Table Of ContentJohn M. Harris Jeffry L. Hirst Michael J. Mossinghoff
• •
Combinatorics
and Graph Theory
Second Edition
123
JohnM.Harris JeffryL.Hirst
DepartmentofMathematics MathematicalSciences
FurmanUniversity AppalachianStateUniversity
Greenville,SC29613 121BodenheimerDr.
USA Boone,NC28608
[email protected] USA
[email protected]
MichaelJ.Mossinghoff
DepartmentofMathematics
DavidsonCollege
Box6996
Davidson,NC28035-6996
USA
[email protected]
ISSN:0172-6056
ISBN:978-0-387-797710-6 e-ISBN:978-0-387-79711-3
DOI:10.1007/978-0-387-79711-3
LibraryofCongressControlNumber:2008934034
MathematicsSubjectClassification(2000):05-0103-01
(cid:2)c 2008SpringerScience+BusinessMedia,LLC
Preface to the Second Edition
Therearecertainrulesthatonemustabidebyinordertocreatea
successfulsequel.
—RandyMeeks,fromthetrailertoScream2
While we maynotfollowthe preciserulesthatMr.Meekshadinmindforsuc-
cessful sequels, we have made a number of changes to the text in this second
edition.Inthenewedition,wecontinuetointroducenewtopicswithconcreteex-
amples,we providecompleteproofsofalmosteveryresult,andwepreservethe
book’sfriendlystyleandlivelypresentation,interspersingthetextwithoccasional
jokesandquotations.Thefirsttwochapters,ongraphtheoryandcombinatorics,
remain largely independent, and may be covered in either order. Chapter 3, on
infinite combinatorics and graphs, may also be studied independently,although
many readers will want to investigate trees, matchings, and Ramsey theory for
finitesetsbeforeexploringthesetopicsforinfinitesetsinthethirdchapter.Like
the first edition, this text is aimed at upper-division undergraduate students in
mathematics, though others will find much of interest as well. It assumes only
familiarity with basic prooftechniques, and some experiencewith matrices and
infiniteseries.
Thesecondeditionoffersmanyadditionaltopicsforuseintheclassroomorfor
independentstudy.Chapter1includesanewsectioncoveringdistanceandrelated
notionsingraphs,followinganexpandedintroductorysection.Thisnewsection
also introducesthe adjacencymatrix ofa graph,anddescribesits connectionto
important features of the graph. Another new section on trails, circuits, paths,
and cycles treats several problemsregardingHamiltonian and Eulerian paths in
graphs,anddescribessomeelementaryopenproblemsregardingpathsingraphs,
andgraphswithforbiddensubgraphs.
Several topics were added to Chapter 2. The introductory section on basic
countingprincipleshasbeenexpanded.Earlyinthechapter,anewsectioncovers
multinomial coefficients and their properties, following the developmentof the
binomialcoefficients.Anothernew section treats the pigeonholeprinciple,with
applicationstosomeproblemsinnumbertheory.ThematerialonPo´lya’stheory
ofcountinghasnowbeenexpandedtocoverdeBruijn’smoregeneralmethodof
countingarrangementsinthepresenceofonesymmetrygroupactingontheob-
jects,andanotheractingonthesetofallowedcolors.Anewsectionhasalsobeen
addedonpartitions,andthetreatmentofEuleriannumbershasbeensignificantly
expanded. The topic of stable marriage is developedfurther as well, with three
interesting variations on the basic problem now covered here. Finally, the end
of the chapterfeatures a new section on combinatorialgeometry.Two principal
problemsserveto introducethisricharea:a niceproblemofSylvester’sregard-
ing lines produced by a set of points in the plane, and the beautiful geometric
approachtoRamseytheorypioneeredbyErdo˝sandSzekeresinaproblemabout
theexistenceofconvexpolygonsamongfinitesetsofpointsintheplane.
In Chapter 3, a new section develops the theory of matchings further by in-
vestigating marriage problemson infinite sets, both countable and uncountable.
Anothernew section toward the end of this chapter describesa characterization
of certainlargeinfinitecardinalsby usinglinearorderings.Manynewexercises
have also been added in each chapter, and the list of references has been com-
pletelyupdated.
Thesecondeditiongrewoutofourexperiencesteachingcoursesingraphthe-
ory,combinatorics,andsettheoryatAppalachianStateUniversity,DavidsonCol-
lege,andFurmanUniversity,andwethanktheseinstitutionsfortheirsupport,and
ourstudentsfortheircomments.WealsothankMarkSpenceratSpringer-Verlag.
Finally,wethankourfamiliesfortheirpatienceandconstantgoodhumorthrough-
outthisprocess.Thefirstandthirdauthorswouldalsoliketoaddthat,sincethe
originalpublicationofthisbook,theirfamilieshavebothgainedtheirownsecond
additions!
May2008 JohnM.Harris
JeffryL.Hirst
MichaelJ.Mossinghoff
Preface to the First Edition
Threethingsshouldbeconsidered:problems,theorems,and
applications.
—GottfriedWilhelmLeibniz,
DissertatiodeArteCombinatoria,1666
Thisbookgrewoutofseveralcoursesincombinatoricsandgraphtheorygivenat
AppalachianStateUniversityandUCLAinrecentyears.Aone-semestercourse
forjuniorsatAppalachianStateUniversityfocusingongraphtheorycoveredmost
of Chapter 1 and the first part of Chapter 2. A one-quarter course at UCLA on
combinatorics for undergraduates concentrated on the topics in Chapter 2 and
includedsomepartsofChapter1.AnothersemestercourseatAppalachianState
foradvancedundergraduatesandbeginninggraduatestudentscoveredmostofthe
topicsfromallthreechapters.
There are rather few prerequisites for this text. We assume some familiarity
with basic proof techniques, like induction. A few topics in Chapter 1 assume
somepriorexposuretoelementarylinearalgebra.Chapter2assumessomefamil-
iaritywithsequencesandseries,especiallyMaclaurinseries,attheleveltypically
coveredinafirst-yearcalculuscourse.Thetextrequiresnopriorexperiencewith
moreadvancedsubjects,suchasgrouptheory.
While this book is primarily intended for upper-division undergraduate stu-
dents,webelievethatotherswillfinditusefulaswell.Lower-divisionundergrad-
uateswithapenchantforproofs,andeventalentedhighschoolstudents,willbe
able to follow muchofthe material,andgraduatestudentslookingfor an intro-
ductiontotopicsingraphtheory,combinatorics,andsettheorymayfindseveral
topicsofinterest.
Chapter1focusesonthetheoryoffinitegraphs.Thefirstsectionservesasan
introduction to basic terminology and concepts. Each of the following sections
presentsa specific branchof graphtheory:trees, planarity,coloring,matchings,
and Ramsey theory. These five topics were chosen for two reasons. First, they
representabroadrangeofthesubfieldsofgraphtheory,andinturntheyprovide
thereaderwithasoundintroductiontothesubject.Second,andjustasimportant,
thesetopicsrelateparticularlywelltotopicsinChapters2and3.
Chapter 2 develops the central techniques of enumerative combinatorics: the
principle of inclusion and exclusion, the theory and application of generating
functions,thesolutionofrecurrencerelations,Po´lya’stheoryofcountingarrange-
mentsinthepresenceofsymmetry,andimportantclassesofnumbers,including
theFibonacci,Catalan,Stirling,Bell,andEuleriannumbers.Thefinalsectionin
thechaptercontinuesthethemeofmatchingsbeguninChapter1withaconsider-
ationofthestablemarriageproblemandtheGale–Shapleyalgorithmforsolving
it.
Chapter 3 presents infinite pigeonhole principles, Ko¨nig’s Lemma, Ramsey’s
Theorem,andtheirconnectionstosettheory.Thesystemsofdistinctrepresenta-
tivesofChapter1reappearininfiniteform,linkedtotheaxiomofchoice.Count-
ingisrecastascardinalarithmetic,andapigeonholepropertyforcardinalsleads
to discussions of incompleteness and large cardinals. The last sections connect
large cardinals to finite combinatorics and describe supplementary material on
computability.
FollowingLeibniz’sadvice,wefocusonproblems,theorems,andapplications
throughout the text. We supply proofs of almost every theorem presented. We
try to introduce each topic with an application or a concrete interpretation, and
weoftenintroducemoreapplicationsintheexercisesattheendofeachsection.
In addition,we believe thatmathematicsis a fun and livelysubject, so we have
triedtoenlivenourpresentationwithanoccasionaljokeor(wehope)interesting
quotation.
We would like to thankthe Departmentof MathematicalSciencesat Appala-
chianStateUniversityandtheDepartmentofMathematicsatUCLA.Wewould
especiallyliketothankourstudents(inparticular,Jae-IlShinatUCLA),whose
questionsandcommentsonpreliminaryversionsofthistexthelpedustoimprove
it.Wewouldalsoliketothankthethreeanonymousreviewers,whosesuggestions
helpedtoshapethisbookintoitspresentform.We alsothankSharonMcPeake,
astudentatASU,forherrenderingoftheKo¨nigsbergbridges.
Inaddition,thefirstauthorwouldliketothankRonGould,hisgraduateadvi-
sor at EmoryUniversity,for teachinghim the methodsandthe joysof studying
graphs,andforcontinuingtobehisadvisorevenaftergraduation.Heespecially
wants to thank his wife, Priscilla, for being his perfectmatch, and his daughter
Sophieforaddingcolorandbrightnesstoeachandeveryday.Theirpatienceand
supportthroughoutthisprocesshavebeenimmeasurable.
ThesecondauthorwouldliketothankJudithRoitman,whointroducedhimto
settheoryandRamsey’sTheoremattheUniversityofKansas,usinganearlydraft
ofherfinetext.Also,hewouldliketothankhiswife,Holly(theotherProfessor
Hirst),forhavingtheinfinitetolerancethatsetsherapartfromthenorm.
ThethirdauthorwouldliketothankBobBlakley,fromwhomhefirstlearned
aboutcombinatoricsas an undergraduateat TexasA & M University,andDon-
aldKnuth,whoseclassConcreteMathematicsatStanfordUniversitytaughthim
muchmoreaboutthesubject.Mostofall,hewouldliketothankhiswife,Kris-
tine,forherconstantsupportandinfinitepatiencethroughoutthegestationofthis
project,andforbeingsomeonehecanalways,well,counton.
September1999 JohnM.Harris
JeffryL.Hirst
MichaelJ.Mossinghoff
Contents
PrefacetotheSecondEdition vii
PrefacetotheFirstEdition ix
1 GraphTheory 1
1.1 IntroductoryConcepts . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 GraphsandTheirRelatives . . . . . . . . . . . . . . . . . 2
1.1.2 TheBasics . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 SpecialTypesofGraphs . . . . . . . . . . . . . . . . . . 10
1.2 DistanceinGraphs . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.1 DefinitionsandaFewProperties . . . . . . . . . . . . . . 18
1.2.2 GraphsandMatrices . . . . . . . . . . . . . . . . . . . . 21
1.2.3 GraphModelsandDistance . . . . . . . . . . . . . . . . 26
1.3 Trees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.1 DefinitionsandExamples . . . . . . . . . . . . . . . . . 31
1.3.2 PropertiesofTrees . . . . . . . . . . . . . . . . . . . . . 34
1.3.3 SpanningTrees . . . . . . . . . . . . . . . . . . . . . . . 38
1.3.4 CountingTrees . . . . . . . . . . . . . . . . . . . . . . . 43
1.4 Trails,Circuits,Paths,andCycles . . . . . . . . . . . . . . . . . 51
1.4.1 TheBridgesofKo¨nigsberg . . . . . . . . . . . . . . . . . 52
1.4.2 EulerianTrailsandCircuits . . . . . . . . . . . . . . . . 55
1.4.3 HamiltonianPathsandCycles . . . . . . . . . . . . . . . 60
1.4.4 ThreeOpenProblems . . . . . . . . . . . . . . . . . . . 67
1.5 Planarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
1.5.1 DefinitionsandExamples . . . . . . . . . . . . . . . . . 74
1.5.2 Euler’sFormulaandBeyond . . . . . . . . . . . . . . . . 78
1.5.3 RegularPolyhedra . . . . . . . . . . . . . . . . . . . . . 80
1.5.4 Kuratowski’sTheorem . . . . . . . . . . . . . . . . . . . 83
1.6 Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
1.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 86
1.6.2 BoundsonChromaticNumber . . . . . . . . . . . . . . . 88
1.6.3 TheFourColorProblem . . . . . . . . . . . . . . . . . . 93
1.6.4 ChromaticPolynomials. . . . . . . . . . . . . . . . . . . 97
1.7 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
1.7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 102
1.7.2 Hall’sTheoremandSDRs . . . . . . . . . . . . . . . . . 104
1.7.3 TheKo¨nig–Egerva´ryTheorem . . . . . . . . . . . . . . . 109
1.7.4 PerfectMatchings . . . . . . . . . . . . . . . . . . . . . 111
1.8 RamseyTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
1.8.1 ClassicalRamseyNumbers. . . . . . . . . . . . . . . . . 116
1.8.2 ExactRamseyNumbersandBounds . . . . . . . . . . . . 118
1.8.3 GraphRamseyTheory . . . . . . . . . . . . . . . . . . . 124
1.9 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
2 Combinatorics 129
2.1 SomeEssentialProblems . . . . . . . . . . . . . . . . . . . . . . 130
2.2 BinomialCoefficients . . . . . . . . . . . . . . . . . . . . . . . . 137
2.3 MultinomialCoefficients . . . . . . . . . . . . . . . . . . . . . . 144
2.4 ThePigeonholePrinciple . . . . . . . . . . . . . . . . . . . . . . 150
2.5 ThePrincipleofInclusionandExclusion. . . . . . . . . . . . . . 156
2.6 GeneratingFunctions . . . . . . . . . . . . . . . . . . . . . . . . 164
2.6.1 DoubleDecks. . . . . . . . . . . . . . . . . . . . . . . . 166
2.6.2 CountingwithRepetition . . . . . . . . . . . . . . . . . . 168
2.6.3 ChangingMoney . . . . . . . . . . . . . . . . . . . . . . 171
2.6.4 FibonacciNumbers . . . . . . . . . . . . . . . . . . . . . 177
2.6.5 RecurrenceRelations . . . . . . . . . . . . . . . . . . . . 181
2.6.6 CatalanNumbers . . . . . . . . . . . . . . . . . . . . . . 185
2.7 Po´lya’sTheoryofCounting. . . . . . . . . . . . . . . . . . . . . 190
2.7.1 PermutationGroups . . . . . . . . . . . . . . . . . . . . 191
2.7.2 Burnside’sLemma . . . . . . . . . . . . . . . . . . . . . 196
2.7.3 TheCycleIndex . . . . . . . . . . . . . . . . . . . . . . 200
2.7.4 Po´lya’sEnumerationFormula . . . . . . . . . . . . . . . 202
2.7.5 deBruijn’sGeneralization . . . . . . . . . . . . . . . . . 209
2.8 MoreNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
2.8.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 218
2.8.2 StirlingCycleNumbers . . . . . . . . . . . . . . . . . . 227
2.8.3 StirlingSetNumbers . . . . . . . . . . . . . . . . . . . . 231
2.8.4 BellNumbers . . . . . . . . . . . . . . . . . . . . . . . . 237
2.8.5 EulerianNumbers . . . . . . . . . . . . . . . . . . . . . 242
2.9 StableMarriage . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
2.9.1 TheGale–ShapleyAlgorithm . . . . . . . . . . . . . . . 250
2.9.2 VariationsonStableMarriage . . . . . . . . . . . . . . . 255
2.10 CombinatorialGeometry . . . . . . . . . . . . . . . . . . . . . . 264
2.10.1 Sylvester’sProblem . . . . . . . . . . . . . . . . . . . . 265
2.10.2 ConvexPolygons . . . . . . . . . . . . . . . . . . . . . . 270
2.11 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
3 InfiniteCombinatoricsandGraphs 281
3.1 PigeonsandTrees . . . . . . . . . . . . . . . . . . . . . . . . . . 282
3.2 RamseyRevisited . . . . . . . . . . . . . . . . . . . . . . . . . . 285
3.3 ZFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
3.3.1 LanguageandLogicalAxioms . . . . . . . . . . . . . . . 290
3.3.2 ProperAxioms . . . . . . . . . . . . . . . . . . . . . . . 292
3.3.3 AxiomofChoice . . . . . . . . . . . . . . . . . . . . . . 297
3.4 TheReturnofderKo¨nig . . . . . . . . . . . . . . . . . . . . . . 301
3.5 Ordinals,Cardinals,andManyPigeons . . . . . . . . . . . . . . 304
3.5.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . 304
3.5.2 OrdinalsandCardinals . . . . . . . . . . . . . . . . . . . 308
3.5.3 PigeonsFinishedOff . . . . . . . . . . . . . . . . . . . . 312
3.6 IncompletenessandCardinals . . . . . . . . . . . . . . . . . . . 318
3.6.1 Go¨del’sTheoremsforPAandZFC. . . . . . . . . . . . . 318
3.6.2 InaccessibleCardinals . . . . . . . . . . . . . . . . . . . 320
3.6.3 ASmallCollageofLargeCardinals . . . . . . . . . . . . 322
3.7 WeaklyCompactCardinals . . . . . . . . . . . . . . . . . . . . . 324
3.8 InfiniteMarriageProblems . . . . . . . . . . . . . . . . . . . . . 327
3.8.1 HallandHall . . . . . . . . . . . . . . . . . . . . . . . . 328
3.8.2 CountablyManyMen . . . . . . . . . . . . . . . . . . . 330
3.8.3 UncountablyManyMen . . . . . . . . . . . . . . . . . . 336
3.8.4 EspousableCardinals . . . . . . . . . . . . . . . . . . . . 340
3.8.5 PerfectMatchings . . . . . . . . . . . . . . . . . . . . . 343
3.9 FiniteCombinatoricswithInfiniteConsequences . . . . . . . . . 344
3.10 k-criticalLinearOrderings . . . . . . . . . . . . . . . . . . . . . 347
3.11 PointsofDeparture . . . . . . . . . . . . . . . . . . . . . . . . . 348
3.12 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
References 355
Index 369
