C a y l e y C o l o r Graph of a Group P r e s e n t a t i o n

30

4

ChapL.

now l e a d s t o a n o t h e r -b -b -b v e r t e x i n ClSw-" , and t h e w a l k a ... "2 2"1 1 ' m -1 correspondins to r , r e t u r n s u s t o cl.

corresponding t o

in

q

cl+w-v

( i i ) Suppose t h a t e d g e s c o l o r e d

C1

to

Ci

and

Then t h e r e e x i s t s that

3 , 9 E Cl

I t now follows t h a t , f o r

is obtained. =

hr

.

#

r,

t h e cle-

hl

( t h i s can be

I,/\)

a Cayley color g r a p h o f

by Theorem 4 - 1 3 ) ,

( T h i s " s h r i n k i n g " may be d e s c r i b e d by a d j o i n i n g , t o G

,

the additional relations

h2 = h j =

= e.)

I n g e n e r a l , given a Cayley c o l o r graph i? of

group

I'

,

CA(r)

w h e t h e r a sub-

i s normal o r n o t , w e o b t a i n a S c h r e i e r ( r i g h t )

!I

t h e v e r t i c e s a r e t h e r i g h t cosets o f

coset g r a p h a s f o l l o w s : -_I

r ,

so

U. 3 '

(i.e. the right cosets) are the

T/i2

the defining relations f o r

...

-1

(as above) normal i n

62

l e a d from

c F Cl.)

hl qhl

such t h a t

and r e s t o r i n g t h e e d g e s c o l o r e d

done unambiguously,

hl

(Ariain a s s u m e

By s h r i n k i n g t h e s e c o m p o n e n t s , e a c h t o a

(G).

single vertex,

# j.

i s n o t n o r m a l i n !~

0

ments of t h e f a c t o r g r o u p components of

1

C.

and t h e r e i s a n e d g e d i r e c t e d f r o m

6 E A , i f and o n l y i f 6 F 9-1S29'.) T h a t Rgd t h e r i g h t c o s e t s of

Qg t o

i2g'

,

in

labeled with

Qg6 = S2g' ( i . e . i f a n d only i f is a r i g h t cosct f o l l o w s f r o m t h e f a c t t h a t

R

in

I'

partition

r .

N o t e t h a t a Schreier

c o s e t g r a p h may a c t u a l l y be a p s e u d o g r a p h , a s l o o p s a n d / o r m u l t i p l e e d g e s may r e s u l t f r o m t h i s p r o c e s s .

R

For t h e s p e c i a l c a s e

=

{el,

t h e S c h r e i e r coset g r a p h is j u s t t h e C a y l e y color g r a p h

CA(r). S e v e r a l of t h e i d e a s d i s c u s s e d a b o v e a r e i l l u s t r a t e d i n F i g u r e

4-5.

Note that

R = Z3

S3 ,

i s normal i n

Z2 x Z4

Note t h a t t h e s u b g r o u p of o r d e r

i n Figure 4-6.

i s normal i n

Z2

Z4

,

see

S 3 = D3

Z2 x

( i n Figure 4-5);

normal h e r e , e i t h e r .

D4.

but not i n

o b v i o u s way t o t h e g r o u p s

Zn

and

A4.

but not i n

A s a f u r t h e r example, c o n t r a s t t h e groups

and

6

,

n

2

3.

[ g ,461

and *

(96 , g )

in

Cc\(T)

r

For example,

of o r d e r

2

,

r is not we a d o p t

t h e s t a n d a r d convention of r e p r e s e n t i n g t h e t w o d i r e c t e d edges (g ,g6)

, as

T h i s comment e x t e n d s i n a n Dn

t h e s u b g r o u p g e n e r a t e d by

For a g e n e r a t o r

D4

g e n e r a t e d by

2

by a s i n g l e u n d i r e c t e d e d g e

Sect.

4-3

31

Properties

r = (12)

*

rs

rs

1

13)

s3’z3

A4

F i g u r e 4-5.

3

r = (13) s = (1234)

rs

sr rs2

z2

x

z4:

r 2 = s4 = r s r s - l = e

D ~ :r 2 = s4 = ( r s ) 2 = e

F i g u r e 4-6

I n F i g u r e 4-7 w e g i v e t h e S c h r e i e r c o s e t g r a p h f o r in

!’

= D4

(see Figure 4-6).

F i g u r e 4-7.

i2 =

{e,r},

The C a y l e y C o l o r Graph of a Group P r e s e n t a t i o n

32

W e close t h i s s e c t i o n w i t h a theorem due t o G r o s s

Thm. 4 - 1 4 .

Chapt. 4

[G4].

Every connected r e g u l a r graph of even d e g r e e u n d e r l i e s a S c h r e i e r c o s e t graph.

Products.

4-4.

W e now d e v e l o p a r e l a t i o n s h i p b e t w e e n t h e d i r e c t p r o d u c t f o r Recall t h e following

g r o u p s and t h e C a r t e s i a n p r o d u c t f o r g r a p h s . from g r o u p t h e o r y : Def.

4-15.

Let

rl

1

with

E 12.

h d

r2

and

Il

ri

Then

subgroup of

b o t h b e s u b g r o u p s o f t h e same g r o u p

I2 Il

= {el

and

gh = hg

I 2 ={qhlg c

x

g F l1

for all

,

l l

il

called t h e direct product of

I ,

,

is also

6 i

h

and

r2. If

=

T2

= (

rl

then

r2

,,..

(kl

l1

km+l

=

(kl

,kmlwl

. - -=

=

, . . . , k n l ~ r + l -, ...

,knlwl

=

w

- *

*..

=

= =

w

for

II

,

x

and

wr + s rfs

e!, I J

1

J

j

5 n>

called t h e standard presentation

!'

2' T h i s b i n a r y o p e r a t i o n may b e e x t e n d e d t o t h e c l a s s of a l l i Y 1

rl

g r o u p s , by n o t i n g t h a t t i t y of

r2'i

and d e f i n i n g

r;

, T2

=

l V 1

ri

=

rl

{(g , e 2 )lg E

= { ( e , , h ) Ih E

{(q , h ) lq E

(ql , h l ) ( g 2 , h 2 ) = (q1g2 , h l h 2 )

,el

rl

,h E

,e2

i s t h e iden-

i s t h e i d e n t i t y of I,,)

with

giving t h e group operation.

A l s o r e c a l l t h e f o l l o w i n g ( s e e , f o r e x a m p l e [ B M l , P. Thm.

,

= k.k.k_lkT1 = e

1c i - m

for all

is a presentation for

e)

4-16.

3481):

(The F u n d a m e n t a l Theorem of F i n i t e A b e l i a n G r o u p s ) : Let

I = Z m i = 2

be a f i n i t e a b e l i a n g r o u p of o r d e r

I'

1

'm

2

, ... , r

. * .

and

'm r ir

where 1

m. = n ;

i=l decomposition is unique.

m

i

n;

divides

furthermore,

then

m

i-1

this

rl},

S e c t . 4-4

Products

rn

( W e assume

rn Def.

4-17.

> 1

,

33

unless

n = 1

,

i n which c a s e

r = 1.)

=

r

The number

t h e abelian group Theorem 4 - 1 6

is called the

o f Theorem 4-15

of

1'.

c o m p l e t e l y s p e c i f i e s t h e s t r u c t u r e o f f i n i t e abe-

The n e x t t h e o r e m s p e c i f i e s , a s a c o r o l l a r y , a C a y l e y

l i a n groups.

W e f i r s t e x t e n d t h e de-

c o l o r graph f o r every f i n i t e abelian group.

f i n i t i o n o f C a r t e s i a n p r o d u c t f o r g r a p h s t o Cayley color g r a p h s , i n t h e n a t u r a l way.

Def.

4-18.

The C a r t e s i a n p r o d u c t ,

CA

1

(rl)

C a y l e y color g r a p h s i s g i v e n b y : v(C,

(rl))

1 (9; ,g;)

(i)

x

V(C

(r2));

A2 by a n e d g e c o l o r e d g1

in

=

g;

V(C,

o of t w -

(rl)

1 (9, , g 2 )

and

,

CA2(r,)

X

cn2(r2)) = is joined to

h

i f and o n l y i f e i t h e r :

and

g2h

gi

,

for

h

a generator

and

g l h = g;

,

for

h

a generator

=

A2

or ( i i ) g 2 = gi

in F i g u r e 4-8 and

shows

Al.

CA1(z3)

x

cA2(z2)

Z2 = ( y l y 2 = e ) .

Figure 4-8

I

where

Z 3 = ( x l x 3 = e!

The C a y l e y C o l o r Graph of a Group P r e s e n t a t i o n

34

Thrn. 4 - 1 9 .

Let

C

i'

(I'i)

4

be t h e C a y l e y c o l o r g r a p h a s s o c i a t e d w i t h

PI

presentation

f o r group

r1 ,

be t h e s t a n d a r d p r e s e n a t l o n f o r

F i r s t we note t h a t

Proof

Chapt.

V(C,

1

,.

1 ,2

1 =

l1

I

( I l )

A

*.

P

Let

Then

CP2(l2))

=

11)) ' v(c ( 2)) = v(cP(ll 1 2 ) ) . We now show 1 p2 t h a t t h e e d g e s e t s of t h e two C a y l e y c o l o r g r a p h s c o i n -

v(cp

cide ( i n colored directed adjacency.) (1)

,<ji)

Let

edqe c o l o r e d

some

1

i

generator of

so t h a t

5

(gl , q 2 ) h e 301ned t o (gi by a n for h in Cp(ll 12). Then h = k 1 ' n. If 1 I_ i 2 m , t h e n h 1 s a

rl ,

gi = ylh

c o l o r e d e d g e in

and

gi = g 2 ;

and DP(!'l

y,

r2)

i.e.

t h i s directed,

i s al.so i n c p l ( " l ) '
i t h e r

or

0.

(Furthermore,

y ( C ) = *,M(G)

i f a n d only i f

- 0

y,(C)

G

Fi

i f and o n l y i f

is a c a c t u s w i t h v e r t c x -

disjoint cycles.) Dcf.

A ___ c a c t u s i s a connected

6-30.

block is Def.

c y c l e or an cdge.

d

A splittinq

6-31.

trec

Tor

T

G - C(T)

( p l a n a r ) qraph i n which every

tree o f G

a connected graph

Thm.

is a spdnning

h a s odd s i z e .

?'he f o l l o w i n y c h a r a c t e r i z a t i o n Junqerman

G

s u c h t h a t a t most o n e c o m p o n e n t of

i s due,

independently,

to

[ J 7 1 a n d Xuong [X2].

6-32.

A graph

G

i s u p p e r imbeddable i f a n d o n l y i f

G

has

a splitting tree, T h u s , f o r e x a m p l e , w e g e t a n i m m e d i a t e p r o o f of T h e o r e m 6 - 2 5

m e r e l y by t a k i n q

T = K

1,n-1. h a s qiven a s u f f i c i e n t c o n d i t i o n f o r upper

"11

Ncbesky irnbeddability.

F i r s t , w e need

_ Def. _6_ - 3 3_ . A graph

every

G

is said to b e l o c a l l y connected i f , f o r

v E V(G),

t h e sct

NG(v)

of v e r t i c e s

Genus Formulae f o r G r a p h s

S e c t . 6-5

adjacent t o induced by Thm.

6-34.

If

v

i s non-empty

NG(v)

67

and t h e s u b g r a p h o f

G

is connected.

is c o n n e c t e d and l o c a l l y c o n n e c t e d , t h e n

G

G

is

upper imbeddable. A l t h o u g h n o w o r k a b l e f o r m u l a i s known f o r t h e g e n u s of an a r b i t r a r y graph, maximum g e n u s . graph

Xuong [ X l l d e v e l o p e d t h e f o l l o w i n g r e s u l t f o r Let

CO(H)

d e n o t e t h e number of c o m p o n e n t s of

o f o d d s i z e , and f o r

H

< ( G ) = min

G

connected set

f,o

(G

-

E(T)),

t h e minimum b e i n g t a k e n o v e r a l l s p a n n i n g t r e e s Thm. 6 - 3 5 .

T

The maximum g e n u s of t h e c o n n e c t e d g r a p h

of

G.

G

Then:

is given

by

H:

Q:

0

F i g u r e 6-3

6-5.

Genus F o r m u l a e f o r G r a p h s

P r i o r t o t h e work of Jungerman a n d Xuong, Theorems 6-25, 6-26,

6-27,

and 6-28 a n d t h e work of R i n g e i s e n [ R 6 ] r e f e r r e d t o

above g a v e t h e o n l y known n o n - t r i v i a l

f o r m u l a s for maximum g e n u s .

68

I m b e d d i n g P r o b l e m s i n Graph T h e o r y

Chapt. G

N o t v e r y many more f o r m u l a s a r e known f o r t h e g e n u s p a r a m e t e r ;

we

l i s t some of t h e s e below.

F o r m o s t of w h a t e l s e was known u p t o

1 9 7 8 , see T a b l e 1 of S t a h l

[S91.

Thm.

6-36.

( R i n g e l [RlO] ; B e i n e k e a n d H a r a r y y(Qn)

+

1

=

[BHl])

~ " - ~ ( n - 4 ) n, 1 2 .

Thm. 6 - 3 7 .

Thm.

6-38.

( R i n q e l and Y o u n q s

Thm.

6-39.

(White [W51; see a l s o [ R Y S ] , f o r t h e case

Th.

6-40.

[RYl])

2

=

6-42.

(n-2) , 2

(3n-2) (n-1) 2

6-43_.

n

2

2.

,

n 1.1.

( J u n g e r m a n [ J 5 ] ; see a l s o Garman [Gl])

m # 3.

Y ( K ~ ( ~) ) = ( m - 1 ) 2 , Thm.

even,

n

for

(St-ah1 a n d White [ S W l l ) ~ ( ~ 2 n , 2 n , n=)

Thm.

1)

=

( S t a h l a n d White [SWlJ) Y(Kn,n,n-2)

Thm. 6 - 4 1 .

m

(Junyerman and Rinqel

[ J K 4 ] ; see also Gross and A l p e r t

[GA11) Y(Kn(2)) = Thm.

6-44.

,

for

n

2

z ',

(mod 3 ) .

( R i n g e l [R16]) For

Thm. 6 - 4 5 .

( n - 3 ) (n-1) 3

2

5 n

J 5,9

(White [W71) gree, cycles.

q

Let

edges,

Let

(mod 1 2 ) , y ( K 2

H

G

k

have

d e c j r e e less t h a n t w o .

have

Then

= JI -( n - 2 ) (n-3))

6

1 '

v e r t i c e s of p o s i t i v e de-

p

non-trivial

2n ( n

Kn)

x

2

c o m p o n e n t s , and n o 3-

1)

v e r t i c e s a n d maximum + n(ny - p)

y ( G [I11 ) = k

.

S e c t . 6-5

Genus F o r m u l a e f o r G r a p h s

Cor. 6-46.

G

Let

have n o 3 - c y c l e s .

Then

69

y(G[K2])

= y(G[K2])

= 6(G).

The s p e c i a l c a s e o f Theorem 6-45 when and

H =

i? m

is b i p a r t i t e ,

G

m

h a s been g e n e r a l i z e d t o i n c l u d e

2

p

3,

o d d , b y Abu-Sbeih

and P a r s o n s [APl].

see B e h z a d ,

F o r a v e r y r e a d a b l e p r o o f of Theorem 6-36, C h a r t r a n d and L e s n i a k - F o s t e r

[BCLl].

Theorems 6-37

are

a n d 6-39

p r o v e d u s i n g s c h e m e s i d e n t i c a l or s i m i l a r t o t h a t d i s c u s s e d i n t h e next section.

Theorem 6 - 3 8 w i l l b e d i s c u s s e d a t l e n g t h i n C h a p t e r 9 .

Theorems 6 - 4 0 a n d 6 - 4 1

a r e o b t a i n e d b y means o f v o l t a g e g r a p h c o n -

s t r u c t i o n s (see Chapter 1 0 ) .

Theorems 6-42

,

u s i n g c u r r e n t graph c o n s t r u c t i o n s (see Chapter 9 ) 6-43

-

are e s t a b l i s h e d

and 6-44

a s i s Theorem

although t h e l a t t e r i s i n t h e c o n t e x t of branched c o v e r i n g

s p a c e s (see C h a p t e r 1 0 . ) F o r an o u t l i n e o f t h e s u r g i c a l p r o o f o f Theorem 6 - 4 5 , f i r s t w e n o t e t h a t w e may assume

the following:

( t h e g e n e r a l r e s u l t w i l l t h e n f o l l o w from t h e B a t t l e , and Youngs T h e o r e m ) .

W e t h e n show t h a t

G[H]

has

H

with

qH

H a r a r y , Kodama,

can have a t m o s t

G[H]

2pGqH t r i a n g u l a r r e g i o n s i n a n y i m b e d d i n g ( w h e r e and

we offer

t o b e connected

G

G

has order

pG

e d g e s ) , and p r o c e e d t o c o n s t r u c t an imbedding of

r 3 = 2pGqA a n d

r 4 = r - r3.

The e u l e r f o r m u l a shows

t h a t t h i s imbedding w i l l b e minimal, and g i v e s t h e desired f o r m u l a f o r t h e genus. c o p i e s of

To c o n s t r u c t t h e imbedding, w e b e g i n w i t h

e a c h q u a d r i l a t e r a l l y imbedded i n

K2n,2n,

qG

S

(n-1) d e s c r i b e d by R i n g e l ( s e e Theorem 6 - 3 7 ) .

H

see [ W 7 ] ) , w e

q u a d r i l a t e r a l l y imbedded i n t h e d e s i r e d

The c o n s t r u c t i o n t h u s f a r a l l o w s t h e

surface. copy o f

G[KZ],

as

By m a k i n g s u i t a b l e v e r t e x

i d e n t i f i c a t i o n s ( a n d t h i s i s t h e h e a r t of t h e p r o o f : obtain t h e graph

2

qH

e d g e s of e a c h

t o be a d d e d ; e a c h s u c h e d g e c o n v e r t s o n e q u a d r i l a t e r a l

r e g i o n o f t h e i m b e d d i n g of

G [ x ]

i n t o t w o triangular regions.

This gives the r e s u l t . F o r t h e above c o n s t r u c t i o n , o n e c a n compute t h e g e n u s o f t h e r e s u l t i n g s u r f a c e d i r e c t l y , without r e c o u r s e t o any e u l e r - t y p e formula.

The c o n t r i b u t i o n s o t t h e g e n u s a r e o f t h r e e t y p e s :

( i ) q G ( n - l ) ',

r e p r e s e n t i n g t h e c o l l e c t i v e g e n e r a of t h e

qG

2 - m a n i f o l d s w i t h which w e began o u r c o n s t r u c t i o n : (ii) correspondong t o every v e r t e x

s e t s of

2n

v

of

G,

w e make ( d e g G v - l )

v e r t e x i d e n t i f i c a t i o n s each, each set

Chapt. G

Imbedding P r o b l e m s i n Graph 'l'heory

70

requirmq a

" b u n d l e " of

n

( 2 q G - p,)

t h i s contributes

t u b e s j o i i i l i i g two 2 - m a n i f o l d s ; (n

- 1)

t o Lhe g e n u s ;

pG + 1 , r e p r e s e n t i n g t h e c o n t r i b u t i o n o f t h e tubes t a k e n c o l l e c t i v e l y .

W e mention t h a t Bouchet

"generative m-valuations.

m

n(mod 1 2 ) and

"

(mod 6 )

[I3131 h a s s t u d i e d

y(Kn (m))

,

usinrj

He c o n s i d e r e d t h e r e s i d u e c l a s s e s of and d e t e r m i n e d

for

y(Kn(m))

32

of

t h e s e 7 2 c a s e s , by c o n s t r u c t i n g t r i a n g u l a r i m b e d d i n g s . P a r s o n s , P i s a n s k i , a n d J a c k s o n ( [PPJI]) and "wrapped q u a s i - c o v e r i n g s " Thm. 6 - 4 6 a .

Let

G

[ J F P I ] ) employed

t o establish:

have a t r i a n g u l a r imbedding n F N

a r e i n f i n i t e l y many

in

then t h e r e

So;

G(Fn)

such t h a t

has

a

t r i a n g u l a r imbedding. F i n a l l y , w e comment t h a t J a c k s o n

[Jl] h a s c o n s t r u c t e d t r i -

a n g u l a r i m b e d d i n q s , a s b r a n c h e d c o v e r i n g s p a c e s ( s e e S e c t i o n l o ) , for some c o m p l e t e n - p a r t i t e g r a p h s o f t h e f o r m

6-6.

. . ,m) .

K ( (n-Z)m,m,.

Edmonds' P e r m u t a t i o n T e c h n i q u e

Before l e a v i n g t h e t h e o r y of g r a p h imbeddings and c o n s i d e r i n g s p e c i f i c imbedding p r o b l e m s , w e p r e s e n t a p o w e r f u l t o o l f o r s o l v i n g such problems:

t h e Edmonds' p e r m u t a t i o n t e c h n i q u e ( [ E l l ; see a l s o

Youngs [ Y l ] . )

T h i s amounts t o a n a l q e b r a i c d e s c r i p t i o n , f o r a n y 2-

c e l l imbedding of a g r a p h i n t h e p r o o f s of many

Of

G.

the

It i s used,

i n o n e f o r m or a n o t h e r ,

theorems l i s t e d above.

Denote t h e v e r t e x s e t of a c o n n e c t e d g r a p h 11,2

,...,n l . pi:

Let

n.

=

For each

V(i)

IV(i)

1

.

->

V(i)

i E V(G),

let

G

by

V(G)

V ( i ) = {k € V ( G )

b e a c y c l i c p e r m u t a t i o n on

Then t h e r e i s a o n e - t o - o n e

V(i)

,

=

1 [i,kl

E E(G)}.

of l e n g t h

correspondence between

Edmonds ' Permu t a t i o n T e c h n i q u e

S e c t . 6-6.

2 - c e l l i m b e d d i n g s of

Thm. 6 - 4 7 .

a n d c h o i c e s of t h e

G

,...

Each c h o i c e

(p,

G(M)

in a surface

of

G

o r i e n t a t i o n on

pi

g i v e n by: imbedding

such t h a t t h e r e i s an

M,

which i n d u c e s a c y c l i c o r d e r i n g o f

M

[i,kl

at

i

cessor t o

[i,k]

is

[i,pi(k) j ,

,...

(p,

,

determines a 2-cell

,pn)

t h e edges f a c t , given

71

i n which t h e immediate suc-

,pn),

i = 1

, ...

In

t h e r e i s an a l g o r i t h m

which p r o d u c e s t h e d e t e r m i n e d imbedding. g i v e n a 2-cell imbedding

,n.

Conversely,

in a surface

G(M)

with

M

a given o r i e n t a t i o n , t h e r e is a corresponding

, .. . , p n )

(p,

____ Proof:

determining t h a t imbedding.

D*

Let

P*: D*

-f

D*

=

by:

{(a,b)1

[a,bl E E ( G ) 1,

P*(a,b) = (b,pb(a)).

and d e f i n e Then

is a

P*

of d i r e c t e d e d g e s of

p e r m u t a t i o n on t h e s e t

D*

(where each edge o f

i s a s s o c i a t e d w i t h two oppo-

G

sitely-directed directed edges)

and t h e o r b i t s u n d e r

I

d e t e r m i n e t h e ( 2 - c e l l ) r e g i o n s of t h e c o r r e s p o n d i n g

P*

imbedding.

--

G

T h e s e r e g i o n s may t h e n b e " p a s t e d " t o g e t h e r

with (a,b)

matched w i t h

t o form a s u r f a c e

M

(Since every edge

(a,b)

(b,a)

i n which

G

i s 2 - c e l l imbedded.

i n t h e b o u n d a r y of a g i v e n re-

g i o n i s matched w i t h an e d g e

--

--

(b,a)

in the

b o u n d a r y of a n o t h e r ( o r p o s s i b l y t h e same) r e g i o n ,

is closed. not with

Since (a,b)

--

(a,b) M

genus o f . M with

r

i s matched w i t h

is orientable.

i s a c y c l i c permutation,

--

a s i n F i g u r e 6-4

M

M

--

(b,a)

Since each

i s a 2-manifold.)

and Pi

The

may now b e d e t e r m i n e d by t h e e u l e r f o r m u l a ,

g i v e n b y t h e number o f o r b i t s u n d e r

P.

The

#

c o n v e r s e f o l l o w s from s i m i l a r c o n s i d e r a t i o n s . A s an e x a m p l e , c o n s i d e r t h e i m b e d d i n g o f

p i c t e d i n F i g u r e 6-5.

v(1)

Let

= v(2) = V ( 3 ) = {4,5,61;

V(KJI3)

K3,3

= {1,2,3,4,5,61,

in

S1

de-

with

V ( 4 ) = V(5) = V(6) = { 1 , 2 , 3 1 .

Then

12

Imbedding P r o b l e m s i n G r a p h T h e o r y

Chapt. 6

3

F i g u r e 6-4

F i g u r e 6-5

d e s c r i b e t h i s imb

(1) (2)

(3)

(Note t h a t

P * ( 4 , 1 ) = (1,s); P*(3,5) = ( 5 , l ) ; P * ( 4 , 2 ) = (2,5).)

A s a m a t t e r of n o t a t i o n ,

from t h i s p o i n t o n , w e w i l l a b b r e v i a t e

a n o r b i t such a s (1) a b o v e b y : p 4 ( 3 ) = 1,

and

p1(4)

1-5-2-6-3-4;

it i s i m p l i c i t t h a t

= 5.

I t now f o l l o w s t h a t t h e g e n u s of a n y c o n n e c t e d g r a p h ( a n d

h e n c e , by C o r o l l a r y 6-19,

of a n y g r a p h ) c a n be c o m p u t e d , b y selec-

n ting,

f r o m among t h e

T

(ni-l) !

possible permutations

P*,

one

i=l

w h i c h g i v e s t h e maximum number o f o r b i t s , a n d h e n c e d e t e r m i n e s t h e g e n u s of t h e g r a p h ( c o m p o n e n t ) . i m b e d d i n g must b e a 2 - c e l l by C o r o l l a r y 5 - 1 5 ,

r

( S i n c e , b y Theorem 6-11,

a minimal

i m b e d d i n g , it c o r r e s p o n d s t o some

w i l l b e minimal f o r t h i s imbedding.)

P*;

The

Sect. 6-7.

Imbedding G r a p h s on P s e u d o s u r f a c e s

13

o b v i o u s d i f f i c u l t y i n a p p l y i n g t h i s p r o c e d u r e i s t h a t of s e l e c t i n g

a suitable

from t h e

P*

( u s u a l l y ) v a s t number o f p o s s i b l e o r d e r e d

n - t u p l e s of l o c a l v e r t e x permutations Each

pi

pi. i s c a l l e d a r o t a t i o n , and P

c a l l e d a r o t a t i o n scheme f o r

=

*,..., p n )

is

(pL,p

G.

S t a h l h a s s t u d i e d " p e r m u t a t i o n p a i r s " as a p u r e l y c o m b i n a t o r i a l g e n e r a l i z a t i o n of graph imbeddings, and h i s powerful approach s u f f i c e s t o e s t a b l i s h many of t h e c l a s s i c a l r e s u l t s ( T h e o r e m s 6 - 1 8 and 6 - 2 1 ,

f o r e x a m p l e ) as w e l l a s t o o b t a i n new i n f o r m a t i o n a b o u t

t h e genus of t h e and

amalgamation of g r a p h s ;

see [ S l O ] , [ S 1 2 ] ,

[Sl3],

[S141.

6-1.

Imbedding Graphs on P s e u d o s u r f a c e s

I n S e c t i o n 5-5 w e i n t r o d u c e d t h e p s e u d o s u r f a c e s

...,n t ( m t ) ) .

S ( k ; n (m ) ,

1

R e c a l l t h a t f o r a n y i m b e d d i n g of a g r a p h

pseudosurface

w e assume t h a t e a c h s i n g u l a r p o i n t o f

S',

o c c u p i e d by a ( s i n c j u l a r ) v e r t e x o f

G.

The number

2

-

ni(mi-l)

g i v e s t h e c h a r a c t e r i s t i c of

S',

denoted by

is

S'

2k

-

t

1

1

in a

G

X I

(sl).

i=l

Def.

6-48.

x' ( G ) , of

The p s e u d o c h a r a c t e r i s t i c ,

largest i n t e g e r which

G

G

x"(G),

G

f o r a l l pseudosurfaces

c a n be imbedded.

acteristic, that

x'(S')

a graph

is the in

S'

The g e n e r a l i z e d p s e u d o c h a r -

is t h e l a r g e s t integer

x ( S " ) such

imbeds i n t h e g e n e r a l i z e d p s e u d o s u r f a c e

S".

A s u r f a c e can be c o n s i d e r e d as a ( d e g e n e r a t e ) p s e u d o s u r f a c e ,

and a p s e u d o s u r f a c e a s a ( d e g e n e r a t e ) g e n e r a l i z e d p s e u d o s u r f a c e . Hence w e h a v e

The s e c o n d i n e q u a l i t y may be s t r i c t ( i . e . p s e u d o s u r f a c e s may be m o r e e f f i c i e n t , f r o m t h e p o i n t o f view of m a x i m i z i n g c h a r a c t e r i s t i c , imbedding g r a p h s i n t o ) : f o r example shows.

( N o w , see P r o b l e m 6 - 1 0 . )

x' (K5) =

1,

as F i g u r e 6-6

for

14

Imbedding P r o b l e m s i n Graph T h e o r y

Chapt. 6

Figure 6-6

Petroelje

[P4] h a s found

t h a t many of t h e b a s i c t h e o r e m s f o r

i m b e d d i n g g r a p h s i n s u r f a c e s c a r r y o v e r for p s e u d o s u r f a c e s .

For

example : Thm. 6 - 4 9 .

Let

G

b e a c o n n e c t e d g r a p h m i n i m a l l y imbedded i n t h e

pseudosurfaces Thm. 6-50.

If

S';

t h e n t h e i m b e d d i n g must be 2 - c e l l .

i s c o n n e c t e d , t h e n x ' ( G ) 5 p-q/3; equality G h a s a t r i a n g u l a r imbedding i n

G(p,q)

h o l d s i f and o n l y i f

some p s e u d o s u r f ace. P e t r o e l j e also d e v e l o p e d a n a n a l o g u e of Edmonds

technique f o r pseudosurfaces

,

permutation

and f o u n d t h e f o l l o w i n g f o r m u l a e

(among o t h e r s ) : Thm. 6-51.

x'(Kn

,n,n,n

)

= 2n(2-n).

( T h i s i s c o n s i s t e n t w i t h Theorem 6 - 4 2 t h a t , f o r n # 3 , 2 Y ( K n,n ,n ,n ) = ( n - 1 ) , w i t h r = r3: see a l s o P r o b l e m 9 - 1 0 . ) Thm.

6-52.

x ' ( K ~ ~ , ~= ~ Z(m+n-mn) , ~ )

-

R i n g e i s e n a n d White [ R W l ] showed:

r(m-l),

where

2m22n r )

21.

O t h e r T o p o l o g i c a l P a r a m e t e r s f o r Graphs

6-8

Sect.

m

I n t h e cases w h e r e n

(mod 4 ) ,

2

and

n

75

a r e b o t h even o r e i t h e r

m

or

the pseudocharacteristic agrees with the character-

i s t i c o f Theorem 6 - 3 7 ; i n a l l o t h e r c a s e s , x'(K,,~) = x(K ) = 1. m, n T h a t i s , i n t e r m s o f p s e u d o c h a r a c t e r i s t i c t h e s e i m b e d d i n g s a r e more e f f i c i e n t t h a n t h o s e f o r t h e g e n u s case. Thm. 6-54.

x'(Qn

+ Kr)

=

n-2 (n+r-4)2

-

r

,

for

0

5 r 5 n, n 2

2.

Generalized pseudosurface imbeddings w i l l have r e l e v a n c e i n Chapter 12.

6-8.

Other Topological Parameters f o r Graphs

W e h a v e s e e n t h a t , i f a g r a p h i s n o t p l a n a r , w e c a n s t i l l make

a " p r o p e r " d r a w i n g o f t h e g r a p h i n some s u r f a c e a n d / o r p s e u d o surface.

Two common t o p o l o g i c a l p a r a m e t e r s ( o t h e r t h a n g e n u s ) w h i c h

a r i s e i f m o d i f i e d d r a w i n g s are a l l o w e d a r e t h e t h i c k n e s s a n d c r o s s i n g number. Def.

6-55.

The t h i c k n e s s

of a g r a p h

O(G)

G

i s t h e minimum

number o f p l a n a r s u b g r a p h s whose u n i o n i s

G.

union is u s u a l l y taken over spanning subgraphs)

(The

.

For sample formulae w e have: Thm. 6-56. _____

( B e i n e k e a n d H a r a r y [ B H 2 ] ; A l e k s e e v a n d Gonchakov [ A G l ] ; Vasak

[Vll)

O(Kn)

Thm. 6-57. -__

(Beineke B(Kn,n)

Thm. 6-58. -___

=

=

(Kleinert O(Qn)

=

r

-

n+5

L T ~ [Kl])

n+l

-

6,

n+7

1

,

except t h a t

Imbedding Problems i n Graph ‘I’heory

76

Chapt. 6

Some t h i c k n e s s r e s u l t s h a v e b e e n o b t a i n e d f o r s u r f a c e s o f p o s i t i v e genus. Def.

6-59.

thickness The -

of a graph

ijn(G)

G

i s t h e minimum

number of s u b g r a p h s , e a c h i m b e d d a b l e on union i s Thus

The n o n o r i e n t a b l e t h i c k n e s s

8(G) = iiO(G).

is defined s i m i l a r l y , for

ill(Kn

-

Cn)

On(G) ( n

I

=

=

n+2-\ I

LTJ

n- 1

( i i i ) 01(K4 ( n )

,

for

n

a n odd p r i m e .

n+l tT] 1-

The c r o s s i n g number

v(G)

of a g r a p h

G

i s t h e minimum

number o f p a i r w i s e i n t e r s e c t i o n s of i t s ( o p e n ) e d g e s , among a l l d r a w i n g s o f

G

i n t h e plane.

One m i q h t s a y t h a t t h e c r o s s i n g number t e l l s u s , i f w e i n s i s t upon d r a w i n g

G

on

So,

0)

[A51)

= -

(ii) “1(K3(n))

E m . 6-63.

whose

Nn.

Thm. 6-62. ( A n d e r s o n [A31, [ A 4 ] , ~-

(i)

Sn,

G.

j u s t how b a d t h i s d r a w i n g m u s t b e .

For

Sect.

6-8

77

O t h e r T o p o l o g i c a l Parameters f o r Graphs

t h i s p a r a m e t e r , e x a c t v a l u e s a r e scarce: w e m e n t i o n t h e f o l l o w i n g bounds (see [ G g ] ) :

n

-

10.

m

-

6.

A s an i n d i c a t i o n o f t h e k i n d of t e c h n i q u e s t h a t m i g h t b e em-

ployed, w e prove t h e following: v ( K ~ , ~ , =~ 2)

Thm. 6 - 6 6 . -__

Proof: graph o f

K3,2,2'

Problem 6 - 3 ) . drawing of graph (If

G,

r

v ( K ~ , ~ , =~ )x.

Suppose

=

Thus

i s a sub-

K

313 (by t h e h i n t f o r

> 1 y ( K 3, 2 , 2 ) x 2 1. C o n s i d e r s u c h a n o p t i m a l

in K 3,212 with p = 7

r3

Since

t h i s g i v e s rise t o a p l a n e

So;

+

x,

q = 16

+

2x,

t h e c o n f i g u r a t i o n i n F i g u r e 6-7a,

x 1. 1,

must e x i s t s i n c e

r # r3.

and

which

must c o r r e s p o n d t o t h e con-

f i g u r a t i o n i n F i g u r e 6-7b,

which c a n n o t o c c u r i n any

complete t r i p a r t i t e graph.)

Hence

2 q = 32

+

4x

x F i g u r e 6-7

4

+

3(r-1) = 3r

31 + 3 3

-

+

25 2 = 3

1,

+

4 3

and X,

9

+ x

Thus

= q

-

p = r

2 < 1 x, 3 - 3

-

2

so t h a t

1. x 2 2.

I m b e d d i n g P r o b l e m s i n Graph T h e o r y

Figure

6-8

shows t h a t

x 2 2,

Chapt. 6

to complete t h e proof.

F i g u r e 6-8

A n a t u r a l e x t e n s i o n of

the c o n s t r u c t i o n i n F i g u r e 6 - 8 g i v e s

the follow in9 : Thm. 6-67. -

Thm. __

6-68.

V(Km,n,r

)

:.

V(Kn,n,n

)

5 a3 ( n - 2 ) ( n - 1 ) 2 ( n + 2 ) ;

+

f(m,n)

f(m,r)

+

f(n,r),

where

f(x,y)

=

equality holds for

n = 1,2. The e x a c t results b e l o w a r e d u e t o U c i n e k e a n d R i n g e i s e n ([HRl]

Thm.

and [ R B l ] ) :

6-68a.

The f o l l o w i n g C a r t e s i a n p r o d u c t s h a v e c r o s s i n g n u m b e r s

as i n d i c a t e d :

(i)

v(C3

(ii)

v(C4

(iii) V(K4

x

1

Cn)

=

n,

for

2

n

3;

4;

Cn) = 2n,

for

n

= 3n,

for

n - 3.

Cn)

-

O t h e r e x a c t r e s u l t s h a v e b e e n f o u n d f o r c r o s s i n g n u m b e r s on s u r f a c e s of p o s i t i v e g e n u s .

Def. __- 6 - 6 9 .

The c r o s s i n g number

vn(G)

of a g r a p h

G

is the

minimum number of p a i r w i s e i n t e r s e c t i o n s of i t s ( o p e n ) edges,

among all d r a w i n g s of

G

in

'n'

#

Applications

S e c t . 6-9

Thus

v ( G ) = vo ( G )

vn(G) ( n > 0 ) Thm.

6-70.

Thm.

6-71.

.

The n o n o r i e n t a b l e c r o s s i n g number

is d e f i n e d s i m i l a r l y , f o r

(Guy and J e n k i n s

(Gross Let

79

Nn*

[GJl])

[G5]) =

(n-1) (n-4) 4

p r i m e power;

,

where

is a

1 (mod 4 )

n

then

W e d e d u c e f r o m Theorem 4 - 2 2

t h a t t h e graphs

Cayley g r a p h s f o r a l l f i n i t e H a m i l t o n i a n p-groups

are 414 ( i n f a c t , as w e

Qnx K

w i l l see i n C h a p t e r 7 , t h e s e g r a p h s a r e o f minimum g e n u s f o r t h e s e ) = 1 + n2", From P r o b l e m 6 - 1 1 w e see t h a t y ( Q n x K 484 and f r o m P r o b l e m 11-8 w e w i l l l e a r n t h a t t h e c o r r e s p o n d i n g nonn+l ) = 2 + n2 o r i e n t a b l e genus i s T ( Q n x K 414

groups.)

.

Thm.

6-72.

( K a i n e n a n d White [KWl]) Let

with

h = y(Qn n

2

x

K4,4)

-

m,

and

k =

T(Qn

x

K4,4)

-

2m,

then

0;

6-9.

Applications

F o r a p p l i c a t i o n s of t h e f o u r b a s i c t o p o l o g i c a l p a r a m e t e r s d i s c u s s e d i n t h i s c h a p t e r , c o n s i d e r t h e problem o f p r i n t i n g an e l e c t r o n -

i c c i r c u i t on a c i r c u i t b o a r d .

I f t h e associated graph

p l a n a r , one board w i l l s u f f i c e , w i t h o u t m o d i f i c a t i o n .

is

G If

G

is not

p l a n a r , a t l e a s t f o u r a l t e r n a t i v e s are available t o a v o i d s h o r t circ u i t s ( t h e c h o i c e d e p e n d i n g upon r e l e v a n t c o n s i d e r a t i o n s o f an e n g i n e e r i n g and/or economic n a t u r e ) :

1.)

t h e c i r c u i t can b e accom-

m o d a t e d by d r i l l i n g h o l e s t h r o u g h t h e b o a r d : imun number o f h o l e s ;

2.)

y(G)

g i v e s t h e min-

some o f t h e v e r t i c e s c a n be p r i n t e d on

b o t h s i d e s o f t h e b o a r d , w i t h c o n n e c t i o n s made t h r o u g h t h e b o a r d

80

Imbedding P r o b l e m s i n Graph Theory

Figure 6-9

Chapt.

6

S e c t . 6-10

Problems

d l

b e t w e e n c o r r e s p o n d i n g i m a g e s o f t h e s a m e v e r t e x ; h e r e we s e e k t h e minimum

n

such t h a t

n(2));

S(0;

can b e imbedded i n t h e p s e u d o s u r f a c e

G

h o w e v e r , i t may o c c u r t h a t n o s u c h

P r o b l e m 6-9.)

e x i s t s (see

n

I f a g i v e n v e r t e x c a n a p p e a r a r b i t r a r i l y o f t e n , i;ith

c o n n e c t i o n s made t h r o u g h t h e b o a r d among c o r r e s p o n d i n g i m a g e s o f t h e same v e r t e x , a n d i f i n a d d i t i o n h o l e s c a n b e d r i l l e d as i n t h e n w e s e e k t h e v a l u e o f t h e p a r a m e t e r \ ' (G), f o r maximum efficiency;

3.)

If s e v e r a l c i r c u i t b o a r d s are u s e d ,

I.),

e a c h con-

t a i n i n g a p l a n a r p o r t i o n o f t h e c i r c u i t , and jumpers are run betiieen s u c c e s s i v e b o a r d s t o c o n n e c t c o r r e s p o n d i n g i m a g e s o f t h e same j u n c -

'.'(G); 4 . )

then w e are s t u d y i n g t h e p a r a m e t e r

tion,

If the circuit

( w i t h no holes l ' e t

i s s t a m p e d on o n e s i d e o f o n e c i r c u i t b o a r d

d r i l l e d ) a n d i f w h e r e v e r t w o c o n n e c t i o n s cross e x t r a n e o u s l y tt,.o

pass t o

h o l e s a r e now d r i l l e d t o a l l o w o n e c o n n e c t i o n t o t e m p o r a r i l : .

t h e o t h e r s i d e o f t h e b o a r d , e n a b l i n g i t t o " c r o s s " t h e s e c o n d con-

v(G)

nection while avoiding a s h o r t c i r c u i t , it is t h e parameter t h a t d i c t a t e s economy o f e f f o r t h e r e . A s an example,

consider the modified wheatstone bridyo cii-cuit

o f F i g u r e 6-9 ( a ) ; t h e a s s o c i a t e d q r a p h i s ( e ) c o r r e s p o n d r e s p e c t i v e l y to: I'(K3,3)

G-1.

)

=2,

and

\

(K3, 3)

G =

=

1,

K ,

F i i l u l - e s 6-3 (b)-

3,3'

'

( i i 3 j) = 1 ,

v ( K ~ , ~ = )1.

Show t h a t t w o i j r a p l i s arc h o i n e o i i i o r \ ~ h i ci n tlic. t ~ r a t ' h -t h e o r e t are ical s e n s e i f a n d only i f t h c i r r e a l i z a t i o n s i n I 2.

-

'-

Thm. 7-13.

~(€1,)

=

Proof:

1

+

2n-2 ( n - 2

By Theorem 2 19, Problem 2-4, and a trivial in-

duction argument,

Hn

is a bipartite graph.

a quadrilateral imbedding for using Corollary 6-15.

For

Hn,

Eln,

We produce

and compute

let

p (n)

and

y(Gq) ,(n)

denote the number of vertices and edges respectively. gree 2n, 2nnM(n' . of

is regular of deit is a simple matter to compute q(n) =

p(") = 2"M'");

Then

and since

Hn

Let the statement S(n) be: there is an imbedding Hn for which r = r4 = n2n-lM(n), including two

disjoint sets of

2n-2M(n)

mutually vertex-dis joint

quadrilateral regions each, both sets containing all 2 " ~ ' ~ ) vertices of Hn' We claim that S(n) is true for all n 1. 2, and we verify this claim by mathematical induction. That S ( 2 )

is true is apparent from Figure

7-4

(which shows an imbedding of C4 x C 6 in S1), with the regions designated by (1) making up one set, and those designated by (2) making up the other. assume n

2 2.

S(n)

to be true and establish

We now

S(n+l) ,

for

Chapt. 7

The (;enus of a Group

90

(2)

Figure 7-4

For the graph

lIn+l, we start with

LmIl+l copics

of t i n , minimally imbedded as described by S(n). We parti tion the corresponding surfaccs irito m n+l of one orientation, and mn+l copies of the reverse orientation, corresponding to the vertex set partition of the bipartite graph

C2m

n+l

.

rrom each copy, two

joins of p (n) edges each must be made, both to copies of opposite orientation, in order to construct I1n+l. From the statement S(n), it is clear thaC these two joins can be made, each one over iny four edges each.

2"-'M(")

tubes carry-

(Attach one end of a tube in the

intcrior 01 each region designated by (1) for one join; u s e the recjions designated by (2) for the second join.)

Each new region formed by this process is a quadrilateral. In this fashion the required

2m,+l

joins can be made to

imbed H n + l , with r = r4. Now form one set of regions by selecting opposite quadrilaterals from each tube added in alternate joins in this construction. Form the second set by selecting the remaining quadrilaterals on the same tubes. It is clear that the two sets oL regions thus selected are disjoint, and that each contains ( 2 ) . (mn+l)(2n-2M(n)) = 2n-lM fn+l) mutually vertex-dis joint 2n+lM(n+l) quadrilaterals; both sets contain all 2m (n) vertices of Hn+l. Furthermore, r (n+l) n+lr + Or, where Ar = ( 2 m n + l ) (2n-2M(n)) ( 2 ) , where 2mn+l joins have been made, with 2n-2 M ( n ) tubes p e r join, and a net increase in r of 2 per tube. Hence, ~

Sect.

Genus F o r m u l a e f o r Groups

7-2

and w e h a v e e s t a b l i s h e d t h a t Therefore,

S(n)

91

f o l l o w s from

S(n+l)

holds, f o r a l l

S(n).

n 2 2.

W e c a n now compute:

#

For t h e s p e c i a l c a s e w h e r e n

m ,

M(n)

=

Cor.

7-14.

K2

K2r

H(") n

The g e n u s of

m

Furthermore, i f

Hn( 2 )

is the

w i t h Theorem 6-36 Cor.

7-15.

m . = m,

i = 1

,...

,n ,

w e have

and:

= 2

i s q i v e n by:

i n t h e above formula, s i n c e

2n-cube,

C4 =

and w e o b t a i n t h e r e s u l t (compare

:

y(Q2n) = 1

+ 2 2n-2 ( n - 2 )

F o r f u r t h e r r e s u l t s c o n c e r n i n g t h e g e n u s of r e p e a t e d C a r t e s i a n p r o d u c t s of b i p a r t i t e g r a p h s , see [W61, a n d a l s o P i s a n s k i ,

Now, l e t

for

); '1

H ( ~ ) , m

graph n

2.

C o r . 7-16.

[P5]. be t h e a b e l i a n g r o u p w i t h minimal C a y l e y color

2

2;

i.e.

and

Z2m,

rAm)

(m) = '2m

'n-1'

Then w e have: y(rLm)) = 1

+

2n-2(n-2)m

n

.

The r e a d e r may w i s h t o combine Theorems 4 - 1 6 ,

4-19,

o b t a i n genus formulae f o r a d d i t i o n a l a b e l i a n groups. u s i n g t h e n o t a t i o n of Theorem 4 - 1 6 ,

consider

where

a n d 7-13 t o

For example,

mr

i s even.

Using c u r r e n t graph c o n s t r u c t i o n s , r a t h e r t h a n t h e s u r g e r y t e c h n i q u e s

of Theorem 7 - 1 3 ,

Jungerman a n d W h i t e [JWl] f o u n d t h e g e n u s o f

"most" of t h e remaining f i n i t e a b e l i a n groups: summarizes t o i n c l u d e p r e v i o u s r e s u l t s as w e l l .

t h e n e x t theorem

The Genus of a Group

92

Thm. 7-17.

Chapt.

7

x zm x . . . ~ Z , where f o r 2 5 i 2 r ml 2 "r m. divides mi-l (and mr > 1, unless 11'1 = 1.) Let N ( T ) = 1 - (r-2) lrl/4: then

r

Let

= Z

(i) If

r

(ii) If

=

1,

r = 2

then and

mr

y ( T ) = 0, =

then

2,

y ( r ) = 0.

(iii) If N ( T ) is an integer, m r > 3, either mr even or r # 3 , then (iv) If N ( T ) is an integer, then y ( T ) 5 N ( 1 ' ) .

mr

r 1, and y(r) = N(l ) . and

= 3,

1

r # 3,

c

The argument of Theorem 7-12 can be modified to assist in the computation of the genus for certain hamiltonian groups; the following results are due to Himelwright [ H 7 ] : n

+

Thm. 7-18.

y ( Q x (Z,)

Thm. 7-19.

y ( Q x Zm

Cor. 7-20.

The groups Q xZm x ( Z 2 ) 8 , asymptotic to the order.

) =

n 2"

(Z2)n )

x

=

1.

mn2n + 1,

for

for

m

m

odd.

odd, have genus

By Theorems 4-16 and 4-22, if G is a hamiltonian group, then Zm x ... x Zm x ( z ~ ) ~where , the mi are odd (i = 1, 1 r , r) and m.L Imi-1 (i = 2, ... , r). Himelwright has also shown:

G = Q x

...

Thm. 7-21.

The genus of the hamiltonian group

zm

r 1 (r

x

(z,)n (n

is asymptotic to

Q x Z

x

-

1)

2n(r + n

m,I

...

E

i=l

x

mi,

if

+ 1.

There are many open questions in this area. If a generalization of Theorem 1-13 for products of arbitrary (not necessarily even) cycles could be found, then the genus of any abelian (and also of any hamiltonian) group could be easily computed. What is y(Sn)? y(An)? The following theorem produces upper bounds for these group genera.

Genus F o r m u l a e f o r Groups

S e c t . 7-2

Thm. 7 - 2 2 .

If

{q,

i s f i n i t e a n d i s m i n i m a l l y g e n e r a t e d by

,...

Proof:

and s a t i s f i e s a t l e a s t t h e r e l a t i o n s

,gn}

P 4

Select

f o r all

g E G.

=

-1 -1 (gql,qql r g g 2 r g g 2 ,

. . ., s g n g g , l )

An o r b i t c o n t a i n i n g t h e d i r e c t e d e d g e p - l ( a ) = agy2: agi

continues with

qqi = q' regions:

(If mi = 2,

m..

and

g'gi

= g,

-1

(a,aqi

)

hence t h i s o r b i t

m. gil = e

A

corresponds t o t h e r e l a t i o n length

I

a l g o r i t h m (see

Then, u s i n g E d m o n d ' s

w e compute o r b i t s as f o l l o w s :

Theorem 6 - 4 2 ) , (i)

93

and h a s

w e draw edges f o r b o t h obtaining

2-sided

f o r e a c h s u c h r e g i o n , t h e t w o s i d e s may b e

i d e n t i f i e d a n d t h e arrows r e m o v e d , so t h a t t h e region i s destroyed b u t t h e genus i s u n a f f e c t e d . ) ( i i ) An o r b i t c o n t a i n i n g t h e d i r e c t e d e d g e tinues with

P

agi

( a ) = agigi+l;

(a,ag.)con-

hence t h i s o r b i t n

corresponds t o t h e r e l a t i o n has length

( IT

j=1

g,)k = e

and

J

A s t h e r e are no o t h e r o r b i t s , w e

nk.

find: n

r =

1

rm

i=l

i

+ I nk

t h e e u l e r f o r m u l a now g i v e s t h e g e n u s

y

t h e o r e m f o r t h i s i m b e d d i n g of

for this

presentation

y ( c A ( r ) )5 Y.

P

for

r.

CA(T),

Hence

of t h e

~ ( 1 ' )5

#

The Genus o f a Group

94

Chnpt. 7

W e n o t e t h a t a n e q u i v a l e n t f o r m u l a w a s obtained by B u r n s i d e

[BIG, p . 3981 i n a d i f f e r e n t c o n t e x t . Theorem 7 - 2 2 g i v e s )(GI exactly, for I = 2 , A4 , S4 , A5 , o r Z j Z3. We a l s o obtain the followiny t w o c o r o l l a r i c s : Cor.

7-23.

2

y(sn) Proof:

(ni2) ‘

1 + ___ ( n 2 - 5 n + 2 ) , n

Take

erators f o r Cor.

...

s = (1 2 3

Sn;

s

then

n

2.

.>

n)

t = ( 12) n-1

and

t2 = (st)

=

a s gen-

#

= e.

7-24.

i

Proof:

For

n

(1 n-1) ( 2 n ) For

n

8

s = (1 2

odd, generate

even,

generate

(n-1)( n - 3 ) ! (n2 -6n+4),

+

An,

. ..

s n-l

with

n-2)

n-1) ( 2 n )

= t 2 =

and

( s t ) ” - l= c

odd

t =

and

sn-2 - t

and

An‘

s = (1 2

...

n

=

( s t ) n= e.

t = (I 2 ) ( 3 n ) (see [ 9 1 ) . #

The two f o r m u l a s g i v e n a b o v e f o r a l s o f o u n d by B r a h a n a

Sn

and

r e s p e c t i v e l y were

An

[ B 1 5 ] , u s i n g a d i f f e r e n t method and i n a

s l i g h t l y different context.

see [WE].

For r e l a t e d r e s u l t s ,

T h e r e h a s b e e n much a c t i v i t y i n t h e s t u d y of t h e g e n u s p a r a m e t e r f o r g r o u p s , i n r e c e n t y e a r s , p e r h a p s a t l e a s t i n p a r t m o t i v a t e d by C h a p t e r 7 of t h e f i r s t e d i t i o n of t h i s book. s e n t e d Theorem 7-16. Thm. 7-25.

W e have a l r e a d y pre-

Here is a c o n t i n u e d s a m p l e of t h i s r e s e a r c h .

(Proulx [P9])

y(~,)

5 1 +

$,

n

for

T h i s improves on C o r o l l a r y 7-23,

for

even,

n

n

even.

2

4. For

n

odd,

a s h a r p e r bound t h a n t h a t of C o r o l l a r y 7 - 2 3 w a s d e v e l o p e d i n [WSl: b u t t h e n e x t r e s u l t s d e t r a c t from t h e i n t e r e s t t h a t m i g h t a c c r u e t o t h e improvement of b o u n d s f o r

y(Sn):

Sect. 7-3

95

Related Results

Thm. 7 - 2 6 .

( P r o u l x [PSI ) Y(S5) = 4.

(The p r o o f i s n o t e a s y ! ) Thm. 7 - 2 7 .

(Tucker [T4] ) Y(Sn) = 1 +

&,

for

n

+ 5 3 3 6 , for

n

2

168.

We a l s o f i n d

Thm. 7-28.

(Tucker [T4] ) y(An)

During and White

1

t h e p e r i o d 1972-77,

Gross

[G6],

168 Gross a n d Lomonaco [ G L l ] ,

[W81 showed t h a t c e r t a i n m e t a c y c l i c a n d d i c y c l i c g r o u p s

are toroidal. Baker

c

Then i n 1977 P r o u l x [ P S I c o m p l e t e d work b e g u n by

i n 1931 [B3], t o c l a s s i f y

all t o r o i d a l

g r o u p s i n a major e f f o r t .

The c l a s s i f i c a t i o n c o n s i s t s o f n i n e t e e n p r e s e n t a t i o n s o n t w o g e n e r a -

t o r s , t e n p r e s e n t a t i o n s on t h r e e g e n e r a t o r s , a n d o n e p r e s e n t a t i o n on four generators.

7-3.

Related Results

I n 1 9 7 7 B a b a i [B2] s o l v e d a p r o b l e m p o s e d i n t h e f i r s t e d i t i o n o f t h i s book, when h e e s t a b l i s h e d : Thm. 7-29. Since

If

Sn

I'l

i s a s u b q r o u p of

r2,

then

i s i s o m o r p h i c t o a s u b g r o u p of

An+*,

y (rl)

5

y(T2).

we obtain

F o r e x a m p l e , w e l e a r n from Theorem 7-25 t h a t

y ( A 7 ) ->- 4 . D e s p i t e t h e f a c t t h a t t h e r e a r e i n f i n i t e l y many p l a n a r g r o u p s

( M a s c h k e ' s Theorem) and i n f i n i t e l y many t o r o i d a l g r o u p s ( P r o u l x ' s c l a s s i f i c a t i o n ) T u c k e r [ T 3 ] showed, i n 1978:

The Genus o f a Group

96

Thm. 7-3L.

For each

4 - 2,

t h e r e a r e a t m o s t f i n i t e l y many q r o u p s

{(r)

such t h a t

1

Chapt. 7

= g.

The f o l l o w i n c j 1 9 8 1 r e s u l t , also d u e t o T u c k e r

[ T 5 ] , is e v e n

more s u r p r i s i n g : Thrn. 7-32.

There is e x a c t l y o n e g r o u p o f cjenus t w o . It has order 96 and p r e s e n t a t i o n ( x , y , z ~ 2x= y 2 = z2 = ( x y ) 2 =

4

( y z ) 3 = (xz)8 = y ( x z ) y(xz)

4

.

e>

=

The ( u n i q u e ) g r o u p of y e n u s t w o i s t h e a u t o m o r p h i s m g r o u p of t h e generalized Petersen graph W a t k i n s [FGWl]

see F r u c h t , G r a v e r , a n d

G(8, 3 ) ;

.

We m e n t i o n t h a t o t h e r d e f i n i t i o n s of t h e " q e n u s of a q r o u p " appear i n t h e l i t e r a t u r e , due t o Levinson [ L 2 ] and B u r n s i d e lBl8J.

meters

by

pctrameter, qards

r

yL(7),

For

and

yB(!')

l i k e the parameter

y(l')

Sk;

b r a n c h e d ) cover o f some

respectively.

Sk

but

is always an

(1.e.

Sn

r

( p o s s i b l y b r a n c h e d ) c o v e r of some S n'

v i a i t s a c t i o n on the Riemann s u r f a c e

meler, i L i s a l w a y s t h e case t h a t

The L e v i n s o n

GA(I')

minimally im-

II'I-fold ( p o s s i b l y

t h e imbedding i s index o n e . )

M a c h l a c h l a n and B u r n s i d e p a r a m e t e r s a l s o r e g a r d fold

[Ml] ,

a d v o c a t e d i n t h i s c h a p t e r , re-

as being d e p i c t e d b y a Cayley graph

bedded on a s u r f a c e

Machlachlan

a f i n i t e group, w e denote t h e s e para-

I

y,(i'),

,

Sk

as an

The

1 r 1-

b u t represent t h e group Sk;

f o r t h e Burnside para-

The f o u r p a r a m e t e r s are

n = 0.

r e l a t e d a s f o l l o w s , where t h e e q u a l i t y h o l d s e x c e p t f o r c e r t a i n

small

I':

zhm. 7-33.

1 a f i n i t e g r o u p , ~ ( 1 ' ) 5 y L ( l ' ) = yM(r) 5 yB(r); all a r e bounded a b o v e by t h e bound of Theorem 7 - 2 2 .

For

Thus t h e most e f f i c i e n t g c n u s , among a11 t h e s e , i s t h a t g i v e n by

y(r).

We remark t h a t b o t h t h e i n e q u a l i t i e s of Theorem 7 - 3 3 c a n T = (ZZn) 4 , for n 2 indicate:

be s t r i c t , a s t h e groups Thm.

7-34-.

Let

I'

(i)

y(T) = 1

(ii)

v,(I')

( i i i ) y,(l')

=

2

n

(Z2n)4,

+

8n

2;

then

4

yM(T) = 1 t 16n 1

+

3 4n (611-5)

4

97

The Characteristic of a Group

Sect. 7-4

7-4.

The Characteristic of a Group

If we allow nonorientable surfaces also, as our "drawing boards" for"picturing"groups, then we are led naturally to the parameter of

this section. surface S = N

S

is

Recall from Section 5-3 that the characteristic of a i(

(S) = 2 - 2 k ,

if

S = Sk;

x

(S)

= 2

-

k,

if

k'

Def. 7-35.

The characteristic of a graph G, denoted by x ( G ) , is the maximum surface characteristic X ( S ) such that G imbeds in

S.

For example, a graph has characteristic two if and only if it is planar:

K5

and

have characteristic one:

K6

K7

has character-

istic zero: and so forth. Def. 7-36.

The characteristic of- a r _group ___

x (r)

=

is given by:

max{X (GA(T))I ,

where the maximum is taken over all generating sets for

A

r.

Thus a group has characteristic two if and only if it appears in Maschke's list (Theorem 7-3.) Here are some less obvious sample results. Thm. 7-37.

(White [WlO]) Let r be finite and abelian:

x (r)

0,

otherwise.

then

'Thm. 7 - 3 8 . _____

( W h i t e [WlO])

(1'1,

Let

G)

=

I, w i t h

lf a n d o r i l y if

where 7-39.

2k+l - b2n+l =

7-40.

~~

7-41.

I 168,

n

n

' -

Wc o b s e r v e t h a t ,

(An )

168,

2 168

0

=

e ) ,

.

)

( 1

= -1.

for

n

*

-n'

.

A

84

1G8,

(Sn )/,(A n ) =

\,

Show: that a n y p r e s e n t a t i o n

for

P

All].

t o r s , n e i t h e r c o n l x of o r d c r

2;

t h e q u a t e r n i o n s , has

Q,

P

h a s e x a c t l y t w o cjencra-

and thaL

e x a c i l y t h r e e g c n e r a t o r s , e a c h of o r d e r

(C ( Q ) )

ISn:

Prohleins

a t lcast t w o cjenerators; t h a t ? f

7-2. )

)

1

=

(S ) n

7-5.

1

=

-1 abab

ur

hnvlii, we see that K 2,212 is a Cayley color graph for Z6. (Here, x = 1 and y = 2, with e = 0; y is clearly redundant, but its presence as a generator enables us to picture Z 6 with K 2 , 2 , 2 . ) We try for an index one imbedding. This requires a quotient graph K having four directed edges (corresponding to 1,5,2,4) and one region in its quotient manifold. But then K has two edges; no such K is possible. Next we try for an index two imbedding. This requires a quotient manifold having two quadrilateral regions, and hence a

Sect. 9-4

Extending the Theory

137

quotient graph with four undirected, and eight directed, edges. The quotient graph is shown in Figure 9-9, imbedded in its quotient manifold, the sphere.

R1

Figure 9-9

Then :

0:

1 , 5 , 4, 2

2:

3,

4:

2,

1, 0 , 4

3:

4 , 5 , 1, 2

5 , 3 , 2, 0

5:

0,

gives a triangular imbedding of The orbits are: 0-1-2

1, 3 , 4

K2,2,2'

1-3-2

0-2-4

1-5-3

0-4-5

2-3-4

0-5-1

3-5-4.

Figure 9-10 shows this imbedding of

in

3, 5, 0

1:

K

2,212

So.

Figure 9 - 1 0

(the octahedral graph)

Q u o t i e n t Graphs and Q u o t i e n t M a n i f o l d s

138

S u p p o s e now t h a t w e t a k e K

2,212

I

= S

Chapt. 9

then 3 = ((23), (12), (132) j; An i n d e x o n e i m b e d d i n g 2

is a g a i n a C a y l e y c o l o r g r a p h .

possible,

i n t h i s c a s e , a s shown by F i g u r e 9 - 1 1 ,

whcrc t h e

F i g u r e 9-11. q u o t i e n t graph i s given.

Note t h a t t h e v e r t e x of d e g r e e o n e h a s a

c u r r e n t of o r d e r t h r e e g o i n g i n t o i t ; t h e two s i n g u l a r e d g e s c a r r y c u r r e n t s of o r d e r t w o , a n d t h e v e r t e x of d e g r e e t h r e e s a t i s f i e s t h e KCL:

(23)(12)(132) = (1) (2)(3) = e. There i s one coset, = S3. W e h a v e :

e:

(23), (12), (132), (123)

(12):

(132)r e , (23), (13)

(23):

e

(13):

(123), (132), (12)I (23)

, (123), (13), (12)

(123):

(13), (23), e l (132)

(132):

(12), (13), (123), e .

The o r b i t s a r e :

-

e-(23)-(123)

(12) (13)- (23)

e- (12)-(23)

(12)- (132)-(13)

e- (132)- (12)

(23)- (13)-(123)

e- (123)-(132)

(13)-(132)- (123)

F i g u r e 9-12 shows t h i s i m b e d d i n g , w h i c h a g r e e s w i t h F i g u r e 9-10,

except for labeling.

Sect. 9-4

Extending the Theory

139

Figure 9-12

K (for the 414 quaternions) given in Figure 7-3. We find the quotient graph for this imbedding and choice of groups. Looking at the dual, we find 2 2 2 3 that e l y I x , and x y have the same boundary: x, x y , x , y. 2 2 2 2 Furthermore , = {eryIx ,x y] is a subgroup of Q , since x = y 2 3 3 3 "x = {x,x y,x ,xy} is a and x y = y , with y4 = e. Also, .I , and each of these regions (in coset of Q with respect to the dual) has boundary: x2 y r x I y 13x Hence this imbedding of K = 2.) We "mod (for Q) is of index two ( ( Q : f i ] = / Q ( / I ? 414 out" regions (in the dual) with identical boundaries, and form the quotient graph, in its quotient manifold S1 , as indicated in Figure 9-13. To make the appropriate identification, we use the general relationship: Example 2:

Consider the quadrilateral imbedding of

.

to match the edge g in the boundary of the region for coset The computations for this situation are: (",x) (i I X

2

-1

Y)

3 = ( 1x,x )

-1

= (\)lY)

((Q,x3)-1 = (\Qx,x) -1 2 (iiX,Y) = (Rx,x y).

X.

140

Q u o t i e n t Graphs a n d Q u o t i e n t N a n i f o l d s

2 X Y

xL-lx3

C

Chapt.

2 X

{

;

[ 3 x

Y

X

Y

F i g u r e 9-13

Y

9

141

Extending the Theory

Sect. 9-4

Finally, we find a quadrilateral imbedding of K Z 1 in (with r = r4 = 105). It is an easy exercise to check that

Example 3: S43

this imbedding is compatible with the euler formula.

Let us hope

1 = Z for the best, and try for an index one imbedding with 21' Then we seek a quotient manifold with r = r = 1. This requires

a quotient graph

K

with

20

directed, or

20

For a quadrilateral imbedding, we would like

10

undirected, edges.

K

to be regular, of

A likely candidate is K = K5. Now we must make an as' * : K* Z21 - 0, satisfying the KCL: then we must signment inducing a single orbit, of length make a selection (pl,p2,...,p5)

degree

4.

+

20,

for

K5.

We find the assignment, using the fact that

eulerian; the selection is made, using the fact that (i.e.

S3

will be the quotient manifold), with

Figure 9-14. a

b

e

C

Figure 9-14 The local vertex permutations are:

The single orbit is:

a:

(b,c,d,e)

b:

(a,c,d,e)

c:

(a,b,d,e)

d:

(a,b,c,e)

e:

(a,c,d,b)

K5

is

yM(K5) = 3

r = r20 = 1;

see

Quotient Graphs and Quotient Manifolds

142

Chapt. 9

a-b-c-d-e-b-a-c-b-d-c-e-d-a-e-c-a-d-be,

giving rise to: 0:

1,2,3,4,5,10,19,7,18,9,17,6,16,8,20,15,14,13,12,11

1:

2,3,4,5,6,11,20,8,19~10,18,7,17,9,0,16,15,14,13,12

which in turn gives r = r4 = 105, for K21 on S 4 3 1 using Edmonds' permutation technique. Although this imbedding may be of little general interest, it does give further indication of the economy of effort possible by seeking quotient graphs in quotient manifolds, rather than graphs in manifolds. Moreover, minimal imbeddings have been found by first finding non-minimal imbeddings of related graphs. For example, Youngs [Y4] on

S

imbeds

K12s+7

on

S

12s + 1 3 s + 3

i

to find

KIZs+10

12s + 1 3 s + 4

9-5.

The General Theory

As we have seen, the standard proof-technique for establishing a genus or maximum genus formula is to first use the Euler formula p - q + r = 2

-

2k,

describing a 2-cell imbedding of the graph G in the surface S k l to find a bound for the parameter under the study. For y ( G ) , we get a lower bound (see, for example, Corollaries 6-14 and 6-15); for we get an upper bound (Theorem 6-24). The second step is to construct an imbedding of G attaining the bound obtained. Inductive constructions are particularly useful for graphs which can be defined as repeated Cartesian products (as in the proofs of Theorems 6-36 and 7-13). A local vertex permutation scheme, such as that discussed in Section 6-6, is frequently helpful (as in the proofs of yM(G)

Sect. 9-5

The General Theory

Theorems 6-25, 6-37, and 6-39).

143

Ad hoc techniques have occasionally

been successful (see the proof of Theorem 6-45). But the most powerful and efficient technique (when it applies, and this is reasonably often!) is the method of quotient graphs and quotient manifolds. In the preceding sections of this chapter, we have discussed this theory, as introduced by Gustin and developed by Youngs, and its application to triangular imbeddings of complete graphs. We have also hinted at a generalization of this theory, and to this we now devote out attention. The material in this section is based on work by Jacques [J2]. Given a finite group

I

with a set

h

of generators for

r ,

recall that the Cayley color graph

C,(l) has vertex set I', with a directed edge -- labeled with generator h . -- if and -1 !f A, unless only if g ' = gc 3 ) ,

=

151

Then

GA(Sn)

5n+2),

Sk

in

with

,

-

r n = nn! =

The s i d e c a r r y i n g c u r r e n t

( T h i s i n d e x o n e imbedding

[W8], and g i v e s a n u p p e r bound f o r t h e appears i n f o r n even.) g e n u s o f t h e s y m m e t r i c g r o u p Sn ,

F i g u r e 9-22 A p p l i c a t i o n 5:

Take

r

M(Z7/Z7)

=

"t

= Z7,

and

A =

{1,2,3}.

i n F i g u r e 9-23 g i v e s

Gn(Z7)

Then = K7

in

S3,

F i g u r e 9-23

r 3 = 7 a n d r I = 3. ( T h i s i n d e x o n e imbedd i n g solves t h e r e g u l a r p a r t of t h e p r o b l e m , f o r

with

finding

y(Kl0);

see S e c t i o n 9-3.

152

Quotient Graphs and Quotient Manifolds

Chapt. 9

New Applications

9-7.

Further connections between the theory and imbedding problems can be explored. For example: Thm. 9-17.

G

is a quotient graph for an imbedding of a graph

K

If

o € index one or two, then

Proof:

=

yM(K)

'L FP (~K )

.

8-

This is a direct consequence of Theorem 6-24.

#

Every Cayley graph has an index one imbedding; in fact Gd(l') has ( I d ? * 1-1) ! index one imbeddings. When are these minimal? A partial answer is provided in: Thm. 9-18.

Let the finite group

...

I

and

An},

I

n

If

subset of

Cor. 9-19.

(G*

(r

)

Let

=

n-1 (

71

i=l in the

=

c

9-20.

Let

n

3

=

,

n

has an index one imbedon

or if

where

Sk

n

=

4

k = 1 t

and no proper

contains a redundant generator, then

A

(zn)n- 1

,

I

n 2 3,

where

E~

with

is the

3 =

{

E

,~ ...

.

I

tn-l,

(n-1)-tuple having

ith position and 0 elsewhere. Then n- 1 .1+" (11-31 , equality holds for

r

3

k.

)-I

-

Cor.

,

A =

is of order

tSi

GA(T)

r = rn = 21rI

(n-3).

I

Then

'

ding, with

be generated by

where each

6 . = e.

IT

i=1

T

1

n = 3,4.

= (3,3 13'3) in the notation of Coxeter and [CMl] (i.e. r is the non-abelian group of order 2 7 , generated by s,t and subject to the relations 2 s 3 = t3 = (st)3 =(s-'t13 = el; let A = {s,t, (st) 1; then y ( G 3 ( r ) ) = 1. Moreover, r is a toroidal group.

Moser

Cor. 9 - 2 1 .

153

New Applications

Sect. 9 - 7 .

Let with

7 ~

=

A

,

the alternating group of degree four, 1; then

= I ( 1 2 3 ) ,( 3 2 4 ) , ( 1 3 4)

, (G (A4)) = 1. (However, A4 is not toroidal, as there exist planar Cayley graphs for A 4 . ) In Section 7 - 2 we saw that we could find the genus of any abelian group ,’ = Z x Zm , where m r i s even zm2 x “1 r

...

...

divides m 1-1 . i = 2 , , r.) The argument used Cm x Cm x Cm = GA(T), for the standard 1 2 r presentation for r , and Theorem 7-13. The techniques employed fail if even one order m is odd. The theory of quotient graphs i and quotient m a n i f o l d s provides a new approach to the genus problem for general cycles, and hence to the genus of an arbitrary finite (assuming mi the fact that

abelian group.

...

(See also Theorem 7 - 1 7 ) .

For example,

M((Z3

where Z 3 , Z2n , and respectively, shows that

9-24,

t

Zln

y

>:

Z2m

Z2m)/(Z3

2 .

Zn

x

Zzm))

are generated by

we conjecture equality for this formula. Figure 9-25, where

x

Also,

r

,

in Figure s

,

M ( (Z3)n/(Z3)n)

is a generator for the ith copy of

shows that

Figure 9-24.

and

Z3,

in

154

Quotient Graphs and Quotient Manifolds

Chapt. 9

Figure 9 - 2 5

b

a

f

d

b C

e d

b

a

f

Figure 9-26

9-8. *9-1.)

9-2.)

Problems

Show that seven "different" minimal imbeddings of K 5 are compatible with the formula of Theorem 9-1. (For example, r3 = 4, r8 = 1: r4 = 5 give two "different" imbeddings of K~ on s1 , since the regions are distributed differently). How many of the seven can you actually find? (Hint: not all seven exist!) Find the quotient manifold and the quotient graph f o r the given (for r = z 8 ) in quadrilateral imbedding of K 4,4

*9-3.)

Figure 9-26. There are twelve different quotient graphs S ( r / f i ) for in s1 with Q # {el; i.e. twelve different K 2,2,2,2 Find these twelve ordered pairs ( r , Q ) for K 2! 2 r 2 I 2 ' ordered pairs and then obtaln an M ( r / i I ) for each nontrivial index.

Problems

Sect. 9-8

9-4.)

155

Use the theory of quotient graphs and quotient manifolds to find a quadrilateral imbedding of

9-5.)

*

9-6.)

K5. Use the theory of quotient graphs and quotient manifolds to find a hexagonal imbedding (r = r6) for K 3 , 3 .

Generalize Example 3 of Section 9-4, to find r = r4n imbcddings of K4n ( 4 n + l )+ I ' where 4n+l is prime. Is this an infinite class of graphs? (In fact, only the parity of

9-7.) 9-8.)

4n+l

is required.)

Prove Theorem 9-15. Find , (Q,) in three ways:

(1) by a construction, such as that employed in the proof of Theorem 7-13; (2) by selecting an appropriate (p, , p2 , , p ) and using 2"

...

Edmonds' algorithm (see Theorem 6-47) to show that every orbit has length 4; (3) by finding an appropriate M ( T / \ ' ) . 9-9.)

*

9-10.)

We see from Theorem 6-25 that K5 has a 2-cell imbedding Show that the theory of this chapter is of no in S 3

.

aid in find this imbedding. Find a triangular imbedding for surface

(n # 3 . )

K

n,n,n,n

in a suitable

This Page Intentionally Left Blank

157

CHAPTER 10 VOLTAGE GRAPHS

In t h e p r e c c c d i n g c h a p t e r , f o l l o w i n g t h e h i s t o r i c a l d e v e l o p m e n t of G u s t i n , Youngs, R i n g e l , J a c q u e s , e t a l , w e g e n e r a t e d g r a p h i m b e d d i n g s by means o f c o n s t r u c t i n g ( s i m p l e r ) c u r r e n t g r a p h i m b e d d i n g s . These l a t t e r were v a r i o u s l y c a l l e d " q u o t i e n t g r a p h i n q u o t i e n t Inanif o l d " and " r e d u c e d c o n s t e l l a t i o n .

Each a p p r o a c h h a s t h e a d v a n t a g e

"

o f economy o f c o n s t r u c t i o n ; e a c h a p p r o a c h h a s t h e d i s a d v a n t a g e o f g e n e r a t i n g r e g i o n s by v e r t i c e s and v e r t i c e s by r e g i o n s .

Moreover,

t h e mode of " g e n e r a t i o n " i s n o t made e x p l i c i t , i n a way t o g i v e maximum a i d t o i n t u i t i o n . These d e f e c t s a r e c o r r e c t e d b y t h e v o l t a g e g r a p h t h e o r y i n i t i a t e d by G r o s s [G3] i n 1 9 7 4 and by i t s i n t e r p r e t a t i o n i n t h e c o n t e x t o f b r a n c h e d c o v e r i n g s p a c e s ( s e e p a p e r s by Gross a n d A l p e r t : [GA2],

[GAl],

and [AG2] . )

I n t h i s c h a p t e r w e p r e s e n t j u s t enough c o v e r i n g s p a c e t h e o r y f o r t h e i m m e d i a t e c o n t e x t ( f o r more d e t a i l s , see [ M 3 ] ) t r o d u c e v o l t a g e g r a p h s , w i t h examples. Map-coloring

and t h e n i n -

W e t h e n r e v i s i t t h e Heawood

Finally, we

Theorem f r o m t h i s a d v a n t a g e o u s v i e w p o i n t .

describe the strong tensor product construction f o r graphs;

this is

an i t e r a t i v e p r o c e s s t h a t o f t e n p r o d u c e s a n i n f i n i t e tower of g r a p h i m b e d d i n g s f r o m o n e v o l t a g e g r a p h i m b e d d i n g a t t h e base.

10-1. Def.

10-1.

A function

Covering Spaces

-

0 :X

-f-

from o n e t o p o l o g i c a l s p a c e t o

X

another is c a l l e d a covering projection i f every point h a s a neighborhood

x C X

i.e.

p

c a l l y onto maps to

-

Y.

$

Ux

maps e a c h component o f Ux.

If

Y

E

X

and

homeomorphically o n t o

which i s e v e n l y c o v e r e d ; p

-1

(Ux)

? c 2 Y,

we say t h a t

x

We call

x

homeomorphi-

i s such t h a t

a covering space f o r

X.

Y

p

lifts

158

V o l t a g e Graphs

Chapt.

10

A s t a n d a r d r e s u l t i n t h e t h e o r y of c o v e r i n g s p a c e s i s t h d t

Ip-l(x) = n,

x E X.

i s i n d e p e n d e n t of t h e c h o i c e o f

~

then

1s c a l l e d an n - f o l d

p

1

Ip-'(x)

If

Here a r e

covering projection.

some s t a n d a r d e x a m p l e s of c o v e r i n g s p a c e s and c o v e r i n g p r o j e c t i o n s :

0:

R

t , s i n t ) . Then p "wra s" t h e a r o u n d t h e u n i t c i r c l e S1 i n f i n i t e l y many times,

where

Sl,

K

real line

b i ( t )

w i t h each half-open

= (cos

[r, r

interval

+

S1

"covering"

277)

exactly

once. I f we pick (1,O) E S1 and ( s a y ) U = { ( x , y ) E s1 1 -1 - -1 2 < y ' -1 l , i s e v e n l y c o v e r e d - a s (! 2 then "(LO) u(l.o)

Q (2ni i€I (2ni

-

-

ii/6,

n:6,

271i

211i

+

+

71/61

=

and e a c h open i n t e r v a l

11/6)

i s c l e a r l y homeomorphic t o

U

(1,O).

The c o v e r i n g p r o j e c t i o n o f t h e p r e c e e d i n q e x a m p l e f a i l s t o b e n-fold,

f o r cach

n

n F N

However, € o r c v c r y

N.

it i s a simple

m a t t e r t o c o n s t r u c t an n - f o l d c o v e r i n g p r o j e c t i o n . Example 2 :

Define

p

: S1

+

S1

by

n,

ip(z) = z

where

z

wraps

S1

t i m e s around i t s e l f , and

n

-

-

f o r example

(I

is

S1

r e g a r d e d a s a complex number. (Recall Chat, i f z = cis8 = n ( c o s o , s i n O ) , t h e n z = c i s nB = ( c o s n 6 , s i n n t i ) . ) Thus

p

-1 (1,O) con-

s i s t s p r e c i s e l y of t h e n t h r o o t s of u n i t y . The above e x a m p l e i s s i g n i f i c a n t f o r t h e s e q u e l , as it d e s c r i b e s how r e g i o n b o u n d a r i e s o f a d e s i r e d g r a p h i m b e d d i n q w i l l p r o j e c t t o

t h o s c of a v o l t a g e g r a p h i m b e d d i n g , i n t h e s i m p l e s t o f cases ( c o r r e s p o n d i n g t o t h e KCL Example 3 :

-

Recall t h a t

is t h e p r o j e c t i v e p l a n e

see S e c t i o n 9-2 So

-

holding.)

R3),

is the sphere ( i n

and t h a t

N1

( n o n o r i e n t a b l e s u r f a c e of g e n u s o n e , or

sphere with one crosscap.)

Define

p : So

+

N1

by

antipodal

identification; i.e.

o(x,y,z) = (-x,-y,-z);

then

(1

is a 2-fold

covering p r o j e c t i o n .

I n t u i t i v e l y , w e could regard

ti

as f i x i n g the

bottom hemisphere

and

d e p r e s s i n g t h e t o p hemisphere i n " r e v e r s e

fashion; the antipodal i d e n t i f i c a t i o n along t h e equator t h e n "sews on" t h e c r o s s c a p . (This takes place i n R4, n o t R3 . )

overlapping

"

S e c t . 10-1

159

Covering Spaces

The f o l l o w i n g r e s u l t i s w e l l known

( b y t h e Riemann-Hurwitz

Theorem.)

Thm. 10-2. -___

If

S

-

i s an n - f o l d c o v e r i n g p r o j e c t i o n f o r

S

s u r f a c e s , t h e n t h e s u r f a c e c h a r a c t e r i s t i c s are r e l a t e d by:

x(S)

=

nu(S).

This relationship is quite natural: in

S,

with

5 .w i t h

p

np

-

+ r

q

then

= ,y(S),

vertices,

nq

e d g e s , and

-

x ( S ) = n p - nq + n r = n ( p

+

q

r)

if

p

-1

is 2-cell

G

imbedded

i s 2 - c e l l imbedded i n

(G)

r e g i o n s , so t h a t

nr

nX(s).

=

T h u s , f o r e x a m p l e , o n l y t h e t o r u s c a n cover t h e t o r u s ( s i n c e h = 1

k = 1.)

forces

covering space is

An i m p o r t a n t g e n e r a l i z a t i o n of t h e c o n c e p t o f

r e q u i r e d , t o c o v e r t h e c a s e w h e r e t h e KCL d o e s n o t h o l d .

(See also

and i t s t w o c o r o l l a r i e s . )

Theorem 9 - 1 4

Def.

A function

10-4.

: X

p

from one t o p o l o g i c a l s p a c e t o

X

a

_ I _

a n o t h e r i s called a branched c o v e r i n g p r o j e c t i o n

-

(and

i s a b r a n c h e d c o v e r i n g s p a c e o f X) i f t h e r e e x i s t s a f i n i t e set B X such t h a t t h e r e s t r i c t e d f u n c t i o n X

2 -

p:

p-l(B)

and

-f

B

p o i n t s of

X

-

i s a covering projection.

B

are c a l l e d b r a n c h p o i n t s .

-.

p:

some c a r d i n a l number

n

branching a t

-1 ( U b

-

b

Ib})

in

Ub

-

Ub

-

b,

is n-fold,

{b]

the for

called the multiplicity of

.

where X.

+

The

b E B

a s u f f i c i e n t l y small n e i g h b o r h o o d of

Ub

restricted f u n c t i o n

p

For

Ub

i s a component of n = 1,

(If

t h e n t h e r e i s no

branching.) Standard examples h e r e are : Example 4 : give

P:

Let D

+

D

a c o m p l e x number,

D = { ( x , y ) E R2

by

p ( z ) = zn

Then

p

I

+

x2

y2 < 11,

w h e r e , as b e f o r e ,

“wraps“

D

t h e u n i t d i s k , and z

around i t s e l f

i s r e g a r d e d as n times,

except t h a t t h e o r i g i n i s f i x e d (and t h u s i s a branch p o i n t of multiplicity

n. )

Example -5 :

Define

p:

p(l,O,@) = (l,n",$).

So

+

Then

i n s p h e r i c a l c o o r d i n a t e s , by

So, p

10

Chapt.

Voltage Graphs

160

"wraps"

around i t s e l f

So

times,

n

e x c e p t t h a t t h e n o r t h a n d s o u t h p o l e s are b o t h f i x e d ; e a c h i s t h u s a b r a n c h p o i n t of m u l t i p l i c i t y

n.

Both of t h e e x a m p l e s a b o v e a r e i m p o r t a n t f o r w h a t i s t o f o l l o w ,

a s t h e f o r m e r g i v e s t h e p r o t o t y p e d e s c r i p t i o n of t h e p r o j e c t i o n o f r e g i o n s i n ( p o s s i b l y b r a n c h e d ) c o v e r i n g s of o n e g r a p h i m b e d d i n g by

-

another

-

the local picture

w h i l e t h e l a t t e r i s o n e i n s t a n c e of

t h e p r o j e c t i o n between t h e c o r r e s p o n d i n q a m b i e n t s u r f a c e s

-

the

qlobal picture.

10-2. be a p s e u d o g r a p h .

K

Let

a s s o c i a t e two o r i e n t e d set

K*

= { ( u , ~J )u v

Dcf. 1 9 - 5 . __-

edges

E E(K)

V o l t a g e Graphs With e a c h e d g e

e = (u,v)

and

uv E E ( K ) , w e -1 e = ( v , u ) , and w e

1.

A voltage graph is a t r i p l e

where

(KIT,@),

r

i s a group, and (I: K* ~ ( e - l )= ( @ ( e ) ) - ' f o r a l l e E K*.

pseudograph,

W e remark t h a t t h e p s e u d o g r a p h

q u o t i e n t graph

Def. 1 0 - 6 .

S(l'/S?)

K

satisfies

i s c l o s e l y related t o t h e

of S e c t i o n 9-3.

The c o v e r i n g g r a p h

Kx

r

4,

V(K)x

r

edges

( u , g ) ( v , g ( I ( e ) ) of

and each edge

for

e = uv

Kx$r

(K,l','$)

of

,

K

has vertex set determines t h e

for all

Irl -

f o l d covering space of

covering space of

K

K;

r .

g

For p s e u d o g r a p h s r e g a r d e d a s t o p o l o g i c a l s p a c e s , t h e n

i s an

is a

K

Kx

@

r

i n f a c t , every regular

c a n b e o b t a i n e d i n t h i s manner (see

[GTZ] )

.

( F o r a d d i t i o n a l i n f o r m a t i o n on c o v e r i n g p r o j e c t i o n s of g r a p h s , see W a l k e r [ W l ] , F a r z a n a n d Waller [FWl] , C l a r k e , Thomas, a n d Waller [CTWL]

,

S i t [S5], and B i g g s [Bll]. )

F o r a n y walk (K.

r,b),

w e set

w: e l , e2 , . . . , e m b e g i n n i n g a t

v E V(K)

in

Sect. 1 0 - 2

V o l t a g e Graphs

161

m

and d e f i n e t h e l o c a l g r o u p a t

rV

Then

r

[r:

thus

and

v,

if

NOW l e t S,

(P,

(K,r,@)

i s g i v e n by

K x I'

v E V(K).

@

be 2 - c e l l imbedded i n an o r i e n t a b l e s u r f a c e

... if

f o r each

g

Thus

r @

f o r any

t h e number of

(K,r,$),

a s d e s c r i b e d a l g e b r a i c a l l y by t h e r o t a t i o n scheme P = of P to Kx r P2 I , P P L W e d e f i n e t h e lift

I

follows:

Kx

,

a r e con-

Tv

is connected.

K

components o f t h e c o v e r i n g g r a p h

[T: I v l

and

a r e i n t h e same component of

v

i s i n d e p e n d e n t of

v}.

Tu

and m o r e o v e r u

For a c o n n e c t e d v o l t a g e g r a p h

10-7.

Thm.

!'v]

if

by:

i s a c l o s e d walk a t

w

i s a s u b g r o u p of

j u g a t e s u b g r o u p s of K;

I

= {$(w)

Tv

v

.

4)

pv(vlu) = (v,w),

E .'l

then

(See F i g u r e 10-1.)

Then

d e t e r m i n e s a 2 - c e l l i m b e d d i n g o f e a c h component of

P

The power o f v o l t a g e g r a p h t h e o r y i s t h a t t h e s i m p l e r i m -

bedding of

K

b e l o w g i v e s much i n f o r m a t i o n a b o u t t h e more c o m p l i -

c a t e d imbedding of

Kx

of t h e imbedding o f t h e o r d e r of walk i n

as

K

@(w)

K

in

$

r

above. on

r

S

To see t h i s , l e t

i n d u c e d by

where

w

=

PI

el

e2

and l e t

, . . . ,em

c o n s i s t i n g of t h e o r d e r e d b o u n d a r y of

is u n i q u e u p t o i n v e r s e s and c o n j u g a c y , t h e o r i e n t a t i o n of

R

/R/

R

R.

be a region

lRl@

be

is a closed (Since

@(w)

i s i n d e p e n d e n t of w.) We then

and o f t h e i n i t i a l v e r t e x of

have t h e f o l l o w i n g c e n t r a l r e s u l t , due t o Gross a n d A l p e r t [GAl]:

162

Chapt.

10

b e a v o l t a g e g r a p h w i t h r o t a t i o n scheme

P

Voltage Graphs

V

( i n K)

F i g u r e 10-1

Thm. 1 0 - 8 . ___-

Let

(KrT,g)

and

P

I

P

t h e l i f t of

to

Kx I .

-

d e t e r m i n e 2 - c e l l i m b e d d i n g s of orientable surfaces

S

and

Q,

S

K

Let

P

and

KX

and $-

P

I

on t h e

respectively.

Then

t h e r e e x i s t s a ( p o s s i b l y branched) covering p r o j e c t i o n

S

p:

S

such t h a t :

-1

(K) = Kx 1'; @ (ii) i f b i s a b r a n c h p o i n t of m u l t i p l i c i t y n , t h e n b i s i n t h e i n t e r i o r of a r e g i o n R s u c h / R / *= n ; that (1)

p

(iii) i f

R

is a r e g i o n of t h e i m b e d d i n g of

i s a k-yon,

then

p

-1 (R)

has

p o n e n t s , e a c h o f w h i c h is a of t h e c o v e r i n g i m b e d d i n g of

W e remark t h a t , f o r mapped b y

p

onto

R,

/RIQ, = n,

p:

D

+

which com-

k ( R ( @- yon r e g i o n Kx

Q,

r.

e a c h component of

e s s e n t i a l l y by

K

Irl/\Rl$-

D, p ( z ) =

p

-1(R) i s n z , as

d e s c r i b e d i n Example 4 o f S e c t i o n 1 0 - 1 .

Note t h a t Theorem 1 0 - 8

of Theorem 9 - 1 3 t h e following:

(iii) i s j u s t t h e v o l t a g e graph analog

for c u r r e n t graphs.

Now compare D e f i n i t i o n 9-3 w i t h

Sect. 10-2

V o l t a g e Graphs

Def. 1 0 - 9 .

:)(w) = e

If

region

of

R

Law

S

ii:

on

K

+

bounding

S,

s a t i s f i e s t h e Kirchoff

R

i f the

(KVL);

KVL

holds f o r a l l regions

w e s a y t h a t t h e i m b e d d i n g of

(K,I',$)

satisfies the

then

KVL.

( i . e . t h e r e i s no

is a covering projection

S

(K,T,O)

KVL.

I f t h e i m b e d d i n g of

10-10.

w

f o r t h e c l o s e d walk

we say t h a t

s a t i s f i e s the Cor.

r

i n t h e v o l t a g e g r a p h i m b e d d i n g , so t h a t

R

lRi,$ = 1, Voltage

in

163

branching.) I n e i t h e r c a s e ( b r a n c h i n g or n o t ) , i f the voltage graph b e d d i n g of

Kx

9

r

K

m

i s an i n d e x

m,

has order

i s s a i d t o have i n d e x

(K,T,@)

m,

then

and t h e im-

i m b e d d i n g ( c o m p a r e Theorem 9-10

( 3 ).) V o l t a g e g r a p h t h e o r y i s e v e n more g e n e r a l t h a n t h e " g e n e r a l theory" of Section 9-5, be a C a y l e y g r a p h .

in t h a t the covering graph

Kx

R

c e r n e d w i t h t h e Cayley g r a p h c a s e ; t h i s w i l l arise f o r g r o u p of

r

and v o l t a g e g r a p h

(K,Q,$)

-

where

.

S c h r e i e r c o s e t graph f o r 9 i n 1' Then Kx@R D = {@(k)1 k K*} (reduced a s i n Section 9-5.)

(X,T,@); t h e n

v o l t a g e graph j o i n t c o p i e s of

need n o t

I'

0.

I n t h i s b o o k , h o w e v e r , w e are p r i m a r i l y c o n -

C, ( 1')

.

In the simplest case

-

Kx

@

r

K*

$:

=

-

K

a sub-

-

1'

a

where

CA(T),

O r , w e could use

would c o n s i s t o f

index 1

+

lrl/lQl

h a s one v e r t e x ,

r

= R, the

two a p p r o a c h e s c o i n c i d e , and t h e c o n s t r u c t i o n i s q u i t e c l e a r : each

g C T,

then

V(KX

r)

we identify = V(K)X

r

2

(v,g)

r

and

with

g

(where

dis-

for

V(K) = {v}),

E(Kx@T) 5 I [ g , g @ ( e ) ]1 g E

and

r,

e = ( u , v ) E K*}. F i n a l l y , w e remark t h a t e v e n v o l t a g e g r a p h t h e o r y h a s b e e n q e n e r a l i z e d ; see Gross a n d T u c k e r [ G T 2 ] .

Also, see B o u c h e t [ B 1 4 1

( a f o l d i s a 1analog of t h e 0-dimensional branching.) And, see

f o r g r a p h i m b e d d i n g s as c o v e r i n g s p a c e s w i t h f o l d s diraensional

P a r s o n s , P i s a n s k i , and Jackson

( [ P P L : l ] a n d [ J P P l ] ) f o r g r a p h imbed-

d i n g s a s b r a n c h e d c o v e r i n g s p a c e s , where t h e r e s t r i c t i o n s o f t h e b r a n c h e d c o v e r i n g s t o t h e imbedded g r a p h s a r e "wrapped q u a s i - c o v e r ings.

"

164

V o l t a g e Graphs

10-3.

Chapt. 1 0

Examples

C o n s i d e r t h e i n d e x one v o l t a g e g r a p h ( a " b o u y u e t " of two

Example 1:

c i r c l e s ) of F i g u r e 1 0 - 2 ,

shown imbedded on t h e t o r u s

W e take

S1.

I< -Ic

Figure 10-2 A = {a,b}

r

for

=

Sl

(IrI 2 5),

abelian

CA(T) = Kx

C o r o l l a r y 1 0 - 3 , w e see t h a t torus as w e l l .

Irl.

so t h a t t h e

t h e r e i s no branching.

h o l d s ; t h u s , by C o r o l l a r y 1 0 - 1 0 ,

r

@

KVL

Then, by

w i l l b e imbedded on t h e

t h i s imbedding w i l l have

By Theorem 1 0 - 8 ,

r = r4

=

Thus F i g u r e 1 0 - 2 c o m p l e t e l y d e t e r m i n e s an i n f i n i t e f a m i l y of

q u a d r i l a t e r a l imbeddings on t h e t o r u s .

We m e n t i o n t h r e e s p e c i a l

c a s e s b e low. a)

Let

a = 1

b = 2

and

bedding i s s e l f - d u a l

r

in

= Z5;

then

GA(I') = K5.

This im-

and i s t h e g r o u n d case f o r s e v e r a l i n f i n i t e

f a m i l i e s of i m b e d d i n g s a p p e a r i n g i n t h e l i t e r a t u r e :

(i)

-

imbedding i n

has a self-dual

S = S

n (4n-3) (see [WS]); h e r e t h e v o l t a g e g r a p h imbedding i s j u s t t h e

K4n+l

normal form r e p r e s e n t a t i o n f o r

( i i ) . ) The c o v e r i n g p r o j e c t i o n i s indeed

-

x(S) = 2

( i i ) The imbedding of m e r s i o n of number for

-

2 n ( 4 n - 3 ) = ( 4 n + l ) (2-211) K5

S1

on

k = (p-1) ( p - 4 ) / 4

=

(4n+l)x(S).

c a n b e augmented t o an i m -

attaining the toroidal crossing

on S1

Kg ( 2 ) v ~ ( K ~ ( ~= ) 10. )

see Theorem 6-71.

( s e e Theorem 5-5 n ( 4 n + l ) - f o l d , and

S = S

Similarly, and

p

v ~ ( K ~ ( ~= ) p) ( p - 1 ) / 2 ,

a p r i m e power

3

1 (mod 4 ) ;

Sect.

Ex amp les

10-3

( i i i ) T h e imbedding of

K5

on

165

c a n also b e m o d i f i e d t o

S1

G5 = K

o b t a i n a g e n u s imbedding f o r t h i s e x t e n d s t o show

y (G,)

less a 1 - f a c t o r ;

5,5

1

I, which

= i(n-1) (n-4) /4

will

be u s e f u l i n C h a p t e r 13. We o b s e r v e t h a t t h e r o t a t i o n scheme f o r r e a d i l y o b t a i n e d from F i g u r e 1 0 - 2 .

K5

S1

in

can be

First, we find

A s i s customary f o r i n d e x one imbeddings, w e i d e n t i f y

(v,g)

N e x t , as K5 h a s no loops o r as permuting t h e P ( v , g ) - 69 r a t h e r t h a n t h e e d g e s a t g. Finally, taking

w i t h g i n t h e c o v e r i n g graph. m u l t i p l e edges, we regard each neighbors of

a = 1

and

g

we get:

b = 2,

(In practice,

t h e above i s w r i t t e n down d i r e c t l y f r o m t h e f i g u r e . )

The c o v e r i n g p r o j e c t i o n i s 5 - f o l d : a n d o n e 4-gon

for

K

in

t e n e d g e s , a n d f i v e 4-gon b)

Now l e t

a = (1,O)

GA(T) = Cm x Cn

and

S1

t h e s i n g l e v e r t e x , two e d g e s ,

l i f t respectively t o five vertices,

regions f o r b = (0,l)

for

and f i n d s a s e l f - d u a l ,

Kx

4

r

r

= K5

= Z

( a l s o i n S1.)

x Zn;

quadrilateral,

then toroidal

i s i l l u s t r a t e d i n F i g u r e 10-3. The c o v e r i n g s p a c e n a t u r e o f t h i s i m b e d d i n g i s r e a d i l y a p p a r e n t : imbedding.

The case

m = n = 3

t h e s i n g l e v e r t e x , two e d g e s , and o n e 4-gon now l i f t , r e s p e c t i v e -

l y , t o n i n e v e r t i c e s , e i g h t e e n e d g e s , and n i n e 4-gons, t h i s nine-fold

projection

p.

under

166

Chapt. 10

Voltage Graphs

Figure 10-3

c)

Now l e t

I' = II(S,)

= (a,b

1

So

-lb4

R

2,

S1,

of

R2

i n t h e sense t h a t

thus

GA(T)

b)

x

t h e funda-

2,

G A ( l ' ) h a s a n imbed-

Gn(I')

a s b e i n g imbedded

a s o n e of t h e t h r e e r e g u l a r t e s s e l l a t i o n s of t h e

(Recall t h a t

plane.

e ) = 2

( i f w e a l l o w t h e v e r t e x set to have l i m i t p o i n t s ) ,

b u t it i s more n a t u r a l t o c o n s i d e r in

=

I n t h i s case

mental group of t h e t o r u s . d i n g on

aba

in

R2

is t h e universal covering space f o r

R2

c o v e r s any s p a c e which c o v e r s

a l s o covers e a c h

CA(Zm

Zn)

SI;

imbedding

a b o v e , i n a n a t u r a l way.

W e mention t h a t

n(Sl)

i s also t h e g r o u p of o n e of t h e

s e v e n t e e n w a l l p a p e r d e s i g n s ( i .e. a p l a n a r c r y s t a l l o g r a p h i c group.)

The v o l t a g e g r a p h t h e o r y i s a p p l i c a b l e t o e a c h of t h e

seventeen p l a n a r i n f i n i t e wallpaper groups ( a s p r e s e n t e d , f o r e x a m p l e , i n [ B 1 7 ] ) ; of t h e s e i m b e d d i n g s i n one branched c o v e r s o f

of

So,

So,

R2,

s i x are index

s i x a r e i n d e x t w o b r a n c h e d covers

o n e i s an i n d e x two u n b r a n c h e d c o v e r of

i n d e x t w o u n b r a n c h e d c o v e r s of

S1,

So,

t h r e e are

and t h e p a t t e r r ! d e s c r i b e d

by

!I

167

Examp 1es

Sect. 10-3

(S1)

Example 2 : ____-_

i s an i n d e x o n e u n b r a n c h e d c o v e r o f

s l i g h t l y , t o o b t a i n Figure 10-4.

Now m o d i f y F i g u r e 1 0 - 2

b

V

sl'

V

-6

Figure 10-4

r

We still require with

11'1 2

r =

index one

t o b e a b e l i a n , b u t now t a k e

Then

7.

S1;

r 3 imbedding i n

= {a,b,a+b}

a g a i n i n f i n i t e l y many s u c h

i m b e d d i n g s a r e d e t e r m i n e d , o n e f o r e a c h c h o i c e of

dual

e a c h of t h e s e h a s b i c h r o m a t i c c h r o m a t i c number t w o , where

A

i s r e g u l a r o f d e g r e e s i x and h a s a n

G l z ( I')

G = G

Ll

I'

.

( t h e geometric dual

( r ) );

Moreover, G*

has

t h e indicated 2-coloring

o f t h e r e g i o n s of F i g u r e 1 0 - 4 l i f t s t o a 2 - c o l o r i n g of t h e r e g i o n s for a)

GA(T).

For

(This w i l l be u s e f u l i n Chapter 1 2 . )

a = 1

and

b = 2

in

T

= Z7,

Gn(I') = K7;

t h i s is t h e

s a m e famous t o r o i d a l i m b e d d i n g o b t a i n e d b y t h e d u a l F i g u r e 9-3 (see a l s o F i g u r e 9-5.) ( i ) A f a i r l y n a t u r a l e x t e n s i o n of t h s v o l t a g e g r a p h g i v e s

o r i e n t a b l e t r i a n g u l a r imbeddings f o r

s 6 N.

( i i ) By t a k i n g f i r s t b = 5 i n

r =

a = 1

Z13,

and

=

14,

where

s = 0.)

and t h e n

a = 2

and

t o r o i d a l imbeddings are o b t a i n e d f o r

t w o complementary g r a p h s i n N(1,l)

b = 3

for a l l

K12s+7,

(The d u a l i s b i c h r o m a t i c o n l y f o r

N(y,y')

K13;

t h i s shows t h a t

i s t h e least i n t e g e r such

168

V o l t a g e Graphs

t h a t e v e r y graph

of o r d e r a t l e a s t

G

see a l s o

and

[A21

a = 1

ding f o r If

i s two

(fj1(K13)

=

(ii).)

6-60

c)

S

see B e i n e k e [ B 6 ] and R i n g e l [ R 1 3 ] ; see a l s o Theorem

2;

If

. 4 shows

in

It also

[ACl].)

K13

t h a t t h e t o r o i d a l t h i c k n e s s of

b)

Sy , G

irnbeddable i n

10

is not

N(y,y') -

(G

( v , y ' ) bi-imbeddable (See [ A w l ] ;

Chapt.

and

b

I' = Z

for

2

=

w e o b t a i n a g e n u s imbed-

8'

K 4( 2 ) * b = (0,l)

a = (1,O) and K3 ( 3 )

bedding f o r

r

for

= Z3

%

a genus i m -

Z3,

results.

V o l t a g e g r a p h c o n s t r u c t i o n s are b y n o means u n i q u e f o r a g i v e n F o r e x a m p l e , t h e d u a l of F i g u r e 10-4 s e r v e s a s an

g r a p h imbedding.

I' = Z ) f o r an i m b e d d i n g of K 3 313 t h e d u a l of t h i s imbeddinq t h e n s e r v e s

i n d e x two v o l t a g e g r a p h ( u s i n g on

S1

r = r

having

= 3;

6 a s an index t h r e e voltage graph (again using

l a r l y imbed Theorem 4.2

on

K3(3)

in

S1

! = Z3)

[KRWI].

W e a l s o mention t h a t t a k i n g

+

but replacing

a

b

imbedding f o r

K3(3)

5

with

a = 1 and

in

r

= Zg,

b = 2

as

K3(3)

= 9,

e x a m p l e of a b r a n c h e d c o v e r i n g i n t h i s s e c t i o n : wraps around a t r i a n q l e W e keep

a

(1,O)

=

below

R

9

r = r Z 7= 2

(iii)); t h i s g i v e s and i s o u r f i r s t e a c h 27-gon

t i m e s , since

b = (O,l),

and

in Figure 10-4,

g i v e s an

( a s s e e n from Theorem 1 0 - 8

t h e maximum g e n u s of

d)

to triangu-

T h i s i s t h e g r o u n d case o f

again.

b u t now i n

Ga(r)

t h e r e s u l t i n g t r i a n g u l a r i m b e d d i n g of

above

/RIib = 9.

r

wj.11

= :i4

x Z4;

b e of

i n t e r e s t -in S e c t i o n 1 2 - 7 . e)

I n [ S 2 0 1 t h e f o l l o w i n g c o n j e c t u r e i s a t t r i b u t e d t o Grunbaum: a graph

G

dual graph number

G*

A,(G*)

(consider the (consider

if

h a s an o r i e n t a b l e t r i a n g u l a r i m b e d d i n g , t h e n t h e h a s a T a i t c o l o r i n g ( t h a t is, h a s edge chromatic = 3.)

G*

G* = P ,

We r e m a r k t h a t

G

c a n n o t b e allowed l o o p s

of F i g u r e 8 . 2 i n [ B C L l ] ) o r m u l t i p l e e d g e s t h e Petersen graph, i n

S1)

and t h a t t h e

c o n j e c t u r e i s f a l s e a l s o f o r n o n o r i e n t a b l e t r i a n g u l a r imbeddings (consider

G = K6

in

N1,

where

G* = P . )

However t h e

Ex amp1es

S e c t . 10-3

169

c o n j e c t u r e i s t r u e , as f o r m u l a t e d , f o r 3 - d e g e n e r a t e g r a p h s

a s a n e a s y i n d u c t i o n a r g u m e n t shows. y(G*) = 2: factorable

x,(G*)

if

It is a l s o t r u e i f

i s c u b i c and b i p a r t i t e ,

G*

( s e e , f o r e x a m p l e , Theorem 8 - 7

= 3.

Moreover, any

G,

then

in

[BCLl])

GA(T) t r i a n g u l a t i n g

i s 1-

G*

S1

,

so t h a t

as a

c o v e r i n g s p a c e o f F i g u r e 1 0 - 4 h a s a n e d g e c o l o r i n g i n d u c e d by the labels

a, b,

a

and

+

now c o l o r t h e d u a l e d g e s

b;

t o a g r e e w i t h t h e c o l o r s on t h e e d g e s of

t h e y cross.

G

( i n G*) Thus

t h e v o l t a g e graph of F i g u r e 10-4 a l s o p r o v i d e s an e a s y a l g o r i t h m i n f i n i t e l y many t o r o i d a l t r i -

f o r GrGnbaum's c o n j e c t u r e , f o r angulation s, Example 3: n # 3,

The g r a p h s

K4(n)

Y ( K ~ ( ~ )= ) ( n - 1 )

have genus

as e s t a b l i s h e d by Jungerman

, for

[ J S ] and i n d e p e n d e n t l y ( f o r

( S e e Theorem 6 - 4 2 . )

e v e n ) b y Garman [ G l l .

2

n

Jungerman d i d n o t

a n n o u n c e Y ( K ~ ( ~ ) ) a, l t h o u g h h e d i d r e p o r t t h a t Y ( K 4 ( 3 ) ) > 4. The i n d e x t h r e e ( t h i s i s n o t a C a y l e y g r a p h c o n s t r u c t i o n ) t o r o i d a l v o l t a g e graph of Figure 10-5, =

5,

using

( R e f e r t o Theorem 1 0 - 8

r

= Z3,

shows t h a t

v(K4 (3)) r3 = 6 a n d r g = 4 . K 3( 3 ) ( i i i ) : t h e r e a r e f o u r branch p o i n t s -

by f i r s t i m b e d d i n g

in

S5

with

0

a

F i g u r e 10-5 i n d i c a t e d by d o t s i n s i d e r e g i o n s d e t e r m i n i n g o n e 9-gon;

-

e a c h of m u l t i p l i c i t y t h r e e a n d

t h e t w o regions having no branch p o i n t each

l i f t to three triangles.)

I t i s e a s i l y v e r i f i e d t h a t e a c h 9-gon

bounded b y a h a m i l t o n i a n c y c l e i n

K3(3).

is

(For example, t h e clock-

w i s e b o u n d a r y o f t h e l i f t of t h e s h a d e d r e g i o n

R

is:

170

V o l t a y e Graphs

Chnpt. 1 0

(bO,aO, c O , b l , a l , c l , b 2 , a 2 , ~ 2 ) - w h e r e , f o r e a s e o f n o t a t i o n , w e w r i t e ( v , g ) a s vg; t h e f a c t t h a t [ R [ += 3 a s s u r e s t h a t t h e l i f t of R wraps around

R

t h r e e times u n d e r b r a n c h e d p r o j e c t i o n b y

Now

b j . )

add a new v e r t e x i n t h e i n t e r i o r o f e a c n of t h e f o u r 9-cjons,

and

join e a c h new v e r t e x t o d l l n i n e b o u n d a r y v e r t i c e s , € o r t h e regLon In The r e s u l t i s an imbeddinrj of K 4(3) w i t h r3 = 3 3 and rg = 1. Note t h a t t h c c o n s t r u c t i o n i s s5 r e a d i l y augmented t o g i v e a t r i a n g u l a r ( a n d h e n c e g e n u s ) i m b e d d i n g for t h e c o m p l e t e 4 - p a r t i t e g r a p h K 4 , 3 , 3 , 3 ' containinq t h a t vertex.

Example 4 :

For

a q r a p h and

G

w e mean t h a t m u l t i g r a p h replaced by the again

11

w h i c h r e s u l t s when e a c h e d q e

uv.

edges

a n a t u r a l n u m b e r , by n - f o l d

n

S = S2).

e x t e n s i o n q i v e n below (where

-

for

T

abelian

uv

of

The c o n s t r u c t l o n of k i g u r e 1 0 - 4

(n

=

1 1

- 7)

and

A

W e take K

G

G

is

suggests

= ia,b,a+b]-

a bouquet of n i n e

a+b

Figure 10-6

circles in

S2,

Gi,(i')

on

Sn+l,

r = r3,

with

c o v e r i n g s p a c e i s an

r = r3

KVL,

and b i c h r o m a t i c d u a l .

having bichromatic dual.

The g r o u n d c a s e

( j i v e s a n o r i e n t a b l e t r i a n g u l a r imbeddinq €or 3 - f o l d a = 1, b = 2 ) .

Junqerman

a l l o w e d ) for m-fold

K,,

The

i m b e d d i n g of t h e 1 8 - r e g u l a r 3 - f o l d K,

(n = 7)

(using

[JSI h a s f o u n d g e n u s i m b e d d i n g s ( n o d i y o n s f o r a l l m and n ( i n b o t h t h e o r i e n t a b l e

and t h e n o n o r i c n t a b l e c a s e s .

)

Sect. 10-4

171

The Heawood Map-coloring Theorem ( a g a i n )

The s p h e r i c a l i n d e x two v o l t d g e g r a p h o f F i g u r e 1 0 - 7

Example 5 :

a l s o appears i n

it uses

[SWl];

I

= 2

n

t o imbed

K

n,n

= K

2(n)

in

n- 1

F i g u r e 10-7 where k = n ( n - 1 ) / 2 , with n ; now a p p l y Theorem 1 0 - 8 ( i i i ). ) Moreo v e r , e a c h r e g i o n boundary i s a h a m i l t o n i a n c y c l e . ( F o r example, Sk

( a s a branched covering s p a c e ) ,

r = r2 n = n

(each

lRl$ =

t h e r e g i o n c o v e r i n g t h e shaded d i g o n h a s c l o c k w i s e b o u n d a r y ( a O , b l , al,b2,a2,

. ..

,

a ( n - 1 ) ,bO).)

augmented t o g i v e an the case

n = 3

r

=

Thus t h i s imbedding i s r e a d i l y

r3

imbedding f o r

K3(n)

i n Example 2 c ) a b o v e ) , so t h a t

(we discussed

Y (K3 (nl 1

= n (n-1) / 2 ;

t h i s p r o o f i s f a r s h o r t e r ( a n d more e l e g a n t ) t h a n t h o s e of e i t h e r [RY5] o r [ W 5 1 .

r = r3

I t i s e a s y ( s e e Problem 1 0 - 1 )

imbedding f o r

t o show t h a t e v e r y

has bicnromatic dual; t h i s w i l l be

K3( n ) important i n Section 12-1. Moreover, t h e s e i m b e d d i n g s t h u s a l l s a t i s f y t h e Grllnbaum c o n j e c t u r e .

( I n f a c t , t h e t h r e e t y p e s of e d g e s

i n t h e complete t r i p a r t i t e graph

G = K

edge c o l o r i n g f o r t h e d u a l

10-4.

3(n)

d e t e r m i n e a n a t u r a l 3-

G*.)

The Heawood Map-coloring Theorem

(again)

W e r e c o n s i d e r t h e c r u x of t h e p r o o f of t h e Heawood m a p - c o l o r i n g

theorem:

t h e c o n s t r u c t i o n of g e n u s i m b e d d i n g s o f c o m p l e t e g r a p h s by

t h e u s e of c u r r e n t g r a p h s ; o u r c o n t e x t now w i l l b e t h a t of b r a n c h e d c o v e r i n g s p a c e s , and w e u s e t h e v o l t a g e g r a p h c o n s t r u c t i o n .

This

v i e w p o i n t t r e a t s o n l y t h e r e g u l a r p a r t of e a c h case; t h e a d d i t i o n a l

v o l t a g e Graphs

172

Chapt.

adjacency p a r t s continue t o be handled a s before [GTl]

,

[W121 , a n d [ R 1 4 I .

(see S e c t i o n 9-3,

)

R e c a l l t h e c u r r e n t g r a p h of F i g u r e 9 - 3 , the torus.

10

The d u a l of F i g u r e 10-4

f o r imbedding

K,

on

( i n p a r t i c u l a r , see Example 2 a )

i s the correspondinq voltage graph.

From e i t h e r t h e s i n g l e r e g i o n

of F i g u r e 9-3 or t h e s i n g l e v e r t e x o f F i g u r e 1 0 - 4 , w e r e a d t h e r o t a -

tion

po:

scheme

(1,3,2,6,4,5);

P = (p,

(Note t h a t t h e

,pl

t h i s generates the e n t i r e rotation

, . . . , p,)

KCL/KVL

and i n t u r n t h e t r i a n g u l a r i m b e d d i n g .

holds.)

In g e n e r a l , t h e c u r r e n t graph

+

p = 1, q = 6 s

3,

and

r = r3 = 4 s

t h e c u r r e n t g r a p h , i s imbedded i n (since the

K(s)

for

h o l d s ) and i s

KVL

+

2;

Ss+l.

(12s

+

with 12s+7' t h e v o l t a g e g r a p h , a s was I' = 2

The c o v e r i n g i s u n b r a n c h e d 7 ) - f o l d , g i v i n g an imbedding

as b e f o r e , w i t h p = 1 2 s 2 1 2 s +2s+l q = ( 1 2 s + 7 ) ( 6 s + 3 ) , and r = r 3 = ( 1 2 s + 7 ) ( 4 s + 2 ) . of

in

K12s+7

S

The c u r r e n t g r a p h for with

I' = Z 7

r3 = 7

and

K10

(see, f o r example,

r7

= 3

+

7,

it is used,

a p p e a r s i n F i g u r e 9-23;

K7

[W12]) t o imbed

(determined by t h e v o r t i c e s

in

S3

with

x,

y,

and

S3

by

K7 + F3 =

T h i s i s r e a d i l y augmented t o a t r i a n g u l a t i o n of

K10 - { x y , x z , y z } ;

(as given i n

K12s+7

h a s as i t s d u a l a v o l t a g e g r a p h f o r

S e c t i o n 9-3)

z.)

t h e n t h e a d d i t i o n a l a d j a c e n c y p a r t o f t h e con-

s t r u c t i o n a d d s t h e t h r e e m i s s i n g e d g e s over o n e e x t r a h a n d l e . The c o r r e s p o n d i n g v o l t a g e g r a p h i s g i v e n i n F i g u r e 1 0 - d .

Figure 10-8 The r o t a t i o n

po = ( 1 , 6 , 4 , 3 , 2 , 5 )

i s e a s i l y read f r o m e i t h e r f i g u r e .

The o u t e r r e g i o n o f F i g u r e 1 0 - 8 s a t i s f i e s t h e

KVL

Sect. 10-5

(1

+

4

+

2

S t r o n g Tensor P r o d u c t s

=

in

0,

173

s o b y Theorem 10-8 ( i i i ) it l i f t s , w i t h -

Z7),

o u t b r a n c h i n g , t o seven t r i a n g l e s i n t h e c o v e r i n g imbedding of

K7.

Each o f t h e t h r e e monogons, h o w e v e r , h a s v o l t a g e sum o f o r d e r

1 R / $ = 7 , s o t h a t e a c h i s c o v e r e d , w i t h b r a n c h i n g , by o n e s e v e n i s imbedded i n S3 a s a sided region ( a heptagon.) Thus K, branched covering space over

Moreover,

So.

s i n c e each voltage

generates

Z7,

each heptagonal region boundary i s a h a m i l t o n i a n

cycle for

K7.

T h u s , i f o n e v e r t e x i s a d d e d i n t h e i n t e r i o r of e a c h

h e p t a g o n , a t r i a n g u l a r i m b e d d i n g of

K,

+ E,

results.

The a d d i t i o n -

a l adjacency c o n s t r u c t i o n i s as b e f o r e . The t h r e e v o r t i c e s f o r e a c h c u r r e n t g r a p h i n case 1 0 ( s e e [ R 1 4 ] ) become monogons i n t h e v o l t a g e g r a p h i m b e d d i n g , w i t h t h e s i n g l e boundary edges always c a r r y i n g a v o l t a g e which g e n e r a t e s by a h a m i l t o n i a n c y c l e for

s

as f o r t h e c a s e

'12 s+7' bounded

( 1 2 s + 7 ) -gon,

Thus e a c h monogon i s b r a n c h - c o v e r e d b y a s i n g l e

and t h e c o n s t r u c t i o n proceeds

K12s+7,

0.

=

F o r a d i s c u s s i o n of a l l t w e l v e c a s e s i n t h i s v o l t a g e g r a p h / b r a n c h e d c o v e r i n g s p a c e s e t t i n g , see Gross a n d T u c k e r [ G T l ] .

10-5.

S t r o n g Tensor Products

In [GRWl] t h e following product operation f o r graphs w a s i n t r o duced: Def.

10-11.

F o r two g r a p h s product

G1

edge set u2vz Thus

GI

3

G1

G2

or

c o n s i s t s of

the strong tensor

G2,

has v e r t e x set

{ ( u , , u 2 ) (v, , v 2 )

E(G2))

G2

and

(u1V1

1 V(G1) I

1

(ul

E(G1)

V(G1)

x

= v1

and

and

V(G2)

u 2 v 2 E E(G2))1.

d i s j o i n t c o p i e s of

t o g e t h e r with a l l t h e t e n s o r product edges.

and

G2

,

The f o l l o w i n g t w o p r o -

p e r t i e s of t h e s t r o n g t e n s o r p r o d u c t f o l l o w f r o m t h e d e f i n i t i o n :

Thm. 1 0 - 1 3 .

G1

G2

G2 [ G 1 ] ,

i s a subgraph of t h e l e x i c o g r a p h i c p r o d u c t w i t h e q u a l i t y i f and o n l y i f

G1

i s complete.

174

V o l t a g e Graphs

The t h e o r e m b e l o w f r o m

[GRWl]

C h a p t . 10

i s e s p c c i a l l y germane i n t h e

p r e s e n t work : Thm. 10-14.

Let

h a v e a t r i a n g u l a r imbedding i n an o r i e n t a b l e

G2

surface, with bichromatic dual; l e t

he connected

G2

a n d b i c h r o m a t i c , w i t h maximum d e g r e e a t m o s t two. Then

'5'

G1

h a s a t r i a n g u l a r imbedding i n an o r i e n t -

G2

able s u r f a c e , w i t h b i c h r o m a t i c d u a l . I n C h a p t e r 1 2 w e w i l l f i n d t h a t o r i e n t a b l e t r i a n g u l a t i o n s of strongly r e g u l a r graphs w i t h bichromatic d u a l s can b e p a r t i c u l a r l y useful.

We s h a l l see t h a t e a c h

we take

G1

= K2,

Kn(m)

is s t r o n g l y regular; thus i f

Theorems 10-12 a n d 10-14 w i l l combine t o p r e s e r v e

t h i s u s e f u l n e s s under t h e s t r o n g t e n s o r product o p e r a t i o n . F o r t h e p r e s e n t , w e o b s e r v e t h a t e a c h c o v e r i n g i m b e d d i n g of Figure 10-4

(with

g i v e s rise t o an i n f i n i t e c o l l e c t i o n

GA(I') = G2)

of b i c h r o m a t i c d u a l t r i a n g u l a t i o n s , v i a r e p e a t e d a p p l i c a t i o n o f

-

Theorem 1 0 - 1 4

f o r each graph

G1

In p a r t i c u l a r , w e can t a k e

G1

as i n t h a t theorem. and e x t e n d Examples

= K2

orientable bichromatic dual triangulations and

K

2a,

r e s p e c t i v e l y t o o b t a i n , f o r each nonnegative i n t e g e r

2b, a n d 2 c

1-

3 ( 3. 2 K )

for

K

.

k,

k ' K 4 ( 2 k + l )' 7(2 )

F i n a l l y , w e remark t h a t Theorem 1 0 - 1 3 c o m b i n e s w i t h Theorem

6-45 t o y i e l d t h e s u r p r i s i n g l y g e n e r a l f o r m u l a o f : Thrn. 1 0 - 1 5 .

Let

k

have

G

v e r t i c e s of p o s i t i v e degree,

p

n o n t r i v i a l components, and n o 3 - c y c l e s .

q

edges,

Then

Y ( K ~ ~ ! ? ?G ) = k + n ( n q - p ) .

10-6. 10-1.)

Show t h a t t h e i d e n t i t y map Show t h a t i f

p: X

X

and

R2,

-f

X

i s always a covering

pxq:

are c o v e r i n g p r o -

q: Y

+

Y

X x Y

-+

X x Y,

.

(pxq) ( X , Y ) = ( p x , q y ) . Regard t h e t o r u s as S1 = C x C ,

circle i n

X 5

+

j e c t i o n s , t h e n so i s 10-3.)

i:

-

projection. 10-2.)

Problems

Show t h a t

-

where

C

where

is the unit

R2 = R x R , R x C ,

and

Sect.

10-6

S1 = C

175

Problems

x

C

all c o v e r

S1.

Try t o v i s u a l i z e p

,

i n each

case. 10-4.)

Show t h a t e v e r y o r i e n t a b l e t r i a n g u l a r i m b e d d i n g f o r

K

3 (n)

h a s bichromat i c d u a l . 10-5.)

Find t h e v o l t a g e g r a p h c o r r e s p o n d i n g t o F i g u r e 9-11.

10-6.)

F i n d t h e v o l t a g e g r a p h c o r r e s p o n d i n g t o F i g u r e 9-13.

10-7.)

What i m b e d d i n g

( a n d of w h a t g r a p h ) c o v e r s t h e v o l t a g e g r a p h

i m b e d d i n g of F i g u r e 1 0 - 9

(r'

= Z8)?

Is t h e r e b r a n c h i n g ?

Where, a n d of w h a t m u l t i p l i c i t y ?

3

F i g u r e 10-9

*

10-8.)

Answer t h e q u e s t i o n s o f P r o b l e m 1 0 - 7 , b u t now f o r F i g u r e 10-10,

with

r

= Z 12.

Then show t h a t

Figure 10-10

y(K 6,6,3)

=

7.

176

*

10-9.)

V o l t a g c Graphs

Chapt. 1 0

t o show t h a t

E x t e n d t h c c o n s t r u c t i o n of P r o b l e m 1 0 - 8 ,

Y ( K 2 n ,2 n , 2n-2

2 ) = 2(n-1) ,

for

n

1 , 2 (mod 3 ) .

(Scc

a l s o [SWl].)

10-10.)

Show t h a t i f w e b e g i n w i t h an a r b i t r a r y 2 - c e l l

a Cayley c o l o r graph

I

and u s e v o l t a g e g r o u p

I;'\ 10-11.)

imbedding o f

as a v o l t a g e qraph imbedding,

Ct!(I)

,

t h e c o v e r i n g s p a c e c o n s i s t s of

d i s j o i n t c o p i e s of t h e b a s e c o n f i g u r a t i o n .

A graph

G

of s i z e

i s s a i d t o be c o n s e r v a t i v e (see

q

Bange, B a r k a u s k a s , and S l a t e r [BBSl]) i f t h e e d g e s c a n b e

1

o r i e n t e d and d i s t i n c t l y l a b e l e d w i t h

, 2 , . . . ,q

so t h a t

a t e a c h v e r t e x t h e sum of t h e numbers on t h e i n w a r d l y d i r e c t e d e d g e s e q u a l s t h a t on t h e o u t w a r d l y d i r e c t e d e d g e s . Show t h a t i f s u c h a g r a p h h a s a 2 - c e l l

t h e n it s e r v e s as a n i n d e x one a u r r e n t g r a p h , w i t h

r = 1,

r

c u r r e n t s from of

Gi,(: )

for

r = r n-1 G

n - 4, :I

n :> 2 y + 1.

= Zn,

determined,

the special cases that,

imbedding w i t h

n = 2q Kn

i: = {1

for

+

S t u d y t h e imbedding

, 2 , . .. ,y } .

1, n = 2y

+

2.

Consider

Use t h e f a c t

i s c o n s e r v a t i v e ( s e e [BBSl]) t o f i n d

imbeddings f o r

,

K

n L 1 ,2

(mod 4 ) .

If

n -n+l

i s c u b i c , show t h a t , i n g e n e r a l , t h e c o v e r i n g i m b e d d i n g

i s minimal f o r

GL ( 1 ' ) .

10-12. )

Prove Theorem 10-12.

10-13. )

P r o v e Theorem 1 0 - 1 3 .

CHAPTER I 1 NONORIENTABLE GRAPH IMBEDDINGS

W e r e i t e r a t e t h a t o u r p r i m a r y m o t i v a t i o n f o r t h i s e n t i r e work

is t o depict graphs

-

a n d i n p a r t i c u l a r q r a p h s o f g r o u p s - on s u r -

f a c e s ( l o c a l l y 2 - d i m e n s i o n a l d r a w i n g b o a r d s ) , u s u a l l y as e f f i c i e n t l y

I n k e e p i n g w i t h our d e s i r e t o s t u d y s t r u c t u r e s t h a t

as p o s s i b l e .

s p a c e , w e h a v e c o n c e n t r a t e d on t h e o r i e n t -

e x i s t i n three-dimensional able surfaces

Sk ( k

orientable surfaces




e q u a l i t y h o l d s i f a n d o n l y if a non-

2;

o r i e n t a b l e t r i a n g u l a r imbedding can b e f o u n d f o r Proof:

Let

i s 2-cell,

-

q

+

b y Theorem 1 1 - 3 .

-

with equality i f

and o n l y i f

T(G)

+

r = r 3'

2.

Thus

-

G

If

where

-

r 5 p

> q/3

then

q

=

-

p

-

-

p

q/3

+

2,

X(NT(G))

2 >

(using

2

q/2 - p

2

=

-

?(G)

2q > 3r

i m b e d d i n g on


3r

n o t 2-cell,

11-8.

p

then

A s i n t h e o r i e n t a b l e case,

q/3 - p

Cor.

be imbedded i n N ? ( G )

G

G.

+

2;

p

2

3,

and h a s n o t r i a n g l e s ,

e q u a l i t y h o l d s i f and o n l y

i f a n o n o r i e n t a b l e q u a d r i l a t e r a l i m b e d d i n g can b e f o u n d for

G.

(The p r o o f i s e n t i r e l y a n a l o g o u s t o t h a t o f C o r o l l a r y 11-7.) Cor. 11-9.

Let

be n o n o r i e n t a b l y , minimally, 2-cell

Kn

imbedded i n

( T h e p r o o f is a n a l o g o u s t o t h a t o f Theorem 9 - 1 . ) T h e n o n o r i e n t a b l e a n a l o g s t o Theorem 6-18 a n d C o r o l l a r y 6-19

are a l s o f a l s e , a s y(2K7) = 2y(K seen,

Y(K7)

7

=

G = 2K7

= 2,

3,

Y(2K7)

so t h a t

shows:

< 5, Y(2K7)

s i n c e , by C o r o l l a r y 6-19, b y Theorem 11-2.

# 2?(K7).

B u t , as w e h a v e

What i s t r u e f o l l o w s

i n t w o d e f i n i t i o n s and o n e t h e o r e m , d u e t o S t a h l and B e i n e k e [SBl]:

Nonor i e n t d b l e Graph Imb e d d i n 9s

180

Def. 1 1 - 1 0 . ____-

The i n a n i I o l d number o f a g r a p h max{2-2.( (G) ,

Def.

11-11.

A graph

Let

G1

G

’((G)

=

11fG)

# 2 -

:

2y(G).

,:

If

G

i s o r i e n t a b l y simple,

then

Nonorientable Covering Spaces

R e c a l l from Example 3 o f S e c t i o n 1 0 - 1 t h a t t h e s p h e r e c o v e r i n g s p a c e of t h e p r o j e c t i v e p l a n e .

2-fold

,i(G)

be a g r a p h w i t h b l o c k s ( o r components)

, G2 , . , , ,Gk.

11 - 2 .

p(G)

IS

2-; ( G )1 .

is o r i e n t a b l y s i i n p l e i f

G

that is, if Thm. 1 1 - 1 2 .

G

C h a p t . 11

is a

So

I n g e n e r a l (see

Stahl [ S S ] , f o r c x a m p l e j , t h e r e i s a 2 - f o l d c o v e r i n q p r o j e c t i o n p: Sk Nk+l, f o r e v e r y n o n n e g a t i v e i n t e g e r k. Thus e v e r y non+

o r i e n t a b l e s u r f a c e h a s an o r i e n t a b l e c o v e r i n g s u r f a c e .

Trivially,

each nonorientable s u r f a c e has a t l e a s t one n o n o r i e n t a b l e covering s u r f a c e , namely i t s e l € ( s e e Problem 10-1.)

see an e x a m p l e o f a n o n o r i e n t a b l e s u r f a c e

I n S e c t i o n 11-4 w e s h a l l (N3)

with i n f i n i t e l y

many n o n o r i e n t a b l e c o v e r i n g s u r f a c e s

“n+2, n -- 7 ) . I n c o n t r a s t , i f t h e b a s e s p a c e i s an o r i e n t a b l e s u r f a c e , t h e n

e v e r y covering s u r f a c e must b e o r i e n t a b l e a l s o .

-

where e a c h base s p a c e i s o r i e n t a b l e

-

Thus i n C h a p t e r 1 0

each covering space is

o r i e n t a b l e and h e n c e u n a m b i g u o u s l y d e t e r m i n e d by i t s c h a r a c t e r i s t i c . I n t h i s c h a p t e r , h o w e v e r , s i n c e t h e b a s e s p a c e w i l l a l w a y s b e nono r i e n t a b l e , i f t h e c o v e r i n g s u r f a c e h a s even c h a r a c t e r i s t i c , t h e n i t s o r i e n t a b i l i t y c h a r a c t e r needs t o be a s c e r t a i n e d t o s p e c i f y t h e

surface uniquely. I n t h e n e x t s e c t i o n w e s h a l l s e e how t o d o t h i s , i n t h e c o n t e x t of n o n o r i e n t a b l e g r a p h imbeddings.

11-3.

N o n o r i e n t a b l e V o l t a g e Graph I m b e d d i n g s

‘To e x t e n d t h e v o l t a g e g r a p h t h e o r y t o n o n o r i e n t a b l e i m b e d d i n g s , w e augment t h e r o t a t i o n scheme

P

o f S e c t i o n 10-2 t o a p a i r

(PIX)

Sect. 11-3

N o n o r i e n t a b l c V o l t a g e Graph I m b e d d i n g s

':

c a l l e d an i m b e d d i n g scheme:

K*

181

g i v e s a voltage graph

Z2

.

The r e g i o n b o u n d a r i e s a r e computed a l m o s t a s i n t h e -1 ( s e e S e c t i o n 6-6), e x c e p t t h a t s o m e t i m e s Pv ( v , u ) orientable case (K,

)

Z2,

i s u s e d i n s t e a d of let

P

and l e t by:

1)

(K, I ,

-

b e t h e l i f t of

l(e) =

see [S8] a n d [SWl] f o r d e t a i l s .

Pv(v,u);

be a v o l t a g e g r a p h w i t h i m b e d d i n g scheme to

P

( c ) , f o r each l i f t

t o b e t h e l i f t of

,I

(P

to

)

Kxi,l

e

e E E(K).

of

Kx I .

A:

define

;

(Kx

Define

9

Now

,A)

(P

r

) *

Z2

-f

A)

(P,

Then t h e c o n c l u s i o n s of Theorem

9

1 0 - 8 and C o r o l l a r y 1 0 - 1 0 c a n b e shown t o f o l l o w v e r b a t i m .

Thm. 1 1 - 1 3 .

(K, i',

Let

scheme Kx 1'.

(6 , i f

and

(P , A )

Let

9

be a v o l t a g e graph w i t h imbedding

(li)

,A)

(P

beddings of

and

K

respectively.

and Kx i'

b

, i) d e t e r m i n e

on t h e s u r f a c e s

Q

61:

S

+

S

(iii) i f

=

R

n 1.

then

p

-1

(R) h a s

t h e c o v e r i n g i m b e d d i n g of I f t h e imbedding o f S

f

S

such t h a t

R

i s a r e g i o n of t h e imbedding of

i s a k-gon,

p:

n, t h e n

~

p o n e n t s , e a c h of w h i c h i s a

Cor. 1 1 - 1 4 .

S

such t h a t :

i s i n t h e i n t e r i o r of a r e g i o n

~

im-

and

S

is a b r a n c h p o i n t of m u l t i p l i c i t y

b

1

to

2-cell

Then t h e r e e x i s t s a ( p o s s i b l y b r a n c h e d )

covering projection

(ii) i f

(P , A)

t h e l i f t of

(P

f , q3)

(K,

1 T1//RI(li k 1 RI -gon

Kx@l'.

9

which

K

comregion of

s a t i s f i e s t h e KVL,

is a covering projection

(i.e.

then

t h e r e i s no

branching. ) Of c o u r s e , t h e i m b e d d i n g scheme

only when

S

scheme

alone s u f f i c e s , and

P

is nonorientable;

as o b s e r v e d a b o v e .

If

S

o r i e n t a b i l i t y c h a r a c t e r of Def. 11-15.

if S

,A )

i s used i n p r a c t i c e

is orientable, the rotation

is n e c e s s a r i l y o r i e n t a b l e also,

i s n o n o r i e n t a b l e , w e must determine t h e

-

t o t h i s end w e g i v e :

S;

For a v o l t a g e graph K

(P S

is s a i d t o b e

(K,

T , n),

11-trivial

if

a c l o s e d walk q(c) = e

in

c I .

in

182

N o n o r i e n t a b l e Graph Imbeddings

Thm.

11-16.

Chapt.

Under t h e h y p o t h e s i s of Theorem 1 1 - 1 3 ,

-

surface

S

11

the derived

i s o r i e n t a b l e i f and o n l y i f e v e r y

@ - t r i v i a l c l o s e d walk i n

is a l s o

K

A-trivial.

W e mention t h a t Garman [ G l ] h a s f u r t h e r e x t e n d e d t h e t h e o r y of v o l t a g e g r a p h s , a s o u t l i n e d i n t h e p r e v i o u s and p r e s e n t c h a p t e r s , t o

In p a r t i c u l a r , Theorems 1 0 - 7 ,

p s e u d o s u r f a c e imbeddings. and 1 1 - 1 6 ,

and C o r o l l a r i e s 1 0 - 1 0 and 1 1 - 1 4

pseudosurface;

in t h i s case,

11-13,

10-8,

a l s o apply f o r

S

a

i s n e c e s s a r i l y also a p s e u d o s u r f a c e

S

( o r a g e n e r a l i z e d p s e u d o s u r f a c e ; see D e f i n i t i o n 5 - 2 6 . )

Examples

11-4.

Example 1:

Let

m : 2 (mod 4 ) .

F i g u r e 11-1 p r e s e n t s a p r o j e c t i v e

p l a n e imbedding of a p s e u d o g r a p h loops:

X (e)

$(e) = 1

= 1

f o r each loop

a s indicated.

Let

Kxb Z n

Then

m/2

w i t h o n e v e r t e x and

K

e.

I' = Zn

(n

even)

,

and

i s a graph with v e r t e x set

F i g u r e 11-1 {(v,i)I 0 < i < by

m/2

below,

n-11

i n which

edges f o r each IRI+ = n/2,

i.

(v,i)

and

For each region

s o t h a t t h e r e g i o n s of

+

1)

of

K

(v, i R

Kx$ Zn

a r e joined imbedded

imbedded above

Examples

Sect. 11-4

a r e a l l n-gons; (m/2)2

=

m

183

i n f a c t each i s a hamiltonian cycle.

There a r e

Thus i f w e p l a c e a new

such r e g i o n s i n a l l (above.)

v e r t e x i n t h e i n t e r i o r of e a c h s u c h r e g i o n , j o i n i t by n o n - i n t e r s e c t i n g e d g e s t o a l l t h e v e r t i c e s on i t s b o u n d a r y , and t h e n d e l e t e a l l t h e o r i g i n a l e d g e s of

Km, n is a l s o

a q u a d r i l a t e r a l i m b e d d i n g of

Kx Z n ,

9

It i s clear t h a t every

results.

it i s i n t o

b y Theorem 1 1 - 1 6 ;

$ - t r i v i a l c l o s e d walk i n

K

so t h a t t h e c o v e r i n g imbedding i s o r i e n t a b l e ,

A-trivial,

Sk,

k = (m-2) (n-2)/4.

where

g i v e s a p a r t i a l p r o o f o f Theorem 6-37;

t h e approach

This

appears i n

[SWlI. Example 2 . (

~

Y1

N3.

= n

).

In F i g u r e 11-2 w e t a k e A

7)

and

K

=

{a,b,a+b}

for

r

abelian

a s a b o u q u e t o f t h r e e c i r c l e s imbedded i n

The s i n g l e v e r t e x h a s b e e n g i v e n a n o r i e n t a t i o n a s i n d i c a t e d .

a

b

F i g u r e 11-2 Then t h o s e e d g e s w h i c h are c o h e r e n t l y o r i e n t e d , a s i n d u c e d b y t h e v e r t e x o r i e n t a t i o n , are a s s i g n e d A = 0 ( t h e r e are n o n e ) ; a l l o t h e r edges

-

i n t h i s case t h e t h r e e e d g e s b o u n d i n g t h e h e x a g o n

a s s i g n e d X = 1. but not

A-trivial;

b y Theorem 11-16. The s p e c i a l c a s e

N9'

Then t h e c l o s e d w a l k

a

+

b

thus the covering surface Moreover,

n = 7

-

(a+b) S

x ( S ) = n x ( S ) = -n,

produces a s e l f - d u a l

is

-

are

@-trivial

is nonorientable, so t h a t

imbedding o f

-

S = N

K7

n+2 on

184

Non o r i e n t a b l e Graph Imbeddi n y s

Example 3 .

Now m o d i f y F i g u r e 1 1 - 2

Chapt. 11

s l i g h t l y , t o g i v e F i g u r e 11-3.

a

b F i g u r e 11-3 s t i l l on

The c o v e r i n g i m b e d d i n g , the 12-regular 2 - f o l d e d g e s of of

A

qives

K

= {1,2,3,n-3,n-2,n-l}

7(2)

On

i s now t r i a n g u l a r and of Nn+2, F i n a l l y , change 9 (i.e. relabel the

see Problem 1 1 - 4 1

N3;

of c h a r a c t e r i s t i c

GA(I')

where

on

K

GA(T').

-2n;

t o give 1-fold triangulations

f o r example, c o n s i d e r

ancl s t i l l

7.

n

GA(Z2n),

The c a s e

n

7

=

N16.

11-5.

The Heawood M a p - c o l o r i n g Theorem, Nonorientable Version

If w e apply Corollary 11-7 t o t h e graphs Lemma 1 1 - 1 7 .

Y(Km)

1.

{(m-3k(m-4)j,f o r

m

2

G = K

= 3.

,

we obtain:

3.

W e h a v e n o t e d t h a t e q u a l i t y does n o t h o l d f o r Y(K,)

m

m

= 7

and t h a t

However, e q u a l i t y d o e s h o l d i n e v e r y o t h e r case:

Thm. 1 1 - 1 8 This

on

s , of c o u r s e , e s t a b l i s h e d b y f i n d i n g an i m b e d d i n g of

).

Km

h = ((m-3) (m-4)

The f i r s t p r o o f w a s b y R i n g e l l R 8 1 , w i t h Nht o u t t h e b e n e f i t of c u r r e n t g r a p h s . L a t e r p r o o f s do employ c u r r e n t g r a p h t h e o r y ( i n p a r t i c u l a r , t h e t h e o r y of c a s c a d e s ) , a n d a g a i n s p l i t n a t u r a l l y i n t o t h e r e s i d u e cases of

m

modulo 1 2 ( s e e [ R 1 4 ] ,

[LYl],

Sect. 11-6

[J41 . )

185

Other R e s u l t s

Lemma 1 1 - 1 7 and Theorem 1 1 - 1 8 c o m b i n e t o form t h e n o n o r i e n t -

a b l e v e r s i o n of t h e complete g r a p h theorem: Thm.

11-19.

I _

(m-3) (m-4) =* 1 i(K7) = 3 ,(Km)

Now w e l e t (7

+

v/49-24n)‘/2;

Mn

=

here

1,

for m

i n Theorem 8 - 1 3 ,

Nh

;3

and

m # 7;

recalling that

f(n) =

n = 2-h:

To c o m p l e t e t h e p r o o f o f t h e Heawood M a p - c o l o r i n g Theorem, nono r i e n t a b l e v e r s i o n , w e need t h e f o l l o w i n g : Lemma 1 1 - 2 1 .

);(Nh)

[ f ( 2 - h ) ] = i ( 7 + J1+24h)/2It

Proof;

Consider

Nh,

and now c o n s i d e r also

so t h a t N. (K,)

*

j((Nh)

2

X(N?(~,))

h

# 2.

NY(K,)

m

=

Note t h a t

[f(2-h)], T(Km)

*

m

)

If(2-h)

Define

Now

2 y(Nh).

h # 2.

Km

h,

imbeds i n

[f(2-h)],

=

5

so t h a t

I.

S i n c e F r a n k l i n [ F 3 ] showed t h a t

x(N2)

=

6,

we c a n now c o m b i n e

L e m m a s 11-20 a n d 1 1 - 2 1 t o r e s t a t e Theorem 8-10. Thm. 1 1 - 2 2 .

j((Nh)

=

[(7

+

11-6.

-h)/21,

for

h # 2;

x ( N 2 ) = 6.

Other R e s u l t s

W e g i v e a few o f the a n a l o g s t o t h e o r i e n t a b l e r e s u l t s of

Chapter 6 .

F o r m o s t of t h e o t h e r n o n o r i e n t a b l e g e n u s r e s u l t s known

by 1 9 7 8 , see T a b l e 2 o f S t a h l [ S 9 ] . Thm. 1 1 - 2 3 .

(Ringel [R12])

186

C h a p t . 11

N o n o r i e n t a b l e Graph I m b e d d i n g s

Thm.

(Jungcrrnan [J6])

11-24.

?(Qn)

+ 2n-2(n-4), n

= 2

i(Q,)

and

-

T(Q,)

except t h a t

2,

= 3

11.

Thm. 1 1 - 2 5 .

Thm. 1 1 - 2 6 .

11-27.

Thm.

( S t a h l a n d W h i t e [SWl]) n ;' 4

For

Bouchet Section 6-5) of

m

Y (Kn, n , n - 4 )

=

(n-2) (n-3)

(see

[B13] h a s a l s o u s e d h i s " g e n e r a t i v e m - v a l u a t i o n s "

to study

7 (Kn

(m) )

.

He c o n s i d e r e d t h e r e s i d u e c l a s s e s

n (mod 6 ) and c o n s t r u c t e d t r i a n g u l a r i m b e d d i n g s f o r

and

of t h e s e

and e v e n ,

36

cases, thus determining

f o r those

Y(K,(~))

18

18

cases. We now t u r n o u r a t t e n t i o n t o t h e maximum n o n o r i e n t a b l e g e n u s parameter. Def.

11-28.

The maximum n o n o r i e n t a b l e g e n u s , graph

G

i s t h e maximum

imbedding i n

h

yM(G)

f o r which

,

of a c o n n e c t e d

G

h a s a 2-cell

Nh.

I n c o n t r a s t t o t h e o r i e n t a b l e case, t h e v a l u e o f t h i s p a r a m e t e r

( s e e R i n g e l [R15]

is r e a d i l y c a l c u l a t e d f o r each connected graph

G

a n d S t a h l [ S 8 ] ) ; r e c a l l t h a t t h e B e t t i number

B(G)

= q

-

p

+

1.

T h i s means t h a t e v e r y c o n n e c t e d g r a p h h a s a n o n o r i e n t a b l e 2 - c e l l imbedding w i t h

r = 1.

The f o l l o w i n g t h e o r e m , a l s o d u e t o S t a h l , g i v e s an a n a l o g t o

Duke's Theorem ( 6 - 2 1 ) ______ Thm. 1 1 - 3 0 .

and t o C o r o l l a r y 6 - 2 2 .

A connected graph

and o n l y i f

T(G)

G

5h

h a s a 2-cell

i U(G).

imbedding i n

Nh

if

Problems

Sect. 11-7

187

We give a nonorientable analog to the famous Kuratowski Theorem (6-6); this result is due to Glover, Huneke, and Wang [GHWl] and to

Archdeacon [A7]. Thm. 11-31.

A graph

G

imbeds in the projective plane

N1

if and

only if, for each graph H in a prescribed list of 103 graphs, G contains no subgraph homeomorphic with H. It has long been conjectured that there is a finite number of graphs obstructing imbedding into each surface, whether orientable or nonorientable (by Theorem 6-6 there are two for 11-31 there are 103 for

So;

by Theorem

Recently Seymour [S4a] has affirmed

N1.)

this conjecture: Thm. 11-32.

For each closed 2-manifold M I there is a finite set SM of graphs such that a graph G imbeds in M if and only if G contains no subgraph homeomorphic from at least one member of sMM'

Finally, we mention that Pisanski [P6] has expanded the surgery techniques he used to generalize work of

[W6]

(see Theorem 7-13,

for exam2le) in the orientable case, to apply to the nonorientable case as well.

11-7.

Problems

11-1.) 11-2.)

Prove Corollary 11-8. Prove Corollary 11-9.

11-3.)

Describe the 2-fold

11-4.)

k 1 1. Relabel the edges of K in Figure 11-3, to yield the imbedding suggested in Example 3. G I ) ( Z2n)

11-5.)

1):

Sk

-+

Nk+l

of Section 11-2, for

What imbedding, and of what graph, covers the voltage graph imbedding of Figure 11-4 ( 1 = Z9)? Is there branching? Where, and of what multiplicity?

188

Nonorientablc Graph Imbeddings

Chapt. 11

1

Figure 11-4 11-6,)

**

11-7.)

11-8.) 11-9.)

11-10.)

Answer the questions of Problem 11-5, but now for

=

Zl0.

Try to prove Theorem 11-19 by the use of nonorlentable voltage graphs. n+l Show that v(Qn x K ) = 2 + n2 I n ;1. 414

Show that the 2-metacyclic group r = 2n ' D m (a,blam = b2n = abab-l = e ?, for n odd and 1, is toroidal. (Hint: use Theorem 7-3 and the voltage graph imbedding that results when the right side arrow is reversed in Figure 10-2.) What other toroidal groups can you find, using the hint of Problem 1 1 - 9 ?

189

CHAPTER 12 BLOCK DESIGNS

Block d e s i g n s a r e c o m b i n a t o r i a l s t r u c t u r e s o f i n t e r e s t i n t h e i r own r i g h t , w i t h a p p l i c a t i o n s t o e x p e r i m e n t a l d e s i g n a n d t o s c h e d u l I i e f f t e r [I151 w a s t h e f i r s t t o o b s e r v e

i n g problems.

imbeddings of c o m p l e t e g r a p h s d e t e r m i n e B I B D s w i t h 3

= 2

(and sometimes

)

c o r r e s p o n d e n c e between

Alpert

= 1.)

BIBDs w i t h

t i o n systems f o r complete qraphs. i2 =

2 ( a n d sometimes

s t r o n g l y regular graphs.

,A2

=

x

3

and

[All e s t a b l i s h e d a o n e - t o - o n e

k = 3

and

X

= 2

and t r i a n g u l a -

I n [Wll] t h i s c o r r e s p o n d e n c e i s

e x t e n d e d t o P B I B D s on two a s s o c i a t i o n c l a s s e s w i t h and

that certain k

1)

k = 3,

X1

= 0

and t r i a n g u l a t i o n s y s t e m s f o r

The g r o u p - d i v i s i b l e d e s i g n s o f H a n a n i

[Ill]

a r e used t o c o n s t r u c t t r i a n g u l a r imbeddings ( i n g e n e r a l i z e d pseudos u r f a c e s ) f o r t h e groups

i n e a c h case p e r m i t t e d b y t h e

Kn(m),

Conversely, t r i a n g u l a r imbeddings o f K are n (m) c o n s t r u c t e d ( b y o t h e r m e a n s ) w h i c h l e a d t o new g r o u p - d i v i s i b l e

e u l e r equation. designs.

A process,

usinq t h e s t r o n y t e n s o r product o p e r a t i o n f o r

g r a p h s , i s d e v e l o p e d f o r " d o u b l i n g " a cjiven P B I B D o f an a p p r o p r i a t e form.

12-1.

Def.

12-1.

A

Balanced I n c o m p l e t e Block D e s i g n s

( v , b , r , k , A)-balanced incomplete block design

(BIBD)

i s a s e t of

v

o b j e c t s and a c o l l e c t i o n of

b

s u b s e t s of t h e o b j e c t s e t , e a c h s u b s e t b e i n g c a l l e d a block, satisfying:

(i) e a c h o b j e c t a p p e a r s i n e x a c t l y (ii) each block contains e x a c t l y

(iii)

k

r (k

blocks: v)

objects;

e a c h p a i r of d i s t i n c t o b j e c t s a p p e a r s t o g e t h e r i n exactly

X

blocks.

190

C h a p t . 12

Block D e s i g n s

If

k = v,

t h e d e s i g n would b e c o m p l e t e ;

complete designs a r c

t r i v i a l t o c o n s t r u c t , b u t have l i t t l e a p p l i c a t i o n . t h e design is "incomplete.

r, k,

f o r m i t y c o n d i t i o n s on

Hence f o r

k

a:

v,

The " b a l a n c e " comes f r o m t h e t h r e e u n i -

I'

and

respectively.

A

A t r i v i a l e x a m p l e of a E I E D ,

v > 1

f o r each

and

0

k < v,

:'

i s o b t a i n e d by t a k i n g t h e b l o c k s t o b e a l l t h e k - s u b s e t s of t h e v-set;

then

b =

[z]

,

\;I:]

r =

,

X

and

ik-2] v- 2

=

.

I n g e n e r a l , e l e m e n t a r y c o u n t i n g a r g u m e n t s ( s e e Problems 1 2 - 1 and 1 2 - 2 )

e s t a b l i s h t h e f o l l o w i n g well-known

Thm. 1 2 - 2 .

result:

(v, b, r, k, X )-BIBD e x i s t s , then:

If a

( i ) v r = bk; ( i i ) X(v-1) = r ( k - 1 ) . T h e s e n e c e s s a r y c o n d i t i o n s h a v e a l s o b e e n shown t o b e s u f f i c i e n t , for

k

=

( a n d f o r some o f t h e cases

3,4,5

and f o r f i x e d

k

1,

and

12-2.

with

v

k = 6,7)

by H a n a n i [ H l ] ,

l a r g e e n o u g h , by W i l s o n [W151.

B I B D s a n d Graph I m b e d d i n g s

What H e f f t e r and A l p e r t o b s e r v e d i s t h a t a t r i a n g u l a r i m b e d d i n g of

i n an a p p r o p r i a t e s u r f a c e ( t h i s i s p o s s i b l e e x a c t l y when

Kn

n E 0,l (mod 3 ) ,

s e e S e c t i o n s 9-1 a n d 1 1 - 5 ) , w i t h t h e

n > 1;

r e g i o n s d e t e r m i n i n g t h e b l o c k s i n t h e n a t u r a l f a s h i o n , s e r v e s as an since:

(i)

every vertex i s adjacent

(n, n(n-1)/3,

n-1,

3, 2)-BIBD,

t o exactly

-

o t h e r v e r t i c e s and h e n c e i s i n e x a c t l y

regions:

n

1

n

-

1

(ii) each region contains e x a c t l y t h r e e v e r t i c e s , by

a s s u m p t i o n ; and ( i i i ) e a c h p a i r o f d i s t i n c t v e r t i c e s c o n s t i t u t e s a n edge ( s i n c e t h e graph i s c o m p l e t e ) and h e n c e b e l o n g s t o e x a c t l y t w o b l o c k s ( t h e two r e g i o n s c o n t a i n i n g t h a t e d g e i n t h e i r b o u n d a r y . ) C o n v e r s e l y , a B I B D on 2-fold

t r i p l e system)

v

objects with

k

=

3

and

X

d e t e r m i n e s a t r i a n g u l a r imbedding o f

a generalized pseudosurface,

as f o l l o w s .

= 2

(a

KV

in

Each b l o c k becomes a 3 -

s i d e d 2 - c e l l r e g i o n , w i t h v e r t i c e s l a b e l l e d b y t h e o b j e c t s of t h e block.

Since

X

= 2,

so t h a t a 2-manifold

e a c h p a i r o f v e r t i c e s a p p e a r s e x a c t l y twice

-

( p o s s i b l y w i t h s e v e r a l components) r e s u l t s from

t h e s t a n d a r d i d e n t i f i c a t i o n p r o c e s s of c o m b i n a t o r i a l t o p o l o g y .

Sect. 12-2

191

BIBDs and Graph Imbeddings

Then identify identically labelled vertices, to form a generalized pscudosurface triangular imbedding for Thm. 12-3.

K\,.

We summarize, in:

The 2-fold triple systems on v objects are in one-to-one correspondence with triangular imbeddings of generalized pseudosurfaces.

If the triangular imbedding of

Kv

Thm. 12-4.

Steiner triple systems on

in

has bichromatic dual, then

the 2-fold triple system splits naturally into two systems (Steiner triple systems, having

Kv

k = 3

v

and

1-fold triple

h

=

1):

objects are in two-to-one

correspondence with triangular imbeddings of Kv in generalized pseudosurfaces having bichromatic dual. Letting k = 3 and I, = 1 or 2 in Theorem 12-2, and invoking the work of iianani (or others) mentioned following that Theorem, we obtain: Thm. 12-5.

v

(1) Steiner triple systems on only if

v = 2, 3 (mod 6 ) ;

(ii) 2-fold tri2le systems on only if

objects exist if and

v

v

objects exist if and

0, 1 (mod 3 ) .

For independent verification of (ii) above, we follow Apert in

observing that triangular imbeddings of Kv (see Sections 9-3 and 11-5) give 2-fold triple systems on v objects for all v 0, 1 ~

[GRWl] it is observed that the orientKn, n 3 (mod 12), a l l have bichromatic dual; thus Steiner triple systems are independently produced for these values of n. (mod 3 ) ,

v # 1.

Moreover, in

able genus imbeddings for

12-3.

Examples

Example 1:

Refer to Example la of Section 10-3; the quadrilateral im-

bedding of

K5

in

S1

obtained there yeilds a

This is atypical in that

k > 3

and

X > 2,

(5,5,4,4,3)-BIBD.

but it does indicate

that the scope of the connection between BIBDs and graph imbeddings is even wider than indicated in Section 12-2. This imbedding is a geometric realization of the "all 4-subsets of a 5-set" abstract design.

Block D e s i g n s

192

Chapt. 1 2

Example 2: -___

R e f e r t o Example 2a o f S e c t i o n 1 0 - 3 ; t h e t r i a n g u l a r im-

bedding of

K7

in

S1

( 7 , 1 4 , 6 , 3 , 2)-

obtained there yields a

BIBD a n d , s i n c e t h e d u a l is b i c h r o m a t i c , t w o ( 7 , 7 , 3 , 3 , l ) - U I U D s . One of t h e s e l a t t e r is g i v e n a b s t r a c t l y i n T a b l e 1 2 - 1 ; 0

1

3

1

2

4

2

3

5

3

4

6

4

5

0

5

6

1

G

O

2

T’able 1 2 - 1 t h e b l o c k s may b e r e a d o f f from t h e s e v e n r e g i o n s c o v e r i n g t h e uns h a d e d r e g i o n i n F i g u r e 10-4

( n o t e t h a t t h e y a r e a l s o h a l f of t h e The b l o c k s may a l s o b e r e g a r d e d as

“blocks“ l i s t e d i n Section 9-2.)

t h e l i n e s of t h e Fano P l a n e ( t h e f i n i t e p r o j e c t i v e p l a n e of o r d e r

two.)

The d e s i g n c o u l d b e employed t o s c h e d u l e f i r e m e n ( s a y ) i n a

w e e k l y s c h e d u l e ( t h r e e men p e r d a y , c t c . ) v o l t a g e g r a p h of F i g u r e 1 0 - 4

to

K12+7

which is

not

The e x t e n s i o n o f t h e

(12s+7, ( 1 2 s + 7 ) .

gives a

of b i c h r o m a t i c d u a l f o r

(4s+2),

12s+6, 3, 2)-BIBD,

s > 0;

h e n c e w e o b t a i n n o a d d i t i o n a l S t e i n e r t r i p l e s y s t e m s from

t h i s figure.

~Example 3:

R e f e r t o Example 3 o f S e c t i o n 11-4: t h e t r i a n g u l a t i o n of

2-fold

K7

in

Example

4:

R e f e r t o E x a m p l e 4 of S e c t i o n 10-3;

3-fold

K7

in

N9

S8

obtained there gives a

o b t a i n e d t h e r e g i v e s one

and, s i n c e t h e dual is bichromatic, t w o

12-4. F o r v a l u e s of Theorem 1 2 - 2

( 7 , 2 8 , 1 2 , 3 , 4)-BIBD.

t h e t r i a n g u l a t i o n of ( 7 , 4 2 , 1 8 , 3 , 6)-BIBD

( 7 , 2 1 , 9 , 3 , 3,)-BIBDs.

S t r o n g l y R e g u l a r Graphs

v , b , r, k ,

X

n o t m e e t i n g t h e c o n d i t i o n s of

( a n d p e r h a p s e v e n f o r t h o s e t h a t d o ) , it i s n a t u r a l t o

attempt a construction of a related design.

For t h i s reason, p a r t i -

a l l y b a l a n c e d i i i c o m p l e t e b l o c k d e s i g n s were i n t r o d u c e d by Bose a n d

Sect. 12-4

Nair

S t r o n g l y R e g u l a r Graphs

193

The p a r t i a l b a l a n c e o c c u r s i n t h a t t h e r e a r e

[BNl].

attention primarily to the case

?

A s t h e a s s o c i a t i o n classes

2.

=

P

(for the case

a p p r o p r i a t e b a l a n c e w i t h i n each c l a s s concept.

x

Let

y

and

e i t h e r non-adjacent

(h

-

l), adjacent t o y y

=

(i

_____ Def. 1 2 - 6 .

=

-

t o e n s u r e an

we f i r s t define t h i s latter

be d i s t i n c t v e r t i c e s i n a graph or adjacent

1)

=

(h = 2).

b e t h e numher of v e r t i c e s w h i c h a r e n o n - a d j a c e n t

y (i = j

are determin-

= 2)

ed by a d j a c e n c i e s w i t h i n a " s t r o n g l y r e g u l a r g r a p h "

and

adjacent t o

x

y (i = 2,

but not to

x ( i = 1, j

but not to

=

Let

t o both

G,

h p. . (x,y) 17 x and

j = l),

2 ) , or a d j a c e n t t o b o t h

x

j = 2).

I f a graph

v,

yet

G

i s r e g u l a r o f degree

G

# Kv

or

1, 2 ,

then

n 2 and i s o f o r d e r h p . . ( x , y ) i s indepen11 y, f o r h , i, j =

and i f

Kv,

d e n t of t h e c h o i c e o f

x

and

is s a i d t o be a s t r o n g l y r e g u l a r graph.

G

I t i s w e l l known t h a t t h e e i g h t c o n d i t i o n s i n v o l v i n g

y

1

t

v a l u e s , one f o r each " a s s o c i a t i o n class;" w e restrict o u r

lambda

x

and

c a n b e r e p l a c e d by two o f them ( s e e P r o b l e m 1 2 - 4 ) :

_____ Thm. 1 2 - 7 .

If

G

i s a r e g u l a r g r a p h which i s n e i t h e r c o m p l e t e n o r

e m p t y , t h e n G i s s t r o n g l y r e g u l a r i f and o n l y i f h x and y , for h = 1 , 2 . ~ ~ ~ ( x ,i sy i)n d e p e n d e n t of h For s t r o n q l y r e g u l a r g r a p h s , i t i s convenient t o w r i t e P ij h f o r p . . ( x , y ) , a s t h e choice of x and y i s i m m a t e r i a l ( e x c e p t 11 for h = 2, and a d j a c e n t i n G , t h a t t h e y must b e a d j a c e n t i n G , for

h = 1.)

Two v e r t i c e s

( o b j e c t s ) of a s t r o n g l y r e g u l a r g r a p h a r e

s a i d t o b e f i r s t a s s o c i a t e s i f t h e y are n o n - a d j a c e n t i t h associates

( i = 1, 2 ) ,

and s e c o n d

Thus e a c h v e r t e x h a s e x a c t l y

associates i f they a r e adjacent. and

n1

+

-

n2 = v

n.

1.

A s a major c l a s s o f examples o f s t r o n g l y r e g u l a r g r a p h s , w e

give : Thm. 1 2 - 8 .

The r e g u l a r c o m p l e t e n - p a r t i t e g r a p h s a l l strongly regular, Proof: Since

Clearly m > 1,

G G

for

m, n

2

G = K

is r e g u l a r , of degree i s n o t complete;

n (m)

arc

2.

since

n2 = m ( n - 1 ) . n1 > 1, G

194

Block D e s i g n s

Chapt.

i s n o t empty. F i n a l l y , w e observe t h a t 2 p22 = m(n-2).

pi,

=

12

m(n-1) #

arid

F o r t h e e x d m p l e s of S e c t i o n 1 2 - 8 , If we start with

w i l l be crucial.

t h e following observation

--

Gl

and t h e s t r o n g l y r e q u l d r

Kr

G1

then t h e s t r o n g t e n s o r product

G2 = Kn(m)

&

is s t r o n g l y

G2

r e q u l a r a l s o , by Theorems 10-12 a n d 1 2 - 8 . O t h e r s t a n d a r d e x a m p l e s of s t r o n g l y r c g u l a r g r a p h s i n c l u d e

and

G = L(Kn)

of g r a p h

G = I,(K2(m)),

where

(We m e n t i o n in p a s s i n q t h a t

il.

denotes t h e l i n e qraph

L(I1)

Km

L(K2(m)) =

s e e Problem 2 - 1 3 . )

12-5.

Def. 1 2 - 9 .

A

P a r t i a l l y B a l a n c e d I n c o m p l e t e B l o c k Desicjns

1 , X 2 ) - -p a r t i a l l y balanced incomplete

( v , b, r , k ;

b l o c k d e s i g n ( P B I B D ) i s a s e t of

v

oblects, pairwise

a s s o c i a t e d i n t o two a s s o c i a t i o n c l a s s e s ( a s d e t e r m i n e d by a s t r o n g l y r e g u l a r g r a p h c o l l e c t i o n of

b

G

of o r d e r

v)

and a

s u b s e t s of t h e o b j e c t s e t , e a c h s u b -

set being c a l l e d a block, s a t i s f y i n g : r

( i ) each o b j e c t appears i n e x a c t l y

blocks;

k (k

( i i ) each block c o n t a i n s e x a c t l y

c:

v)

objects;

( i i i ) e a c h p a i r of i t h a s s o c i a t e s a p p e a r t o g e t h e r i n exactly Again, t h e r e q u i r e m e n t

Ti

blocks

k

v

corresponds t o t h e incomplete-

n c s s of t h e d e s i g n ; t h e r e q u i r e m e n t

n1n2 =

_I

G

#

Kv

-

or

Kv

e n s u r e s that

s o t h a t t h e PBIBD d o e s n o t c o l l a p s e t o a EIBD

0,

1 2 ,

( k = 1, 2 ) .

and t h e r e q u i r e m e n t t h a t

G

-

unless

be s t r o n g l y r e g u l a r i s an

d t t e m p t t o r e s t o r e some of t h e b a l a n c e t o t h e e x p e r i m e n t t h a t w a s

lost whcn o n e

X

c o u l d n o t b e found f o r

d l l

pairs

x,y.

We now g i v e a d d i t i o n a l t e r m i n o l o g y , some of w h i c h a p p l i e s t o both Def.

D I B D s and P B I U D s .

12-10.

A PBlBD i s s a i d t o b e group-divisible

r e g u l a r graph

G

i f the strongly

upon which t h e d e s i g n i s b a s e d h a s a

p a r t i t i o n of i t s v e r t e x s e t i n t o

n

y r o u p s of

m

S e c t . 12-5

195

P a r t i a l l y Balanced I n c o m p l e t e Block D e s i g n s

v e r t i c e s e a c h so t h a t t w o v e r t i c e s are f i r s t associates i f and o n l y i f t h e y are i n t h e s a m e group. C l e a r y t h i s i s p o s s i b l e i f and o n l y i f Def.

12-11

G = K

n(m)

A g r o u p - d i v i s i b l e PBIBD i s a t r a n s v e r s a l d e s i g n i f each b l o c k c o n t a i n s e x a c t l y o n e v e r t e x from e a c h g r o u p .

Clearly t h i s requires Def.

12-12.

A design

k

=

n.

is said to be resolvable i f

( B I B D o r PBIBD)

t h e s e t of

s e t s , of

b

r

b l o c k s c a n be p a r t i t i o n e d i n t o

v/k

sub-

b l o c k s e a c h , each s u b s e t c o n t a i n i n g each

ob je c t e x a c t l y o n c e . F o r e x a m p l e , i f w e t a k e t h e e d g e s of a s t r o n g l y r e g u l a r g r a p h G

( k = 2 ) , t h e r e s u l t i n g PBIBD i s r e s o l v a b l e i f and o n l y

a s blocks

if

is 1-factorable.

G

(mn, m2n(n-1)/2,

m(n-1)

,

In particular,

G = K

2;

which i s r e s o l v a b l e (see

0,

l)-PBIBD,

H i m e l w r i g h t and W i l l i a m s o n I H W 2 1 )

m

= 1,

t h e B I B D on

G = K

n(1)

n(m)

g i v e s an

mn i s e v e n ( f o r i s r e s o l v a b l e i f and o n l y i f n

i f and o n l y i f

= Kn

i s e v e n ) and i n f a c t i s a t r a n s v e r s a l d e s i g n i f and o n l y i f

The t r a n s v e r s a l d e s i g n o n

n = 2.

f o r example, is used i n d u p l i -

K2(2w+l),

cate bridge scheduling t o ensure t h a t ,

in

rounds, each north-

2wtl

s o u t h c o u p l e p l a y s e x a c t l y o n e r o u n d a g a i n s t e a c h east-west c o u p l e . Def.

12-13.

( B I B D or P B I B D )

A design

is s a i d t o be z-resolvable

i f t h e b l o c k s e t c a n be p a r t i t i o n e d s o t h a t e a c h s e t i n t h e p a r t i t i o n contains each v e r t e x e x a c t l y

z

times.

Thus, a 1 - r e s o l v a b l e d e s i g n i s r e s o l v a b l e .

Def. 1 2 - 1 4 . _____

Two b l o c k d e s i g n s

D1

and

D2

-

a r e s a i d t o be iso-

morphic i f t h e r e e x i s t s a one-to-one O1

O2

between t h e i r o b j e c t s e t s such t h a t

...

,

xk}

{ t ) ( x l ), t i ( x 2 ) ,

...

0:

Ix,,

Clearly i f v, b ,

r, k, A

correspondence

+

x2,

D1

(or

and

X1

and

D2

is a block i n

, O(xk)

}

D1

if

is a block i n

and o n l y i f D2.

are isomorphic, t h e i r p a r a m e t e r s

X2)

must a g r e e ; t h e c o n v e r s e i s n o t

19G

C h a p t . 12

Block Designs

t r u e , a s w e s h a l l see i n S e c t i o n 1 2 - 6 . F o r a n e x a m p l e of a l l t h e s e d e f i n i t i o n s , c o n s i d e r t h e d e s i g n s of T a b l e s 12-2 and 12-3; map

sending

'-I

( 6 , 8, 4,

The

i

c l e a r l y t h e s e arc isomorphic, under t h e

t o the i t h letter of t h e alphabet,

1 3)

Kn

determined

d e t e r m i n e s i n t u r n an

(n2,2nb,2n-2,k;O,X)-PBIBD as f o l l o w s . The l i n e g r a p h L ( K 2 ( n ) ) - Kn x Kn ( s e e P r o b l e m 2 - 1 3 ) i s s t r o n g l y r e g u l a r , a n d t h e C a r t e s i a n product

Kn

x Kn

a t e l y among

2n

{ ( i , j )11 5 j 1' -o,(G) =

G

is

[[3(G)/2].

The next result could be useful in seeking a genus imbedding

for a self-complementary graph:

Thm. 14-52,

239

Self-dual Maps

Sect. 14-7

(Clapham [C51) The number of triangles in a selfcomplementary graph of order if p p(p-2) ( p - 4 ) / 4 8 , p(p-1) (p-5)/48, if p numbers are attained.

p

is at least

0 (mod 4), 1 (mod 4).

and at least These minimum

Finally, we comment that self-complementary graphs determined by the quadratic residues of a finite field have been used to give lower bounds for the Ramsey numbers [GG1] and Burling and Reyner

r(k,k); BR21.

see Greenwood and Gleason

See also Clapham [C61 , for a

more general construction; he finds a self-complementary graph of order 113 containing no K7, giving the improved bound r(7,7) 2 114.

14-7.

Self-dual Maps

The “self-complementary‘‘property for a graph depends only upon the abstract structure of the graph itself. To obtain an analog in terms of a geometric realization f o r the graph, we first imbed the graph on a surface, form the dual graph for this imbedding, and then compare the original graph with its dual. (In a sense, duality is a higher-dimensional analog of complementation:

in taking a dual, we

fix 1-dimensional subsets, while interchanging 2- and 0-dimensional subsets; in taking a complement, we fix 0-dimensional subsets, while interchanging 1-dimensional subsets and “(-1)-dimensional“ subsets

-

the “non-edges. ” ) Def. 14-53.

Let inap

PI = M ( G , p )

then

is said to be self-dual if

M

have dual map

M* = M(G*,p*);

G

G*.

For example, the maps of Figures 5-4 ( b ) , 7-4, and 10-3, and the map of Example la in Section 10-3 are all self-dual. The first and last of those are special cases of: Thm. 14-54.

(Heffter [H6], Biggs, [ E 9 1 , White [W9], Pengelley [P3], Stahl, [ S 7 ] , and Bouchet [ B l Z ] ) The complete graph has a self-dual imbedding if and only if n E 0 or

1 (mod 4). (Compare with Theorem 14-43.)

Kn

Chapt. 14

Map Automorphism Groups

240

[W9].

W e now o u t l i n e t h e d e v e l o p m e n t of

L e t the f i n i t e abelldn

...

b e g e n e r a t e d by = f a 1 , bl , a2 , b2 , , ah , bhl , group where no g e n e r a t o r h a s o r d e r 2 . Take as t h e r e d u c e d c o n s t e l l a t i o n

(see S e c t i o n 9-5)

(C')*

i n Theorem 5 - 5 ) :

t h e normal form f o r

a r e g u l a r polygon of

4h

.. .

Sh ( h

-

1)

( a s given

s i d e s , with clockwise

.

'rhen a l , bl, a -1 l , b -1 l , , a h , b h , ah-1 , bh-1 (C')* h a s one v e r t e x , 2h e d g e s , and o n e r e g i o n o n S h and s a t i s f i e s t h e

boundary KCL

-

s i n c e e a c h c u r r e n t a p p e a r s e x a c t l y o n c e , on a l o o p , and

abelian.

Also,

Figure 14-6,

(C')*

i s an i n d e x o n e q u o t i e n t m a n i f o l d ; h = 2.

f o r the case

(The c o n d i t i o n t h a i n o g e n e r a t o r

h a v e o r d e r two e n s u r e s t h a t p r o p e r t y 4 (Theorem 9 - 1 0 ) o t h e r four p r o p e r t i e s hold t r i v i a l l y h e r e . ) and 9-13, space

-

,

and some r o u t i n e c a l c u l a t i o n ,

bedding of

C, (

)

on

S , , (h-l)+l,

w i t h no b r a n c h i n g

bedding has

r = r4h =

-

over

1 1;

holds; the

T h e n , by Theorems 9 - 1 1

(C')*

which i s a

determines an im-

j

1-fold c o v e r i n g

M o r e o v e r , t h e c o v e r i n g im-

Sh.

and, i n € a c t :

V

V

Figure 14-6 Thm. 1 4 - 5 5 .

'rhe map

M ( G r s ( I ' ) , i,)

is

see

c o n s t r u c t e d above i s s e l f - d u a l .

Sect.

14-7

Self-dual Maps

Thus for each

h - 1,

241

the normal form for

Sh

determines a

variety of self-dual imbeddinas of Cayley graphs (see Problem 14-18.) This has many ramifications. Thm. 14-56.

For h - 1 , there is a self-dual imbedding of some graph G of order p on Sp(h-l)+l if and only p - 4h+l.

if

Proof:

G

(1) ror the necessity, we let

dual imbedded on

Sp (h-l)+l;

(Theorem 5-14) gives

q = 2ph

be self-

the euler equation

-

p(p-1)/2

p 2

and

4h+l. ror the sufficiency, choose

(11) \

=

1:

Cor. 14-57.

h.

, 2h1; that is, al

= Z

= 21-1,

p

bl = 21,

Now apply the construction of Theorem 14-55.

on the torus if and only if

The finite abelian group

1

self-dual imbedding f o r some a generating set property that if Cor. 14-59.

1

Thm. 1 4 - 6 0 .

If

= Z

n

4

(n

and

P

There is a self-dual imbedding of some graph order

Thm. 14-58.

...

11, 2, 1 1

..

~

A 14

for E A,

divides

m(n-1

imbedding.

,

of

is self-dual (1.e. has a Ga ( 1 ) )

if there exists

of even order with the -1 then 6 f? A. I-

is se f-dual if and only if

1)

G

p 2 5.

then

K

n (m)

n

L

4.

has a self-dual

with A* consisting of I' mn' less all multiples of n; then (since the multiples of n induce the graph nKm, the complement of Kn (m)) Since 4 divides m(n-1) , A has G,, ( 1 ' ) = K n (m)' even order; moreover, if m n / 2 E T , mn/2 5? A. Thus Proof:

Take

I'

= Z

the construction of Theorem 14-55 applies. Now Theorem 14-54 follows as an immediate corollary. additional corollaries, we have:

#

For

,

Map Automorphism Groups

242

Cor.

14-61.

Cor. 14-62.

K

m

has a s e l f - d u a l imbedding, for

mrm

For

Chapt. 14

n r 3 (mod 4) ,

if and only if

m

Kn(m) is even.

0 (mod 4).

has a self-dual imbedding

Cor. 1 4 - 6 3 .

K has a self-dual imbedding if and only if mlmlm is even.

Cor. 1 4 - 6 4 .

The 1-skeleton of the n-dimensional octahedron, has a self-dual imbedding for

n

odd.

m

K

n(2)

In [Sll] Stahl (see also Bouchet [B12]) established the following:

Thm. 14-65.

If m-1 = n

0 (mod 4), then

K

imbedding.

has a self-dual

n (m)

(And, in [all QuittG added: Thm. 1 4 - 6 6 .

If

n

1

0 (mod 8) and m : 2 , 3

(mod 4 )

I

has a self-dual imbedding. )

then

Kn (m)

1 (mod 4 ) , the complete maps constructed in the proof of Theorem 1 4 - 3 5 are self-dual.

Thm. 1 4 - 6 7 .

For

n

Thm. 1 4 - 6 8 .

The fundamental group

rr(Sk), k

I,

has a self-dual

imbedding in the plane. Thm. 14-69.

The finitely generated abelian group dual imbedding if and only if T # Z2

has a selfor =3*

All of the self-dual imbeddings considered thus far have been Stahl [ S l l l considered the non-orientable

into orientable surfaces. case as well: Thm. 14-70.

n ' 1, has a nonorientable selfK n (m)' dual imbeddinq if any one of the following holds:

The graph

(1)

m : 0 (mod 4)

and

n > 2;

Paley Maps

Sect. 14-8

(ii) m (iii) m

2 (mod 4) =

(iv) m of

1

and

n

and

243

n $ 2 (mod 4);

0 (mod 4), n # 4;

1 (mod 2), n Z, 2 (mod 4 ) ,

2,

and

n

is not a power

(m,n) # ( 1 , 3 ) , ( 1 , 5 ) , or

(3,3).

Thm. 14-71.

The fundamental group '(Nh), h imbedding in the plane.

Thm. 14-12.

The finitely generated abelian group I has a nonorientable self-dual imbedding if and only if 171 2 6.

14-8.

1,

has a self-dual

Paley Maps

The properties for graphs and maps we have been discussing are symmetry properties; in the self-complementation or self-duality case, the symmetry is external ( G compares with G or G* respectively), while in the context of symmetrical maps, the symmetry is internal (M = (G,p) compares with itself.) In this section we attempt to tie these three properties together. First we recall, from Definition 3-3, that two permutation groups ( l ' , X ) and (r',X') are equivalent if there exist a bijection 6: X -t X ' and a group isomorphism i! : r + r ' such that, for each x E X and y E T I @(y) (ii(x)) = G ( y ( x ) ) ; that is @(y)C = By, so that $(y) = 13yB-l and Q, is induced by B, which may be regarded as a relabelling. For example, ( ( i ( G ) , V(G)) and ( G ( G ) , V(G ) ) are equivalent, with P as the identity function, inducing I$ as the identity function also. A s a less trivial example, if If = M ( G , p ) is a map, with dual map M* = M(G*,p*) , then ((;(M) , D*) and (G(M*), (D*)*) are equivalent, under B: D* (D*)* assigning, to each s E D*, the unique S* E (D*)* 'crossing' s (recall that D* = { (u,v) Iuv E E ( G ) } ) . However, ((;(MI, V(G)) and (G(M*), V ( G * ) ) need not be equivalent; in fact, it is quite possible that / V ( G ) 1 # I V ( G * ) I . We combine the symmetry properties of self-complementation, self-duality, and symmetricality, and proceed with the development as in [W131. -+

244

Def.

Chapt.

Map Automorphism Groups

14-73.

T h e map

14

i s s a i d t o bc s t r o n y l y s y m m e t r i c

14 = M ( G , . )

if: -

G

(i)

G;

=

(ii)

(;*

= C;

( i ii )

M

is syriunetrical

(iv)

c(M)

and

a r e e q u i v a l e n t , under

fL(M*)

V(G*)

: V(G)

an anti-isomorphism

.

We need o n e p r e l i m i n a r y r e s u l t , b e f o r c c h a r a c t c r j z i n g s t r o n g l y s y m m e t r i c maps a s t o o r d e r ; Thm. 1 4 - 7 4 .

compare Theorem

( J o r d a n ; see B u r n s i d e [B18, p .

1,

'Yhm. 1 4 - 7 5 .

thcn

1721)

Tf

n = IX~ - 6

F r o b e n i u s g r o u p of d e g r e e n(n-1) / 2 = j

14-33.

n

is a

,X)

(

and o r d e r

i s a p r i m e power.

?'here e x i s t s a s t r o n g l y s y m m e t r i c map of o r d e r

n

and o n l y i f Proof:

M = M(G,

Let

(A)

map, w i t h

n = IV(G)

1

-

1 (mod 4 )

is symmetrical,

Since

so t h a t

1

n

+(2) = n ( n - 1 ) / 4 ;

is self-dual,

M

gives

1 (mod 8 ) .

M

Since

W e show t h a t

n

0

M

( b y Theorem

f = n,

-

2n

= 2 / C ( G )1 = n(n-1)/2,

thus

on

C,

is

G

But s i n c e

(n-1)/2,

imbed

M

so t h a t

with

Sk,

f

and t h e e u l c r

n(n-1)/4

=

2

-

2k,

i s symmetrical,

and

t i v e p e r m u t a t i o n g r o u p of o r d e r n - 9.

Since

I .

is vertex-transitive

G

e q u a t i o n (Theorem 5 - 1 4 ) G(M)

E(G)

n

see a l s o 'Theorem 14-43.

Now l e t

1 (mod 4 ) .

regions.

I

e =

and h e n c e r e g u l a r of d e g r e e

14-18)

n

=

if

1 (mod 8 ) .

be a s t r o n g l y s y m m e t r i c

)

and

e

self-complementary,

or

n

i s a p r i m e power c o n g r u e n t t o

G(M)

n(n-1)/2

is a t r a n s i and d e g r e e

i s a F r o b e n i u s group and

G(M)

t h c n a p p l y Theorem 1 4 - 7 4 , t o see t h a t , i n f a c t ,

n

is

a p r i m e power. So, l e t (u) = u

L

and

EG(M)

,

u # v E V(G)

with

, ( v ) = v.

u v E E(G) ,

If

t h e i d e n t i t y p e r m u t a t i o n , by Lemma 1 4 - 1 4 . then we apply the anti-isomorphism equivalent to between

G(M)

C(M*).

and

Then G(M*)

,

I

such t h a t then If

giving

i

is

uv f! E ( G ) G(M)

i n d u c e s an isomorphism -1 i ) = I rT

so t h a t 1 (

.

,

Sect.

14-8

245

P a l e y Maps

-1

(u)) = ( (u)) = z(U) = (u); similarly, ( ) also f i x e d , ( v ) . B u t , s i n c e L , i s a n a n t i - i s o m o r p h i s m and uv f E(G) , , ' ( u )I ( v ) e E(G*). IIence, by Lemma 1 4 - 1 4 a g a i n , ( L ) is the identity i n CL (M*) ; b u t s i n c e i s a g r o u p isomorphism, i i s t h e and G(M) 1s a F r o b e n i u s g r o u p . i d e n t i t y i n 6 (M) Thus

(

) ) (

(

For t h e c o n v e r s e , l e t

(B)

prime and

- G

Gn

(

r

E N.

n)

,

1 (mod 8 )

,

p

a

We c o n s t r u c t a C a y l e y g r a p h

where

n

n i n the Galois f i e l d element for

n = pr = (Zp)r

GF(pr).

GF(pr)/

-

t h e a d d i t i v e group

Take

so t h a t

x

as a p r i m i t i v e

x

generates the multi2 4 n-3 }, t h e p l i c a t i v e g r o u p , a n d l e t I * = {l,x ,x , . . . , x n set o f a l l s q u a r e s i n G F ( p r ) . ( E q u i v a l e n t l y , uv C E ( G n ) i f and o n l y i f

Cayley graph

v Gn

-

u

is a square i n

GF(Pr) . )

The

i s c a l l e d a P a y l e y g r a p h (see [ P l l ,

where t h e i d e a s b e h i n d t h i s c o n s t r u c t i o n w e r e i n t r o -

We r e m a r k t h a t P a l e y g r a p h s a r e d e f i n e d f o r

duced.)

a l l prime powers

pr

1 (mod 4 ) ,

c i s e l y t h e c a s e s f o r which

-

u n d i r e c t e d edges a r e well-def pr

~

s i n c e t h e s e a r e pre-

is a square

so t h a t

a r e s e l f - d u a l imbeddings p o s s i b l e . by r n ( S ) = x 2 I' , Next w e d e f i n e r n : A * + :I,

1 (mod 8 )

t h a t Mn

rn

induces vertex r o t a t i o n s

i n a c c o r d a n c e w i t h D e f i n i t i o n 14-23;

-

r : il*

n

+

so

M ( I n, j n , r n ) i s a C a y l e y map- w h i c h w e now

=

c a l l a P a l e y map;

n

-

b u t o n l y i n t h e case

ned;

LA:,

and t h e p e r m u t a t i o n

g i v e n by

i n a c c o r d a n c e w i t h t h e r e m a r k s p r e c e e d i n g Theorem 1 4 - 2 8 . We o b s e r v e t h a t P a l e y maps a r e i n f a c t d e f i n e d f o r a l l

n

= pr

case pr

1 (mod 4 ) , b u t w e c o n t i n u e t o s p e c i a l i z e t o t h e r . n = p - 1 (mod 8 ) . We c l a i m t h a t , f o r n =

1 (mod 8 ) , t h e P a l e y map

s t r o n g l y symmetric.

Mn

= M ( l n , . l n , r n1

1s

246

Chapt. 14

Map Autornorphism Groups

(i) The permutation 1 , : (Zp) (ZP)' given by [')(v) = xv is an anti-automorphism for Gn - since w = 2k u + xZk if and only if ~'(w)= xw = x(u + x ) = xu + x 2k+1 = [:(u) t x2k+1; note the use of the distri-+

butive law in the field GF(pr) - so that Gn is selfcomplementary. We remark that Paley graphs, being Cayley graphs of odd order, not only have a 2-factor (as required by Theorem 14-46), but in fact are 2-factorable (the edge set partitons into 2-factors): see Problem 4-14. We also observe that the anti-automorphism ii has exactly one fixed point (the vertex 0) and one orbit of length n - 1 : 0 (mod 4) , as required by Theorem 14-44. (ii) To show that M(Gn,p) is a self-dual map, we study the corresponding imbedding of Gn into Sk, r 2 where n = p = 8m + 1 and k = 8m -7m, for m = 1, 2, 3, 5, 6, 9, This imbedding is an (8m+l)fold covering space (no branching) of a voltage graph imbedding which is the following alternative normal form .a(Note that the values for sm: ala2. a2ma;1a;1. 2m' n = 8 m + l , h = m are consistent with Theorem 14-56 and that Theorem 14-69 is also illustrated by this construction.) The 2m directed edges bounding this 4mgon are labelled with generators from An by the c i - 2m. This assignment a . 1-f x (1-1)(4m-2), 1 1 assignment describes each region boundary (and each vertex rotation) in the covering space, the map (Gn,p), by the voltage graph theory; in particular, the single 4m-gon below lifts to 8m+l 4m-gons above. Figure 10-3 (letting b in Figure 10-2 be 1, -a = x 2) depicts the entire situation for m = 1 , using x2 = 2x + 1 in G F ( ~ : ) al I+ 1 = 01, a2\+ x2 = 21; x = 10; for just the voltage graph for the case m = 2, replace the labels in Figure 14-6 with: 1, x6, x 12, x 2 , -1, -x 6 , -x 12 , -x 2 We now utilize a method introfiuced by Bouchet [B12] for showing self-duality. Each region in the covering space imbedding has a unique lift of the directed edge a l from the normal-form voltage graph in its boundary;

. .. .

..

.

.

247

Paley Maps

Sect. 14-8

is this lift is a side (g,g+l) in Gn, where g uniquely determined for that region; label the region

g*. It is now routine to check that, for each r g 6 (Zp) , the region g* (g* E V(Gt) ) has neighbors 2 k N(g*) = { ( g + w ( - x ) ) * 1 0 - k 2 4m-11 in G;, where 2 2 In fact, the dual is a Cayley map w = (1-x )/(l+x ) . 2 where r*(W ) = -x w&(b E An), Mn * = M n (In , w ' ~ ,r;), n

with

.

r* to that of rn n being taken in the sense opposite r Thus if w is a square in GF(p ) , then G* = G On n n the other hand, if w is a non-square, then G* n = Gn =

.

Gn* (iii) Using the distributive law in rn r = (Zp) ;

GF(n)

again,

we readily see that the permutation

of

to a group automorphism of

thus Theorem

extends

A;

14-27 applies, to show that Mn(Tn,An,rn) is a symmetrical map. In fact, G(Mn) is a Frobenius group with Frobenius kernel the regular normal subgroup I'n and Yg: I'n + I'n, 'g (h) = g+h I g E rn = V(Gn ) 1 Frobenius complement the cyclic group stablizing the {

generated by the autonorphism c0 extending 2 (See Theorems 14-32 and 14-15.) The rn, cO(h) = x h.

vertex

0,

other vertex stabilizers are conjugate, with (G(Mn))g -1 generated by r, = y < y O f course, IG(M,) I = 9 g 0 9 ' /!'nl lAnl = 2/E(Gn)1 . Finally; we emphasize that G(Mn) contains a subgroup isomorphic with T n , as required by Theorem 14-24, 0: (G,)

,

this subgroup must be proper, in

by Theorem 14-48.

(iv) If

2 2 w = (1-x ) / (l+x )

the anti-isomorphism

I;:

V(Gn)

+

is a non-square , then V(GR)

[~(g)= g* G(M;)

,

gives an equivalence between -1 since if u* = Iitul? , where N

given by G(Mn) and is one of the

generating automorphisms c0 or yg ( G E rn) of G(M,) N* preserves oriented region one readily checks that boundaries in Mi. If, however, w E On, then we take ~;(g)= (xg)*, to again see that

G(Mn)

equivalent, under an anti-isomorphism B.

and

G(MA)

are #

,

Map Automorphism Groups

248

The Paley maps Mn = ( n, tional properties of interest.

Chapt. 1 4

n, rn) constructed above have adcliWe use the following facts & o u t

Galois fields, taken from Storer [S171. Thm. 1 4 - 7 6 .

x

Let

be

il

primitive element for

GF(pr),

where

pr = 2f+l and f is even. L e t ( i , ] ) denote the number of ordered pairs (s,t) such that x 2s+i + 1 =

t

-

f-1;

(1)

(0 0)

=

(f-2)/2;

(ii)

(0 1)

=

(1,O)

0 L s

.2t+’,

Thm. 1 4 - 7 7 .

For

n = 4m+

i :

then:

=

(1,l) = €/2.

the Paley graph

I

1

p22 = m

with parameters

and

Cn

is strongly regular,

7

p;2

= m-1

f = 2111. From Theorem 1 4 - 7 6 (i) we see that ( 0 , O ) = f-2)/2 = m-1. Now clearly x2’+l = x2t if and only if x 2 ~ + 2 k + ~ 2=k 2 X 2t+2k; it follows that p 2 2 = m-1 €or Gn. Similarly, we use Theorem 1 4 - 7 6 (ii) to deduce that p22 1 (1,l) = f/2 = m. # From

Proof:

n = 4m+l = 2f+l,

we find

This result overlaps with the next: Thm. 1 4 - 7 8 ,

If

G

is self-complementary and has a symmetrical map

then

G

Proof:

is strongly regular. In fact, for any graph

having a symmetrica

G(M) is transitive on edges and is well-defined. But if G is selfcomplementary, then G ( G ) is transitive on non-edges

map

M,

since

G

2

G ( M ) 5 G(G), p22

also, so that

1

p22

is well-defined too.

Thus the Paley graphs Gnl for n = pr dates for determining association classes for

~

#

1 (mod 8 ) , are candiPBIBDs, and in fact

the Paley maps give such designs: Thm. 1 4 - 7 9 .

Let

x

be a primitive element for

the Paley map

Mn,

for

n = 8m+l,

(8m+1,8m+1,4m,4m;X1,X2)-PBIBDl

GF(n) , where n = pr; yields an

where

X1

=

2m

and

Sect. 14-8

249

Paley Maps

2m-1 = 2m-1 =

Proof:

if

and

xi tl =

is a square In GF(n), while 2m if x 2+1 is a non-square.

See Problem 14-20.

#

It can be shown that, in a Paley map, either each region bounary consists of neighbors N(v) , v E (ZP)l, or each region boundary consists of neighbors in the complementary graph, so that the designs

constructed above a l s o exist in a natural, topology-free, context. The topological context, however, does facilitate the next observation:

d u a l of

these designs are also self-dual, in a very strong sense.

The

a design has as its objects the blocks of the oriqinal

design, and a block in the dual design contains all those objects of the dual design which represent blocks of the original design to which one fixed object of the original design belongs. Thm. 14-80.

Given a Paley map, the dual of the design of the map and the design of the dual of the map coincide, and both are isomorphic to the design of the Paley map itself. Proof:

See Problem 14-21.

#

We make several additional comments relevant to Theorem 14-75. Firstly, we observe that self-dual imbeddings of self-complementary graphs need not be unique for a given order. need not be a prime power,

Moreover, the order

(We are temporarily relinquishing the

symmetricality condition for strongly symmetric maps.)

For the first

A = ~5;1,6,11,16,211 in i' = Z 2 5 , S O that G(,(T) example, take i s the composition C5 [C,]. Since C5 = C5 and G [ H ] = G[H] i n general (see Problem 2-51, C 5 [ C 5 1 is self-complementary. dual imbedding is constructed by the use of Theorem 14-55.

A selfSince C,[C,l is not strongly regular, the map M is not symmetrical, by Theorem 14-78; thus M differs from the Paley map M25 of the same order. In fact, G ( M ) 2 I = 225 and i s equivalent to G(M*) (both are Frobenius groups) under the anti-isomorphism CJ (4) = (2g)*. (Every map M of a Cayley graph GA(T) covering a normal-form voltage graph has r j ' - G(M) ; in this case the fact that there are no additional automorphisms follows from the observation that each region boundary contains repeated vertices.) We mention that Theorem 14-55 also provides a self-dual imbedding, but not a

250

Chapt. 14

Map Automorphism Groups

symmetrical map, for the Paley graph

G25.

For the second example, we take ?> = { 1 3 : 5 , 2 0 , 1 5 ; 1 , 4 , 1 6 ; 3 , 1 2 , - 1 7 ; 7,28,-18;11,-21,-191 in -- Z65; ! , ( g ) = 29 gives an anti-automorphism, so that G , ( i ’ ) is self-complementary. Again a self-dual imbedding is constructed by Theorem 14-55, and again the map fails to be symmetrical, by Theorem 14-78. Again, G ( M ) and G ( M * ) are equivalent under the anti-isomorphism ( ~ ( g=) (2g)*, and both are Frobenius groups isomorphic to

I

=

ZG5.

The non-prime-power order

is possible, because the autonorphism group is not large enough for the Jordan theorem (Theorem 14-74) to apply. Next we present another sufficient condition to give primepower order (see Problem 14-22.) Thm. 14-81.

2 22 = 2m, p22 = 2m-1), M = (G,,,) a symmetrical map yielding an (Emtl,8mtl14rn,4 m ; 2m-l12m)-PBIBD. ‘Then 1 V (G)1 = 8m+l Let

be strongly regular (with

G

p1

is a prime power. We remark that, for the Paley graphs QG,)

Gn(n = p

consists not only of the map automorphisms

r

1 (mod 4 ) ) ,

G(Mn) as in

(111)

of the proof of Theorem 14-75, but also of the field automorphisms, generated by (G(M),

Thm.

I

’

(Zp)r

:

(ZP)r ,

O ( g ) = gp;

in fact,

G(G,)

=

>:

14-82.

(Carlitz [C3])

The automorphism group of the Paley S

graph 0

-

s

is

Gn

G(G,)

= fg

2kgp +a [ 0

1 - b ~

2 k 5 4m-1,

L

-

r-1, a E (Zp) 1 .

We use this result to study the reflexibility of the Paley maps. Def. 14-83.

If a map that

is symmetric but not reflexible, we say

M

M

is chiral.

Thm. 14-84. The Paley map thus Proof:

Mn

If

must have Since

Mn

is reflexible if and only if

is chiral i f and only if

0’

Mn

is reflexible, with

r = 2s and O s ( g ) fixes both 0 and

= gPs

1,

n = 9;

n # 9. n = prl

then we

giving a reflection we must have

Sect. 14-8

Paley Maps

s (x2 )

251

2 x2Ps = p . 2 s - 3 (since L j 0 - (1,x X Thus p2’-3 = 2ps, so that p divides 3 ; hence p = 3 and s = 1 (or else 3 divides 1); that is, n = 9. Conversely, (g) = g3 is readily seen to be a

,...,

=

reflection for

M9;

refer to Figure 10-3.

As we have seen, the Paley maps

#

- defined for prime powers con-

gruent to 1 (mod 8) - have several interesting properties: they are regular (symmetrical), self-dual imbeddings of strongly regular, self-complementary graphs which produce self-dual block designs, for example. Now we see how closely we can approximate these properties by similar maps, for all other prime power orders. For pr 1 5 (mod 8 ) , the Paley graphs Gn are defined, and are both self-complementary and strongly regular, as before. The euler equation disallows self-dual surface imbeddings, S O we utilize pseudosurfaces. Again we use a normal-form voltage graph but, for n = 8m+5, it is a (4m+2)-gon and thus has two vertices; see Figure 14-7 for the case m = 1. The (8m+5)-fold covering imbedding has

1

Figure 14-7. 16m+10 vertices, each of degree 2m+l. For each g 6 GF(n), we identify the two vertices (a,g) and (b,g); the result is a symmetrical, self-dual pseudosurface imbedding of Gn, f o r m > O . consists of five disjoint loops. For the pseudosurface (G; = C; theory of voltage graphs, see Garman [ G l ] .) In fact, the map is a r n (6) = x4 , having Cayley map PI(Tn,An,rn), with rn: :A A;, +

252

Map Autornorphisin G r o u p s

two c y c l e s . or

-

if

G

The d u a l

w

is

G*

w = (1-x

if

C

Chapt.

4

4

14

is a square,

) /(l+x )

The autornorphism g r o u p i s F r o b e n i . u s ,

i s a non-square.

w i t h F r o b e n i u s k e r n e l 2 ( Z ) r . The s t a b l i z i e r o f v e r t e x 0 i s P 2 g e n e r a t e d by C J -x. g , w h i c h a l t e r n a t c s b e t w e e n t h e t w o o r b i t s of the vertex rotation at results i f

0.

(8m+5,8m+5,4m+2,4m+2;2m,2m+l)-PBIBD

An

is non-square;

l+x4

otherwise,

the two

values inter-

change. For

pr

3 (mod 4 ) , -1 = x 2m+1

fined, since

the Paley graphs

'rn,

d e f i n e t h e Paley tournament if

g2 :

- g1

is a square i n

I n s t e a d , we

(g1,g2) E E(Tn)

if and o n l y

of c o u r s c , w e s t i l l t a k e

GF(n);

as our v e r t e x set.

(ZP)l-

~ { ( g l , g 2 ) ,( y 2 , g 1 ) i

by:

a r e no l o n g e r de-

Gn

not

a square.

is

( p r = 4ni+3)

g1 # g2

Then

gives

E(T,) I = 1, s o t h a t w e do h a v e a t o u r n a m e n t . The u n d e r l y i n g u n d i r e c t e d q r a p h i s t h u s c o m p l c t e , a n d h e n c e d e q e n e r fl

ately "strongly regular.

"

The Paley t o u r n a m e n t s a r e s e l f - c o n v e r s e , I: gl-, xg.

under t h e anti-automorphisrn

metrical imbeddinqs o f t h e associated e a c h of l e n c j t h where

rn(g)

=

2m+1;

having

K4m+3

t h e s e a r e C a y l c y maps

M ( 1'

,

xg.

regions,

- i0 i ,

rn),

(4m+3,4m+3,2m+l ,2 m + l , m ) - B I R D S .

W e now m o d i f y t h i s map t o f o r m a

('The l a t t e r a r e Iladamard d e s i g n s . ) p s e u d o s u r f a c e i m b e d d i n c j of

e x a c t l y one edge corresponding t o

Each region c o n t a i n s

X4m+3;

1 E (Zp)

,

i n one o f t h e t w o

p o s s i b l e scnses ( c l o c k w i s e o r c o u n t e r c l o c k w i s e . )

In fact, this

d i s t i n c t i o n d e t e r m i n e s t h e 2 - c o l o r i n g of t h c d u a l .

g* ( g Cr ( Z )r)

i f either:

The r e g i o n i s

( i ) (g,g+l)

P r e g i o n i n t h e c l o c k w i s e s e n s e , or ( i i ) (g-a,g-a-1) r e g i o n i n t h e clockwise s e n s e , where

a

=

bounds t h e

bounds t h e

2/ ( ~ ~ ~ - T 1h )u s. e a c h

a p p e a r s e x a c t l y t w i c e a s a r e g i o n label, a n d i f t h e s e

the self-converse

tournament

Tn,

w i t h each v e r t e x neighborhood

q*

o n e c o n s i s t s of t h o s e v e r t i c e s d o m i n a t e d b y

o t h e r c o n s i s t s of t h o s e d o m i n a t i n g F i n a l l y , we consider 1

i m b e d d i n g of

p a r t i t i o n e d i n t o t w o sets (corresponding t o t h e

fication):

is a square

(1

=

x2r-1 )

pseudo-

M o r e o v e r , i f t h e edge d i r e c t i o n s are

K4m+3.

carried. o v e r i n t o t h e d u a l , t h e n w e h a v e a s e l f - d u a l

N(g*)

g

pairs o f

n

v e r t i c e s i n t h e d u a l are i d e n t i f i e d , w e o b t a i n a s e l f - d u a l s u r f a c e imbecldinq of

directly,

Hence w e o b t a i n o n e

(4m+3,8m+6,41n+2,2m+l,2m)- B I R D a n d t w o

assigned label

8m+6

n, n T h u s t h e i m b e d d i n g s c a n n o t be s e l f - d u a l

although the dual i s bichromatic.

self-dual

w e f i n d sym-

I n B i g g s [B9]

p = 2, in

g*,

identithe

g*. so t h a t

GF ( 2 r ) ,

p r = 2r.

Here n o t e v e n

so o u r p r e v i o u s

Sect. 14-8

Paley Maps

However, if we use the planar

constructions seem not to apply. voltage graph of Figure 14-8, with covering imbedding of a 2-fold

253

Kn,

n

=

,'2

with

we obtain an n-fold n

(n-1)-gons and

/;]

If each digon is closed (by identifying its two edges), a syrmnetrical, self-dual imbedding of the "strongly regular" Kn ( s e l f -

digons.

complementary i n the 2-fold Kn) results. (In fact, r*, = r for n this case.) The concomitant design 1s an (n,n,n-l,n-l,n-2)-BIBD which, of course, can a l s o be constructed by taking complements of singletons a s blocks.

... Figure 1 4 - 8

We close this lengthy section by discussing conditions under Self-duality, strong regularity, and designs do not enter into the characterizations given, but as it is Paley maps being characterized, these additional properties persist - except that self-duality fails €or n . 5 (mod 8). For proofs of the following two theorems, refer to [W13] or to Section 14-9.

which the Paley maps are, in some sense, unique.

Thm. 14-85.

If M = ( G , r i ) is vertex-transitive and if ( G ( M ) , V ( G ) ) is self-equivalent under an anti-automorphism 0 of G, then

M

is isomorphic to a Cayley map

)I,) = /V(G)[

Thm. 14-86.

,

/ A * ] =(/V(G)

I-

M(I',A,r), where

1)/2.

There exists a symmetrical imbedding of a self-complementary graph C, of order n, with ( G ( t . l ) , V ( G ) )

Map Automorphism Groups

254

Chapt. 1 4

self-equivalent under an anti-automorphism I\ of if and only if n is a prime power Congruent to 1 (mod 4). Moreover, if (i2 E ( M ) , the maps are essentially unique, with at most six exceptions. Cor. 1 4 - 8 7 .

G,

For each prime p 1 (mod 4 ) , there exists a unique symmetrical imbedding of a self-complementary graph G of order p having G ( M ) = G(G).

Proof:

The Paley maps give existence. But if G(M) then K 2 and B n R - ' (for each IY. E G ( M ) and where [i is an anti-automorphism of G ) are both i n G ( M ) , so that Theorem 1 4 - 8 6 (and its proof) show

= G(G),

that

M

is a Paley map.

#

The strength of this uniqueness claim is illustrated by Theorem 1 4 - 4 5 : the Paley maps of orders 13 and 17, for example, are unique among 5 , 6 0 0 and 1 1 , 2 2 0 , 0 0 0 self-complementary graphs respectively, with respect to being symmetrical, with G ( M ) = G ( G ) . Cor. 1 4 - 8 8 .

If there exists a symmetrical imbedding of a s e l f complementary graph G of order n, with G ( M ) = G ( G ) , then (with at most six exceptions) n is a prime congruent to 1 (mod 4 ) .

As in the proof of Corollary 1 4 - 6 7 ,

we find by Theorem 1 4 - 8 6 that n is a prime power congruent to 1 (mod 4); moreover, the maps (with at most s i x exceptions) are unique and thus are Paley maps. But if n = pr with r -' 1, then G(M) is a proper subgroup of G(G), by Theorem 1 4 - 8 2 ; thus r = 1, and n is

Proof:

prime

.

#

14-9.

1

n Il (ni-l)! i=l

14-1.)

Show that

14-2.)

Prove Theorem 14-5. Show that G ( M ) = A 4 ,

14-3.)

IR(G)

=

Problems

for

M

as in Figure 1 4 - 2 .

Sect.

14-9

Problems

255

P r o v e t h a t t h e f o l l o w i n g a r e e q u i v a l e n t , f o r a map

14-4. )

(G,,)

M =

:

(i) M

is reflexible; and i t s mirror image

(ii) M

(iii) t h e r e e x i s t s an

E V(G)

V

:i

G(G) such t h a t , f o r a l l

E

= ( t i 1

I ) , ~ ( ~ )

r

-1 -1 CY

V

14-5.)

Show t h a t t h e map o f F i g u r e 1 0 - 3

14-6. )

P r o v e Theorem 1 4 - 1 3 .

14-7.)

Verify t h e e n t r i e s

14-8. )

Show t h a t ICL*(M)

1

.

is reflexible.

i n Table 14-2.

/G*(M)1

divides

Thus a n o n o r i e n t a b l e map if

are equivalent;

M

1.

= 4jE(G)

1.

( S e e Theorem 5 - 2 5 . )

is d e f i n e d t o be symmetrical

M

I f w e augment F i g u r e 8 - 7 ( a ) i n t h e

K6

o b v i o u s way t o imbed

41E(G)

N1,

in

i s t h e r e s u l t i n g map

symmetrical?

*

Prove o r d i s p r o v e :

14-9. )

If

is a strongly regular graph,

G

0

then there e x i s t s a rotation i s a r e g u l a r map. 14-10. )

Let

M

(Hint:

be t h e imbedding o f

l a of S e c t i o n 1 0 - 3 .

IG(M)

1

= 20)

I

for

K5

Show t h a t

on M

M = (G,p)

but not reflexible

S1 g i v e n i n Example is symmetrical (i.e.

(i.e.

Show t h a t t h e c o r r e s p o n d i n g d e s i g n Section 12-3)

so t h a t

G

consider t h e Petersen graph.)

is chiral.)

M

(Example 1 o f

D

h a s automorphism g r o u p

G(D)

Does

2 S5.

t h i s c o n t r a d i c t Theorem 1 4 - 1 3 ?

*

14-11.)

P r o v e Theorem

14-12.)

P r o v e Theorem 1 4 - 3 7 .

14-13.)

P r o v e Theorem 1 4 - 3 8 .

14-11.)

P r o v e Theorem 1 4 - 3 9 .

14-15.)

Show t h a t

*

map f o r

14-36.

M(Zgr {1,2,4},

3 ( 3 )' F o r what o t h e r v a l u e s o f

14-16.)

i s a symmetrical

(1,2,4,8,7,5))

U s e t h i s map t o show t h a t

K

n

does

K

3 (n)

('3,3,3,6 ) = 7. have a symmetrical

map ? 14-17. )

L e t map

a description of where

b e d u a l t o map

M* = M ( G * , [ i * )

i n t e r m s of

IJ*,

D* = [ ( u , v ) Iuv € E ( G )

1

and

p

M = M(G,p).

and

: D*

+

(u,v) = (v,u).

Find

D*, (See

B i g g s [BlO]. T4-18. )

Show t h a t

14-19.)

*

14-20.

P r o v e Theorem 1 4 - 5 5 . yM(Gn)

g r a p h of o r d e r )

=

(n-1) ( n - 4 ) / 8 ,

n = pr

P r o v e Theorem 1 4 - 7 9 .

Z

1 (mod 8 . )

where

Gn

i s t h e Paley

256

Map A u t o m o r p h i s r n C r o u p

*14-21.)

Prove T h e o r e m 1 4 - 8 0 .

14-22.)

P r o v e Theorem 14-81.

?4-23.)

P r o v e Theorem 14-85.

14-24.)

P r o v e Theorem 1 4 - 8 6 .

*

Chapt.

14

251

CHAPTER 15 CHANGE RINGING

T h e ancient and continuing art of change ringing, or campanology

(how the British ring church bells), is here studied from a mathematical viewpoint.

An "extent" on

n

bells is regarded as a hamilton-

ian cycle in a Cayley color graph for the symmetric group bedded on an appropriate surface.

Sn, T'hus - perhaps surprisingly

im-

-

graphs, groups, and surfaces combine to model something musical. After making this model explicit, we discuss two ringing methods for variable n (Plain Bob for all (mod 4)); and a new method for n methods

(n = 4)

n and Grandsire for n 3 odd is introduced. All minimus

and five doubles methods

(n = 5)

are depicted,

one of these being the new "No Call" Doubles. The material here also appears in IW141.

15-1.

Definitions

In the science of change ringing (sometimes called campanology), the central problem is to ring a full extent on signate the

n

bells by the natural numbers

n

bells.

1, 2,

...

,

We den,

arranged in order of descending pitch. Def. 15-1.

Bell 1

is called the treble, bell

ringing of the

n

n

the tenor.

Each

bells, once each in some order, is

called a change; the special change called rounds. An extent consists of changes, satisfying:

1, 2,

n!

...

+ 1

,

n

is

successive

(i) the first and last change are both rounds; (ii) no other change is repeated (so that each possible change other than rounds is rung exactly once); (iii)

from one change to the next, no bell moves more than one position in its order of ringing (positions 1 and

n

are

not

adjacent, unless n = 2);

Change Ringing

258

Chapt. 15

no bell rests in the same position for more than two successive changes; each bell (except perhaps the treble) does the same amount of "work" (hunting, plain hunting, dodging, etc.); the method employed is palindromic, or self-reversing (this will be made more precise in Section 15.2.) Conditions (iv) and (v) are occasionally relaxed in practice, but the other conditions appear to be inviolable. 1 venture to guess that rule (i) is for musicality, rule (ii) for thoroughness, rule (iii) for mechanical considerations (dealing with how the bells are hung and rung) , rules (iv) and (v) to keep the performance interesting for the ringers, and rule (vi) to ease the memory burden for the ringers. Typically the number of bells is in the range 3 2 n 5 12, with names assigned to the extents as in Table 15-1. The odd-bell names reflect the maximum number of pairs of bells that could be exchanged, in their order or ringing.

n

name

n!

+

1

Singles Minimus

7 25

Doubles Minor

721

Triples Major Caters

5,041 4 0 ,321 362,881

11

Royal Cinques

3,628,801 39,916,801

12

Maximus

479,001,601

3

4 5

6

7 8

9 10

121

Table 15-1

As an extent of Major takes approximately eighteen hours to

ring (the ringers are allowed no visual aids to memory, so this must surely rank as one of the "major" physical

intellectual feats of

Sect. 15-1

mankind),

259

Definitions

extents on more than eight bells are clearly beyond the

limits of human endurance; even on eight bells they are extremely rare. Much more tractable, an extent of triples requires under three hours of concentrated ringing.

For this reason peals are often

attempted, and often succeed; a peal consists of at least 5000 and,

5 1n 7, exactly 5041 successive changes satisfying rules (i) through (vi) above (but not (ii), if n < 7.) For n < 7, a peal consists of several extents strung together; for n = 7 a for

peal fi an extent; and for n 3 7, a peal is a partial extent. If tower bells are being rung, then each ringer operates one bell (a huge amount of practice is required for the beginner to learn to ring his or her bell even individually, let alone in rounds with the other bells, let alone through all the changes of a peal or an extent); if handbells are employed, then each ringer rings two bells. There is also a bell-ringing machine (designed and constructed by John Carter, a gunsmith and bell-ringer, and currently on loan to the Science Museum in London); it mechanically rings twelve hand bells suspended in a cabinet, after being "programmed" for method. As generally an even number of bells is provided, for the odd bell methods a "covering" bell

-

always the tenor

-

rings last in

every change, in forgiven violation of rules (iv) and (v). It might even be argued that the stability and regularity thus produced is musically pleasinq. Change ringing, as described above, began in England in the middle of the seventeenth century and remains almost exclusively a British art.

Today there are over five thousand bell towers in

Great Britain, about a half dozen in Canada and a dozen in the United States, and perhaps twice that in Australia. On July 27 and 28, 1963, the first extent of Plain Bob Major was successfully rung on tower bells at the Loughborough Bell Foundry in England. On

December 27 and 28, 1977, the same extent was accomplished on handbells in a private home in Farnham, Surrey, England. Change ringing has been popularized, to an extent, by Dorothy Sayers in her novel The Nine Tailors [ S 4 ] and, more recently, as part of the festivities at the Royal Wedding of Prince Charles and Princess Diana. Only recently have mathematicians begun to study change ringing from their vantage point, although curiously the early composers of peals and extents were doing coset decomposition in symmetric groups a full century before Lagrange.

The reader could

260

Chapt. 15

Change Ringing

consult works by Rankin

[R2], Fletcher [ F ’ l ]

,

Dickinson [D2], Price

[P7], Budden [B17], and White [WlO]. For more on the history and practice of change ringing, the reader should seek out Wilson [Wl7], Camp [CZ], and the weekly publication “The Ringing World” (published in Guildford, Surrey, England.)

15-2.

Notation

It would be easy to confuse the number of a given bell with the number of the position it occupies in a particular change.

To avoid

this confusion we regard each change as a function from the set of n

positions to the set of

n

bells, so that domain elements de-

scribe positions and range elements describe bells.

Thus a change f (n), would indicate that bell f ( 1 ) is to be rung first, bell f(2) second, and so on. Rounds is given by r(i) = i, 1 c i - n. In order to satisfy condition (iii) of Section 1, we describe a potential transition from one change to the next by a permutation on the set of n positions which either fixes a given position or interchanges it with an adjacent position. Conversely, each transition satisfying (iii) is of this type. Thus each transition is represented by an involution (that is, by a product of disjoint transpositions) in the symmetric group Sn. Now let t(n) give the number of possible transitions for n bells, and let F(n) give the nth Fibonacci number (where F ( 0 ) = F ( 1 ) = 1) ; then we have the following well-known result: f,

recorded as

Thm. 15-2.

For

f (1), f (2),

n 1 2,

Proof:

... ,

t(n) = F(n)

- 1.

It is easy to verify that

t (2) = 1

and that

k and cont(3) = 2 , so we assume the result for n sider t(k), k - 4 . Since there are t(k-1) allowable transitions fixing position k , t(k-2) allowable transitions exchanging positions

k

and

k

-

1

and at least

one other pair, and the single transposition we have:

(k-1,k),

261

Notation

Sect. 15-2

t(k) = t(k - 1)

-

=

F(k

=

F(k)

1)

-

+ t(k - 1 +

+

- 2)

F(k

-

1 2)

- 1 +

1

1 #

Now, if d l , d2 , in an extent, then the

... , dk

represent the first k transitions (k+l)st change is the function rdld2,..dk,

where - in accordance with the standard notation for function composition - we take products in Sn from right to left. Let (Sn); then we have the immediate but absolutely central

result: Thm. 15-3.

An "extent" on n bells, satisfying conditions (i)(iii and using transition rules from A , can be rung if and only if

C.(Sn)

is hamiltonian.

d E: has order two, CA(Sn) has no directed edges by convent on (see, for example, White [w101): hence Since each

hamiltonian cycles in C,(Sn) coincide with those in the underlying Cayley graph G , j (Sn). We retain the edge colors of Cii(Sn), however, as they are valuable in constructing the extent itself from the hamiltonian cycle and in checking the palindromic condition (vi). Each method is based upon a principle, which usually (but not always) consists of

2n

changes.

juxtaposed transitions

The word in

Sn

describing the

2n

- 1

of the principle should be a palindrome:

dld2...dn-1dndn-l...d2dl. These ideas are illustrated in Figure 5-1, with

A

= {

(12), (23))

for the case n = 3, The hamiltonian cycle and the extent it gives rise to shown in Table 15-2 describe the interior clockwise region boundary of the spherical imbedding of C A ( S 3 ) depicted. The extent is called "quick six"; the clockwise boundary of the exterior region produces "slow six", the only other extent possible on 3 bells.

(Not surprisingly, "slow six" is just "quick six" in the

opposite order.)

If we let

a = (12)

(denoted by solid edges) and

b = (23) (denoted by dashed edges), then "quick six" is completely described by the identity word (ab)3 in S 3 and "slow six" by (ba)3 . In the former case the relevant palindrome is ababa; in the latter it is babab.

262

Chapt. 15

Change Ringing

\ \ \

\

Figure 15-1

Table 15-2

15-3.

General Theory

In [R4], Rapaport provided an inductive construction, to establish the following: Thm. 15-4.

Let then

1)

= {

(12) (12)(34) (56).. . I (23) (45) (67).

Ci.(Sn)

is hamiltonian.

.. 1

for

Sn;

Sect. 15-3

263

General Theory

In [wlo], we required only conditions (i), (ii), and (iii) for an extent, and gava the following two ramifications of the Rapaport construction: Thm. 15-5.

An extent on

n

bells exists, for all

n.

Moreover,

only three transition rules are required. Thm. 15-6.

For

n

odd an extent exists which also satisfies condi-

tion (iv) at all but position n, where (at the worst) no bell rests for more than four successive changes. In [WlO] we also displayed an imbedding of C A ( S 4 ) , for Rapaport's !i, in the projective plane. (This is essentially Figure 15-2 below.) In this imbedding the three left cosets of D 4 bound regions which link together nicely to trace out the in S4

hamiltonian cycle giving the extent, in extension of the idea behind Figure 15-1. This suggests the efficacy of picturing extents on surfaces, and as we naturally wish to do this as efficiently as possible, we make the following definition: Def. 15-7.

The characteristic of an extent (on n transition rules

A

)

bells, with

is the maximum characteristic of

all surface imbeddings for

C A(Sn).

As certain transition rules (the "calls"

-

so called because

the conductor calls them at the appropriate time, while all other transitions are trusted to the memory of the ringers - known as "bobs" and "singles") are used infrequently in a particular extent (a single is perhapsemployed only twice among the n! transitions, for example) we take A to include only the other generators for this has the effect of avoiding unnecessary clutter in Cn(Sn). Sn; Thus, in the event that A does not qenerate Sn, a characteristic imbedding of an extent may be on a disconnected surface. For example, the Rapaport

..,

A = {a,b,c}

...

(where a = (12),

and c = (23)( 4 5 ) (67) ) b = (12)(34)(56). involves no "call" generators and the relators

for Sn (n 5) (ab)*, (ac)6-, and

(bc)" bound regions for an imbedding (orientable if and only if n 1 3 (mod 4)) of C, (Sn) of characteristic (n-1)! (3-n)/ 6 , which is thus a lower bound for the characteristic of the extent. single relator

(abc)n-l

Or, the

can be taken to bound regions for an

Change R i n g i n g

264

C ,( S n )

o r i e n t a b l e i m b e d d i n g of

b u t t h i s i s less e f f i c i e n t .

Chapt.

of characteristic

Note t h a t

n ( n - 2 ) ! (3-n) / 2 ,

Sn,

does generate

.'

15

a r e known t o d o s o ; see C o x e t e r a n d Moscr

a

and

bc

For

n = 4 , t h e imbedding i n t h e p r o j e c t i v e p l a n e cannot b e improved

(as C , (S4)

c o n t a i n s a homeomorph of

,

K3,3)

JCM1, p.

as

631.

so t l i a t t l i e c h a r a c t e r -

i s t i c of t h i s e x t e n t i s e x a c t l y o n e . W e now c o n s i d e r

c

=

(23) (45) (67)

P l a i n Bob -b

c

and

n

i s always

regular

n-gon

n

2n(n-1)

changes.

1

.. . )

2n

n -5,

Finally, for

...

and,

(bc)",

(bd)

- (n-2)

! (n2

-

5n

+

,

is used

d

tlie c a l l g e n e r a t o r s ( t h e

the single

2

,

+

n! and

2) /4;

f =

1 c h a n g e s of bound

(cd)"-'

r e g i o n s f o r dn always n o n o r i e n t a b l e imbedding o f characteristic

...

t o g i v e a "touch" of

n 2 6,

for

1,2,4,6,

Thc g e n e r a t o r

are added t o produce t h e f u l l

Here t h e r e l a t o r s

extent.

changes.

of t h e s e b l o c k s t o g e t h e r ,

e = ( 2 3 ) (56) (78)

...,

various

the

( a s c a n be s e e n from i t s a c t i o n o n a

Dn

with v e r t i c e s labeled, i n order,

g i v i n g a b l o c k of

(56) (78)

[or

Here t h e h u n t i n g g r o u p g e n e r a t e d by

bells.

7,5,3),

bob

b = ( 1 2 ) (34) (56)

...,

to link

-

where

d = ( 3 4 ) ( 5 6 ) (78)

and

e x t e n t s on

,

= 'b,c,cl~

"

...,

C, ( S n )

t h i s is for

n

of

-

4.

This

a p p e a r s t o b e t h e most u s e f u l i m b e d d i n g f o r P l a i n Bob e x t e n t s , a s t h e relator

e x a c t l y p o r t r a y s t h e p r i n c i p 1 . e embodied i n t h e

(bc)"

h u n t i n g group, and r e p l i c a t e d i n t h e l e f t cosets o f t h a t g r o u p , Aaain we n o t e t h a t

generates

.'

as

Sn,

bc

a r e known t o

a n d bd

do so. Next c o n s i d e r

ib,c,gi,

i. =

c = ( 2 3 ) ( 4 5 ) ( 6 7 ) ...,

and

Grandsire e x t e n t s on

n

where

b = ( 1 2 ) (34) ( 5 6 ) . . . ,

...

g = ( 1 2 ) ( 4 5 ) (67) (odd)

for the various

Here w e see t h a t

bells.

(a,c) =

i s used as a l i n k i n g permutation. The b o b f o r G r a n d s i r e i s c o n s t r u c t e d f r o m L , b u t t h e s i n g l e i s t h e new h =

Dn

and

g

(45) (67) (89)

... .

we observe t h a t

(mod 4 ) ) ,

a l t e r n a t e l y by I ( 1 2 k ) 13

b

and

1k 1 ni;

But t h e n (cb)"

T o see t h a t

(An,q) = S

g

g c = (123)

and t h a t c o n j u g a t i n g

[ C M l , p . 661

W e use t h e relators Cc,(Sn)

to generate 3 n-2 ( g c ) ,( g b ) ,

(orientable for

n o n o r i e n t a b l e and i n t w o components f o r single

3

An.

and

( t h e l a t t e r i s u s e d i n t h e p r i n c i p l e f o r G r a n d s i r e ) t o bound

r e g i o n s f o r a n i m b e d d i n q of

istic

n

( a n d t h e n t a k i n g irwerses) p r o d u c e s

t h i s i s known

n'

is connected ( f o r

Cr,(Sn)

- ( n - 1) ( n - 3 ) ! ( n 2 h

i s odd,

so t h a t

-

5n

+

(A,,h)

3) /3.

n For

= Sn.

n

3 (mod 4 )

,

1 (mod 4 ) ) o f c h a r a c t e r n : 1 (mod 4 ) t h e

Sect. 15-3

General Theory

265

Finally, we endeavor to improve on the Rapaport generating set, so as to satisfy condition (iv) universally. Mer construction is based upon the following lemma: Lemma 15-8:

A connected graph regular of degree three has a

hamiltonian cycle if there is a set and a set

of polygons

P

of squares, each set partitioning the

Q

vertex set of the graph and such that no member of contains every vertex of a member of For the Rapaport (12)(34)(56)...,

of

n!/12

(bc)")

=

i a,b,c 1 ,

where

a = (12), b =

c = (23)(45)( 6 7 ) ..., the set P 6 (ac) (or, we could use (n-1)!/2

and

12-gons

and the set

0 consists of

squares

n!/4

P

Q.

is composed

2n-gons (ab)2 The

.

Rapaport construction of a hamiltonian cycle uses b edges to combine 12-gons from P (or a edges to combine 2n-gons from k h (or P); thus the cycle consists of partial words (ac) b(ca) (bc)ka(cb)h) and we see that condition (iv) is uniformly satisfied, if

n

is odd.

If we modify d

=

J:

. .,

(34)(56)(78).

applies, to show that

This strengthens Theorem 1 5 - 6 . to

.[d,b,cj as in Plain Bob (so that

=

,',I

instead of

(12)), then Lemma 15-8

still

is hamiltonian (it is clear that

C!,(Sn)

A'

generates Sn, since A does.) But now the hamiltonian cycle cdc which fixes the bell found by Rapaport always involves

... ...,

in position one unduly.

To retain the constant movement of Plain Bob (that is, to avoid a = (12),

the "stagnation" of Rapaport's

which fixes

but to avoid the dawdling in position one, we now try where

...,

h = (45)( 6 7 ) (89)

first check that

h"

argument fails if

n

determined by (say)

and where

generates

hamiltonian cycle in which

and b

-

1

0

-

2

bells)

= {b,c,hl,

is odd.

Sn = (ch, (bc)2m-3 )

is even.) (bc)"

n = 2m

n A"

We

(this

Lemma 15-8 then applies, with P 2 by (ch) Then we obtain a

.

alternates throughout; as

b

moves

every position but the last, which is moved by both c and h, we see immediately that condition (iv) is satisfied. Moreover, there is an efficiency here (over Plain Bob), as only three generators are used; there are no bobs or singles.

However, conditions (v)

and (vi) have not been checked, and indeed it appears difficult to do so from this inductive construction. The special case n = 5 will be considered in detail in Section 5.

Change R i n g i n g

266

Chapt.

15

The i m b e d d i n g of t h e " n o c a l l " e x t e n t d e s c r i b e d a b o v e is nono r i e n t a b l e , of c h a r a c t e r i s t i c

- (n-1) 2 (n-4)

(n-3) !/ 4 .

Thus t h e v a r i o u s e x t e n t i m b e d d i n g s w e h a v e c o n s t r u c t e d i n t h i s n!/k,

s e c t i o n h a v e had c h a r a c t e r i s t i c s a s y m p t o t i c t o k

where

E 12,3,4,6~.

15-4.

Four-bell

E x t e n t s (Minimus)

The e l e v e n minimus e x t e n t s [ S S l ] a r e a l g e b r a i c a l l y d e s c r i b e d and g r a p h i c a l l y d e p i c t e d i n t h i s s e c t i o n .

We s e e f r o m Theorem 15-2

t h a t t h e r e are f o u r a l l o w a b l e t r a n s i t i o n r u l e s f o r f o u r b e l l s ;

they

are : a

=

( 1 2 ) (34)

b = (23)

- ---- -- -

c = (34)

. . . . . . ..

d = (12)

sfttiftt-

Each e x t e n t i s d e s c r i b e d , i n T a b l e 1 5 - 3 , in

S4

by t h e i d e n t i t y word

g i v i n g t h e hamiltonian c y c l e i n t h e Cayley c o l o r graph.

S i n c e a l l Cayley c o l o r g r a p h s are v e r t e x - t r a n s i t i v e ,

the i n i t i a l

v e r t e x of t h e h a m i l t o n i a n c y c l e c a n be a r b i t r a r i l y s e l e c t e d ; t h e r e m a i n d e r of t h e c y c l e i s t h e n u n i q u e l y d e t e r m i n e d . Minimus E x t e n t

Algebraic Description

Double B o b

3 3 ( ( a b ) ac) 2 3 (abad(ab) ) (abadabac ) 3

Canterbury

( a b c d c b a b )3

Reverse C a n t e r b u r y

( d b ( a b )2 d c )

P l a i n Bob R e v e r s e Bob

Double C a n t e r b u r y

(dbcdcbcd)

S i n g l e Court

( d b ( a b )2 d b ) 2 3 (ab(cb) ab)

Reverse Court Double C o u r t

( d b ( c b )2 d b )

S t . Nicholas

(dbadabdc)

Reverse S t . Nicholas

(abcdcbac) T a b l e 15-3

3

Sect. 15-4

Four-bell Extents (Minimus)

261

Obviously, each word uses twenty-four symbols, but it is worth noting that only Plain Bob requires as few as four letters to be written down in its algebraic description. We observe that none of the minimum extents satisfies a l l six of the conditions given in all fail (v), and only the first three listed satisfy

Section 15-1:

(iv). Now follow the graphical pictures of the corresponding Cayley cjraphs. Figure 15-2 serves directly for Plain Bob Minimums and Reverse Court Minimus; it also applies to Reverse Bob and Single Court, with istic one.

d

replacing

c.

These four extents a l l have character-

Figure 15-3 pictures Double Court and Double Canterbury;

these two extents have characteristic two. planar extents.)

(They are the only

Figure 15-4 depicts Reverse St. Nicholas, St.

Nicholas, Reverse Canterbury, Canterbury, and Double Bob; these are immersions and not imbeddings, as the redundant generator a is allowed to diagonalize six quadrilaterals. In the language of topological graph theory, this immersion shows that, for A = {a,b,c,d}, the crossing number of

is at most six.

CA(S4)

a

a Figure 15-2

268

Change R i n g i n g

F i g u r e 15-3

Chapt.

15

Sect. 15-4

Four-bell Extents (Minimus)

Figure 15-4

269

Chanqe Ringing

270

15-5.

Chapt.

15

Five-bell Extents (Doubles)

There are sixteen doubles extents given in [ S S l ] ; there are undoubtedly many more. Theorem 15-2)

The

seven possible transitions (see

are: a

=

(12)( 3 4 )

b

=

(23)(45)

-i-/+k;*t

c = (34)

. .. . . .. .

d = (23)

++++++++

e

=

f =

(12) (12) ( 4 5 ) (45)

.

We shall not need to represent

e

=

,+>F>,,>

-------for the extents discussed in this

section, which are: Plain Bob = (((ab)4 ac) 3 (ab)4 ad) 3

i

Grandsire Stedmdn

=

(fb(ab)4 fb(ab)3fb) 2fb(ab)4 fb(ab) 3fg) 2 (fba(f (bf)2ab(fb)2a)4f (bf)2abfd)2

No-Call

=

((ag)3 (ab)3ag(ab) 2 (ag)2ab) 5

Figure 15-5

=

(

shows

CA(S5)

.

for Plain Bob Doubles

(

I? =

a,b,cl) imbedded on the nonorientable surface N 5 , of characteristic -3. Indeed Plain Bob Doubles & characteristic ,,-3, as the graph which results when each decagon is contracted to a vertex is K less a 1-factor, for which the resulting quadrilateral

{

616

imbedding is necessarily optimal, and contraction cannot decrease characteristic. If we start the hamiltonian cycle at the distinguished vertex in Figure 15-5 and call the

bob

d

as indicated by the

superimposed edges, then the 121 rows of the extent can be readily obtained: see Problem 15-2.

Sect.

15-4

Five-bell E x t e n t s ( D o u b l e s )

F i g u r e 15-5

271

Chanqe R i n q i n q

272

C (A5),

F i g u r e 15-6 s h o w s orientable surface

N6,

-

15

Ckiapt.

d,b,€

i m b e d d e d o n the non-

-4.

of c h a r a c t e r i s t i c

The f i r s t

59

s y m b o l s of t h e i d e n t i t y w o r d f o r G r a n d s i r c D o u b l e s t r a c e a17 b u t o n c edge of a hamiltonian c y c l e f o r

C

on

(A5)

another, d i s j o i n t , surface

(Thus

N6.

C

C (A5)

on t h i s second s u r f a c e , and t h e s i n g l e

g

on

has t w o components,

(S5)

The samc h a m i l t o n i a n p a t h

As.)

o n e f o r e a c h coset o f

9

The s i n q l e

N6.

t h e n s h u n t s u s a c r o s s t o a s e c o n d c o p y o f t h e Lame

i b

repeated

returns us to the inLtial

The r e s u l t i s a h a m i l t o n i a n c y c l e i n

v e r t e x on t h e f i r s t s u r f a c e .

d e p i c t e d on a t o p o l o g i c a l space c o n s i s t i n g of d i s -

CL , , , , ( S 5 ) ,

c o n n e c t e d s u r f a c e whose two h o m e o m o r p h i c c o m p o n e n t s dre j o i n e d by a p a i r of l i n e s e q m e n t s . F i g u r e 15-6 a l s o s e r v e s f o r manner, i f we u s e

d

i n t h e same

Stcdnian D o u b l e s ,

f o r the single instead of

e x t e n t s have c h a r a c t e r i s t i c a t l e a s t

-8.

W e n o t e t h a t t h e same s u r f a c e ( t w o c o p i e s of 2 3 2 f o r S t . Simon D o u b l e s - ( ( a b ( a f ) a b a c ) a b ( a f ) a b a d ) c

,

i s u s e d n i n e t i m e s (of 1 2 0 ) a s p a r t of

bob t h r e e t i m e s ; N6

y6), Li

and

can be used except that

is u s e d a s a

t h e hamiltonian c y c l e , due t o t h e a c t i o n o f

d o e s n o t r e s p e c t t h e cosets of from one

'i'hus b o t h t h e s e

g.

A5

-

c,

t h a t i s , it t r a n s f e r s f r e q u e n t l y

to the other,

15-6.

A New composition

F i n a l l y , F i g u r e 15-7 shows

C

(S5),

Ii=

i a,b,g /,

imbedded on

The t h e n o n o r i e n t a b l e s u r f a c e NlO, 2also o f c h a r a c t e r i s t i c - 8 . 3 3 2 i d e n t i t y word ( ( a g ) ( a b ) a g ( a b ) ( a g ) a b ) 5 f o r "No-call'' D o u b l e s

( s o named by t h e a u t h o r , a s n o b o b s o r s i n g l e s a r e r e q u i r e d ) t r a c e s o u t a h a m i l t o n i a n c y c l e in

Ct(S5).

( I n f a c t , the c y c l e w a s found

i n t h i s f i g u r e f i r s t , as s u g g e s t e d by t h e f i v e - f o l d t h e n the word w a s w r i t t e n down f r o m t h e p i c t u r e ; d i c a t e d verter,,

f o r example.)

e x t e n t a r e given i n Table 15-4.

The c o r r e s p o n d i n g

symmetry, and

start a t 121

t h e in-

rows o f t h i s

Sect. 15-6

A New Composition

h

Figure 1 5 - 6

273

274

Change R i n g i n g

Figure 15-7

Chapt.

15

Sect. 15-6

275

A N e w Composition

"No C a l l " Doubles 12345

45123

23451

51234

34512

21435

54213

32541

15324

43152

21453

54231

32514

15342

43125

12543

45321

23154

51432

34215

12534

45312

23145

51423

34251

21354

54132

32415

15243

43521

21345

54123

32451

15234

43512

51324

34152

12435

45213

23541

14253

42531

25314

53142

31425

41523

24351

52134

35412

13245

45132

23415

51243

34521

12354

54312

32145

15423

43251

21534

53421

31254

14532

42315

25143

35241

13524

41352

24135

52413

35214

13542

41325

24153

52431

53124

31452

14235

42513

25341

51342

34125

12453

45231

23514

15432

43215

21543

54321

32154

14523

42351

25134

53412

31245

41253

24531

52314

35142

13425

41235

24513

52341

35124

13452

14325 14352

42153

25431

53214

31542

42135

25413

53241

31524

41532

24315

52143

35421

13254 12345

Table 15-4

276

Chanqe Ringi.ng

Chapt. 15

From the table it is a simple matter to check not only that conditions (i) through (iv) arc satisfied (of coursc, (i)-(iii) must hold, by Theorem 15-3, if we designate the first (and last) vertex of the hamiltonian cycle as rounds; this is always possible, since Cayley graphs are vertex-transitive. Moreover, we get immediate algebraic verification of (iv) by analyzing the identity word for No-call Doubles: a alternates throughout, moving every position but 5, and both b and g move position 5 ) , but that condition (v) holds as Well: each bell does the same amount of work as every other. In particular, the path of bell 1 in column i is exactly mimicked by that of bell 4 in column i + 1, bell 2 in column i + 2, bell 5 in column i + 3, and bell 3 in column i + 4, 1 -5 i 2 5 (arithmetic in Z 5 , writing "5" for "O".) Unfortunately, condition (vi) just fails, as agagagabababagababagaga is not palindromic. There are four pairs of symbols (such as the third "g" and the second "b") that could be switched to obtain a palindrome, but sadly each such switch destroys the much more crucial hamiltonian property. Perhaps the efficiency of "No-call" - in that only three transition rules are required (no other doubles extent uses this few) - would argue its merit, in spite of the failure of the final condition. But our final paragraph indicates that such argument is not needed! John Elliott , in his article "Doubles Principles" [ E 2 ] , generated by computer eight doubles extents not requiring bobs or singles. Each of these is palindromic, and in fact satisfies all conditions (i) - (iv): but it appears that none employs fewer than five generators. In the June 12, 1981 issue of "The Ringing World," a writer identified only as S. J. P. commented that the Central Council of Church Bell Ringers relaxes rule (iv) by allowing a bell to rest in the same position €or up to four successive changes and abolishes rule (vi) - removed "some years ago" - altogether! This latter, the writer continues, allows "any old rubbish to qualify as a method. "

15-7. 15-1.)

Problems

The planar imbedding of Figure 15-3 is a 12-fold branched covering space of an index two voltage graph imbedding; find this latter configuration.

Problems

Sect. 15-7

15-2.)

** 15-3.) ** 15-4.) ** 15-5.)

277

Use Figure 15-5 and the corresponding identity word in to write the rows for Plain Bob Doubles.

s5

Find the characteristic of Plain Bob Minor. Find the characteristic of Plain Bob Major. Find

F(n),

(We know that

the characteristic of Plain Bob on F ( 4 ) = 1, F ( 5 ) = - 3 ,

n

bells.

and in general

F(n) - -(n-2) L (n2-5n+4)/4. 15-6.)

If the principle for an extent on n bells corresponds to a subgroup of Sn (as is the case for Plain Bob), then contracting each coset of the principle to a single vertex produces a Schreier coset graph. Discuss the ramifications, for the calculation of the characteristic of the extent.

15-7.)

Show that the lead-ins (coset leads) for Plain Bob Minor To what extent correspond to a 1-factorization of K 6 .

(i.e. other minor extents, other even values of this generalize, and why?

n)

does

This Page Intentionally Left Blank

279

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P.)

305

INDEX OF SYMBOLS Graphs

16

5

G

Surfaces

Groups

20 20 G,, (I

1 cp (I' 1 c, (r 1

23

36

25 27

25

4

k' Nk

Mn

yo

83 30

I'/n

x (r)

124

42 42 108

S'

73

S I'

73

Y (Sk) M ( r / $2

42 130

x (Sk) x (Nk)

47,102

X (Mn)

108

47,108

73

x' ( S '

73

73

X(S")

73

'k (Mn

119

[XI

130

tS*I

130

119

124

9

15

9

10 10

10

rl rl rl

: r2 +

r2 r2

rl [ r21

15

17 17,32 17

173

X

130

Pi

70

143

P*

71

245 245

n'

10

18

m'

55

18

n'

10

Qn

11

K

10

mln 'n K

PI'.

10 9

*

rPn

11

n' (Z,)" Q

18 88 35

Index of Symbols

306

Groups

Surfaces

G(M)

221

cI* ( M ) G(D)

224

z( S )

122

M

221

Mn

245

'1I ( ' k ) K C L

213

K V L

103

225

130

5

130 25

i' r

47

5 9

143

R"

40

9

143

B

40

114

25

D

39

118

18

D

159

9,75

44

on

40

86

S2

76

86

S (k;nl (ml). .) 5 4

76

161

S(O;n(2))

76

161

M

78

161

.

120 40 42

79

166

144

65

222

144

64

222

144

52

222

n

146

9

234

V

147

10

237

117

10

245

117

7

260

117

7

260

121

7

261

7 5

5 5

D YX

X

157,220 157 157 167 161,219

71

161

71

207

Index of Symbols

Graphs

307

Groups

Surfaces 221

70

0

47

(G,P)

220

R (G)

-

220

229

25

M M(T,A,r) i '

233

71

M*

239

M(G*,O*)

239

70,205 5 5

130

224

71 160 13 66 67

82 168 168 177

180 186 189 189 189 189 189 193 193 194 205

207 205 205

211 211 214 215

252

Note: The number following the symbol gives the first page on which the symbol appears. In the top half of the list, corresponding symbols are displayed alongside one another. The six distinguished symbols correspond to the six relationships depicted in Figure 0-1.

This Page Intentionally Left Blank

309

INDEX OF DEFINITIONS additional adjacency part 1 adjacent edges, adjacent vertices, anti-automorphism of a graph I arc , associated bipartite graph associates first, second , automorphism of a Cayley color graph 1 of a graph, of a group, of a map, automorphism group of a Cayley color graph 1 of a graph, of a map, balanced incomplete block design ( R I B D ) bell tenor , treble, Betti number of a graph, bichromatic dual, bipartite graph, block of a design, block of a graph, n-book , bounded subspace, branch points, branched covering projection, branched covering space, bridge , brin, cactus , capping, Cayley color graph, Cayley graph, Cayley map, Cartesian product of two Cayley color graphs , of two graphs, of two permutation groups 2-cell imbedding, change ,

136 5 5 237 60 207 193 193 26 16 231

221 27 16 221 189 257 257 65 167 11 189 63 57 40 159 159 159 37 145 66 46 24 36 2 29 33 10 17 46 257

characteristic of a closed 2-manifold, 47 of an extent, 263 of a graph, 97 of a group, 97 of a pseudosurface, 54 chromatic number 102 of a graph, of a group, 124 of a hypergraph, 212 of a pseudosurface, 120 of a surface 102 1-chromatic number, 117 class function, 222 closed 2-manifold, 40 closed neighborhood, 19 closed walk, 7 n-trivial, 181 cochromatic number of a graph, 122 122 of a surface, complement, 9 complete bipartite graph, 10 complete design, 190 complete graph, 10 complete map, 233 complete n-partite graph, 11 component, 8 composite graph, 19 composition 10 of two graphs, of two permutation 17 groups I connected design, 197 connected graph, 8 n-connected graph, 53 conservative graph, 176 constellation, 144 quotient , 144 144 reduced , covering graph, 160 covering projection, 157 branched, 159 158 n-fold , 157 covering space, 159 branched , 123 n-critical graph, crossing number, 76,78 11 n-cube 9 cubic graph, 130,143 current , current graph, 130 cut vertex, 3 7 ,6 3 8 cycle, cycle rank, 65

Index of Definitions

310

k-degenerate graph, degree of a vertex, of a permutation group, design balanced incomplete block ( B I B D ) , complete, connected, genus of, group divisible, isomorphic, Latin square, partially balanced incomplete block (PBIBD)

,

resolvable, z-resolvable, transversal, dicyclic qroup, digraph, strongly connected, unilaterally connected, weakly connected, direct product of two groups, of two permutation groups 1 directed edge, directed graph, distance, dodecahedron, doubles , dual , bichromatic, edge of a graph, of a hypergraph, -automorphism, -automorphism group, directed , -isomorphic graphs, -isomorphism, induced, multiple, singular, n-edge colorable graph, n-edge critical graph, elementary subdivision, equivalent permutation groups 1 equivalent rotations, extended map automorphism group I extent , characteristic of, doubles, Grandsire, minimus ,

1 8

5 5 189 190 197 214 194 195 199 194 195 195 195 99 6 25

25 26 32

17 6 6 8 50 270 52 167

5 205 20

20 6 20

20

6 130 104 115 58 16 221 224 257 263 270 264 266

no-call doubles, plain bob, planar, eulcr characteristic, euler polyhedral identity, eulerian graph, evenly covered, face, n-factor, n-factorable, finite presentation, finitely generated presentation, finitely related presentation, first associates, n-fold covering projection, 2-fold triple system, forest , four color theorem, free group, Frobenius group, fundamental group, generalized pseudocharacteristic of a graph, generalized pseudosurface, generator , minimal set, redundant, genus of a block design, of a graph, of a group, of a hypergraph, of a surface, girth, Grandsire, graph, anti-automorphism of, automorphism group of, automorphism of, Betti number of, bipartite, Cayley, Cayley color, characteristic of, chromatic number of, cochromatic number of, complement of, complete , complete bipartite, complete n-bipartite, composite, connected, n-connected, conservative, covering , n-critical,

272 264 267 47 44 8 157 44 37 37 24 24 24 193 158 190 54

102 87 234 4 73 54 23 29 29

214

61 83

211 42 114 264

6 237 16

16

65

11

36 24 97 102 122 9 10 10

11

19

8 53 176 160 123

Index of Definitions

graph (continued) cubic , 9 current , 130 cycle rank of, 65 k-degenerate, 118 directed, 6 n-edge colorable, 104 n-edge critical, 115 edge isomorphic, 20 eulerian, 8 generalized pseudocharacteristic of, 73 genus of, 61 girth of, 114 hamiltonian, 8 identity , 16 imbedding of, 60 imbedded , 43 ,60 infinite , 6,84 isomorphic, 9 labeled, 5 line, 13 locally connected, 66 manifold number of, 180 maximum genus of, 64 orientably simple, 180 Paley, 245 n-partite, 11 perfect, 12 planar , 47 planar infinite, 84 plane , 47 3-polytopal, 53 urime, 19 pseudocharacteristic of , 73 quotient, 128,130 Gegular , 9 regular complete npartite , 11 relatively prime, 19 Schreier coset, 30 self-complementary, 237 strongly regular, 193 totally disconnected, 10 trivial , 19 underlying, 6,83 vertex arboricity of, 118 vertex partition number of, 119 voltage , 160 graphical regular representation, 238 group characteristic of, 97 chromatic number of, 124 dicyclic , 99 edge-automorphism, 20 extended map auto224 morphism ,

311

free, Frobenius, fundamental, genus of, graph automorphism, hamiltonian, hunting , identical permutation, infinite dihedral, isomorphic permutation, local , map automorphism, permutation, planar , vertex partition number of, group divisible PBIBD,

87 234 4 83 16 35 264 16 87 15 161 221 15 83

hamiltonian graph, hamiltonian group, hexahedron, homeomorphic from, homeomorphic with, hunting group, hypergraph, adjacency in, chromatic number of edge of, genus of, incidence in, maximum genus of, r-uniform, vertex degrees in, vertices of,

8 35 50 58 58 264 205 205 212 205 211 205 212 205 205 205

124 194

icosahedron, 50 identical permutation 16 groups I identity graph, 16 1-imbeddable, 117 imbedded graph, 43,60 imbedding of a graph, 60 of a pseudograph, 43 2-cell. 46 index of, 129,145,163 irregular, 144 178 maximal , minimal , 61 nonorientably minimal, 177 quadrilateral, 62 178 simplest , triangular, 62 imbedding scheme, 181 immersions, 267 5 incident , index of an imbedding 129,145 ~~

163

312

Index of Definitions

induced edge-automorphism group I induced subgraph, infinite dihedral group, infinite genus, infinite graph, irregular imbedding, isomorphic designs, isomorphic graphs, isomorphic permutation group I join , Kirchoff's Current Law (KCL) Kirchoff Voltage Law (KVL) klein bottle, labeled graph, Latin square designs, lead-ins , length of a region, length of a walk, lexicographic product, lifts, line graph, local group, locally connected graph, loop I

20

7 87 85 6,84 144 195 9 15 10

130 163 41 5 199 277 48 7

10 157 13 161 66 6

manifold number of a 180 graph, 2-manifold, 39 closed, 40 41 nonorientable, orientable, 41 vertex partition number of, 119 n-manifold, 40 manifold, quotient, 123,130 220 map 1 221 automorphism group of, automorphism of, 221 Cayley I 229 chiral , 250 complete, 233 224 mirror image of, nonorientable symmetrical, 228 ,255 Paley , 245 233 periods of, reflexible, 224 self-dual, 239 strongly symmetric, 244 symmetrical, 226 maximal imbedding, 178

maximum genus of a graph, of a hypergraph, maximum nonorientable genus , minimal generating set, minimal imbedding, minimums , mirror image of a map, mgbius strip, multigraph, multiple edge, multiplicity of branching,

64 211 186 29 61 266 224 41 6 6 159

neighborhood of a vertex, 19 no-call doubles, 272 nonorientable genus, 177 nonorientable 2-manifold, 41 nonorientable symmetrical 228,255 map I nonorientably minimal 177 imbedding , object set, objects, octahedron, open walk, open unit disk, orbits, order of a graph, order of a permutation group I orientable 2-manifold, orientably simple graph, Paley graph, Paley map, Paley tournament, palindromic method, partially balanced incomplete block design (PBIBD), path I peal I perfect graph, periods of a map, permutation group, degree of, equivalence, order of, regular , transitive, plain bob, planar BIBDs , planar extent, planar graph, planar group, planar infinite graph, plane graph,

15 189 50 7 39 15,222 5

15 41 180 245 245 252 258 194 8 259 12 233 15 15 15 15 15 16 264 215 267 47 83 84 47

Index of Definitions

Platonic solids, 52 point, singular, 54 polyhedron, 49 49 reqular , 53 3-polytopal graph, presentation, 24 finite, 24 finitely generated, 24 finitely related, 24 standard , 32 prime graph, 19 principle, 261 product cartesian (Cayley color graphs) , 33 Cartesian (graphs), 10 Cartesian (permutation 17 groups) , direct (groups), 32 direct (permutation 17 groups) , lexicographic, 10 173 strong tensor , 17 wreath , projective plane, 41 pseudocharacteristic of 73 a graph, 6 pseudographs, imbedded , 43 54 pseudosurface, characteristic of, 54 120 chromatic number of, generalized, 54 quadrilateral imbedding, 62 quotient constellation, 144 quotient graph, 128,130 quotient manifold, 128,130 rank of an abelian group, 33 reduced constellation, 144 redundant generator, 27 reflection, 224 reflexible map, 224 region , 44 2-cell , 46 length of, 48 regular complete n-partite graph I 11 9 regular graph, regular part, 136 regular permutation group, 15 regular polyhedron, 49 relation , 23 relatively prime graphs, 29 resolvable design, 195 z-resolvable design, 195 rotation, 220 equivalent, 221 rotation scheme, 73,219 rounds , 257

313

Schreier coset graph, 30 second associates, 193 self-complementary graph, 237 self-dual map, 239 side , 145 simplest imbedding, 176 singular edge, 130 singular point, 54 singular vertex, 54 size of a graph, 5 spanning subgraph, 7 splitting tree, 66 stabilizer, 222 standard presentation, 32 Steiner triple system, 191 stereographic projection, 45 strong tensor product, 173 strongly connected digraphs, 2 5 strongly regular graph, 193 strongly symmetric map, 244 subdivision, 58 elementary, 58 subgraph , 7 induced , 7 spanning , 7 sum, 17 surface , 39 chromatic number of, 102 cochromatic number of, 122 genus of, 42 hypergraph chromatic number of, 213 vertex arboricity of, 118 symmetrical map, 226 nonorientable, 2 2 8 ,2 5 5 tenor bell, tetrahedron, thickness, toroidal , torus , toroidal thickness, totally disconnected graph I tournament, Paley trail , transitive permutation group I transversal design, treble bell, tree , splitting, triangular imbedding, triple system 2-fold, Steiner , trivial graph, q-trivial closed walk,

257 50 75,76 168 41 168

10 252 8

15 195 257 43 66 62 190 191 19 181

314

Index of Definitions

underlying graph, r-uniform hypergraph, unilaterally connected digraph, union, upper-imbeddable, valence , vertex , neighborhood of, singular , vertex arboricity of a graph, of a surface, vertex partition number of a graph, of a group, of a closed 2-manifold, voltage g r a p h , vortices, theory of, walk closed, length of, open I weakly connected digraph, wheel , word , wreath product,

6,83 205 25 10

65 146 5 19 54

118 118 119 124 119 160 136 7 7 7 7

26 55 23

17