Prime Cordial Labeling

Transcription

1 Circlation in Compter Science Vol., No., pp: (-0), May 0 Prime Cordial Labeling M. A. Seod Egypt Department of Mathematics Faclty of Science Ain Shams Uniersity A. T. M. Matar Egypt Department of Mathematics Faclty of Science Ain Shams Uniersity R. A. Al-Zraiqi Yemen Department of Mathematics Faclty of Science Ain Shams Uniersity ABSTRACT We sho that some special families of graphs hae prime cordial labeling. We proe that If G is not a prime cordial graph of order m then is a prime cordial graph if, and e proe that, Jelly fish graph, Jeel graph, the graph obtained by dplicating a erte in the rim of the helm and the graph obtained by fsing the erte ith in a Helm graph are prime cordial graphs. Keyords Cordial labeling, prime cordial labeling.. INTRODUCTION In this paper e deal ith only finite simple and ndirected graphs.we shall se the basic notations and terminology of graph theory and nmber theory as gien in [],[]. The notion of a prime cordial labeling as introdced by Sndaram,Ponraj and Somasndaram [] A prime cordial labeling of a graph G ith erte set V is a bijection from } sch that if each edge is assigned the label if and 0 if, then the nmber of edges labeled ith 0 and the nmber of edges labeled ith differ by at most. Sndaram, Ponraj, and Somasndram[] proe the folloing graphs are prime cordial: if and only if ; if and only if or ; (n odd); the graph obtained by sbdiiding each edge of if and only if ; bistars; dragons; crons; trianglar snakes if and only if the snake has at least three triangles; ladders; if n is een and there eists a prime p sch that ; if n is een and if there eists a prime p sch that, and if n is odd and if there eists a prime p sch that. They also proe that if G is a prime cordial graph of een size, then the graph obtained by identifying the central erte of ith the erte of G labeled ith is prime cordial, and if G is a prime cordial graph of odd size, then the graph obtained by identifying the central erte of ith the erte of G labeled ith is prime cordial. [0]Vaidya and Shah proe that the folloing graphs are prime cordial: split graphs of and ; the sqare graph of ; the middle graph of for ; and if and only if. Also Vaidya and Shah [],[] proed folloing graphs are prime cordial: gear graphs for ; helms; closed helms for ; oer graphs for,degree splitting graphs of and the bistar ; doble fans for and ; the graphs obtained by dplication of an arbitrary rim edge by an edge in here ; and the graphs obtained by dplication of an arbitrary spoke edge by an edge in heel here and. S. Babitha,J. Baskar Babjee [] proed are prime cordial graphs. Baskar Babjee and Shobana[] proed sn graphs, ith a path of length attached to a erte; and ith pendent edges attached to a pendent erte of hae prime cordial labeling. M.Seod, M.Salim [] gie a necessary condition for a graph to be prime cordial and a necessary condition for a bipartite graph to be prime cordial and they proe dose not prime cordial labeling for and conjectre that is not prime cordial for all. In the this paper e proe the disjoint of G and is prime cordial in to cases: if n is an odd nmber and m is an een nmber and if n is an een nmber,,here are the nmber of edges labeled ith 0 and the nmber of edges labeled ith respectiely, and m is an odd nmber.and e proe,jelly fish graph, Jeel graph, the graph obtained by dplicating a erte in the rim of the helm and the graph obtained by fsing the erte ith in a Helm graph are prime cordial graphs.. RESULTS. Definition: A prime cordial labeling of a graph G ith erte set V is a bijection from } sch that if each edge is assigned the label if and 0 if, then the nmber of edges labeled ith 0 and the nmber of edges labeled ith differ by at most.. Theorem: Let G be a prime cordial graph ith order n and cases: be the ell-knon bipartite graph.we hae to (i)if n is an odd nmber and m is an een nmber then the disjoint nion of G and is a prime cordial (ii)if n is an een nmber,,here are the nmber of edges labeled ith 0 and the nmber of edges labeled ith respectiely, and m is an odd nmber then the disjoint nion of G and is a prime cordial Proof: Let be gien as in Defnition. and be the ertices of G and be the ertices of.no e define the ne graph called as the disjoint nion of G and ith erte set Copyright 0 Al-Zraiqi et al. This is an open-access article distribted nder the terms of the Creatie Commons Attribtion License.0, hich permits nrestricted se, distribtion, and reprodction in any medim, proided the original athor and sorce are credited.

2 Case(i) g. Consider the bijectie fnction g: define by g, here is the greatest een nmber,sch that +. g No e hae Hence the total nmber of edges labeled 's are gien by and the total nmber of edges labeled 0's are gien. Eample 0 by. Since G is prime cordial of odd order then or Fig..Hence the total nmber of edges labeled ith 's of the graph is and the total nmber of edges labeled ith 0's is or is and. Then is a prime cordial ( See Figre.) Case (ii) g G, e interchange the erte labeled in G by, here is the greatest een nmber,sch that + No e hae g Hence the total nmber of edges labeled 's of are gien by and the total nmber of edges labeled 0's of are gien by. No since G is prime cordial of een order and,either e hae then, or then,,hence the total nmber of edges labeled ith 's of is and,then ( ),or if then and,then ( ). Then is a prime cordial if. Theorem: If G is not a prime cordial graph of order m then is a prime cordial graph if Proof: Let be the first m een nmbers less than or eqal to m+n+.we label the ertices of sch that center of is labeled by and e label the ertices of G by then e fined that is prime cordial 0 Fig : prime cordial labeling of the disjoint nion of G and K, by

3 . Eample Therefore,. Hence is prime cordial K K,. Eample 0 Fig : prime cordial labeling for J(, ) 0 K K,. Definition []: For a graph G the splitting graph of a graph G is obtained by adding for each erte of G a ne erte so that is adjacent to eery erte that is adjacent to.. Theorem: The graph is prime for all n,. Proof: Let be the ertices of K K,0 Fig. SOME FAMILIES OF PRIME CORDIAL GRAPHS. Definition[]: Jelly fish graphs are obtained from a -cycle by joining and ith an edge and appending pendent edges to and pendent edges to. Then are the ertices of and then. Define as follos: For n=. Theorem: is a prime cordial Proof: Let ( ) and Define as follos: 0 is the smallest prime nmber sch that it diides Fig : prime cordial labeling for S' (K,) For the ertices e assign the remaining odd nmbers hich are less than

4 For all Then for any Therefore,. Hence G is prime cordial. Eample If e define as follos: is the greatest een nmber less than or eqal to n+ ritten in the form,,here is a prime nmber. een nmbers hich For the ertices,,,,n+. No e hae to cases We assign the remaining e assign the labeling If, then so G is prime cordial graph see Figre (),(),() If, then,so e assign the erte as hich, are prime nmbers and sch that the smallest nmber immediately folloing the first,see Figres (). 0 Therefore,. Hence is prime cordial graph for n is een.. Eample. Definition []: The jeel graph is the graph ith erte set Fig : prime cordial labeling for S' (K, ) and edge set.. Theorem: The Jeel graph is prime cordial graph if n is een. Proof: Let = and If n=: 0=* =* 0 Fig prime cordial labeling for J

5 .0 Definition []: Dplication of a erte of a graph G prodces a ne graph by adding a erte ith (the set of neighbor ertices to ). In other ords a erte is said to be a dplication of if all the ertices hich are adjacent to are no adjacent to.. Theorem: The graph obtained by dplicating a erte in the rim of the helm is a prime cordial Proof: Let, Let be the graph obtained by dplicating the erte in and let the ne erte be Then and. Define a labeling f: as follos: Case(): n is een, ( ) Fig prime cordial labeling for dplicating the erte k in H H H

6 0 0 Case(): n is odd,, ( ) case(): 0 0, ( ) Case ():

7 ( ) Then hen n is een and then also if then, hen n is odd and then and if then and Therefore, 0. Hence G is prime cordial. Definition: Let and be to distinct ertices of a graph G. A ne graph is constrcted by identifying (fsing) to ertices and by a single erte sch that eery edge hich as incident ith either or in G is no incident ith in G.. Theorem: The graph obtained by fsing the erte ith in a Helm graph is prime cordial Proof: Let, Let be the graph obtained by fsing the erte ith in. Then and. 0 0 Define a labeling f: as follos: eqatio here 0

8 0 0 i i Case(): n is een, 0 0 i i Case(): n is odd 0 0 i i Case():n=,,0,,

9 0 0 Eample.. ere Then hen n is een, hen n is odd Therefore,. Hence G is prime cordial. Theorem: The graph hich is obtained by attaching central erte of a star at one of the ertices of is prime cordial Proof: Let = and. CONCLUSION If G is a prime cordial graph ith order n and ell-knon bipartite We hae to cases: be the We define as : (i) If n is an odd nmber and m is an een nmber then the disjoint nion of G and is a prime cordial If n is odd then ; if n is een then Therefore,. Hence is prime cordial (ii) If n is an een nmber,,here are the nmber of edges labeled ith 0 and the nmber of edges labeled ith respectiely, and m is an odd nmber then the disjoint nion of G and is a prime cordial In the ftre e ill try to constrct other general theorems and fined some other prime cordial families.

10 . REFERENCES [] J. Baskar Babjee and L. Shobana, Prime cordial labeling, Int. Reie on Pre and Appl. Math., (00) -. [] S. Babitha,J. Baskar Babjee.Prime cordial labeling on graphs, World Academy of Science, Engineering and Technology International Jornal of Mathematical, Comptational, Physical, Electrical and Compter Engineering Vol:, No:, 0. [] D. M. Brton, Elmentary Nmber Therory, Seconed Edition, Wm;C.Bron Company Pblishers. 0. [] J.A.Gallian, A dynamic srey of graph labeling, The Electonic Jornal of Combinatorics (0). [] J.A.Gallian, A dynamic srey of graph labeling, Electonic Jornal of Combinatorics (0). [] F. Harary, Graph Theory, (Addison Wesley, Reading, Massachsetts. ). [] R.Ponraj and S.Somasndarma,On the degree splitting graph of a graph,nathional Academy Scinence letter, (00) -. [] M.A.Seod, M.Salim., To pper bonds of prime cordial graphs, JCMCC, (00) -0. [] M. Sndaram, R. Ponraj, and S. Somasndram, Prime cordial labeling of graphs, J. Indian Acad. Math., (00) -0 [0] S. K. Vaidya and N. H. Shah, Prime cordial labeling of some graphs, Open J. Discrete Math., (0) -. [] S. K. Vaidya and N. H. Shah, Prime cordial labeling of some heel related graphs, Malaya J. Mate., () (0) -. [] S. K. Vaidya and N. H. Shah, Some ne reslts on prime cordial labeling, ISRN Combin., 0 Article ID 00, pages, 0. CCS 0 ISSN - Pblished by: CSL Press, USA 0