Problems in combinatorics and graph theory

356 Pages·1985·24.466 MB·English

by  Ioan Tomescu (author), Robert A. Melter (translator)

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