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WorkshopKKKombinatorika:Pewarnaan dan Pelabelan pada GrafJurusan Matematika Universitas Andalas

Padang,25April2019

SLAMIN

UniversitasJember,Indonesia

DISTANCE IRREGULAR VERTEX LABELING OF GRAPH

A Introduction

BDistanceirregularlabeling

C Inclusivedistanceirregular labeling

DOpenProblems

Outline

A

BCD

NotabletowalkbacktoAthrougheachbridgeexactlyonce

Impossiblebecausethereareregionsconnectionbyoddnumbersofbridge

HistoryofGraphTheoryINTRODUCTION

Graphrepresentationof7bridgesinKonigsberg

IfapseudographGhasaEuleriancircuit,thenthegraphisconnectedandthedegree

ofeveryvertexiseven

B

AC

D

B

AC

D

HistoryofGraphTheoryINTRODUCTION

SincethenmanyotherconceptsinGraphTheoryhavesprunguptosolveproblemsindailylife.

DevelopmentofGraphTheory

Graphlabelingisgrowingveryfastwithseveraltypeandbroadrangeofapplications

GraphLabelings

HistoryofGraphTheoryINTRODUCTION

MotivationsIntroduction

• G.Chartrand,M.S.Jacobson,J.Lehel,O.R.Oellermann,S.Ruiz,F.Saba(1988)

• M.Bačá,S.Jendrol’,M.Miller,andJ.Ryan(2007)

Irregularassignmentofgraph

• M. Miller, C. Rodger,R. Simanjuntak(2003)

Distancemagiclabelingofgraphs

• S.Arumugam, N.Kamatchi (2012)

(a,d)-distanceantimagic graphs

DistanceIrregularVertexLabelingIntroduction

DefinitionAdistanceirregularvertexlabelingofthegraphGwith vverticesisanassignmentλ:V→{1,2,…,k} sothattheweightscalculatedatverticesaredistinct.

WeightTheweight ofavertexxin Gisdefinedasthesumofthelabelsofalltheverticesadjacentto x(distance1from x),thatis,

dis(G)Thedistanceirregularitystrength ofG,denotedbydis(G),istheminimumvalueofthelargestlabelk overallsuchirregularassignments.

(Slamin,2017)

DistanceIrregularVertexLabelingIntroduction

DistanceIrregularVertexLabelingIntroduction

BasicConceptdis(G)

ExampleAdistanceirregularvertexlabeling ofthegraphGwith 5 verticeswithdis(G)=2.

2 2 2

11

243

7 5

NecessaryConditiondis(G)

Letu andw beanytwodistinctverticesinaconnectedgraphG.Ifu andw haveidenticalneighbors,i.e.,N(u)=N(w),thenG hasnodistanceirregularvertexlabeling.

Observation1

v1 v2 vn

wu

…

NonDistanceIrregularLabelingdis(G)

• CompletebipartitegraphsKm,n foranym,n ≥2.• CompletemultipartitegraphsHm,n foranym,n ≥2.• Starsonn+1verticesSnforn ≥2.• TreesTn foranyn≥3,thatcontainavertexwithatleasttwoleaves.

Graphsthathavenodistanceirregularlabeling:

OtherNecessaryConditiondis(G)

Letu andw betwoadjacentverticesinaconnectedgraphG.IfN(u)– {w}=N(w)– {u},thenthelabelsofuandwmustbedistinct,thatis, λ(u) ≠λ(w).

Observation2

v1 v2 vn

wu

…

Lowerbounddis(G)

LetG beaconnectedgraphonv verticeswithminimumdegreeδandmaximumdegreeΔandthereisnovertexhavingidenticalneighbors.Then

Lemma

Proof

• Thesmallestweightisδ.• Sincetheweightmustbedistinct,thelargestweightisatleastv+δ– 1.• ThisweightisobtainedfromthesumofatmostΔintegers.• Thusthelargestlabelthatcontributestothisweightmustbeatleast

(Slamin,2017)

Let Knbeacompletegraphwith n≥3 vertices.Thendis(Kn)=n.

CompleteGraphsdis(G)

Theorem

(Slamin,2017)

n(n-1)/2- 1

n(n-1)/2- 2

n(n-1)/2- 3n(n-1)/2- 4

n(n-1)/2- n 2n

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1

Pathsdis(G)

Let Pn beapathwithn≥4 vertices.Then

Theorem

(Slamin,2017)

Cyclesdis(G)

Let Cn beacyclewithn≥ 5 vertices.Then

if n=0,1,2,5mod8.

Theorem

(Slamin,2017)

(N.H.Bong,Y.Lin,Slamin,2017+)

Wheelsdis(G)

LetWn beawheelwithn≥ 5 rimvertices.Then

if n=0,1,2,5mod8.

Theorem

(Slamin,2017)

(N.H.Bong,Y.Lin,Slamin,2017+)

• Everyvertexontherimmusthavedifferentlabel,otherwisetherewillbethesameweightamongthevertices• Thusdis(fn)≥2n.• Labelthe2n verticesoffn asfollows: Thisgivestheweightsforvertices:

• Thelabelingimpliesthatdis(fn)≤2n• Thereforedis(fn)=2n

FriendshipGraphsdis(G)

Let fnbeafriendshipgraphwith n≥3 triangles.Thendis(fn)=2n.

Theorem

Proof

(Slamin,T.Windartini,K.D.Purnomo,2017+)

FriendshipGraphsdis(G)

OtherWheelRelatedGraphsdis(G)

Let Hn beahelmwithn≥3rimvertices.Thendis(Hn)=n

Helm

Let Fln beaflowerwithn≥ 3 rimvertices.Thendis(Fln)=n

Flower

Let Fn beafanwithn≥ 4 rim vertices.Thendis(Fn)= n/2

Fan

(Slamin,T.Windartini,K.D.Purnomo,2017+)

InclusiveDistanceIrregularVertexLabelingdis(G)

DefinitionAninclusive distanceirregularvertexlabelingofthegraphGwith vverticesisanassignmentf :V→{1,2,…,k} sothattheweightscalculatedatverticesaredistinct.

WeightTheweight ofavertexxin Gisdefinedasthesumofthelabelofxandthelabelsofalltheverticesadjacentto x(distance1from x),thatis,

dis(G)

Theinclusive distanceirregularitystrength ofG,denotedbydis(G),istheminimumvalueofthelargestlabelk overallsuchirregularassignments.

(N.H.Bong,Y.Lin,Slamin,2017+)

BasicConceptdis(G)

ExampleInclusivedistanceirregularvertexlabeling ofthegraphGwith 5verticeswithdis(G)=2.

2 2 1

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475

8 6

DoubleStarsdis(G)

Let S(m,n)beadoublestaronm+n+2 vertices.Then

dis

Theorem

(N.H.Bong,Y.Lin,Slamin,2017+)

Starsdis(G)

Let Sn beastar onn+1 verticesandn>1.Thendis(Sn)=n

Theorem

(N.H.Bong,Y.Lin,Slamin,2017+)

2

3

45

n+1 2n

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1

1

Pathsdis(G)

Let Pn beapathonnvertices.Then

Theorem

(M.Baca,A.Semanicova-Fenovcıkova,Slamin,K.A.Sugeng,2017+)

Cyclesdis(G)

Let Cn beacycleonnvertices.Then

Theorem

(M.Baca,A.Semanicova-Fenovcıkova,Slamin,K.A.Sugeng,2017+)

Wheelsdis(G)

LetWn beawheelonnrimvertices.Then

Theorem

(M.Baca,A.Semanicova-Fenovcıkova,Slamin,K.A.Sugeng,2017+)

OpenProblems

Determinethe(inclusive)distanceirregularitystrengthsofsomeparticularfamiliesofgraphs.

OpenProblem1

Expandthe(inclusive)distanceirregularlabelingofgraphstothedistanceatleast2.

OpenProblem2

Characterizetherelationshipbetweeninclusivedistanceirregularlabelingand(a,d)-distanceantimagic labeling.

OpenProblem3

References[1]S.ArumugamandN.Kamatchi.On(a,d)-distanceantimagicgraphs,

AustralasianJournalofCombinatorics54(2012),279-287.[2]M.Baca,S.Jendrol,M.MillerandJ.Ryan,Onirregulartotallabellings,

DiscreteMath.307(2007),1378-1388.[3]M.Baca,A.Semanicova-Fenovcıkova,Slamin,K.A.Sugeng,On

inclusivedistancevertexirregularlabelings,submittedtoEJGTA.[4]N.H.Bong,Y.LinandSlamin,AustralasianJournalofCombinatorics69

(3)(2017),315–322[5]N.H.Bong,Y.LinandSlamin,Oninclusiveandnon-inclusivevertex

irregulard-distancevertexlabelings,submittedtoJCMCC.[6]G.Chartrand,M.S.Jacobson,J.Lehel,O.R.Oellermann,S.RuizandF.

Saba,Irregularnetworks,Congr.Numer.64(1988)187-192.[7]M.Miller,C.RodgerandR.Simanjuntak,Distancemagiclabelingsof

graphs,AustralasianJournalofCombinatorics28.305-315.[8]Slamin,Ondistanceirregularlabelingsofgraphs,FarEastJournalof

MathematicalSciences(FJMS)102(5)(2017)919-932.

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THANK YOU