Malaya J. Mat. ()() 88 96 Product cordial labelig for alterate sake graphs S. K. Vaidya a, ad N B Vyas b a Departmet of Mathematics, Saurashtra Uiversity, Rajkot-6, Gujarat, Idia. b Departmet of Mathematics, Atmiya Istitute of Techology ad Sciece, Rajkot-6, Gujarat, Idia. Abstract The product cordial labelig is a variat of cordial labelig. Here we ivestigate product cordial labeligs for alterate triagular sake ad alterate quadrilateral sake graphs. Keywords: Cordial labelig, Product cordial labelig, Sake graph. MSC: C78. c MJM. All rights reserved. Itroductio We begi with simple, fiite, coected ad udirected graph G = (V(G), E(G)). For stadard termiology ad otatios we follow West[]. If the vertices are assiged values subject to certai coditio(s) the it is kow as graph labelig. A mappig f : V(G) {, } is called biary vertex labelig of G ad f (v) is called the label of vertex v of G uder f. For a edge e = uv, the iduced edge labelig f : E(G) {, } is give by f (e = uv) = f (u) f (v). Let v f () ad v f () be the umber of vertices of G havig labels ad respectively uder f ad let e f ()ad e f () be the umber of edges of G havig labels ad respectively uder f. A biary vertex labelig of graph G is called a cordial labelig if v f () v f () ad e f () e f (). A graph G is called cordial if admits a cordial labelig. The cocept of cordial labelig was itroduced by Cahit[] ad i the same paper he ivestigated several results o this ewly itroduced cocept. A latest survey o various graph labelig problems ca be foud i Gallia []. Motivated through the cocept of cordial labelig, Sudaram et al. [] have itroduced a labelig which has the flavour of cordial lableig but absolute differece of vertex labels is replaced by product of vertex labels. A biary vertex labelig of graph G with iduced edge labelig f : E(G) {, } defied by f (e = uv) = f (u) f (v) is called a product cordial labelig if v f () v f () ad e f () e f (). A graph G is product cordial if it admits a product cordial labelig. May researchers have explored this cocept, Sudaram et al. [] have proved that trees, uicyclic graphs of odd order, triagular sakes, dragos, helms ad uio of two path graphs are product cordial. They Correspodig author. E-mail address: samirkvaidya@yahoo.co.i (S. K. Vaidya), iravbvyas@gmail.com(n B Vyas).
S. K. Vaidya et al. / Product cordial labelig... 89 have also proved that a graph with p vertices ad q edges with p is product cordial the q < p +. Vaidya ad Dai[] have proved that the graphs obtaied by joiig apex vertices of k copies of stars, shells ad wheels to a ew vertex are product cordial while Vaidya ad Kaai[6] have reported the product cordial labelig for some cycle related graphs ad ivestigated product cordial labelig for the shadow graph of cycle C. The same authors have ivestigated some ew product cordial graphs i [7]. Vaidya ad Vyas [8] have ivestigated product cordial labelig i the cotext of tesor product of some stadard graphs. The product cordial labeligs for closed helm, web graph, flower graph, double triagular sake ad gear graph are ivestigated by Vaidya ad Barasara [9]. A alterate triagular sake A(T ) is obtaied from a path u, u,..., u by joiig u i ad u i+ (alterately) to a ew vertex v i. That is every alterate edge of a path is replaced by C. A alterate quadrilateral sake A(QS ) is obtaied from a path u, u,..., u by joiig u i, u i+ (alterately) to ew vertices v i, w i respectively ad the joiig v i ad w i. That is every alterate edge of a path is replaced by a cycle C. A double alterate triagular sake DA(T ) cosists of two alterate triagular sakes that have a commo path. That is, double alterate triagular sake is obtaied from a path u, u,..., u by joiig u i ad u i+ (alterately) to two ew vertices v i ad w i. A double alterate quadrilateral sake DA(QS ) cosists of two alterate quadrilateral sakes that have a commo path. That is, it is obtaied from a path u, u,..., u by joiig u i ad u i+ (alterately) to ew vertices v i, x i ad w i, y i respectively ad addig the edges v i w i ad x i y i. Mai results Theorem.. A(T ) is product cordial where (mod ). Proof. Let A(T ) be alterate triagular sake obtaied from a path u, u,..., u by joiig u i ad u i+ (alterately) to ew vertex v i where i for eve ad i for odd. Therefore V(A(T )) = {u i, v j / i, j }. We ote that {, (mod ) V(A(T )) =,, (mod ). {, (mod ) E(A(T )) =, (mod ). We defie f : V(A(T )) {, } as follows. Case : (mod ) For i : For i : I view of above defied labelig patters we have v f () = v f () =, e f () = e f () + = Case : (mod ) For i :
9 S. K. Vaidya et al. / Product cordial labelig... For i : I view of above defied labelig patters we have v f () + = v f () = + Case : (mod ) For i : For i : For i =, e f () = e f () = I view of above defied labelig patters we have v f () + = v f () = +, e f () = e f () + = Thus, i each case we have v f () v f () ad e f () e f (). Hece, A(T ) is a product cordial graph where (mod ). Remark.. A(T ) is ot product cordial graph for (mod ). Because to satisfy the vertex coditio for product cordial labelig it is essetial to assig label to vertices out of vertices. The vertices with label will give rise to at least + edges with label ad edges with label. Cosequetly e f () e f (). Example.. A(T ) ad its product cordial labelig is show i below Figure. v v v v v u u u u u u u u u u 6 7 8 9 Figure Theorem.. A(QS ) is product cordial where (mod ). Proof. Let A(QS ) be a alterate quadrilateral sake obtaied from a path u, u,..., u by joiig u i, u i+ (alterately) to ew vertices v i, w i respectively ad the joiig v i ad w i where i for eve ad i for odd. Therefore V(A(T )) = {u i, v j, w j / i, j }. We ote that {, (mod ) V(A(QS )) =,, (mod ). E(A(QS )) = {, (mod ), (mod ). We defie f : V(A(QS )) {, } as follows.
S. K. Vaidya et al. / Product cordial labelig... 9 Case : (mod ) For i : For i : I view of above defied labelig patters we have v f () = v f () =, e f () = e f () + = Case : (mod ) For i : For i : I view of above defied labelig patters we have v f () + = v f () =, e f () = e f () = Case : (mod ) For i : For i : For i =, I view of above defied labelig patters we have
9 S. K. Vaidya et al. / Product cordial labelig... v f () + = v f () =, e f () = e f () + = Thus i each case we have v f () v f () ad e f () e f (). Hece A(QS ) is a product cordial graph where (mod ). Remark.. A(QS ) is ot product cordial graph for (mod ). Because to satisfy the vertex coditio for product cordial labelig it is essetial to assig label to vertices out of vertices. The vertices with label will give rise to at least + edges with label ad 6 edges with label. Cosequetly e f () e f (). Example.. A(QS ) ad its product cordial labelig. Figure. v w v w v w v w v w u u u u u u u 6 7 u u u u 8 9 Figure Theorem.. DA(T ) ia a product cordial graph where (mod ). Proof. Let G be a double alterate triagular sake DA(T ) the V(G) = {u i, v j, w j / i, j }. We ote that {, (mod ) V(G) =,, (mod ). {, (mod ) E(G) =, (mod ). We defie f : V(A(QS )) {, } as follows. Case : (mod ) For i : For i : I view of above defied labelig patters we have v f () = v f () =, e f () = e f () + = Case : (mod ) For i : For i :
S. K. Vaidya et al. / Product cordial labelig... 9 I view of above defied labelig patters we have v f () + = v f () =, e f () = e f () = Case : (mod ) For i : f (u ) = For i : For i =, I view of above defied labelig patters we have v f () + = v f () =, e f () = e f () = Thus, i each case we have v f () v f () ad e f () e f (). Hece, DA(T ) is a product cordial graph where (mod ). Remark.. DA(T ) is ot product cordial graph for (mod ). Because to satisfy the vertex coditio for product cordial labelig it is essetial to assig label to vertices out of vertices. The vertices with label will give rise to at least + edges with label ad edges with label. Cosequetly e f () e f (). Example.. DA(T ) ad its product cordial labelig is show i Figure. v v v v v u u u u u u u u u u 6 7 8 9 w w w w Figure w u Theorem.. DA(QS ) is a product cordial graph where (mod ). Proof. Let G be a double alterate quadrilateral sake DA(T ) the
9 S. K. Vaidya et al. / Product cordial labelig... V(G) = {u i, v j, w j, x j, y j / i, j }. We ote that {, (mod ) V(G) =,, (mod ). {, (mod ) E(G) =, (mod ). We defie f : V(G) {, } as follows. Case : (mod ) For i : For i : f (x i ) = f (y i ) = I view of above defied labelig patters we have v f () = v f () =, e f () = e f () + = Case : (mod ) For i : For i : f (x i ) = f (y i ) = I view of above defied labelig patters we have v f () + = v f () = Case : (mod ) For i :, e f () = e f () =
S. K. Vaidya et al. / Product cordial labelig... 9 f (u ) = For i : f (x i ) = f (y i ) = For i =, f (x i ) =, f (y i ) = I view of above defied labelig patters we have v f () + = v f () =, e f () = e f () = Thus, i each case we have v f () v f () ad e f () e f (). Hece, DA(QS ) is a product cordial graph where (mod ). Remark.. DA(QS ) is ot product cordial graph for (mod ). Because to satisfy the vertex coditio for product cordial labelig it is essetial to assig label to vertices out of vertices. The vertices with label will give rise to at least + edges with label ad edges with label. Cosequetly e f () e f (). Example.. DA(QS 8 ) ad its product cordial labelig is show i Figure. v v w v w w v w u u u u u u 6 u u7 8 y x x Figure y x y x y Cocludig remarks The labelig of discrete structures is oe of the potetial areas of research. Here we ivestigate product cordial labelig for some alterate sake graphs. To derive similar results for other graph families is a ope area of research.
96 S. K. Vaidya et al. / Product cordial labelig... Ackowledgemet Our thaks are due to the aoymous referees for their costructive commets o the first draft of this paper. Refereces [] D. B. West, Itroductio to Graph Theory, Pritice-Hall of Idia,. [] I. Cahit, Cordial Graphs: A weaker versio of graceful ad harmoious Graphs, Ars Combiatoria, (987), -7. [] J. A. Gallia, A dyamic survey of graph labelig, The Electroics Joural of Combiatorics, 6(#DS6),. [] M. Sudaram, R. Poraj ad S. Somsudaram, Product cordial labelig of graphs, Bull. Pure ad Applied Scieces(Mathematics ad Statistics), E(), -6. [] S. K. Vaidya ad N. A. Dai, Some ew product cordial graphs, Joural of App. Comp. Sci. Math., 8()(), 6-66. [6] S. K. Vaidya ad K. K. Kaai, Some cycle related product cordial graphs, Iteratioal Joural of Algorithms, ()(), 9-6. [7] S. K. Vaidya ad K. K. Kaai, Some ew product cordial graphs, Mathematics Today, 7 (), 6-7. [8] S. K. Vaidya ad N. B. Vyas, Product Cordial Labelig i the Cotext of Tesor Product of Graphs, Joural of Mathematics Research, ()(), 8-88. [9] S. K. Vaidya ad C. M. Barasara, Further results o product cordial labelig, Iteratioal Joural of Math. Combi., (), 6-7. Received: December 6, Accepted: Jauary, UNIVERSITY PRESS Website: http://www.malayajoural.org/