InternationalJournalofPureandAppliedMathematics - PDF document

InternationalJournalofPureandAppliedMathematics Volume75No.32012,269-278ISSN:1311-8080(printedversion)url:http://www.ijpam.eu
 ijpam.euPETALGRAPHSV.Kolappan1x,R.SelvaKumar21;2VITUniversityVellore,INDIAAbstract:Inthispaperweintroducep-petalgraphs.Weprovethatthenecessaryandsucientconditionforaplanarp-petalgraphGisthatGhasevennumberofpetalseachofsizethree.Wealsocharacterizetheplanarpartialp-petalgraphs.AMSSubjectClassication:05C10KeyWords:petalgraph,p-petalgraph,partialp-petalgraph,planarity1.IntroductionApetalgraphGisasimpleconnected(possiblyinnite)graphwithmaximumdegreethree,minimumdegreetwo,andsuchthatthesetofverticesofde-greethreeinducesa2-regulargraphG4(possiblydisconnected)andthesetofverticesofdegreetwoinducesatotallydisconnectedgraphG.(see[1]).IfG4isdisconnected,theneachofitscomponentsisacycle.Inthispaper,weconsiderpetalgraphswithpetals,P0;P1;:::;Pa1andcomponents,G40;G41;:::;G4r1.ThevertexsetofGisgivenby=1[2,where1=fig;i=0;1;:::;21isthesetofverticesofdegreethree,and2=fjg;j=0;1;:::;a1isthesetofverticesofdegreetwo.ThesubgraphGisthetotallydisconnectedgraphwithvertexset2.ThesubgraphG4isthecycleG4:0;u1;:::;u2a1.ThesetP(G)=P0;P1;:::;Pa1isthepetalset Received:July7,2011c\r2012AcademicPublications,Ltd.url:www.acadpubl.euxCorrespondenceauthor 270V.Kolappan,R.S.Kumar Figure1:AniteandaninnitepetalgraphofG.Considerthepathii+1:::ui+k;k1onthecomponentG4lofG4.Letj22beadjacenttoiandi+k.ThenthepathPj=iji+kiscalledapetalofGinthecomponentG4l.Thepathfromitoi+koflengthpj=minfk;2kgiscalledthebaseofPj.WecallpjthesizeofthepetalPj.ThepetalsizeofGisgivenbyp(G)=maxfpj;j=0;1;:::;a1g.ThevertexjiscalledthecenterofthepetalPj.Theverticesiandi+karecalledthebasepointsofPj.Wereferapetalofnitesizetobeanitepetal.WedenotethenumberofnitepetalsinGasf=r1Pt=0t.ThepetalPj=kjl,wherekandlareindistinctcomponentsofG4iscalledaninnitepetal.Thesizeofaninnitepetalisinnity.WecalltheverticeskandlasthebasepointsofPj.AninnitepetalconnectstwocomponentsofG.ThesetofinnitepetalsconnectingtwocomponentsG4kandG4lisdenotedbyP(G4k[G4l).ThenumberofinnitepetalsinP(G4k[G4l)isdenotedbykl.Thenumberofinnitepetalsinapetalgraphisgivenby1=P0klr1kl.ThenumberofpetalsinapetalgraphGisgivenby=f+1.Figure1presentsaniteandaninnitepetalgraph.WegivebasicdenitionsandprovepreliminarytheoremsinSection2.OurmainresultsaregiveninTheorem10andCorollary11inSection3.2.BasicDenitionsandResultsDenition1.Thesequencefigofvertices0;u1;:::;u2a1ofG4iscalledthevertexsequenceofG.Thesequencefjgofvertices0;v1;:::;va1 PETALGRAPHS271 Figure2:A3-petalgraphofGiscalledthevertexsequenceofthepetalsofG.ThesequencefPjgofpetalsP0;P1;:::;Pa1iscalledthepetalsequenceofG.Notethatthesucesarewrittenintheascendingorder.Denition2.Thedistancel(Pi;Pj)betweentwopetalsPi=kilandPj=k0jl0inG4listhelengthoftheshortestpathbetweentheirbasepointskandk0orlandl0inG4l.LetPi=kilandPj=k0jl0betwopetalsofG.IfPi2P(G4s)andPj2P(G4s[G4t),thenl(Pi;Pj)inG4sisthelengthoftheshortestpathbetweenkandk0inG4s.IfPiandPj2P(G4s[G4t),thenl(Pi;Pj)inG4sisthelengthoftheshortestpathbetweentheirbasepointskandk0inG4s;l(Pi;Pj)inG4tisthelengthoftheshortestpathbetweentheirbasepointsinlandl0inG4t.WheneachbasepointofPiandPjisindistinctcomponentsofG4,thenl(Pi;Pj)=1.Inthispaperweconsiderpetalgraphswhoseinnitepetalsdonotcrossoneanother.Denition3.ApetalgraphGofsizenwithpetalsequencefPjgissaidtobeap-petalgraphdenotedG=Pn;p,ifeverypetalinGisofsizepandl(Pi;Pi+1)=2;i=0;1;2;:::;a1wherethesucesaretakenmodulo.Inap-petalgraphthepetalsizepisalwaysoddbecause,otherwisel(Pi;Pi+1)willnotbe2forsomei.Figure2showsa3-petalgraphwith6petals.Denition4.ApetalgraphGissaidtobeapartialpetalgraphifG4isdisconnected.ThepartialpetalgraphGiscalledapartialp-petalgraphifeverynitepetalinGisofsizepandl(Pi;Pi+1)=2foranypetalPiinanycomponentG4l.Denition5.TwoinnitepetalsPiandPjofP(G4k[G4l)formaninnitepetalpairiftheirbasepointslieonthebasesoftwosuccessivenite 272V.Kolappan,R.S.Kumar Figure3:Non-planarrepresentationofG=P9;3petalsinbothG4kandG4l.Denition6.AcomponentG4kiscalledapendantcomponentifitisconnectedtoonlyonecomponentofGbywayofinnitepetals.Theorem7.IfGisap-petalgraphofordernandsizem,thenn=3andm=4suchthatm3n6,wherea&#x-400;1:Proof.LetGbeap-petalgraphofordernandsizem.Nowj(G)j=impliesthatj(G4)j=2.Hencej(G)j=3orn=3.ThecontributiontotheedgesetofGbyeachpetalis2andbyG4is2.Thus,m=4.Now,3n6=96&#x-400;m,whena&#x-400;1. Theorem8.IfGisap-petalgraph,thenthereareatleastppetalsinG:Proof.Considerap-petalgraphGwithpetals.Therearep1verticesonthebaseofanypetalPj=ijkotherthaniandk.Thisimpliesthattherearep1petalsinGotherthanPjwithoneoftheirbasepointslyinginthebase. 3.Planarityofp-PetalGraphsItcanbeeasilyveriedthatap-petalgraphG=Pn;pisplanarwhenp(G)=1foranyvalueofnaswellas.ThegraphPobtainedfromthePetersengraph PETALGRAPHS273byremovingoneoftheverticesisa3-petalgraphP9;3.ThepetalgraphPisasubdivisionofK3;3andhencenotaplanargraph.Note9.Thesucesofiarealwaystakenmodulo2andthatofjaretakenmodulo.Note10.ForanypetalPj=ijk,j=even(i;j)2.Theorem11.Ap-petalgraphG=Pn;p(p=1)isplanarifandonlyifthefollowingconditionsaresatised.(i)p=3.(ii)isaneveninteger.Proof.Considerthep-petalgraphG=Pn;p,wherepisanoddnumber.LetS:0;u1;u2;:::;u2a1bethevertexsequenceofG.Assumethatatleastoneoftheaboveconditionsisnotsatised.Thefollowingcasesarise:Case1:p=3andisodd.Case1a:=3:LetG=P9;3bethegivenp-petalgraphwiththevertexsequenceS:0,1,2,3,4,5.Partitionthevertexset1(G)into11(G)=f0;u2;u4gand21(G)=f1;u3;u5g.RepresentGusingthefollowingstepssothattheadjacencyoftheverticesofGispreserved:Takethecycle012345containingtheverticesof11(G)[21(G);Subdividetheedges(2i;u2i+2),withtheverticesi,i=0to2respectively.ReferFigure3inpage5.ThisrepresentationofG=P9;3isisomorphictoasubdivisionofK3;3.Case1b:a�3.Partitionthevertexset1(G)intothreesets11(G),21(G)and31(G)suchthat11(G)=f0;u2;u2a2gand21(G)=f1;u3;u2a1gand31(G)hastheremainingvertices.RepresentGusingthefollowingstepssothattheadjacencyoftheverticesofGispreserved:Takethecycle0,1,2,3,2a2,2a1containingtheverticesof11(G)[21(G);Connectthepairsofvertices(0;u3),(1;u2a2),(2;u2a1);Subdi-videtheedges(3;u2a2)withthevertices4,7,:::,i,i+3,i+4,i+7,:::,2a6,2a3andtheedge(2;u2a1)withthevertices5,6,:::,i,i+1,i+4,i+5,:::,2a5,2a4;Connectthepairsofvertices(4;u5);(6;u7);:::;(2a4,2a3);Subdividetheedges(2i;u2i+3),withtheverticesi,i=0to1re-spectively.ReferFigure4inpage6. 274V.Kolappan,R.S.Kumar Figure4:G=Pn;3isnonplanarwhena�3isoddThisrepresentationofG=Pn;3isisomorphictoagraphhavingasubdivi-sionofK3;3asitssubgraph.Case2:p�3andisodd.Case2a:=p.Partitionthevertexset1(G)intothreesets11(G),21(G)and31(G)suchthat11(G)=f0;ua1;ua+1gand21(G)=f1;ua;u2a1gand31(G)hastheremainingvertices.RepresentGusingthefollowingstepssothattheadjacencyoftheverticesofGispreserved:Takethecycle0,1,a1,a,a+1,2a1containingtheverticesof11(G)[21(G);Connectthepairsofvertices(0;ua),(1;ua+1),(a1;u2a1);Sub-dividetheedge(1;ua1)withthevertices2,3,:::;ua2and(a+1;u2a1)withtheverticesa+2,a+3,:::;u2a2;Connectthevertexpairs(2;ua+2),(3;ua+3),:::;(i;ua+i),:::;(a2;u2a2);Subdividetheedges(2i;u2i+a),withtheverticesi,i=0to1respectively.ReferFigure5inpage7.ThisrepresentationofG=Pn;pisisomorphictoagraphhavingasubdivi-sionofK3;3asitssubgraph.Case2b:a�p.Partitionthevertexset1(G)intothreesets11(G),21(G)and31(G)suchthat11(G)=f0,2,2a2gand21(G)=f1,p2,2a1gand31(G)hastheremainingvertices.RepresentGusingthefollowingstepssothattheadjacencyoftheverticesofGispreserved:Takethecycle0,1,2,p2,2a2,2a1containingtheverticesof11(G)[21(G);Connectthepairsofvertices(0,p2),(1,2a2),(2,2a1); PETALGRAPHS275 Figure5:G=Pn;pisnotplanarwhenp�3and=pSubdividetheedge(2;up2)withtheverticesi,i=3top3,theedge(p2,0)withtheverticesp1andp,theedge(2,2a1)withtheverticesp+2,p+3,:::;u2p1;theedge(1,2a2)withthevertices2ap+1,2ap+2,:::;u2a3;Connecttheverticesp,p+2andsubdividetheedgewithp+1,connectthevertices2ap1,2ap+1andsubdivideitwith2ap;Connectthevertices(i1,p+i1)fortheremainingvaluesofi,andSubdividetheedges(2i;u2i+p),withtheverticesi,i=0to1respectively.ReferFigure6inpage8.ThisrepresentationofG=Pn;pisisomorphictoagraphhavingasubdivi-sionofK3;3asitssubgraph.Case3:p�3andiseven.Case3a:=p+1.Partitionthevertexset1(G)intothreesets11(G),21(G)and31(G)suchthat11(G)=f0,2,2a2gand21(G)=f1,p2,2a1gand31(G)hastheremainingvertices.RepresentGusingthefollowingstepssothattheadjacencyoftheverticesofGispreserved:Takethecycle0,1,2,p2,2a2,2a1containingtheverticesof11(G)[21(G);Connectthepairsofvertices(0;up2),(1;u2a2),(2;u2a1);Subdividetheedge(p2;u0)withtheverticesp1,p,theedge(2a1;u2)withtheverticesp+1,p+2,theedge(1;u2a2)withtheverticesp+3,p+4,:::u2a3;Connectthepairsofvertices(p1;u2a3),(p;up+1)andp+2;up+3)
;Connectthevertices(2i,p+2i)fortheremainingvaluesofi(calculationsonsucestakenmodulo2a),andSubdividetheedges(2i;u2i+p),withthevertices 276V.Kolappan,R.S.Kumar Figure6:G=Pn;pisnotplanarwhenp�3anda�pisoddi,i=0to1respectively..ReferFigure7inpage9.ThisrepresentationofG=Pn;pisisomorphictoagraphhavingasubdivi-sionofK3;3asitssubgraph.Case3b:a�p+1.SameasCase2Conversely,letG=Pn;3bea3-petalgraphwithpetalsequencefPig,i=0;1;2;:::;a1,whereiseven.ThecycleG4dividestheplaneintotworegions,theinnerandtheouterregion.Itispossibletodrawthea 2petalsP0,P2,:::,Pa2ofGintheinner(orouter)regionsothattheydonotcrosstheremaininga 2petalsP1,P3,:::,Pa1thatareintheouter(orinner)region.Thisrepresentationofthe3-petalgraphisobviouslyplanar. Corollary12.LetGbeapartialp-petalgraphwithapetals.LetG41;G42;:::;G4rbethecomponentsofGwith1;a2;:::;arnitepetalsrespectively.LetkldenotethenumberofinnitepetalsinP(G4k[G4l)forarbitrarykandl.Ifthefollowingconditionsaresatised,thenthegraphGisplanar.(i)p=3andiseven;(ii)ThenumberofnitepetalsPi1;Pi+1;:::;Pj1inG4kwherePi2P(G4k[G4l)andPj2P(G4k[G4q),possiblyG4l=G4q,iseitherzeroorodd.Proof.LetGbeapartialp-petalgraphasgiven.FromTheorem3anyp-petalgraphisplanarifandonlyifp=3andisaneveninteger.Henceitissuf-cienttoprovethatGisplanarifcondition(iii)issatised.Letusassumethat PETALGRAPHS277 Figure7:G=Pn;pisnotplanarwhenp�3andisoddsuchthat=p+1condition(iii)holdstrue.Considerthe1knitepetalsPi1;Pi+1;Pi+2;:::;Pj1andtheinnitepetalsPi;Pj(ij)suchthatPi2P(G4k[G4l)andPj2P(G4k[G4q).Fromcondition(iii),if1k&#x-399;&#x.986;0,then1kisodd.Now,drawthela1k 2mnitepetalsPi1;Pi+2;:::;Pj1intheinnerregionofG4kandtheremainingja1k 2kpetalsintheouterregionofG4k.ThisrepresentationofG4kisobviouslyplanar.SinceG4kisanarbitrarycomponentofG,weconcludethatGisplanar.ReferFigure8inpage10. Theconversepartisnottrue.Letthepartialp-petalgraphGbeplanar.Therefore,eachcomponentG4k;k=0;1;2;:::;r1isalsoplanar.Fromgivenconditions,p=3andeachiiseven.AssumethatG4kispendant,connectedtoG4l.LetPi=si0sandPj=tj0tbetheinnitepetalpairbetweenG4kandG4lwheresandtareinG4k.LetG04kandG04lbethecomponentsobtainedbyidentifyingiandjtogetanewvertexij.Thepathssijtand0sij0tcanactasnitepetalsinG04kandG04lrespectively.Clearly,eachofthesecomponentsisplanar.Hence,thenumberofnitepetalsotherthansijtinG04kisodd.Similarly,thenumberofnitepetalsotherthan0sij0tinG04lisalsoodd.ThisideaisapplicabletoanypendantcomponentconnectedtoG4k.WhenG4kisconnectedtoG4landthependantcomponentG4q,evenifcondition(iii)isnotsatised,planaritycanbepreservedbydrawingtheplanegraphofG4qintheinnerregionofG4k. 278V.Kolappan,R.S.Kumar Figure8:Apartialp-petalgraphthatsatisescondition(iii)4.ConclusionThepetalgraphisaveryinterestingclassofgraphswhosepropertiesandcharacteristicsareyettobeexploredandmanyconjecturesingraphtheorycanbesolvedwithreferencetopetalgraphs.Further,theauthorswishtoacknowledgethecontributionofDr.Neelainthepreparationofthenaldraftofthispaper.References[1]DavidCariolaro,andGianfrancoCariolaro,Coloringthepetalsofagraph,TheElectronicJ.Combin.,R6(2003),1-11.