Table Of ContentProblem Books in Mathematics
Ralucca Gera · Teresa W. Haynes
Stephen T. Hedetniemi Editors
Graph
Theory
Favorite Conjectures and Open
Problems - 2
Problem Books in Mathematics
SeriesEditor:
PeterWinkler
DepartmentofMathematics
DartmouthCollege
Hanover,NH
USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/714
Ralucca Gera • Teresa W. Haynes
Stephen T. Hedetniemi
Editors
Graph Theory
Favorite Conjectures and Open Problems - 2
123
Editors
RaluccaGera TeresaW.Haynes
DepartmentofAppliedMathematics DepartmentofMathematics
NavalPostgraduateSchool EastTennesseeStateUniversity
Monterey,CA,USA JohnsonCity,TN,USA
StephenT.Hedetniemi
SchoolofComputing
ClemsonUniversity
Clemson,SC,USA
ISSN0941-3502 ISSN2197-8506 (electronic)
ProblemBooksinMathematics
ISBN978-3-319-97684-6 ISBN978-3-319-97686-0 (eBook)
https://doi.org/10.1007/978-3-319-97686-0
LibraryofCongressControlNumber:2018959757
MathematicsSubjectClassification(2010):05Cxx,01-02,01-08,03Fxx
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Contents
Introduction ...................................................................... 1
RaluccaGera,StephenT.Hedetniemi,andTeresaW.Haynes
DesertIslandConjectures....................................................... 7
LowellW.Beineke
BindingNumber,Cycles,andCliques ......................................... 19
WayneGoddard
AConjectureonLaplacianEigenvaluesofTrees............................. 27
DavidP.JacobsandVilmarTrevisan
QueensAroundtheWorldinTwenty-FiveYears............................. 43
WilliamD.Weakley
ReflectionsonaThemeofUlam................................................ 55
RonGraham
UlamNumbersofGraphs....................................................... 63
StephenT.Hedetniemi
ForbiddenTrees.................................................................. 69
DavidSumner
SomeofMyFavouriteConjectures:LocalConditionsImplying
GlobalCycleProperties......................................................... 91
OrtrudR.Oellermann
ThePathPartitionConjecture.................................................. 101
MarietjieFrickandJeanE.Dunbar
TotheMoonandBeyond........................................................ 115
EllenGethner
MyFavoriteDominationGameConjectures.................................. 135
MichaelA.Henning
v
vi Contents
ADeBruijn–Erdo˝sTheoreminGraphs?..................................... 149
VašekChvátal
AnAnnotatedGlossaryofGraphTheoryParameters,
withConjectures ................................................................. 177
Ralucca Gera, Teresa W. Haynes, Stephen T. Hedetniemi,
andMichaelA.Henning
Introduction
RaluccaGera,StephenT.Hedetniemi,andTeresaW.Haynes
1 Conjectures andOpenProblems
Thisbookisthesecondinatwo-volumeseriesonconjecturesandopenproblemsin
graphtheory.Theprimarymotivation,themeandvisionoftheserieswasexpressed
in the Introductionto Volume I, a slightly revised version of which we reproduce
here.
The series has its roots in the idea that conjecturesare central to mathematics,
and that it is useful to periodically identify and survey conjectures in the various
branches of mathematics. Typically, the end results of mathematics research are
theorems,themostimportantandfamousofwhichshowupintextbooks,whichin
turnare taughtto students. Thisoftengivesstudentsthe impressionthattheorems
arethemostimportantthingsinmathematics.Thepopularpressreinforcesthisidea;
when mathematicsis in the newspapersit is most often to reporta proof of some
well-known,unsolvedconjectureorproblem.
However, as every research mathematician knows, progress in mathematics
involvesmuchmorethanprovingtheorems,anditspracticeismuchricher.Math-
ematics research involves not only proving theorems, but also raising questions,
formulating open problems, and stating the conjectures, the solutions to which
become the new theorems. Mathematics research also involves the formation of
new conceptsand methods,the productionof counterexamplesto conjectures,the
R.Gera((cid:2))
DepartmentofAppliedMathematics,NavalPostgraduateSchool,Monterey,CA,USA
e-mail:[email protected]
S.T.Hedetniemi
SchoolofComputing,ClemsonUniversity,Clemson,SC,USA
T.W.Haynes
DepartmentofMathematics,EastTennesseeStateUniversity,JohnsonCity,TN,USA
©SpringerNatureSwitzerlandAG2018 1
R.Geraetal.(eds.),GraphTheory,ProblemBooksinMathematics,
https://doi.org/10.1007/978-3-319-97686-0_1
2 R.Geraetal.
simplificationandsynthesisofdifferentareasofmathematics,andthedevelopment
ofanalogiesacrossdifferentareasofmathematics.
Thethreeeditorsofthisvolumehappentobegraphtheorists,ormoregenerally
discretemathematicians,whoexplainthemajorfocusofthefollowingchapters.In
this collection of papers, the contributing authors present and discuss, often in a
story-telling style, some of the most well-known conjectures in the field of graph
theoryandcombinatorics.
Related to conjectures are open problems. Conjectures are either true or false.
Butwhatcountsastheresolutionofaproblemisoftenlessclear-cut.Nevertheless,
a conjecture clearly specifies a problem—and many problems can be naturally
formulated as conjectures. For example, it is a famous unsolved problem to
determinewhetherornottheclassPofdecisionproblemsisequaltoitssuperclass
NP.ThefamousP=NPproblemisoneofthesevenMillenniumProblemsidentified
bytheClayMathematicsInstitute,whoseresolutioncarriesa$1milliondollarprize
(http://www.claymath.org/millennium-problems/rules-millennium-prizes). For any
problemlikethis,havingayesornooutcome,theassociatedconjecturesareeither
thattheproblemcanberesolvedinthepositiveorthatitcannotbe.Manyormost
mathematicians, for instance, conjecture that P(cid:2)=NP. But a few, including Bela
Bollabás,conjecturethatP=NP.
While most mathematicians are most likely to be known for their theorems,
someareknownfortheirconjectures.FermatandPoincaré,whilefamousfortheir
theorems,are also knownfor their conjectures.Graphtheoristsknowthe nameof
Francis Guthrie only for his conjecturethat planar maps can be coloredwith four
colors[10].
Theworld-famousmathematicianPaulErdösis anexceptionalexample.While
he is known for, among other things, his development of Ramsey theory, the
probabilistic method, and contributions to the elementary proof of the prime
number theorem, he is perhaps equally famous for his conjectures and problems.
He travelled with these, talked about them, worked on them with hundreds of
collaborators,andevenofferedmonetaryprizesforthesolutionsofmanyofthem.
Someofhisgraphtheoryconjecturesarecollectedin[3].Hisconjecturesandprizes
have inspired considerable research and numerousresearch papers, and still more
than20yearsafterhisdeathin1996hisconjecturescontinuetohaveconsiderable
influence.
Notallconjecturesareofequalimportanceorsignificance,andnotallwillhave
the same influence on mathematics research. The resolution of some conjectures
willimpacttextbooksandeventhehistoryofmathematics.Theresolutionofothers
will soon be forgotten. What makes a conjecture significant or important? A few
mathematicianshaverecordedtheirthoughtsonthisquestion.
The famous British mathematician, G.H. Hardy, the early twentieth century
analyst and number theorist, discussed this question in his 1940 essay, A Mathe-
matician’sApology[8],whichisabiographicaldefenseofmathematicsashesaw
andpracticedit.Hardyisoftenrememberedfordiscountingthepracticalityorutility
ofmathematics.
Introduction 3
LaszloLovászhasdiscussedthequestionofwhatmakesagoodconjecture[9].
He says that “it is easy to agree that” the resolutionof a good conjecture“should
advance our knowledge significantly.” Nevertheless, Lovász wants to make room
for some of the conjecturesof Erdös that don’tobviously satisfy this criteria, but
are“conjecturessosurprising,soutterlyinaccessiblebycurrentmethods,thattheir
resolutionmustbringsomethingnew—wejustdon’tknowwhere.”
Lovász also discusses experimental mathematics as a source of conjectures, a
specificexampleofwhichbeingFajtlowicz’sGraffiti[4],acomputerprogramwhich
makesconjectures,manyofwhichareingraphtheory.Itiseasytowriteaprogram
toproducesyntacticallycorrectmathematicalstatements.Thedifficultyinwritinga
mathematical conjecture-making program is exactly how to limit the program to
making interesting or significant statements. When Fajtlowicz began writing his
program he would ask mathematicians what constituted a good conjecture. John
Conway told him that a good conjectureshould be “outrageous.”Erdös, in effect,
refusedtoanswer,tellingFajtlowicz,“Let’sleaveittoRadymanthus.”
We won’t here give a definitive answer to the question: what makes a good
conjecture?Fame isneithera necessarynora sufficientconditionfor a conjecture
to be considered good. Sociology plays some role in fame. The nonexistence of
oddperfectnumbersisprobablymorefamousduetoitsage,datingbacktoEuclid
and later to Descartes, than its importance [11]. But many conjectures of famous
mathematiciansareworkedonbecauseoftheirintrinsicimportancetomathematics.
We certainly expect there to exist little known but significant conjectures. The
history of mathematics contains numerous examples of important research which
was not recognized in its own time. The work of Galois, for instance, is a well-
knownexample.
Our thought is that there may not be any better way of identifying good
conjectures than to ask the experts, people who have tilled the mathematical soil
forsometime,andknowbestwhichseedswillsprout.
There are also internal reasons for the importance of a conjecture, strictly
mathematicalreasons related to the furtheranceof mathematicalresearch, and the
question,howwouldresolutionofagivenconjectureadvancemathematics?
Ofcoursethegoalofmathematicalresearch,andwhatresearchmathematicians
are paidto do,is to advancemathematics.Mathematicsis seen by the publicas a
toolforthesciences—theywouldhavemuchlessinterestinpayingmathematicians
tobeartiststhantheywouldasresearcherswhomayplayaroleinimprovingtheir
lives.
Buthowcanaconjectureplayaroleinadvancingmathematics?Inparticularit
mayseemthatwehaveanewquestiontoaddress:howdoesmathematicsadvance?
A conjecture can be said to advance mathematics if the truth of the conjecture
yieldsnewknowledgeaboutaquestionorobjectofexistingmathematicalinterest.
Furthermore, the advancement of mathematics requires not just new concepts,
conjectures,counterexamples,andproofs(uniquelymathematicalproducts)butalso
effectivecommunication.Lovászwrites:
Conjecture-makingisoneof thecentralactivitiesin mathematics.Thecreation
and dissemination of open problems is crucial to the growth and developmentof
