Covering Walks in Graphs

123 Pages·2014·2.312 MB·English

by  Futaba Fujie, Ping Zhang (auth.)

SPRINGER BRIEFS IN MATHEMATICS
Futaba Fujie
Ping Zhang
Covering Walks in
Graphs
123
SpringerBriefs in Mathematics
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PingZhang
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Futaba Fujie • Ping Zhang
Covering Walks in Graphs
123
FutabaFujie PingZhang
GraduateSchoolofMathematics DepartmentofMathematics
NagoyaUniversity WesternMichiganUniversity
Nagoya,Aichi,Japan Kalamazoo,MI,USA
ISSN2191-8198 ISSN2191-8201(electronic)
ISBN978-1-4939-0304-7 ISBN978-1-4939-0305-4(eBook)
DOI10.1007/978-1-4939-0305-4
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013957930
MathematicsSubjectClassification(2010):05C12,05C38,05C45
©FutabaFujie,PingZhang2014
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Preface
Researchersingraphtheory,includinggraduatestudents,aretheprimaryaudience
for this book. Although it is assumed that the reader is acquainted with the basic
concepts in graph theory, this book is self-contained in that all concepts and
terminologyneededforthetopicareclearlypresentedandillustrated.Thisbookcan
beusedforareadingcourse,aseminarorashortcourseforgraduatestudentswho
areinterestedinEulerianandHamiltonianpropertiesofgraphsaswellascovering
walksofgraphsingeneral.Inaddition,thisbookcontainsthebackground needed
to begin a research program on a variety of topics concerning covering walks in
graphsandprovideseasyaccesstorecentresultsandopenproblemsinthisareaof
research.
Many theorems involving walks in graphs can be traced back to problems that
ledtosomeofthebest-knownandmost-studiedconceptsingraphtheory.Awalkin
agraphG beginsatsomevertexuofG,proceedstoanedgee Duu0 incidentwith
u,thenproceedstou0 andnexttoanedgeincidentwithu0 (possiblye again).This
continuesuntiltheprocedurestopsatsomevertexv,producingau(cid:2)vwalkW.If
W containseveryedgeofG,thenW isanedge-coveringwalk,whileifW contains
everyvertexofG,thenitisavertex-coveringwalk.
Graph theory is considered to have begun in 1736 when the great Swiss
mathematician Leonhard Euler solved the famous Königsberg Bridge Problem,
which asks whether it is possible to walk about the city of Königsberg (located
inPrussiaatthetime)andcrosseachofitssevenbridgesinthecityexactlyonce.
Eventually it was seen that the Königsberg Bridge Problem could be expressed as
aproblemingraphtheory,anareaofmathematics thatdidnotexistin1736.This
ledtotheconceptofEuleriangraphsandlatertothemoregeneralconceptofedge-
coveringwalksingraphs.ThisisthetopicofChap.1.
After presenting Euler’s characterization of Eulerian graphs as those con-
nected graphs containing only even vertices, graphs containing Eulerian trails are
described as those connected graphs containing exactly two odd vertices. Oswald
Veblen’s characterization of Eulerian graphs as those connected graphs that can
be decomposed into cycles is presented. From this theorem, a number of results
and conjectures emanated. One of the recent conjectures is the Eulerian Cycle
DecompositionConjecture.Manyresultsobtainedonthisconjecturearepresented.
Several results are presented that deal with connected graphs containing a
specified number of odd vertices. While connected graphs with odd vertices do
v
vi Preface
not have circuits containing each edge exactly once, they do contain closed walks
containingeachedgeatleastonce.Determiningtheminimumnumberofedgesin
suchaclosedwalkisthefamousChinesePostmanProblem.Thisledtotherecent
studyofirregularEulerianwalksinwhichnotwoedgesareencounteredthesame
numberoftimesinthewalk.
SomeEuleriangraphscontainverticesuhavingthepropertythateverytrailwith
initialvertexucanbeextendedtoanEuleriancircuit.Graphswiththispropertyare
described.Theanalogousresultforconnectedgraphswithtwooddverticesisalso
presented.
Chapter 2 deals with graphs that possess closed vertex-covering walks. This
concept emanates from the so-called Icosian Game of the Irish mathematician
WilliamRowanHamiltonandhisAroundtheWorldpuzzle,whichdealtwithcycles
inadodecahedroncontainingeveryvertex.ThisledtotheconceptsofHamiltonian
cyclesandHamiltoniangraphs.AlthoughnocharacterizationofHamiltoniangraphs
haseverbeenfound,manysufficientconditionsforagraphtobeHamiltonianhave
beendiscovered,thefirstofwhichwasa1952theoremoftheDanishmathematician
GabrielAndrewDirac,whoprovedthatiftheminimumdegreeofagraphisatleast
half of the order of the graph, then that graph is guaranteed to be Hamiltonian.
This was extended somewhat by the Norwegian mathematician Oystein Ore, who
showedthatifthesumofthedegreesofeverytwononadjacentverticesinagraphis
atleastitsorder,thenthatgraphisHamiltonian.Thisledtothestudyoftheclosure
ofgraphsanditsconnectionwithHamiltoniangraphs.
The best-known necessary condition for a Hamiltonian graph is one that states
that every Hamiltonian graph G has the property that when a set of vertices is
removed from G, then the number of components in the resulting graph never
exceeds the number of vertices removed. This observation led to the well-studied
concept of toughness in graphs and its relationship to Hamiltonian graphs. The
famousTravelingSalesmanProblemisalsodescribed.
Ofanumberofoperationsdefinedonagraphthatresultinnewgraphs,twoofthe
mostcommonarethelinegraphandpowersofagraph.Therehavebeennumerous
theoremsdealingwiththoseoperationsthatresultinHamiltoniangraphsandgraphs
withrelatedproperties.
Although many connected graphs do not contain Hamiltonian cycles, every
connected graph contains a closed vertex-covering walk. The major interest here
istheminimumlengthofsuchwalksinagraph,whichistheHamiltoniannumber
ofthegraph.RecentresearchshowedthattheHamiltoniannumberofagraphGcan
bedeterminedbycomputingthesumofthedistancesofconsecutivetermsineach
cyclic ordering of the vertices of G and then finding the minimum of these sums.
ThemaximumofthesesumsistheupperHamiltoniannumberofG andresultson
thistopicarepresentedaswellinChap.2.Furthermore,severalresultsonthesetof
allsuchnumbersobtainedinthismannerarediscussed.
In Chap.3, the emphasis changes from closed vertex-covering walks in a
connectedgraphG totherecentresearchtopicofopenvertex-coveringwalksinG.
Such a walk of minimum length is referred to as a traceable walk and its length
is the traceable number of G. Here too it is seen that the traceable number of G
Preface vii
can be obtained by first computing the sum of the distances of consecutive terms
in each linear ordering of the vertices of G. The minimum value of such a sum is
thetraceablenumberofG;themaximumsuchsumistheuppertraceablenumber
ofG.Specialattentionisgiventothesetwoparametersoftrees.Also,comparisons
between the Hamiltonian and traceable numbers are described as are comparisons
betweentheupperHamiltoniananduppertraceablenumbersofgivengraphs.
One of the differences between closed vertex-covering walks and open vertex-
covering walks in a graph is that, for an open vertex-covering walk, its length
dependsonwhichvertexthewalkbegins(orends).Forthisreason,foreachvertex
vinagraph,sequencesoftheverticesofthegraphwithvastheirinitialtermandthe
resultingsumsofdistancesofconsecutivetermsareconsidered.Theminimumsuch
sum is the traceable number of v. Related concepts such as the maximum vertex-
traceable numbers of graphs, traceably singular graphs, and the total traceable
numbersofgraphsaredescribed.
Thenumerousnewareasofresearchpresentedinthisbookhaveledtoanumber
ofconjecturesandopenproblems,whicharedescribedthroughoutthebook.
Nagoya,Aichi,Japan FutabaFujie
Kalamazoo,MI,USA PingZhang

Acknowledgements
With pleasure, we thank Gary Chartrand for the advice and information he kindly
supplied to us on many topics.In addition, we thank the reviewer for the valuable
input and suggestions he/she provided to us. Finally, we are grateful to Razia
Amzad, SpringerBriefs editor for her kindness and encouragement in writing this
book.Itisbecauseofallofyouthatanimprovedbookresulted.
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