Aals o Pure ad Applied Mathematics Vol. 4, No., 03, 50-59 ISSN: 79-087X (P), 79-0888(olie) Published o 7 October 03 www.researchmathsci.org Aals o Further Results o Divisor Cordial Labelig S. K. Vaidya ad N. H. Shah Departmet o Mathematics, Saurashtra Uiversity, Rajkot 360005, Gujarat, Idia. E-mail: samirkvaidya@yahoo.co.i Govermet Polytechic, Rajkot-360003, Gujarat, Idia E-mail: irav.hs@gmail.com Received 3 October 03; accepted 5 October 03 Abstract. A divisor cordial labelig o a graph G with vertex set V is a bijectio rom V to {,,... V } such that a edge uv is assiged the label i ( u) ( v) or () v () u ad the label 0 otherwise, the umber o edges labeled with 0 ad the umber o edges labeled with dier by at most. A graph with a divisor cordial labelig is called a divisor cordial graph. I this paper we prove that helm H, lower graph Fl ad Gear graph G are divisor cordial graphs. Moreover we show that switchig o a vertex i cycle C, switchig o a rim vertex i wheel W ad switchig o the apex vertex i helm H admit divisor cordial labelig. Keywords: Labelig, Divisor cordial labelig, Switchig o a vertex AMS Mathematics Subject Classiicatio (00): 05C78. Itroductio We begi with simple, iite, coected ad udirected graph G = ( V( G), E( G)) with p vertices ad q edges. For stadard termiology ad otatios related to graph theory we reer to Gross ad Yelle [] while or umber theory we reer to Burto []. We will provide brie summary o deiitios ad other iormatio which are ecessary or the preset ivestigatios. Deiitio.. I the vertices are assiged values subject to certai coditio(s) the it is kow as graph labelig. Accordig to Beieke ad Hegde [3] graph labelig serves as a rotier betwee umber theory ad structure o graphs. Graph labeligs have eormous applicatios withi mathematics as well as to several areas o computer sciece ad commuicatio etworks. Yegaaryaa ad Vaidhyaatha [4] have discussed applicatios o edge balaced graph labelig, edge magic labelig ad (,) edge magic graphs. For a dyamic 50
Further Results o Divisor Cordial Labelig survey o various graph labelig problems alog with a extesive bibliography we reer to Gallia [5]. Deiitio.. A mappig : V( G) {0,} is called biary vertex labelig o G ad () v is called the label o the vertex v o G uder. Notatio.3. I or a edge e= uv, the iduced edge labelig by * () e = () u () v. The v ( i) = umber o vertices o G havig label i uder e ( i) = umber o vertices o G havig label i uder * * : E( G) {0,} is give Deiitio.4. A biary vertex labelig o a graph G is called a cordial labelig i v (0) v() ad e (0) e (). A graph G is cordial i it admits cordial labelig. The above cocept was itroduced by Cahit [6]. Ater this may labelig schemes are also itroduced with mior variatios i cordial theme. The product cordial labelig, total product cordial labelig ad prime cordial labelig are amog metio a ew. The preset work is ocused o divisor cordial labelig. Deiitio.5. A prime cordial labelig o a graph G with vertex set VG ( ) is a bijectio : V( G) {,,3,, V( G) } ad the iduced uctio * : E( G) {0,} is deied by *, i gcd( ( u), ( v)) = ; ( e= uv) = 0, otherwise. which satisies the coditio e (0) e (). A graph which admits prime cordial labelig is called a prime cordial graph. The cocept o prime cordial labelig was itroduced by Sudaram et al. [7] ad i the same paper they have ivestigated several results o prime cordial labelig. Vaidya ad Vihol [8, 9] as well as Vaidya ad Shah [0,, ] have proved may results o prime cordial labelig. Motivated through the cocept o prime cordial labelig Varatharaja et al. [3] itroduced a ew cocept called divisor cordial labelig which is a combiatio o divisibility o umbers ad cordial labeligs o graphs. Deiitio.6. Let G = ( V( G), E( G)) be a simple graph ad : {,,... V( G) } be a bijectio. For each edge uv, assig the label i ( u) ( v) or () v () u ad the label 0 otherwise. The uctio is called a divisor cordial labelig i e (0) e (). A graph with a divisor cordial labelig is called a divisor cordial graph. 5
S. K. Vaidya ad N. H. Shah I [3] authors have proved that path, cycle, wheel, star, K, ad K3, are divisor cordial graphs while K is ot divisor cordial or 7. The divisor cordial labelig o ull biary tree as well as some star related graphs are reported by Varatharaja et al. [4] while some star ad bistar related graphs are proved to be divisor cordial graphs by Vaidya ad Shah [5]. It is importat to ote that prime cordial labelig ad divisor cordial labelig are two idepedet cocepts. A graph may possess oe or both o these properties or either as exhibited below. i) P ( 6) is both prime cordial as proved i [7] ad divisor cordial as proved i [3]. ii) C 3 is ot prime cordial as proved i [7] but it is divisor cordial as proved i [3]. iii) We oud that a 7-regular graph with vertices admits prime cordial labelig but does ot admit divisor cordial labelig. iv) Complete graph K 7 is ot a prime cordial as stated i Gallia [5] ad ot divisor cordial as proved i [3]. Deiitio.7. The helm H is the graph obtaied rom a wheel W by attachig a pedat edge to each rim vertex. It cotais three types o vertices: a apex o degree, vertices o degree 4 ad pedat vertices. Deiitio.8. The lower Fl is the graph obtaied rom a helm H by joiig each pedat vertex to the apex o the helm. It cotais three types o vertices: a apex o degree, vertices o degree 4 ad vertices o degree. Deiitio.9. Let e=uv be a edge o the graph G ad w is ot a vertex o G. The edge e is called subdivided whe it is replaced by edges e = uwad e = wv. Deiitio.0. The gear graph rim edge. G is obtaied rom the wheel by subdividig each o its Deiitio.. A vertex switchig G v o a graph G is the graph obtaied by takig a vertex v o G, removig all the edges icidet to v ad addig edges joiig v to every other vertex which are ot adjacet to v i G.. Divisor Cordial Labelig o Some Wheel Related Graphs Theorem.. H is a divisor cordial graph or every. Proo : Let v be the apex, v, v,, v be the vertices o degree 4 ad u, u,, u be the pedat vertices o H. The V( H) = + ad E( H) = 3. We deie vertex labelig as : V( G) {,,3,,+ } as ollows. () v =, For i k =, 5
Further Results o Divisor Cordial Labelig Assig the labels v i ad u i such that ( vi) = ( ui) ad ( vi ) ( v i + ). Now or remaiig vertices, vk+, vk+,, v ad uk+, uk+,, u assig the labels such that ( vj ) ( v + ) where k j, ( v ) ( v ) ad ( vj ) ( uj ) where k < j. j 3 3 I view o above labelig patter we have, e () =, e (0) =. Thus, e (0) e (). Hece, H is a divisor cordial graph or each. Example.. The graph H 3 ad its divisor cordial labelig is show i Figure. Figure : Divisor cordial labelig o H 3 Theorem.3. Fl is a divisor cordial graph or each. Proo : Let v be the apex v, v,, v be the vertices o degree 4 ad u, u,, u be the vertices o degree o Fl. The V( Fl ) = + ad E( Fl ) = 4. We deie vertex labelig : V( G) {,,3,,+ } as ollows. () v =, ( v ) =, ( u ) = 3, ( v + i ) = 5+ ( i ); i ( u + i ) = 4+ ( i ); i I view o the above labelig patter we have, e (0) = = e (). Thus, e (0) e (). Hece, Fl is a divisor cordial graph or each. 53
S. K. Vaidya ad N. H. Shah Example.4. Divisor cordial labelig o the graph Fl is show i Figure. Figure : Divisor cordial labelig o Fl Theorem.5. G is a divisor cordial graph or every. Proo: Let W be the wheel with apex vertex v ad rim vertices v, v,, v. To obtai the gear graph G subdivide each rim edge o wheel by the vertices u, u,, u. Where each u i is added betwee v i ad v i + or i=,,, ad u is added betwee v ad v. The VG ( ) = + ad E( G ) = 3. We deie vertex labelig : V( G) {,,3,,+ }, as ollows. 3 Our aim is to geerate edges havig label ad 3 edges havig label 0. () v =, which geerates edges havig label. 3 Now it remais to geerate k = edges with label. For the vertices v, u, v, u, assig the vertex label as per ollowig ordered patter upto it geerate k edges with label.,, 3,..., k, 3, 3, 3,..., k 3, 5, 5, 5,..., k3 5,...,...,...,...,...,...,...,...,...,..., k where (m ) m + ad m, k m 0. 54
Further Results o Divisor Cordial Labelig α α+ Observe that (m ) (m ) ad ( ) k i m does ot divide m +. The or remaiig vertices o G, assig the vertex label such that the cosecutive vertices do ot geerate edge label. 3 I view o above labelig patter we have, e () =, 3 e (0) =. Thus, e (0) e (). Hece, G is a divisor cordial graph or each. Example.6. Divisor cordial labelig o the graph G0 is show i Figure 3. Figure 3: Divisor cordial labelig o G 0 3. Switchig o a Vertex ad Divisor Cordial Labelig Theorem 3.. Switchig o a vertex i cycle C admits divisor cordial labelig. Proo: Let v, v,, v be the successive vertices o C ad Gv deotes graph obtaied by switchig o vertex v o G= C. Without loss o geerality let the switched vertex be v. E G = 5. We deie vertex labelig We ote that V( Gv ) : V( Gv ) {,,, } = ad ( v ) as ollows: ( v ) =, ( v + i ) = + i. I view o the above labelig patter we have, e () = 3, e (0) =. Thus, e (0) e (). 55
S. K. Vaidya ad N. H. Shah Hece, the graph obtaied by switchig o a vertex i cycle C is a divisor cordial labelig. Example 3.. The graph obtaied by switchig o a vertex i cycle C 8 ad its divisor cordial labelig is show i Figure 4. Figure 4: Switchig o a vertex i C8 ad its divisor cordial labelig Theorem 3.3. Switchig o a rim vertex i a wheel W admits divisor cordial labelig. Proo : Let v be the apex vertex ad v, v,, v be the rim vertices o wheel W. Let G v deotes graph obtaied by switchig o a rim vertex v o G= W. We ote V G E G = 3 6. To deie vertex labelig that ( v ) = + ad ( ) v : V( Gv ) {,,, } +, we cosider ollowig two cases. Case : = 4 ( v) =, ( v ) = 5, ( v ) =, ( v ) = 3, ( v ) = 4. The e (0) = 3 = e (). 3 4 Case : 5 () v =, ( v ) =, ( v ) = 3, ( v 3) = 6, ( v 4) = 4, ( v 5) = 5, ( v5+ i ) = 6 + i; i 5 I view o the above deied labelig patter or case, 3 6 3 6 I is eve the e (0) = = e (), otherwise e (0) = e () =. Thus i both the cases we have, e (0) e (). Hece, the graph obtaied by switchig o a rim vertex i a wheel W is a divisor cordial labelig. 56
Further Results o Divisor Cordial Labelig Example 3.4. The graph obtaied by switchig o a rim vertex i the wheel W 9 ad its divisor cordial labelig is show i Figure 5. Figure 5: Switchig o a rim vertex i W 9 ad its divisor cordial labelig Theorem 3.5. Switchig o the apex vertex i helm H admits divisor cordial labelig. Proo: Let v be the apex, v, v,, v be the vertices o degree 4 ad u, u,, u be the pedat vertices o H. Let G v deotes graph obtaied by switchig o a apex vertex v o G H V G E G = 3. We deie vertex labelig =. We ote that ( v ) ( v ) { } = + ad ( v ) : V G,,, + as ollows: 3 Our aim is to geerate edges havig label ad 3 edges havig label 0. () v =, which geerates edges havig label. 3 Now it remais to geerate k = edges with label. For the vertices v, v,, vl assig the vertex label as per ollowig ordered patter upto it geerate k edges with label. 3,,,...,, 3, 3, 3,..., 3, 3 5, 5, 5,..., 5,...,...,...,...,...,...,...,...,...,..., k k k k m where (m ) + ad m, k m 0. 57
S. K. Vaidya ad N. H. Shah α α+ Observe that (m ) (m ) ad (m ) i does ot divide m +. The or remaiig vertices vl+, vl+,, v ad u, u,, u assig the vertex label such that o edge label geerate. 3 I view o above labelig patter we have, e () =, 3 e (0) =. Thus, e (0) e (). Hece, the graph obtaied by switchig o the apex vertex i helm H admits divisor cordial labelig. Example 3.6. The graph obtaied by switchig o the apex vertex i helm H ad its divisor cordial labelig is show i Figure 6. k Figure 6: Switchig o the apex vertex i Had its divisor cordial labelig 4. Cocludig Remarks The divisor cordial labelig is a variat o cordial labelig. It is very iterestig to ivestigate graph or graph amilies which are divisor cordial as all the graphs do ot admit divisor cordial labelig. Here it has bee proved that helm H, lower graph Fl ad Gear graph G are divisor cordial graphs. The graphs C ad W are proved to be divisor cordial graphs by Varatharaja et al. [3] while we prove the graphs obtaied by switchig o a vertex i C, switchig o a rim vertex i W ad switchig o the apex vertex i H are divisor cordial graphs. Hece C, W ad H are switchig ivariat graphs or divisor cordial labelig. Ackowledgemet: The authors are highly thakul to the aoymous reeree or kid commets ad costructive suggestios. 58
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