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Problems in combinatorics and graph theory PDF

356 Pages·1985·24.466 MB·English
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PROBLEMS IN COMBINATORICS AND GRAPH THEORY WILEY-INTERSCIENCE SERIES IN DISCRETE MATHEMATICS ADVISORYEDITORS RonaldL.Graham AT&TBellLaboratories,MurrayHill,NewJersey JanKarelLenstra MathematischCentrum,Amsterdam, TheNetherlands Graham,Rothschild,andSpencer RAMSEYTHEORY Tucker APPLIEDCOMBINATORICS Pless INTRODUCTIONTOTHETHEORY OFERROR-CORRECTINGCODES NemirovskyandYudin PROBLEM COMPLEXITY AND METHODEFFICIENCYINOPTIMIZATION ('IhnslatedbyER.Dawson) GouldenandJackson COMBINATORIALENUMERATION GondranandMinoux GRAPHSANDALGORITHMS (TranslatedbyS.Vajda) Fishburn INTERVALORDERSANDINTERVALGRAPHS: ASTUDYOFPARTIALLY ORDEREDSETS Tomescu PROBLEMS IN COMBINATORICS ANDGRAPHTHEORY (TranslatedbyRobertA.Melter) Palmer GRAPHICALEVOLUTION: AN INTRODUCTIONTOTHETHEORYOF RANDOM GRAPHS PROBLEMS IN COMBINATORICS AND GRAPH THEORY IOAN TOMESCU Faculty ofMathematics University ofBucharest Bucharest, Romania Translated from Romanian by ROBERT A. MELTER Department ofMathematics Long Island University Southampton, New York A Wiley-IntersciencePublication JOHN WILEY & SONS NewYork - Chichester - Brisbane - Toronto - Singapore Copyright©1985byJohnWiley&Sons,Inc. Allrightsreserved.PublishedsimultaneouslyinCanada. Reproductionortranslationofanypartofthiswork beyondthatpermittedbySection107or108ofthe 1976UnitedStatesCopyrightActwithoutthepermission ofthecopyrightownerisunlawful. Requestsfor permissionorfurtherinformationshouldbeaddressedto thePermissionsDepartment,JohnWiley&Sons,Inc. LibraryofCongressCataloginginPublicationData: Tomescu,loan. Problemsincombinatoricsandgraphtheory. (Wiley-lnterscienceseriesindiscretemathematics) Translationof:Problemedccombinatoricasiteoria grafurilor. Bibliography:p. l. Combinatorialanalysis. 2. Graphtheory. I. Title. II. Series. QA164.T6713 I985 511 .6 84-2170] ISBN0-471-80155-0 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 l Preface ThisbookisatranslationofPrablemedeCombinatoricd.giTeoriaGrafim‘lor, whichwaspublishedinBucharest, Romaniain 1981.InRomania,graphtheory istaughtinthefacultyofmathematicsandinparticularinwhatisknownasthe chair of informatics (=computer science). Students in preparatory schools which specialize in mathematics and physics also receive instruction in this subject. Thus the selection of problems presented includes some which are quite elementary and self-contained and others which were previously acces- sible only in research journals. The author has used the text to prepare Romanian candidates for participation in International Mathematical Olympiads. Each problem isaccompaniedbyacompleteanddetailed solution together with appropriate references to the mathematical literature. Thisshouldenable mature students to use the book independently. Teachers ofcourses in com- binatorics and graph theory will also find the text useful as a supplement, since important concepts are developed in the problems themselves. Even in the more elementary problems the reader will learn the important concepts and structures in thefield. Revisions in the original Romanian editionhave been made, and about 60 new problems and solutions have been added. The careful translation ofthis material by Professor Melterwill makeavailable to users ofEnglish-language texts a unique collection of problems in an expanding field of mathematics whichhas manyscientific applications. The author and translator are grateful to Ms. Linda Kallansrude for her professional typing and to the staff ofJohn Wiley & Sons for their interest andconcern inthedevelopment ofthis book. IOANTOMESCU Bucharest. Romania FebruaryI985 Contents GLOSSARYOFTERMSUSED ix PART1. STATEMENTOFPROBLEMS 1. COMBINATORIALIDENTITIES THEPRINCIPLEOFINCLUSIONANDEXCLUSION; INVERSIONFORMULAS 10 STIRLING, BELL, FIBONACCI,ANDCATALANNUMBERS 15 PROBLEMSINCOMBINATORIALSETTHEORY 22 PARTITIONSOFINTEGERS TREES 33 PARITY 39 CONNECTEDNESS 41 EXTREMALPROBLEMSFORGRAPHSANDNETWORKS 45 10. COLORINGPROBLEMS 52 11. HAMILTONIANPROBLEMS 12. PERMUTATIONS 58 13. THENUMBEROFCLASSES OFCONFIGURATIONS RELATIVETOAGROUPOFPERMUTATIONS 62 vii viii. Contents 14. PROBLEMSOF RAMSEYTYPE 65 PARTII. SOLUTIONS 69 BIBLIOGRAPHY 335

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