Symmetry in Graphs

527 Pages·2022·4.307 MB·English

by  Ted Dobson, Aleksander Malnič, Dragan Marušič

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 198
EditorialBoard
J. BERTOIN, B. BOLLOBA´S, W. FULTON, B. KRA, I. MOERDIJK,
C. RAEGER, P. SARNAK, B. SIMON, B. TOTARO
SYMMETRYINGRAPHS
Thisisthefirstfull-lengthbookonthemajorthemeofsymmetryingraphs.Forming
part of algebraic graph theory, this fast-growing field is concerned with the study of
highlysymmetricgraphs,particularlyvertex-transitivegraphs,andothercombinatorial
structures, primarily by group-theoretic techniques. In practice, the street goes both
ways and these investigations shed new light on permutation groups and related
algebraicstructures.
Thebookassumesafirstcourseingraphtheoryandgrouptheorybutnospecialized
knowledge of the theory of permutation groups or vertex-transitive graphs. It begins
withthebasicmaterialbeforeintroducingthefield’smajorproblemsandmostactive
research themes in order to motivate the detailed discussion of individual topics that
follows. Featuring many examples and with over 450 exercises, it is an essential
introductiontothefieldforgraduatestudentsandavaluableadditiontoanyalgebraic
graphtheorist’sbookshelf.
TedDobson isProfessorattheUniversityofPrimorska,Slovenia.
AleksanderMalnicˇ isProfessorattheUniversityofLjubljana,Slovenia.
DraganMarusˇicˇ isProfessorattheUniversityofPrimorska,Slovenia,andthefounder
oftheSlovenianschoolinAlgebraicGraphTheory.
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
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J.Bertoin,B.Bolloba´s,W.Fulton,B.Kra,I.Moerdijk,C.Praeger,P.Sarnak,B.Simon,B.Totaro
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Symmetry in Graphs
TED DOBSON
UniversityofPrimorska,Slovenia
ALEKSANDER MALNICˇ
UniversityofLjubljana,Slovenia
DRAGAN MARUSˇICˇ
UniversityofPrimorska,Slovenia
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108429061
DOI:10.1017/9781108553995
©TedDobson,AleksanderMalnicˇ,andDraganMarusˇicˇ2022
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
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Firstpublished2022
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Contents
Preface pageix
1 IntroductionandConstructions 1
1.1 BasicDefinitions 4
1.2 CayleyDigraphs 11
1.3 DoubleCosetDigraphs 19
1.4 OrbitalDigraphs 26
1.5 MetacirculantDigraphs 31
1.6 Exercises 38
2 ThePetersenGraph,Blocks,andActionsofA 45
5
2.1 TheAutomorphismGroupofthePetersenGraph 46
2.2 BlocksandImprimitiveGroups 48
2.3 MoreWaysofFindingBlocks 54
2.4 PrimitiveandQuasiprimitiveGroups 59
2.5 ActionsofA 66
5
2.6 OrbitalDigraphsofA 74
5
2.7 Exercises 82
3 SomeMotivatingProblems 87
3.1 Cubics-Arc-transitiveGraphs 88
3.2 Arc-,Edge-,andHalf-arc-transitiveGraphs 92
3.3 IntroductiontotheHamiltonicityProblem 102
3.4 Vertex-transitiveGraphsThatAreNotHamiltonian 103
3.5 Connected Cayley Graphs of Abelian Groups Are
Hamiltonian 110
3.6 BlockQuotientDigraphs 112
3.7 TheSpiralPathArgumentandFruchtNotation 119
3.8 SemiregularElements 123
3.9 Exercises 128
v
vi Contents
4 GraphswithImprimitiveAutomorphismGroup 132
4.1 PermutationIsomorphismandEquivalentRepresentations 133
4.2 WreathProductsofDigraphsandGroups 141
4.3 TheEmbeddingTheorem 149
4.4 GroupsofPrimeDegreeandDigraphsofPrimeOrder 152
4.5 TheAutomorphismGroupoftheHeawoodGraph 158
4.6 Exercises 163
5 TheEndoftheBeginning 167
5.1 TheAutomorphismGroupofWreathProductGraphs 168
5.2 2-ClosedPermutationGroups 173
5.3 GeneralizedWreathProducts 179
5.4 PrimitiveGroupsandtheO’Nan–ScottTheorem 188
5.5 Covers 197
5.6 Exercises 208
6 OtherClassesofGraphs 214
6.1 TheCoxeterGraph 214
6.2 KneserGraphsandOddGraphs 223
6.3 TheJohnsonGraphs 225
6.4 LeviGraphs 237
6.5 TheGeneralizedPetersenGraphs 248
6.6 FermatGraphs 262
6.7 Exercises 268
7 TheCayleyIsomorphismProblem 272
7.1 GraphIsomorphimsinQuasipolynomialTime 273
7.2 IntroductionandBasicDefinitions 274
7.3 BasicTools 277
7.4 Non-CI-groupsWithRespecttoGraphs 283
7.5 GroupsofOrderaProductofTwoPrimes 286
7.6 Pa´lfy’sTheorem 296
7.7 IsomorphismsofCirculantDigraphs 303
7.8 Exercises 317
8 AutomorphismGroupsofVertex-TransitiveGraphs 322
8.1 NormalCayleyDigraphs 322
8.2 TheTransfer 328
8.3 MoreToolsandAutomorphismGroupsofCirculants
ofOrder p2 333
8.4 AutomorphismGroupsofCayleyDigraphsofZ2 341
p
8.5 AutomorphismGroupsofCirculantDigraphs 346
8.6 Exercises 353
Contents vii
9 ClassifyingVertex-TransitiveGraphs 357
9.1 DigraphsofPrimePowerOrder 357
9.2 GraphsofOrderaProductofTwoDistinctPrimes 365
9.3 CirculantDigraphsofOrdernwithgcd(n,ϕ(n))=1 372
9.4 Exercises 376
10 SymmetricGraphs 378
10.1 The17FamiliesofCubicSymmetricGraphs 379
10.2 CubicSymmetricGraphsofSmallGirth 381
10.3 RigidCells 385
10.4 ValencyPreservingQuotienting 391
10.5 SymmetricGraphsofLargerValencies 395
10.6 Exercises 399
11 Hamiltonicity 401
11.1 HistoricPerspectives 401
11.2 TheLiftingCycleTechnique 406
11.3 Vertex-TransitiveGraphsofParticularOrders 415
11.4 SpecificApproachesforCayleyGraphs 421
11.5 EmbeddingsonSurfaces 425
11.6 AFinalObservation 435
11.7 Exercises 435
12 Semiregularity 437
12.1 HistoricPerspectives 437
12.2 Ordersof(Di)graphs 443
12.3 ValencyofGraphs 444
12.4 TypesofGroupActions 448
12.5 FurtherDirections 449
12.6 Exercises 450
13 GraphswithOtherTypesofSymmetry:
Half-arc-transitiveGraphsandSemisymmetricGraphs 451
13.1 HistoricPerspectives 451
13.2 Half-arc-transitiveGraphs:AGeometricApproach 453
13.3 QuarticHalf-arc-transitiveGraphs 458
13.4 SemisymmetricGraphs 462
13.5 Exercises 465
viii Contents
14 FareYouWell 468
14.1 AlgebraicGraphTheory:TowardsCFSG-FreeArguments 468
14.2 Even/OddAutomorphismsProblem 472
References 474
IndexofGraphs 497
IndexofSymbols 500
SelectAuthorIndex 503
IndexofTerms 506
Preface
Symmetry and its absence, asymmetry, is one of the core concepts in the
arts, science, or daily life, for that matter. In everyday language, symmetry –
from the Greek “συµµ(cid:15)τρια = symmetria,” agreement in due proportion,
arrangement–usuallyreferstoasenseofbeauty,ofproportionalharmony,and
ofbalance.Intryingtopresenttheconceptofsymmetryinasgeneralasetting
aspossible,onecanthinkofitasameasureoftheinnerstructuralstabilityof
theobjectinquestionwhenconfrontedwithpossibleexternalintrusions.When
thelatterexceedtheacceptablerobustnessthresholdandthusbreaktheinnate
symmetry,atransformationoftheobjectoccurs.Givenanewlifeofenriched
complexity, the object finds itself in an environmental absence of symmetry
andstartsaslowbutsteadyascenttoabalancedstateofabrand-newstability.
Inmathematics,thenotionofsymmetry,althoughpresentinmanydifferent
areas,ismostlystudiedbymeansofgroupsandisthereforearguablyaninher-
entlyalgebraicconcept–namely,thesetofalltransformations/symmetriesof
a given mathematical object that preserves its inner structure forms a group.
Knowing the full set of symmetries of an object is important because it
providesthemostcompletedescriptionofitsstructure.
Historically,manyofthemostfruitfultechniquesforstudyinggroupshave
been developed within the framework of permutation groups and, more gen-
erally, group actions, preferably on sets enjoying additional inner structure.
Among these, transitive actions are perhaps the most natural ones for they
represent the building blocks for actions in general. This interplay is nicely
recapturedinthecombinatorialsettingofrelationalstructures,mostnotablyin
graphs,asthesimplestmanifestationofsuchstructures.Beingoneofthecore
conceptsessentialbothtounderstandingnaturalphenomenaandthedynamics
of social systems and to providing a theoretical framework for efficient mass
communication,relationalstructures(andgraphsinparticular)withhighlevels
ofsymmetryaresoughtfortheiroptimalbehaviorandhighperformance.
ix