Malaya Jornal of Matematik 403 4 5 Prime cordial labeling of some wheel related graphs S. K. Vaidya a, and N. H. Shah b a Department of Mathematics, Sarashtra University, Rajkot - 30005, Gjarat, India. b Department of Mathematics, Government Polytechnic, Rajkot - 30003, Gjarat, India. Abstract A prime cordial labeling of a graph G with the vertex set V G is a bijection f : V G {,, 3,..., V G } sch that each edge v is assigned the label if gcdf, fv = and 0 if gcdf, fv >, then the nmber of edges labeled with 0 and the nmber of edges labeled with differ by at most. A graph which admits prime cordial labeling is called prime cordial graph. In this paper we prove that the gear graph G n admits prime cordial labeling for n 4. We also show that the helm H n for every n, the closed helm CH n for n 5 and the flower graph F l n for n 4 are prime cordial graphs. Keywords: Prime cordial labeling, gear graph, helm, closed helm, flower graph. 00 MSC: 05C. c 0 MJM. All rights reserved. Introdction We begin with simple, finite, connected and ndirected graph G = V G, EG with p vertices and q edges. For standard terminology and notations we follow Gross and Yellen [5]. We will provide brief smmary of definitions and other information which are necessary for the present investigations. Definition.. If the vertices are assigned vales sbject to certain conditions then it is known as graph labeling. Any graph labeling will have following three common characteristics:. a set of nmbers from which vertex labels are chosen;. a rle that assigns a vale to each edge; 3. a condition that this vale has to satisfy. According to Beineke and Hegde [] graph labeling serves as a frontier between nmber theory and strctre of graphs. Graph labelings have many applications within mathematics as well as to several areas of compter science and commnication networks. According to Graham and Sloane [4] the harmonios labellings are closely related to problems in error correcting codes while odd harmonios labeling is sefl to solve ndetermined eqations as described by Liang and Bai []. The optimal linear arrangement concern to wiring network problems in electrical engineering and placement problems in prodction engineering can be formalised as a graph labeling problem as stated by Yegnanaryanan and Vaidhyanathan [3]. The watershed transform is an important morphological tool sed for image segmentation. An improved algorithm sing Gracefl labeling for watershed image segmentation is also proposed by Sridevi et al.[]. For a dynamic srvey on varios graph labeling problems along with an extensive bibliography we refer to Gallian [3]. Corresponding athor. E-mail addresses: samirkvaidya@yahoo.co.in S. K. Vaidya and nirav.hs@gmail.com N. H. Shah.
S. K. Vaidya et al. / Prime cordial labeling... 4 Definition.. A mapping f : V G {0, } is called binary vertex labeling of G and fv is called the label of the vertex v of G nder f. Definition.3. If for an edge e = v, the indced edge labeling f : EG {0, } is given by f e = f fv. Then v f i = e f i = nmber of vertices of G having label i nder f nmber of edges of G having label i nder f } where i = 0 or Definition.4. A binary vertex labeling f of a graph G is called a cordial labeling if v f 0 v f and e f 0 e f. A graph G is cordial if it admits cordial labeling. The concept of cordial labeling was introdced by Cahit []. Some labeling schemes are also introdced with minor variations in cordial theme. Prodct cordial labeling, total prodct cordial labeling and prime cordial labeling are among mention a few. The present work is focsed on prime cordial labeling. Definition.5. A prime cordial labeling of a graph G with vertex set V G is a bijection f : V G {,, 3,..., V G } and the indced fnction f : EG {0, } is defined by f e = v =, if gcdf, fv = ; = 0, otherwise. satisfies the condition e f 0 e f. A graph which admits prime cordial labeling is called a prime cordial graph. The concept of prime cordial labeling was introdced by Sndaram et al.[] and in the same paper they have investigated several reslts on prime cordial labeling. Vaidya and Vihol [] as well as Vaidya and Shah [] have discssed prime cordial labeling in the context of some graph operations. Prime cordial labeling for some cycle related graphs have been discssed by Vaidya and Vihol in [0]. Vaidya and Shah [] have investigated many reslts on prime cordial labeling. Same athors in [] have proved that the wheel graph W n admits prime cordial labeling for n. The present work is aimed to investigate some new reslts on prime cordial labeling for some wheel related graphs. Definition.. The wheel W n is defined to be the join K + C n. The vertex corresponding to K is known as apex and vertices corresponding to cycle are known as rim vertices while the edges corresponding to cycle are known as rim edges. We contine to recognize apex of wheel as the apex of respective graphs corresponding to definitions. to.. Definition.. The gear graph G n is obtained from the wheel by sbdividing each of its rim edge. Definition.. The helm H n is the graph obtained from a wheel W n by attaching a pendant edge to each rim vertex. It contains three types of vertices: an apex of degree n, n vertices of degree 4 and n pendant vertices. Definition.. The closed helm CH n is the graph obtained from a helm H n by joining each pendant vertex to form a cycle. It contains three types of vertices: an apex of degree n, n vertices of degree 4 and n vertices degree 3. Definition.0. The flower F l n is the graph obtained from a helm H n by joining each pendant vertex to the apex of the helm. It contains three types of vertices: an apex of degree n, n vertices of degree 4 and n vertices of degree. Main Reslts Theorem.. Gear graph G n is a prime cordial graph for n 4. Proof. Let W n be the wheel with apex vertex v and rim vertices v, v,...,. To obtain the gear graph G n sbdivide each rim edge of wheel by the vertices,,..., n. Where each i is added between v i and v i+ for i =,,..., n and n is added between v and. Then V G n = n + and EG n =. To define f : V G {,, 3,..., n + }, we consider following for cases.
50 S. K. Vaidya et al. / Prime cordial labeling... Case : n = 3 In G 3 to satisfy the edge condition for prime cordial labeling it is essential to label for edges with label 0 and five edges with label ot of nine edges. Bt all the possible assignments of vertex labels will give rise to 0 labels for at most three edges and labels for at least six edges. That is, e f 0 e f = 3 >. Hence, G 3 is not prime cordial graph. Case : n = 4 to,, 4, For n = 4, fv =, fv = 3, fv =, fv 3 = 4, fv 4 = and f =, f =, f 3 =, f 4 = 5. Then e f 0 = = e f. For n = 5, fv =, fv =, fv = 5, fv 3 = 4, fv 4 =, fv 5 = 3 and f =, f = 0, f 3 =, f 4 =, f 5 =. Then e f 0 =, e f =. For n =, fv =, fv =, fv =, fv 3 = 4, fv 4 =, fv 5 =, fv = 0 and f =, f =, f 3 = 3, f 4 = 5, f 5 =, f = 3. Then e f 0 = = e f. For n =, fv =, fv =, fv = 4, fv 3 =, fv 4 =, fv 5 =, fv = 0, fv = 4 and f = 5, f = 3, f 3 =, f 4 = 5, f 5 =, f =, f = 3. Then e f 0 = 0, e f =. For n =, fv =, fv =, fv = 4, fv 3 =, fv 4 =, fv 5 =, fv = 0, fv = 4, fv = and f =, f = 3, f 3 =, f 4 = 5, f 5 = 5, f =, f = 3, f =. Then e f 0 = = e f. For n =, fv =, fv = 4, fv = 5, fv 3 =, fv 4 =, fv 5 =, fv =, fv = 0, fv = 4, fv = and f =, f = 3, f 3 =, f 4 = 5, f 5 =, f =, f = 3, f =, f =. Then e f 0 = 3, e f = 4. For n =, fv =, fv = 4, fv = 5, fv 3 =, fv 4 =, fv 5 =, fv =, fv = 0, fv = 4, fv =, fv 0 = 0, fv = and f =, f = 3, f 3 =, f 4 = 5, f 5 =, f =, f = 3, f =, f =, f 0 = 3, f =. Then e f 0 =, e f =. For n = 4, fv =, fv = 4, fv = 5, fv 3 =, fv 4 =, fv 5 =, fv = 4, fv =, fv = 0, fv = 4, fv 0 =, fv = 0, fv =, fv 3 =, fv 4 = and f =, f = 3, f 3 =, f 4 = 5, f 5 =, f =, f =, f = 3, f =, f 0 =, f = 3, f = 5, f 3 =, f 4 =. Then e f 0 = = e f. For n =, fv =, fv = 4, fv = 5, fv 3 =, fv 4 =, fv 5 =, fv = 4, fv = 30, fv = 3, fv =, fv 0 = 0, fv = 4, fv =, fv 3 = 0, fv 4 =, fv 5 =, fv =, fv = 3, fv = 34, fv = 3 and f = 3, f =, f 3 =, f 4 = 5, f 5 =, f =, f = 33, f = 3, f =, f 0 = 3, f =, f =, f 3 = 3, f 4 = 5, f 4 =, f 4 = 3, f 4 = 35, f 4 = 3, f 4 =. Then e f 0 =, e f =. Now for the remaining two cases let, n n + n n + s =, k =, t = n + 3 3 3 n + m = h o =largest odd nmber not divisible by 3 n +. fv =, fv = 4, fv = 5,, 3 + s + t, h e = largest even nmber not divisible by 3 n, fv +i = i; i s f =, f +i = 3 + i ; i k If k = s, then f k+ = or f n =. Case 3: t = 0n = 0,, 3, 5, m m For m odd, consider x =, x = x 3 = x 4 = and for m even consider, x = x = x 3 = x 4 = m fv s++i = + i ; i x fv s++i = 0 + i, i x 3 f s++i = + i ; i x f s++i = 3 + i ; i x 4 which assigns all the vertex labels for case 3. Case 4: t n =,, n 0 m For m odd, consider x = x = x 3 =, x 4 = m 3 and for m even consider, x = m, x = x 3 = x 4 =
S. K. Vaidya et al. / Prime cordial labeling... 5 m. fv s++i = + i ; i x fv s++i = 0 + i, i x 3 f s++i = + i ; i x f s++i = 3 + i ; i x 4 For the vertices n, n,..., n t+ we assign even nmbers not congrent 0 mod 3 in descending order starting from h e respectively while for n,,,,..., t we assign odd nmbersnot congrent 0 mod 3 in descending order starting from h 0 respectively sch that f v j j i or f v j j+i do not generate edge label 0. Which assigns all the vertex labels for case 4. In view of the above defined labeling pattern for cases 3 and 4, we have e f 0 = and e f =. Ths, we have e f 0 e f. Hence, G n is a prime cordial graph for n 4. Example.. For the graph G 0, V G 0 = 4 and EG 0 = 0. In accordance with Theorem. we have s =, k =, t =, m =, x = x = x 3 = 5, x 4 = 4 and sing the labeling pattern described in case 4. The corresponding prime cordial labeling is shown in Fig.. It is easy to visalise that e f 0 = 30 = e f. 4 5 4 3 40 3 3 34 35 3 3 5 3 0 0 3 4 Fig. 5 4 30 33 3 3 Theorem.. Helm graph H n is a prime cordial graph for every n. Proof. Let v be the apex, v, v,..., be the vertices of degree 4 and,,..., n be the pendant vertices of H n. Then V H n = n + and EH n =. To define f : V G {,, 3,..., n + }, we consider following three cases. Case : n = 3 to For n = 3, fv =, fv =, fv = 4, fv 3 = 3 and f =, f =, f 3 = 5. Then e f 0 = 4, e f = 5. For n = 4, fv =, fv = 3, fv =, fv 3 = 4, fv 4 = and f =, f =, f 3 = 5, f 4 =. Then e f 0 = = e f. For n = 5, fv =, fv = 3, fv =, fv 3 = 5, fv 4 =, fv 5 = 0 and f =, f =, f 3 = 5, f 4 =, f 5 =. Then e f 0 =, e f =. For n =, fv =, fv = 3, fv =, fv 3 =, fv 4 = 0, fv 5 =, fv = and f =, f = 4, f 3 =, f 4 = 5, f 5 =, f = 3. Then e f 0 = = e f. For n =, fv =, fv = 3, fv =, fv 3 = 4, fv 4 = 0, fv 5 = 5, fv =, fv = and
5 S. K. Vaidya et al. / Prime cordial labeling... f =, f =, f 3 =, f 4 = 4, f 5 =, f = 3, f = 5. Then e f 0 =, e f = 0. For n =, fv =, fv =, fv =, fv 3 = 4, fv 4 = 0, fv 5 = 5, fv =, fv = 3, fv = and f = 3, f =, f 3 = 4, f 4 =, f 5 =, f =, f = 5, f =. Then e f 0 = = e f. For n =, fv =, fv = 3, fv =, fv 3 =, fv 4 = 4, fv 5 = 0, fv = 5, fv =, fv = 3, fv = and f =, f =, f 3 = 4, f 4 =, f 5 =, f =, f =, f = 5, f =. Then e f 0 =, e f =. Case : n is even, n 0 fv =, fv = 0, fv = 4, fv 3 =, fv 3+i = + i ; i n 4 f =, f + =, f n = 3, f n + = n +, f i = n i ; i n f i = 5 + 4i; 0 i n f n i = + 4i; 0 i n Case 3: n is odd, n fv =, fv = 0, fv = 4, fv 3 =, fv 3+i = + i ; i n 4 f =, f + = 3, f =, n+ f i = n i ; i n f i = 5 + 4i; 0 i < n f n i = + 4i; 0 i < n In view of the above defined labeling pattern for cases and 3, If n 0mod 3 then e f 0 = and e f =, otherwise e f 0 = and e f =. Ths, we have e f 0 e f. Hence, H n is a prime cordial graph for every n. Example.. The graph H 3 and its prime cordial labeling is shown in Fig.. 3 v 3 0 v 5 4 v v 5 3 v 0 0 v 5 3 v v 3 4 v v v v 4 4 v 5 v 3 3 0 4 5 Fig. Theorem.3. Closed helm CH n is a prime cordial graph for n 5.
S. K. Vaidya et al. / Prime cordial labeling... 53 Proof. Let v be the apex, v, v,..., be the vertices of degree 4 and,,..., n be the vertices of degree 3 of CH n. Then V CH n = n + and ECH n = 4n. To define f : V G {,, 3,..., n + }, we consider following three cases. Case : n = 3, 4 In CH 3 to satisfy the edge condition for prime cordial labeling it is essential to label six edges with label 0 and six edges with label ot of twelve edges. Bt all the possible assignments of vertex labels will give rise to 0 labels for at most for edges and labels for at least eight edges. That is, e f 0 e f = 4 >. Hence, CH 3 is not prime cordial graph. In CH 4 to satisfy the edge condition for prime cordial labeling it is essential to label eight edges with label 0 and eight edges with label ot of sixteen edges. Bt all the possible assignments of vertex labels will give rise to 0 labels for at most seven edges and labels for at least nine edges. That is, e f 0 e f = >. Hence, CH 4 is not prime cordial graph. Case : n = 5, For n = 5, fv =, fv =, fv = 4, fv 3 = 3, fv 4 =, fv 5 = and f = 0, f =, f 3 =, f 4 =, f 5 = 5. Then e f 0 = 0 = e f. For n =, fv =, fv =, fv = 5, fv 3 = 0, fv 4 = 4, fv 5 =, fv = 3 and f =, f = 3, f 3 =, f 4 =, f 5 =, f =. Then e f 0 = = e f. Case 3: n fv =, fv = 4, fv =, fv 3 = 3, fv 4 =, fv 5 =, fv = 0, fv +i = 4 + i ; i n f =, f = 5, f 3 =, f 4 =, f 5 = 3, f =, f +i = 5 + i ; i n In view of the above defined labeling pattern we have e f 0 = n = e f. Ths, we have e f 0 e f. Hence, CH n is a prime cordial graph for n 5. Example.3. The graph CH 0 and its prime cordial labeling is shown in Fig. 3. 0 v 0 0 v v v 5 4 v 5 4 v 3 0 v v 5 v 4 v 3 3 v 4 Fig. 3 3 5 Theorem.4. Flower graph F l n is a prime cordial graph for n 4.
54 S. K. Vaidya et al. / Prime cordial labeling... Proof. Let v be the apex, v, v,..., be the vertices of degree 4 and,,..., n be the vertices of degree of F l n. Then V F l n = n + and EF l n = 4n. To define f : V G {,, 3,..., n + }, we consider following for cases. Case : n = 3 In F l 3 to satisfy the edge condition for prime cordial labeling it is essential to label six edges with label 0 and six edges with label ot of twelve edges. Bt all the possible assignments of vertex labels will give rise to 0 labels for at most for edges and labels for at least eight edges. That is, e f 0 e f = 4 >. Hence, F l 3 is not prime cordial graph. Case : n = 4 to. For F l 4, fv =, fv = 4, fv =, fv 3 =, fv 4 = 3 and f =, f =, f 3 = 5, f 4 =. Then e f 0 = = e f. For F l 5, fv =, fv =, fv = 4, fv 3 =, fv 4 = 0, fv 5 = 3 and f =, f =, f 3 = 5, f 4 =, f 5 =. Then e f 0 = 0 = e f. For F l, fv =, fv =, fv = 4, fv 3 =, fv 4 = 0, fv 5 =, fv = 3 and f = 5, f =, f 3 =, f 4 =, f 5 = 3, f =. Then e f 0 = = e f. For F l, fv =, fv = 3, fv =, fv 3 = 0, fv 4 =, fv 5 = 4, fv = 4, fv = and f =, f = 5, f 3 =, f 4 =, f 5 = 3, f = 5, f =. Then e f 0 = 4 = e f. For F l, fv =, fv = 3, fv =, fv 3 = 4, fv 4 =, fv 5 = 0, fv = 4, fv =, fv = and f =, f =, f 3 =, f 4 = 5, f 5 =, f = 3, f = 5, f =. Then e f 0 = = e f. For F l, fv =, fv = 3, fv =, fv 3 = 4, fv 4 =, fv 5 = 0, fv = 4, fv =, fv =, fv = and f =, f =, f 3 = 5, f 4 =, f 5 =, f = 3, f = 5, f =, f =. Then e f 0 = = e f. Case 3: n is even, n 0 fv =, fv = 0, fv = 4, fv 3 =, fv 3+i = + i ; i n 4 f =, f i = 5 + 4i; 0 i < n f i = n i ; i n f n = 3, f n i = + 4i; 0 i < n For n + 0mod 3 f +i = n + 4i ; i f n + =, f n + = n For n + mod 3 f + = n 3, f + =, f n + = n +, f n + = n For n + mod 3 f + = n, f f n + = n +, f n + = n 3, + = Case 4: n is odd, n fv =, fv = 0, fv = 4, fv 3 =, fv 3+i = + i ; i n 4 f =, f = 3, f i = 5 + 4i; f i = n i ; f n i = + 4i; For n + 0mod 3 f n+ = n +, f f n+ = n 3, + For n + mod 3 + 0 i < n i n 0 i < n 5 n+ + =,
f f n+ n+ + = n 3, f =, S. K. Vaidya et al. / Prime cordial labeling... 55 n+ + = n +, For n + mod 3 f n+ =, f n+ = n + 4i ; i +i In view of the above defined labeling pattern we have e f 0 = n = e f. Ths, we have e f 0 e f. Hence, F l n is a prime cordial graph for n 4. Example.4. The graph F l and its prime cordial labeling is shown in Fig. 4. 0 v v 0 v 0 5 4 v 0 v 5 v 3 v v 3 Fig. 4 3 v v 5 v 3 3 v 4 4 4 5 3 Open problems To investigate necessary and sfficient conditions for a graph to admit a prime cordial labeling. To investigate some new graph or graph families which admit prime cordial labeling. To obtain forbidden sbgraphs characterisation for prime cordial labeling. 4 Conclsion As all the graphs are not prime cordial graphs it is very interesting and challenging as well to investigate prime cordial labeling for the graph or graph families which admit prime cordial labeling. Here we have contribted some new reslts by investigating prime cordial labeling for some wheel related graphs. 5 Acknowledgement The athors are highly thankfl to the anonymos referees for their kind sggestions and comments on the first draft of this paper. References [] L. W. Beineke and S. M. Hegde, Strongly mltiplicative graphs, Discss. Math. Graph Theory, 00, 3-5.
5 S. K. Vaidya et al. / Prime cordial labeling... [] I. Cahit, Cordial Graphs, A weaker version of gracefl and harmonios Graphs, Ars Combinatoria, 3, 0-0. [3] J. A. Gallian, A dynamic srvey of graph labeling, The Electronic Jornal of Combinatorics, 0, #DS. [4] R. L. Graham and N. J. A. Sloane, On additive bases and harmonios graphs, SIAM J. Alg. Disc. Meth., 40, 3-404. [5] J. Gross and J. Yellen, Graph Theory and its Applications, CRC Press,. [] Z. Liang and Z. Bai, On the odd harmonios graphs with applications, J. Appl. Math. Compt., 00, 05-. [] R. Sridevi, K. Krishnaveni and S. Navaneethakrishnan, A novel watershed image segmentation techniqe sing gracefl labeling, International Jornal of Mathematics and Soft Compting, 303, -. [] M. Sndaram, R. Ponraj and S. Somasndram, Prime Cordial Labeling of graphs, J. Indian Acad. Math., 005, 33-30. [] S. K. Vaidya and P. L. Vihol, Prime cordial labeling for some graphs, Modern Applied Science, 400, -. [0] S. K. Vaidya and P. L. Vihol, Prime cordial labeling for some cycle related graphs, Int. J. of Open Problems in Compter Science and Mathematics, 3500, 3-3. [] S. K. Vaidya and N. H. Shah, Some New Families of Prime Cordial Graphs, J. of Mathematics Research, 340, -30. [] S. K. Vaidya and N. H. Shah, Prime Cordial Labeling of Some Graphs, Open Jornal of Discrete Mathematics, 0, -. [3] V. Yegnanaryanan and P. Vaidhyanathan, Some Interesting Applications of Graph Labellings, J. Math. Compt. Sci., 5 0, 5-53. Received: Jly, 03; Accepted: Jly 4, 03 UNIVERSITY PRESS