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INTRODUCTION TO GRAPH THEORY With Solutions
to Selected Problems
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Notation
N = {1, 2, 3, ···}
|S| = thenumberofelementsinthefiniteset S n r = thenumberof r-elementsubsetsofan n-elementset= n! r!(n r)!
B \ A = {x ∈ B|x/ ∈ A},where A and B aresets
i∈I
Si = {x|x ∈ Si forsome i ∈ I},where Si isasetforeach i ∈ I
Inwhatfollows, G and H aremultigraphs,and D isa digraph.
V (G): thevertexsetof G
E(G): theedgesetof G
v(G): thenumberofverticesin G ortheorderof G
e(G): thenumberofedgesin G orthesizeof G
V (D): thevertexsetof D
E(D): thearcsetof D
v(D): thenumberofverticesin D ortheorderof D
e(D): thenumberofarcsin D
x → y : x isadjacentto y,where x,y areverticesin D
x ̸→ y : x isnotadjacentto y,where x,y areverticesin D
G ∼ = H : G isisomorphicto H
A(G): theadjacencymatrixof G
G : thecomplementof G
[A]: thesubgraphof G inducedby A,where A ⊆ V (G)
e(A,B): thenumberofedgesin G havinganendin A andthe otherin B,where A,B ⊆ V (G)
G v : thesubgraphof G obtainedbyremoving v andalledges incidentwith v from G,where v ∈ V (G)
x IntroductiontoGraphTheory
G e : thesubgraphof G obtainedbyremoving e from G,where e ∈ E(G)
G F : thesubgraphof G obtainedbyremovingalledgesin F from G,where F ⊆ E(G)
G A : thesubgraphof G obtainedbyremovingeachvertexin A togetherwiththeedgesincidentwithverticesin A from G,where A ⊆ V (G)
G + xy : thegraphobtainedbyaddinganewedge xy to G,where x,y ∈ V (G)and xy/ ∈ E(G)
N (u)= NG(u): thesetofvertices v suchthat uv ∈ E(G)
N (S)= u∈S N (u),where S ⊆ V (G)
d(v)= dG(v): thedegreeof v in G,where v ∈ V (G)
id(v): theindegreeof v in D,where v ∈ V (D)
od(v): theoutdegreeof v in D,where v ∈ V (D)
d(u,v): thedistancebetween u and v in G,where u,v ∈ V (G)
c(G): thenumberofcomponentsin G
δ(G): theminimumdegreeof G
∆(G): themaximumdegreeof G
χ(G): thechromaticnumberof G
α(G): theindependencenumberof G
G + H : thejoinof G and H
G ∪ H : thedisjointunionof G and H
kG : thedisjointunionof k copiesof G
G(D): theunderlyinggraphof D
nG(H): thenumberofsubgraphsin G whichareisomorphicto H
Cn : thecycleoforder n
Kn : thecompletegraphoforder n
Nn : thenullgraphoremptygraphoforder n
Pn : thepathoforder n
Wn : thewheeloforder n, Wn = Cn 1 + K1
K(p,q): thecompletebipartitegraphwithabipartition(X,Y ) suchthat |X| = p and |Y | = q
3.BipartiteGraphsandTrees71
4.Vertex-colouringsofGraphs97
7.DigraphsandTournaments193
8.Solutionsofselectedquestions225
FundamentalConceptsandBasic Results
1.1TheK¨onigsbergbridgeproblem
InanoldcityofEasternPrussia,named K¨onigsberg,therewasariver, calledRiverPregel,flowingthroughitscentre.Inthe18th century,there weresevenbridgesovertheriverconnectingthetwoislands(B and D)and twooppositebanks(A and C)asshowninFigure1.1. A
Itwassaidthatthepeopleinthecityhadalwaysamusedthemselves withthefollowingproblem:
Startingwithanyoneofthefourplaces A,B,C or D asshowninFigure 1.1,isitpossibletohaveawalkwhichpassesthrougheachoftheseven bridgesonceandonlyonce,andreturntowhereyoustarted?
Noonecouldfindsuchawalk;andafteranumberoftries,peoplebelieved thatitwassimplynotpossible,butnoonecouldproveiteither.
LeonhardEuler,thegreatestmathematicianthatSwitzerlandhasever
Figure1.1
IntroductiontoGraphTheory
produced,wastoldoftheproblem.Henoticedthattheproblemwasvery muchdifferentinnaturefromtheproblemsintraditionalgeometry,and insteadofconsideringtheoriginalproblem,hestudieditsmuchmoregeneralversionwhichencompassedanynumberofislandsorbanks,andany numberofbridgesconnectingthem.Hisfindingwascontainedinthearticle[E](theEnglishtranslationofitstitleis: Thesolutionofaproblem tothegeometryofposition)publishedin1736.Asadirectconsequence ofhisfinding,hededucedtheimpossibilityofhavingsuchawalkinthe K¨onigsbergbridgeproblem.Thiswashistoricallythefirsttimeaproofwas givenfromthemathematicalpointofview.
HowdidEulergeneralizetheK¨onigsbergbridgeproblem?Howdidhe solvehismoregeneralproblem?Whatwashisfinding?
1.2Multigraphsandgraphs
EulerobservedthattheK¨onigsbergbridgeproblemhadnothingtodo withtraditionalgeometrywherethemeasurementsoflengthsandangles, andrelativelocationsofverticescount.Howlargetheislandsandbanks are,howlongthebridgesare,andwhetheranislandisatthesouthornorth ofabankareimmaterial.Thekeyingredientsarewhethertheislandsor banksareconnectedbyabridge,andbyhowmanybridges.
Euler’sideawasessentiallyasfollows:representtheislandsorbanks by‘dots’,oneforeachislandorbank,andtwodotsarejoinedby k ‘lines’ (notnecessarilystraight),where k ≥ 0,whenandonlywhentherespectiveislandsorbanksrepresentedbythedotsareconnectedby k bridges. ThusthesituationfortheK¨onigsbergbridgeproblemisrepresentedbythe diagraminFigure1.2.
Figure1.2
ThediagraminFigure1.2isnowknownasa multigraph. Intuitively, a multigraph isadiagramconsistingof‘dots’and‘lines’,whereeachline joinssomepairofdots,andtwodotsmaybejoinedbynolinesorany numberoflines.Moreformally,wecalla‘dot’a vertex (plural,vertices) andcalla‘line’an edge.
Forinstance,inthemultigraphofFigure1.2,therearefourverticesand sevenedges,whereeachedgejoinssomepairofvertices;vertices A and C arenotjoinedbyanyedges, A and D arejoinedbyoneedge,and B and C arejoinedbytwoedges,etc.
Notethatthesizesandtherelativelocationsofdots(vertices),and thelengthsofthelines(edges)areimmaterial.Onlythe‘linkingrelations’ amongtheverticesandthenumberofedgesthatjointwoverticescount. Thus,thesituationfortheK¨onigsbergbridgeproblemcanequallywellbe representedbythemultigraphofFigure1.3.
B D
Letusgivemoreexamplesofmultigraphwhichrepresentcertainsituationsindifferentnature.
Example1.2.1. Thereweresixpeople: A,B,C,D,E and F inaparty andseveralhandshakesamongthemtookplace.Supposethat
A shookhandswith B,C,D,E and F , B,inaddition,shookhandswith C and F , C,inaddition,shookhandswith D and E, D,inaddition,shookhandwith E, E,inaddition,shookhandwith F .
ThissituationcanbeclearlyshownbythemultigraphinFigure1.4,where peoplearerepresentedbyverticesandtwoverticesarejoinedbyanedge wheneverthecorrespondingpersonsshookhands.
Figure1.3
Example1.2.2. ThediagraminFigure1.5isamultigraphwhichshows theavailabilityofflightsoperatedbyanairlinecompanybetweenanumber ofcities.Theverticesrepresentthecities,andtwoverticesarejoinedby anedgeifthereisaflightavailablebetweenthetwocorrespondingcities.
Example1.2.3. ThediagraminFigure1.6isamultigraphwhichmodels ajob-applicationsituation.Theverticesaredividedintotwoparts: X and Y ,wheretheverticesin X representtheapplicants,whilethosein Y representthejobsavailable.Avertexin X isjoinedtoavertexin Y byan edgeifthecorrespondingapplicantappliesforthecorrespondingjob.
Figure1.4
Figure1.5
Question1.2.1. Givethreeexamplesfromoureverydaylifewherethe situationscanbemodeledbymultigraphs.
ItisnotedthatinthethreemultigraphsshowninFigures1.4to1.6, everytwoverticesarejoinedbyatmostoneedge(thatis,eithernoedges orexactlyoneedge).Thesesituationsaredifferentfromthemultigraph inFigure1.2(orFigure1.3)wherethereareverticesjoinedbymorethan oneedge.Todistinguishthem,wecallthediagramsinFigures1.4to1.6 simplegraphs,orsimply, graphs.ThusthediagraminFigure1.2(or Figure1.3)isamultigraph,butnota(simple)graph.
Letusconsideranotherexample.
Example1.2.4. InthediagramshowninFigure1.7,thereare
• fourvertices: u,v,w and z,and
• eightedges: f1 and f2 joining u and v; e1,e2 and e3 joining w and z; h1 joining v and w; h2 joining u and w; h3 joining v toitself.
Figure1.6
Figure1.7
IntroductiontoGraphTheory
Twoormoreedgesjoiningthesamepairofverticesarecalled parallel edges.Thus,inFigure1.7, f1 and f2 areparalleledges; e1,e2 and e3 are paralleledges.
Anyedgejoiningavertextoitselfiscalleda loop.Thus,inFigure1.7, h3 isaloop.
Remarks.(1)Inthisbook,weshallnotconsider‘loops’inanydiagramof verticesandedgesunlessotherwisestated.Adiagramwiththeexistenceof paralleledgesis not a (simple)graph.Anotherexampleofamultigraph whichis not a(simple)graphisshowninFigure1.8.
(2)Bearinmindthata‘graph’ora‘multigraph’inGraphTheoryis notageometricalfigure.Thuswedonotconsider
• thesizeofa‘dot’,
• thelocationofavertex,and
• theshapeofanedge.
(3)Whenthereisonlyoneedgejoiningapairofvertices,say a and b,we maydenotethisedgeby ab.Forexample,theedge h1 inFigure1.7can alsobedenotedby vw
Wenowgiveformaldefinitionsof‘graph’and‘multigraph’.
Figure1.8
A multigraph G consistsofanon-emptyfiniteset V (G)ofvertices togetherwithafiniteset E(G)(possiblyempty)ofedgessuchthat
(1) eachedgejoinstwodistinctverticesin V (G)and (2) anytwodistinctverticesin V (G)arejoinedbyafinitenumber (includingzero)ofedges.
Thesets V (G)and E(G)arecalledthe vertexset andthe edgeset of G respectively.
Thenumberofverticesin G,denotedby v(G),iscalledthe order of G (thus v(G)= |V (G)|).Thenumberofedgesin G,denotedby e(G),is calledthe size of G (thus e(G)= |E(G)|).
Amultigraph G iscalleda(simple) graph ifanytwoverticesin V (G) arejoinedbyatmostoneedge(thatis,eithertheyarenotjoinedby anedgeorjoinedbyexactlyoneedge).
(a)Itfollowsfromtheabovedefinitionsthat (i) everygraphisamultigraphbutnotviceversaand (ii) noloopsareallowedinanymultigraph.
Whenaconceptisdefinedorastatementismadeformultigraphs,theyarealsovalid,inparticular,forgraphs.
(b)If e istheonlyedgejoiningtwovertices u and v,thenwemaywrite e = uv or e = vu.Theorderingof u and v intheexpressionisimmaterial.
Example1.2.5. Let G bethemultigraphshowninFigure1.8.Then
V (G)= {x,y,z,p,q},
E(G)= {xy,xz,yz,yp,e1,e2,f1,f2,pq},
v(G)=5 and e(G)=9
Let H bethegraphshowninFigure1.4.Then
V (H)= {A,B,C,D,E,F },
E(H)= {AB,AC,AD,AE,AF,BC,BF,CD,CE,DE,EF },
v(H)=6 and e(H)=11
Question1.2.2. Let G bethemultigraphshownbelow.Find V (G),E(G),v(G) and e(G).
Question1.2.3. Let H bethegraphwith V (H)= {a,b,c,x,y,z} and E(H)= {ab,ay,bx,by,cx,cz,xz,yz}.Find v(H) and e(H),anddrawa diagramof H
Matricesandmultigraphs
Asdiscussedearlier,amultigraph G canberepresentedbyadiagram consistingof‘dots’and‘lines’,andcanbedefinedintermsofitsvertexset V (G)andedgeset E(G).Multigraphscanalsoberepresentedbymatrices invariousways.Inwhatfollows,weintroduceoneofthem.
Example1.2.6. Let G bethemultigraphshownbelow,whereitsfour verticesarenamedas v1,v2,v3 and v4. v1 v v v2 3 4
Consideralsothefollowing 4 × 4 matrix A: A =
0201 2010 0103 1030
Canyoufindanyrelationbetween G and A?
Whatisthevalueofthe (1, 2)-entryin A?Itis‘2’.Howmanyedges in G join v1 and v2?Thereare‘2’also.
Howmanyedgesin G join v3 and v4?Thereare‘3’.Whatisthevalue ofthe (3, 4)-entryin A?Itis‘3’also.
Indeed,itisobservedthatthevalueofthe(i,j)-entryin A isthenumber ofedgesin G joining vi and vj ,where i,j ∈{1, 2, 3, 4}.Notethatthevalue ofeach(i,i)-entry(thatis,adiagonalentry)in A is‘0’asthereisnoedge in G joining vi toitself.Wecall A the adjacencymatrix of G.Two verticesareadjacentiftheyarejoinedbyanedge.Evidently,thematrix isdependentonthelabellingofthevertices.
Let G beamultigraphoforder n with V (G)= {v1,v2, ,vn}.The adjacencymatrix of G isthe n × n matrix
A(G)=(ai,j )n×n,
where ai,j ,the(i,j)-entryin A(G),isthenumberofedgesjoining vi and vj forall i,j ∈{1, 2, ··· ,n}.
Question1.2.4. Let G bethemultigraphshownbelow. v1
(i) Find A(G)
(ii) Is A(G) symmetric(i.e., (i,j)-entry= (j,i)-entry)?
(iii) Whatisthesumofthevaluesoftheentriesineachrow(respectively, column)?
(iv) Whatisyourinterpretationofthe‘sum’obtainedin(iii)?
Question1.2.5. Theadjacencymatrixofamultigraph G isgivenbelow:
02101 20100 11032 00300 10200
Drawadiagramof G. Remark.Therearemanywaysofstoringmultigraphsincomputers.The useoftheadjacencymatricesis,perhaps,oneofthemostcommonand convenientways.
Exercise1.2
(1) Let G bethemultigraphrepresentingthefollowingdiagram.Determine V (G), E(G), v(G)and e(G).Is G asimplegraph?
(2) Drawthegraph G modelingtheflightconnectivitybetweentwelvecapitalcitieswiththefollowingvertexset V (G)andedgeset E(G).
V (G)= {Asuncion,Beijing,Canberra,Dili,Havana,KualaLumpur, London,Nairobi,PhnomPenh,Singapore,Wellington, Zagreb}
E(G)= {Asuncion-London,Asuncion-Havana,Beijing-Canberra, Beijing-KualaLumpur,Beijing-London,Beijing-Singapore, Beijing-PhnomPenh,Dili-KualaLumpur,Dili-Singapore, Dili-Canberra,Havana-London,London-Wellington, KualaLumpur-London,KualaLumpur-PhnomPenh, KualaLumpur-Singapore,KualaLumpur-Wellington, London-Nairobi,PhnomPenh-Singapore,London-Singapore, London-Zagreb,Singapore-Wellington,Havana-Nairobi} (NotethatyoumayuseAtorepresentAsuncion,BtorepresentBeijing, CtorepresentCanberra,etc.)
(3) Defineagraph G suchthat V (G)= {2, 3, 4, 5, 11, 12, 13, 14} andtwo vertices s and t areadjacentifandonlyif gcd{s,t} =1.Drawa diagramof G andfinditssize e(G).
(4) Thediagraminpage12isamapoftheroadsysteminatown.Draw amultigraphtomodeltheroadsystem,usingavertextorepresenta junctionandanedgetorepresentaroadjoiningtwojunctions.
DiagramforProblem4
(5) Let G beagraphwith V (G)= {1, 2, , 10},suchthattwonumbers i and j in V (G)areadjacentifandonlyif |i j|≤ 3.Drawthegraph G anddetermine e(G).
(6) Let G beagraphwith V (G)= {1, 2, ··· , 10},suchthattwonumbers i and j in V (G)areadjacentifandonlyif i + j isamultipleof4.Draw thegraph G anddetermine e(G).
(7) Let G beagraphwith V (G)= {1, 2, ··· , 10},suchthattwonumbers i and j in V (G)areadjacentifandonlyif i × j isamultipleof10.Draw thegraph G anddetermine e(G).
(8) Findtheadjacencymatrixofthefollowinggraph G
(9) Theadjacencymatrixofamultigraph G isshownbelow:
Drawadiagramof G.
(10) Fourteamsofthreespecialistsoldierseach(ascout,asignalerand asniper)aretobesentintoenemyterritory.However,someofthe soldierscannotworkwellwithsomeothers.Thefollowingtableshows thesoldiers,theirspecializationsandwhotheycannotworkwith.
Soldier Specialization Cannotcooperatewith
(i) Drawamultigraphtomodelthesituationsothatwemayseehow toform3-manteamssuchthateachspecializationisrepresented andeverymemberoftheteamcanworkwitheveryother.State clearlywhattheverticesrepresentandunderwhatcondition(s)two verticesarejoinedbyanedge.
(ii) Canyouformfour3-manteamssuchthateachspecializationis representedandallmembersoftheteamcanworkwithoneanother?
1.3Vertexdegrees
Let G beamultigraph.
Twovertices u and v in G aresaidtobe adjacent iftheyarejoined byanedge,say, e in G.Inthecasewhen e istheonlyedgejoining u and v,wealsowrite e = uv,andwesaythat (1) u isa neighbour of v andviceversa, (2)theedge e is incidentwith thevertex u (and v)and (3) u and v arethetwo ends of e.
Thesetofallneighboursof v in G isdenotedby N (v);thatis, N (v)= {x|x isaneighbourof v}.
Example1.3.1. Let G bethemultigraphshowninFigure1.9. a b c w u v
Then
(1)thevertices a and b areadjacent,soare b and v,butnot a and c; (2)thevertices a and u arethetwoendsoftheedge au; (3)theedge av isincidentwiththevertices a and v; (4)thevertex a hasthreeneighbours,namely, b, u and v;and (5) N (a)= {b,u,v}, N (b)= {a,v,c}, N (w)= {c},etc.
Let G beamultigraph.Wenowintroduceaveryusefulandimportant numberassociatedwitheachvertexin G
Figure1.9
Givenavertex v in G,the degree of v in G,denotedby dG(v),is definedasthenumberofedgesincidentwith v.
Forsimplicity,weshallreplace dG(v)simplyby d(v)ifthereisnodanger ofconfusion.
Question1.3.1.
(i) Findthedegreeofeachvertexin G ofFigure1.9.
(ii) Find N (x) foreachvertex x in G ofFigure1.9.
(iii) Bydefinition,isittruethat d(v)= |N (v)|?
Example1.3.2. Let G bethemultigraphofFigure1.10.
Observethattherearesevenedgesincidentwiththevertex x.Thus, d(x)=7.Therearenoedgesincidentwiththevertex j.Thus, d(j)=0. Thedegreesoftheverticesin G areshowninTable1.11.
Vertex abcjpwxyz Degree 123004743
Table1.11
Figure1.10
