Applied Combinatorics PDF
Preview Applied Combinatorics
Keller Applied Trotter Combinatorics 2017 Edition 2017 Edition Mitchel T. Keller William T. Trotter Applied Combinatorics Applied Combinatorics Mitchel T. Keller Washington and Lee University Lexington, Virginia William T. Trotter Georgia Institute of Technology Atlanta, Georgia 2017 Edition Edition: 2017Edition Website: http://rellek.net/appcomb/ ©2006–2017 MitchelT.Keller,WilliamT.Trotter ThisworkislicensedundertheCreativeCommonsAttribution-ShareAlike4.0Interna- tionalLicense. Toviewacopyofthislicense,visithttp://creativecommons.org/licenses/ by-sa/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042,USA. Summary of Contents About the Authors ix Acknowledgements xi Preface xiii Preface to 2017 Edition xv Preface to 2016 Edition xvii Prologue 1 1 An Introduction to Combinatorics 3 2 Strings, Sets, and Binomial Coefficients 17 3 Induction 39 4 Combinatorial Basics 59 5 Graph Theory 69 6 Partially Ordered Sets 113 7 Inclusion-Exclusion 141 8 Generating Functions 157 9 Recurrence Equations 183 10 Probability 213 11 Applying Probability to Combinatorics 229 12 Graph Algorithms 239 vii SUMMARYOFCONTENTS 13 Network Flows 259 14 Combinatorial Applications of Network Flows 279 15 Pólya’s Enumeration Theorem 291 16 The Many Faces of Combinatorics 315 A Epilogue 331 B Background Material for Combinatorics 333 C List of Notation 361 Index 363 viii About the Authors About William T. Trotter William T. Trotter is a Professor in the School of Mathematics at Georgia Tech. He was first exposed to combinatorial mathematics through the 1971 Bowdoin Combi- natoricsConferencewhichfeaturedanarrayofsuperstarsofthatera,includingGian CarloRota,PaulErdős,MarshallHall,HerbRyzer,HerbWilf,WilliamTutte,RonGra- ham, Daniel Kleitman and Ray Fulkerson. Since that time, he has published more than120researchpapersongraphtheory,discretegeometry,Ramseytheory,andex- tremal combinatorics. Perhaps his best known work is in the area of combinatorics and partially ordered sets, and his 1992 research monograph on this topic has been veryinfluential. (Hetakessomeprideinthefactthatthismonographisstillinprint andcopiesarebeingsoldin2016.) Hehasmorethan70co-authors,butconsidershis extensivejointworkwithGrahamBrightwell,StefanFelsner,PeterFishburn,HalKier- steadandEndreSzemerèdiasrepresentinghisbestwork. Hiscareerincludesinvited presentationsatmorethan50internationalconferencesandmorethan30meetingsof professionalsocieties. HewasthefoundingeditoroftheSIAMJournalonDiscreteMath- ematicsandhasservedontheEditorialBoardofOrdersincethejournalwaslaunched in 1984, and his service includes an eight year stint as Editor-in-Chief. Currently, he servesontheeditorialboardsofthreeotherjournalsincombinatorialmathematics. Stillhehashisquirks. First,heinsistsonbeingcalled“Tom”,asThomasishismiddle name, while continuing to sign as William T. Trotter. Second, he has invested time andenergyserving fiveterms as department/school chair, oneat GeorgiaTech, two atArizonaStateUniversityandtwoattheUniversityofSouthCarolina. Inaddition, he has served as a Vice Provost and as an Assistant Dean. Third, he is fascinated by computeroperatingsystemsandisalwaysinstallingnewones. Inoneparticularweek, heputelevendifferentflavorsofLinuxonthesamemachine,interspersedwithfour completeinstallsofWindows7. Incidentally,theentireprocessstartedandendedwith Windows7. Fourth,helikestohitgolfballs,notplaygolf,justhitballs. Withoutthese diversions,hemightevenhavehadenoughtimetosettletheRiemannhypothesis. HehashadelevenPh.D.students,oneofwhichisnowhisco-authoronthistext. ix AbouttheAuthors About Mitchel T. Keller Mitchel T. Keller is a super-achiever (this description is written by WTT) extraordi- nairefromNorthDakota. AsagraduatestudentatGeorgiaTech,hewonalengthylist ofhonorsandawards,includingaVIGREGraduateFellowship,anIMPACTScholar- ship,aJohnR.FestaFellowshipandthe2009PriceResearchAward. Mitchisanatural leader and was elected President (and Vice President) of the Georgia Tech Graduate StudentGovernmentAssociation, rolesinwhichheservedwithdistinction. Indeed, aftercompletinghisterms,hisstudentcolleaguesvotedtoestablishacontinuingaward fordistinguishedleadership,tobenamedtheMitchelT.Kelleraward,withMitchas thefirstrecipient. Veryfewgraduatestudentswinawardsinthefirstplace,butMitch istheonlyoneIknowwhohasanawardnamedafterthem. Mitchisalsoagiftedteacherofmathematics,receivingtheprestigiousGeorgiaTech 2008OutstandingTeacherAward,acampus-widecompetition. Heisquicktoexper- iment with the latest approaches to teaching mathematics, adopting what works for himwhilerefiningandpolishingthingsalongtheway. Hereallyunderstandsthelit- eraturebehindactivelearningandtheprinciplesofengagingstudentsinthelearning process. Mitchhaseventaughthismoresenior(somesayancient)co-authorathing ortwoandgothimtotrypersonalresponsesystemsinalargecalculussection. Mitchisofftoafaststartinhisownresearchcareer,andisalreadyanexpertinthe subjectoflineardiscrepancy. Mitchhasalsomadesubstantivecontributionstoatopic knownasStanleydepth,whichisrightattheboundaryofcombinatorialmathematics andalgebraiccombinatorics. AfterfinishinghisPh.D.,Mitchreceivedanothersignalhonor,aMarshallSherfield PostdoctoralFellowshipandspenttwoyearsattheLondonSchoolofEconomics. He ispresentlyanAssistantProfessorofMathematicsatWashingtonandLeeUniversity, andafewyearsdowntheroad,he’llprobablybepresidentofsomething. Onthepersonalside,MitchisthekeeperoftheMathematicsGenealogyProject,and heisagreatcook. Hisdessertsaretodiefor. x
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