Odd Graceful Labeling of Some New Type of Graphs

Transcription

1 Internatonal Journal o Mathematcs Trends and Technology IJMTT) Volume 41 Number 1- January 2017 Odd Graceul Labelng o Some New Type o Graphs Sushant Kumar Rout 1 Debdas Mshra 2 and Purna chandra Nayak 3 1 Department o Mathematcs College o Engneerng and Technologes BhubaneswarInda Department o Mathematcs C. V. Raman College o Engneerng BhubaneswarInda Department o Mathematcs Bhadrak Autonomous) College BhadrakInda Abstract In ths paper some new type o odd graceul graphs are nvestgated. We prove that the graph obtaned by onng a cycle C 8 wth some star graphs S 1r keepng one vertex and three vertces gap between par o vertces o the cycle admts odd graceul labellng. We also prove that the graph obtaned by onng a cycle C 12 wth some star graphs S 1r keepng two three and ve vertces gap between par o vertces o the cycle admts odd graceul labellng. We observed that C 8 ʘ S 4 1r wth one vertex gap C 8 ʘ S 2 1r three vertces gap between par o vertces o the cycle C 8 are odd graceul C 12 ʘ S 6 1r wth one vertex gap C 12 ʘ S 4 1r two vertces gap C 12 ʘ S 3 1r wth three vertces gap and C 12 ʘ S 2 1r wth ve vertces gap between par o vertces are odd graceul graphs. Keywords: Graceul labelng graceul graphs odd graceul graphs star graphs. AMS Subect Classcaton 2010):05C78 1. Introducton All graphs n ths paper are nte smple and undrected. Let G = V E) denotes a graph wth n number o vertces and q number o edges. Here the termnology and notaton we use are ollowed by Harary[1]. The symbols V and E wll denote the vertex set and edge set o a graph G respectvely. The cardnalty o the vertex set s called the order o G denoted by p. The cardnalty o the edge set s called the sze o G denoted by q. A graph wth p vertces and q edges s called a p q) graph. In 1985 Lo[2] ntroduced the noton o edge-graceul graphs. Sn-Mn Lee Kuo-Jye Chen and Yung- Chn Wang[3] ntroduced the k-edge-graceul graphs. Here we ntroduce odd graceul graphs. We provde some dentons and other normaton whch are requred or the present nvestgatons. Denton 1.1 I all the vertces o a graph are assgned some values subect to certan condtons then t s known as graph labellng Denton 1.2 A smple graph G m n) wth m vertces and n edges s graceul there s a labelng l o ts vertces wth dstnct ntegers rom the set { n} So that the nduced edge labelng e dened by e uv) l u) l v)) assgns each edge a derent label. A graph whch admts graceul labelng s called a graceul graph. Denton 1.3 A graph G V E) wth q edges s sad to admt odd graceul labelng : V { } such that the nduced uncton *: E {13-1} dened as *ab) = a) b) s bectve. A graph whch admts odd graceul labelng s called an odd graceul graph. Denton 1.4 A Star graph s a tree consstng o one vertex adacent to all others. We denote here a star graph as S 1r where r number o pendant vertces are connected to one vertex. The study o graph labelng and graceul graphs was ntroduced by Rosa[4].Gnanaoth R.B1991)[5 ] ntroduced the concept o odd graceul graphs and she has proved many results. Kathresan K.M.2008[6] has dscussed odd graceulness o ladders and graphs obtaned romthem by subdvdng each step exactly once. Vadya.S.K. e-tal 2010[7 ]proved the odd graceulness o onng o even cycle wth path and cycle sharng a common edge. Vadya S.K. e-tal 2013[ 8] proved the odd graceulness o splttng graph and the shadow graph o bster. Barrentos Chrstan 2009[ 9] dscussed the odd graceulness o Trees o Dameter 5.For detaled survey on graph labelng and related results we reer to Gallan J.A.2015[10 ] Man results: Theorem-1: Graph obtaned by onng the cycle C 8 wth C 8 ʘ S 4 11 wth one vertex gap our star ISSN: Page 9

2 Internatonal Journal o Mathematcs Trends and Technology IJMTT) Volume 41 Number 1- January 2017 graphs S 1r keepng one vertex gap between par o vertces o the cycle s odd graceul. C 8 ʘ S 4 12 wth one vertex gap s odd graceul Proo: Let a1 a2... be the vertces o the cycle C 8 and a are pendant vertces o the star graph S 1r adacent to a or =12...r. We dene the vertex labellng as : V { where q s the total Case-1: I s odd we dene x) { 1 x a } Let r-pendant vertces o the star graph S 1r are connected to each a keepng one vertex gap between par o vertces o the cycle C 8. Case-2: I s evenwe dene ) r 1) 1 8 and 8 ) 1 ) except ) 1) r 1) 1) 2 1) r 7 r or 7 ) 3. The vertex uncton dened nduces a bectve edge uncton *: { } Thus s an edge graceul labelng o G = C 8 ʘ S 4 1r. Hence C 8 ʘ S 4 1r s an odd graceul graph. Example: C 8 ʘ S 4 1r wth one vertex gap C 12 ʘ S 4 16 wth one vertex gap s odd graceul Theorem-2: Graph obtaned by onng the cycle C 8 wth three star graphs S 1r keepng three vertces gap between par o vertces o the cycle s odd graceul. Proo: Let a1 a2... be the vertces o the cycle C 8. a are pendant vertces o the star graph S 1r adacent to a or =12...r. We dene the vertex labelng as C 8 ʘ S 4 11 wth one vertex gap : V { where q s the total Case-1: I s odd We dene x) { 1 x a } Let r-pendant vertces o the star graph S 1r are connected to each a keepng three vertces gap between par o vertces o the cycle Case-2: I s even We dene ) r and 8 ) 1 ISSN: Page 10

3 Internatonal Journal o Mathematcs Trends and Technology IJMTT) Volume 41 Number 1- January 2017 ) 1) r 1) 1) 2 1) 1 ) q 1) 2 1) 5 r 7 5 r Thus s an edge graceul labelng o G = C 8 ʘ S 1r.Hence C 8 ʘ S 1r s an odd graceul graph. Example: The vertex uncton dened nduces a bectve edge uncton *: { } Thus s an edge graceul labelng o G = C 8 ʘ S 2 1r.Hence C 8 ʘ S 2 1r s an odd graceul graph. Example: C 8 ʘ S 16 s odd graceul. Theorem-4 C 12 ʘ S 6 1r where r > 4) wth one vertex gap between par o vertces o the cycle C 12 s an odd graceul graph. C 8 ʘ S 2 16 wth three vertces gap s odd graceul Theorem-3: Graph obtaned by onng the cycle C 8 wth a star graph S 1r s odd graceul. Proo: Let a1 a2... be the vertces o the cycle C 8. a 1 are pendant vertces o the star graph S 1r adacent to a 1 or =12...r. We dene the vertex labelng as : V { where q s the total Case-1: I s odd We dene x) { 1 x a } Let r-pendant vertces o the star S 1r are connected to a 1 Case-2: I s even Proo: Let a1 a2... a12 be the vertces o the cycle C 12. a are pendant vertces o the star graph S 1r adacent to a or =12...r. We dene the vertex labellng as : V { where q s the total Case-1: I s odd We dene a ) 1 Case-2: I s even We dene ) r 1) 1 and 2n 1) and1 r 1 n 4 The vertex uncton dened nduces a bectve edge uncton *: { } Thus s a graceul labelng o G = C 12 ʘ S 6 1r.Hence C 12 ʘ S 6 1r s an odd graceul graph. We dene ) 1 8 and 8 ) 1 1 ) and 1 r 1 The vertex uncton dened nduces a bectve edge uncton *: { } ISSN: Page 11

4 Internatonal Journal o Mathematcs Trends and Technology IJMTT) Volume 41 Number 1- January 2017 Example: keepng vertces. We dene two vertces gap n between par o ) 2 1 or 1 ) q 2 4 or 4 q 4 1 or 7 q 2 or 10 C 12 ʘ S 6 15 s odd graceul. Theorem-5:C 12 ʘ S 4 1r wth two vertces gap between par o vertces o the cycle C 12 s an odd graceul graph. Proo: Let a1 a2... a12 be the vertces o the cycle C 12. a are pendant vertces o the star graph S 1r adacent to a or =12...r. We dene the vertex labelng as : V { where q s the total Case-1: I s odd We dene a ) 1 For r>2.e. more than two pendant vertces are attached to each vertex o the cyclc graph C 12 keepng two vertces gap n between par o vertces We dene 2 1 or 1 ) ) 4 2 or 4 2r 2 9 or 7 4r 2 or 10 The vertex uncton dened nduces a bectve edge uncton *: { } Thus s an edge graceul labelng o G = C 12 ʘ S 4 1r.Hence C 12 ʘ S 4 1r s an odd graceul graph. Examples: I s even We dene ) and 12 ) 1 2n 1) 2n r 1) 2 1 except n 4 For r=1.e. only one pendant vertex s attached to each vertex o the cyclc graph C 12 keepng two vertces gap n between par o vertces. C 12 ʘ S 4 11 wth two vertces gap s odd graceul We dene ) 1 3 q 2 or 1 or 47 or 10 For r=2.e. only two pendant vertces are attached to each vertex o the cyclc graph C 12 C 12 ʘ S 4 12 wth two vertces gap s odd graceul ISSN: Page 12

5 Internatonal Journal o Mathematcs Trends and Technology IJMTT) Volume 41 Number 1- January 2017 and ) 2 1 ) 2r 2 3 4r or 1 and 1 r or 5 and 1 r or 9 and1 r or 9 and r The vertex uncton dened nduces a bectve edge uncton *: { } C 12 ʘ S 4 1r wth two vertces gap s odd graceul Thus s an edge graceul labelng o G = C 8 U P r.hence C 8 U P r s an odd graceul graph. C 12 ʘ S 4 16 wth two vertces gap s odd graceul Theorem-6 C 12 ʘ S 4 1r wth three vertces gap C 12 ʘ S 3 1r wth three vertces gap between par o vertces o the cycle C 12 s an odd graceul graph. Proo: Let a1 a2... be the vertces o the cycle C 8. a are pendant vertces o the star graph S 1r adacent to a or =12...r. We dene the vertex labelng as *: V { } total number o edges as ollows I s odd we dene a ) 1 where q s the I s even we dene 2r 1 4 a ) 4r a ) 15 and a C 12 ʘ S 4 15 wth three vertces gap s odd graceul Theorem-7 C 12 ʘ S 4 1r wth ve vertces gap between par o vertces o the cycle C 12 s an odd graceul graph. Proo: Let a1 a2... a12 be the vertces o the cycle C 12 and a are pendant vertces adacent to a or =12...r. I q s the total number o edges o the graph G we dene the vertex labelng as : V { } as ollows ISSN: Page 13

6 Internatonal Journal o Mathematcs Trends and Technology IJMTT) Volume 41 Number 1- January 2017 We 1 s odd a ) 1 s even and 12 dene Case-1: For r=1 only one pendant vertex s attached to each vertex o the cyclc graph C 12 keepng ve vertces gap n between par o vertces). We dene 12 ) ) ) 1 C 12 ʘ S 4 1r wth ve vertces gap Case-2 : For r=2 two pendant vertces are attached to each vertex o the cyclc graph C 12 keepng ve vertces gap n between par o vertces). We dene 12 ) 1 1 ) 2 1 or or 1 7 ) 17 or 2 Case-3: For r > 2 more than two pendant vertces are attached to each vertex o the cyclc graph C 12 keepng ve vertces gap n between par o vertces) We dene 12 ) or 4 1 ) 2 3 or 4 a ) q 2 3 or 1 r 7 The vertex uncton dened above nduces a bectve edge uncton *: { } Thus s an edge graceul labelng o G = C 12 ʘ S 4 1r.Hence C 12 ʘ S 4 1r s an odd edge graceul graph. C 12 ʘ S 4 15 wth ve vertces gap s odd graceul Theorem-8: Graph obtaned by onng the cycle C 12 wth a star graph S 1r s odd graceul. Proo: Let a1 a2... be the vertces o the cycle C 12 and a 1 are pendant vertces o the star graph S 1r adacent to a 1 or =12...r. We dene the vertex labelng as : V { where q s the total Case-1: I s odd We dene a ) 1 Let r-pendant vertces o the star S 1r are connected to each a 1 Case-2: I s even We dene ) 1 12 and 12 ) 1 1 ) and 1 r 1 The vertex uncton dened nduces a bectve edge uncton *: { } ISSN: Page 14

7 Internatonal Journal o Mathematcs Trends and Technology IJMTT) Volume 41 Number 1- January 2017.Thus s an edge graceul labelng o G = C 12 ʘ S 1r.Hence C 12 ʘ S 1r s an odd graceul graph. Examples: C 12 ʘ S 16 s odd graceul. Applcaton: Graph labelng has wde applcaton n codng theory communcaton networks optmal crcuts layouts and graph decomposton problems. Reerences: [1]. F. Harary Graph Theory Addson Wesley Readng Mass 1972). [2]. S. Lo On edge-graceul labelngs o graphs Congr. Numer ) pp [3]. Sn-Mn Lee Kuo-Jye Chen and Yung-Chn Wang On the edge-graceul spectra o cycles wth one chord and dumbbell graphs. [4] Rosa. A On certan valuaton o the vertces o a graph theory o graphs. Gordon and Breach.N.Y and Dunod Pars pp [5] R.B.Gnanaoth Topcs n graph theoryph.d ThessMadura Kamra UnverstyMadura1991). [6] K.M. KathresanSome classes o odd graceul graphs ANJAC Journal o Scences 71)2008).5-10 [7] S K Vadya Lekha Bukumar Odd Graceul Labelng o Some New Graphs Publshed by Canadan Center o Scence and Educaton Vol. 4 No. 10; October 2010 pp [8] S K Vadya N H Shah. Graceul and odd Graceul Labelng o Some graphs Publshed byinternatonal Journal o Mathematcs and Sot Computng Vol. 3 No pp [9] CHRISTIAN BARRIENTOS. ODD-GRACEFUL LABELINGS OF TREES OF DIAMETER 5 AKCE J.Graphs.Combn.6No )pp [10]. J.A. Gallan A dynamc survey o graph labelng The electronc ournal o Combnatorcs ). ISSN: Page 15