4-Difference Cordial Labeling of Cycle and
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1 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 5 Number 9 December 7 -Differece Cordial Labelig of Cycle ad Wheel Related Graphs S M Vaghasiya, G V Ghodasara Research Scholar, R K Uiversity, Rajkot 6, Idia H & H B Kotak Istitute of Sciece,Rajkot 6, Idia sarlaspkm@gmailcom gaurag ejoy@yahoocoi Abstact Let G be a (p, q) graph Let k be a iteger with k p ad f : V (G) {,, k} be a map For each edge uv, assig the label f (u) f (v) The fuctio f is called a k-differece cordial labelig of G if vf () vf () ad ef () ef () where vf (x) deotes the umber of vertices labelled with x (x {,,, k}), ef () ad ef () respectively deote the umber of edges labelled with ad ot labelled with A graph with a k-differece cordial labelig is called a k-differece cordial graph I this paper we discuss -differece cordial labelig for cycle, wheel, crow, helm ad gear graph Key words : Differece cordial labelig, differece cordial labelig Subject classificatio umber: 5C78 I this paper we have proved that cycle, wheel, helm, crow ad gear graph are differece cordial graphs Defiitio II A cycle C ( N, ) is closed path with vertices Theorem II Cycle C is a -differece cordial graph Proof Let V (C ) = {v, v,, v } defie labelig fuctio f : V (C ) {,,, } as follows Case : is odd f (vi+ ) = ; i b f (vi ) = ; i b f (vi+ ) = ; i b Case : is eve Subcase : (mod) f (vi ) = ; i f (vi+ ) = ; i f (vi+ ) = ; i f (vi+ ) = ; i f (vi+ ) = ; i b I I NTRODUCTION cosider simple, fiite, udirected graph G = (V, E) R Poraj, M Maria Adaickalam ad R Kala [6] itroduced kdifferece cordial labelig of graphs I [6], they ivestigated k-differece cordial labelig behavior of star, m copies of star ad proved that every graph is a subgraph of a coected k-differece cordial graph I [7], R Poraj ad M Maria Adaickalam discussed the -differece cordial labelig behavior of path, cycle, star, bistar, complete graph, complete bipartite graph, comb, double comb, quadrilateral sake For the stadard termiology ad otatios we follow Harary[] ISSN: -57 II M AIN R ESULTS Page 6
2 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 5 Number 9 December 7 Subcase : (mod) f (v ) =, f (v ) = f (vi+ ) = ; i f (vi ) = ; i 6 f (vi+ ) = ; i I each case cycle C satisfies the coditios for -differece cordial labelig Hece C is a -differece cordial graph f (vi+ ) = ; i Example The -differece cordial labelig of C8 is show i Figure v, v,, v be the rim vertices of W defie labelig fuctio f : V (W ) {,,, } as follows Case : is odd f (vi ) = ; i b f (vi+ ) = ; i b f (vi+ ) = ; i b f (vi+ ) = ; i b Case : is eve f (v ) =, f (v ) =, f (v ) = f (vi ) = ; i b f (vi+ ) = ; i b f (vi+ ) = ; i b I each case wheel graph W satisfies the coditios of -differece cordial labelig Hece W is -differece cordial graph f (vi+ ) = ; i b Example -differece cordial labelig of W is show i Figure Fig Defiitio II The wheel W ( N, ) is a joi of the graphs C ad K ie W = C + K Here vertices correspodig to C are called rim vertices ad C is called rim of W The vertex correspodig to K is called apex vertex Theorem II W is a -differece cordial graph Fig Proof Let v be the apex vertex ad ISSN: Page 6
3 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 5 Number 9 December 7 Defiitio II [] The crow C K ( N, ) is obtaied by joiig a pedat edge to each vertex of C Theorem II Crow C differece cordial graph K is a Proof Let V (C K ) = {v, v,, v, v, v,, v }, where v, v,, v are rim vertices ad v, v,, v are pedat vertices defie labelig fuctio f : V (C K ) {,,, } as follows Case : is odd f (vi+ ) = ; i f (vi ) = ; i f (vi+ ) = ; i f (vi ) = ; i Case : is eve f (vi+ ) = ; i f (vi ) = ; i f (vi+ ) = ; i f (vi ) = ; i I each case the crow graph C K satisfies the coditios of -differece cordial labelig Hece it is -differece cordial graph Example -differece cordial labelig of crow C9 K is show i Figure ISSN: -57 Fig Defiitio II A helm H ( ) is the graph obtaied from the wheel W by addig a pedat edge at each vertex o the rim of W Theorem II H is a -differece cordial graph Proof Let V (H ) = where {v, v,, v, v, v,, v }, v is apex vertex, {v, v,, v } are rim vertices ad {v, v,, v } are pedat vertices defie labelig fuctio f : V (H ) {,,, } as follows Case : is odd f (vi ) = ; i b f (vi+ ) = ; i b f (vi+ ) = ; i b f (vi+ ) = ; i b f (vi+ ) = ; i b f (vi+ ) = ; i b f (vi+ ) = ; i b f (vi ) = ; i b Page 6
4 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 5 Number 9 December 7 Case : is odd Case : is eve f (vi ) = ; i f (vi+ ) = ; i ) = ; i f (vi I each case the helm graph H satisfies the coditios of -differece cordial labelig Hece H is -differece cordial graph f (vi+ ) = ; i Example -differece cordial labelig of helm H9 is show i Figure v = f (vi+ ) = ; i f (vi+ ) = ; i f (vi+ ) = ; i Case : is eve f (vi+ ) = ; i v = f (vi+ ) = ; i f (vi+ ) = ; i f (vi+ ) = ; i I each case the gear graph G satisfies the coditios of -differece cordial labelig Hece G is -differece cordial graph f (vi+ ) = ; i Example 5 -differece cordial labelig of G5 is show i Figure 5 Fig Defiitio II5 A gear graph G ( ) is obtaied from the wheel W by addig a vertex betwee every pair of adjacet vertices of rim of W Theorem II5 Gear G is a -differece cordial graph Proof Let G ={v, v,, v }, where v is apex vertex, {v, v,, v } are the vertices of degree ad {v, v,, v } are the vertices of degree defie labelig fuctio f : V (G ) {,,, } as follows ISSN: Fig 5 Page 65
5 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 5 Number 9 December 7 REFERENCES [] F Harary, Graph theory, Addisio-wesley, Readig, MA (969) [] I Cahit, O cordial ad equitable labeligs of graphs, Util Math, 7(99), [] J A Gallia, A dyemic survey of graph labelig, The Electroics Joural of Combiatorics, 6(), ]DS6 8 [] J Gross ad J Yelle, Graph theory ad its applicatios, CRC Press, (999) [5] R Poraj, S Sathish Narayaa ad R Kala, Differece Cordial Labelig of Graphs, Global Joural of Mathematical Scieces: Theory ad Practical, 5 () [6] R Poraj, M Maria Adaickalam ad R Kala, kdifferece cordial labelig of graphs, Iteratioal Joural of Mathematical Combiatorics, (6), - [7] R Poraj ad M Maria Adaickalam, -differece cordial labelig of some cycle related graphs, Joural of Algorithms ad Computatio, 7 (6), - [8] S M Vaghasiya ad G V Ghodasara, Differece Cordial of Operatioal Graph Related to Cycle, Iteratioal Joural of Advaced Egieerig Research ad Sciece, (6), 6-9 ISSN: Page 66
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