Lucky Edge Labeling of P n, C n and Corona of P n, C n

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1 International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume, Issue 8, August 0, PP ISSN 7-07X (Print) & ISSN 7- (Online) Lucky Edge Labeling of P n, C n and Corona of P n, C n Dr. A. Nellai Murugan Department of Mathematics, V.O.Chidambaram College, Tuticorin, Tamilnadu(INDIA) anellai.vocc@gmail.com R. Maria Irudhaya Aspin Chitra Department of Mathematics, V.O.Chidambaram College, Tuticorin, Tamilnadu(INDIA) aspinvjs@gmail.com Abstract: Let G be a Simple Graph with Vertex set V(G) and Edge set E(G) respectively. Vertex set V(G) is labeled arbitrary by positive integers and let E(e) denote the edge label such that it is the sum of labels of vertices incident with edge e. The labeling is said to be lucky edge labeling if the edge set E(G) is a proper coloring of G, that is, if we have E(e ) E(e ) whenever e and e are adjacent edges. The least integer k for which a graph G has a lucky edge labeling from the set {,,.,k} is the lucky number of G denoted by η(g). A graph which admits lucky edge labeling is the lucky edge labeled graph.in this paper, it is proved that Path P n, Comb P n +, Cycle C n, Crown C n + are lucky edge labeled graphs. Keywords: Lucky Edge Labeled Graph, Lucky Edge Labeling, Lucky Number, 00 Mathematics subject classification Number: 0C78.. INTRODUCTION A graph G is a finite non empty set of objects called vertices together with a set of pairs of distinct vertices of G which is called edges. Each e={uv} of vertices in E is called an edge or a line of G. For Graph Theoretical Terminology, [].. PRELIMINARIES Definition:. Let G be a Simple Graph with Vertex set V(G) and Edge set E(G) respectively. Vertex set V(G) are labeled arbitrary by positive integers and let E(e) denote the edge label such that it is the sum of labels of vertices incident with edge e. The labeling is said to be lucky edge labeling if the edge set E(G) is a proper coloring of G,that is, if we have E(e ) E(e ) whenever e and e are adjacent edges. The least integer k for which a graph G has a lucky edge labeling from the set {,,.,k} is the lucky number of G denoted by η(g). A graph which admits lucky edge labeling is the lucky edge labeled graph. Definition:. A Walk of a graph G is an alternating sequence of vertices and edges v, e, v, e,..v n-, e n, v n beginning and ending with vertices such that each edge e i is incident with v i- and v i. Definition:. If all the vertices in a walk are distinct, then it is called a Path and a path of length n is denoted by P n+. Definition:. A graph obtained by joining each u i to a vertex v i is called a Comb and denoted by P n +. The Vetex set and Edge set of P n + is V[P n + ] = {u i,v i : i n} and E[P n + ] = {(u i u i+ ): i n-} {(u i v i ): i n} respectively. P n + has n vertices and n- edges. Definition:. A closed path is called a Cycle and a cycle of length n is denoted by C n. ARC Page 70

2 Dr. A. Nellai Murugan & R. Maria Irudhaya Aspin Chitra Definition:.6 C n + is a graph obtained from G by attaching a pendent vertex from each vertex of the graph Cn is called Crown.. MAIN RESULTS Theorem:. P n has {a, b} lucky edge labeling graph for any a, b. Proof: Let V[P n ] = { u i : i n }and E[P n ] = { (u i u i+ ): i n- }. Let f: V[P n ] {, } defined by f(u i ) = and i, for n is odd and i, for n is even. f(u i- ) = and i, for n is odd and i, for n is even. whenn is odd, i and when n is even, i ( f * (u i u i+ ) = whenn is odd, i f * (u i- u i ) = and when n is even, i It is clear that lucky edge labeling of P n is {,, }. Hence,P n has lucky edge labeling graph. For example, lucky edge labeling of P 6 is shown in figure and η(p n ) =. Theorem:. Figure and η(p n ) = P n + has {a, b, c} lucky edge labeling graph for any a, b, c. Proof: u u u u u u 6 Let V[P n + ] = {{u i : i n}, {v i : i n}}and E[P n + ] = {(u i u i+ ): i n-} {(u i v i ): i n}. Let f: V[P n + ] {,, } defined by f(u i ) = and i, for n is odd and i, for n is even. f(u i- ) = and i, for n is odd and i, for n is even. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 7 u

3 Lucky Edge Labeling of P n, C n and Corona of P n, C n f(v ) = f(v i ) =, i n- f(v n ) =. whenn is odd, i and when n is even, i ( f * (u i u i+ ) = whenn is odd, i and when n is even, i f * (u i- u i ) = f * (u v ) = f * (u i v i ) = for i n-. f * (u n v n ) =. It is clear that lucky edge labeling of P + n is {,,, }. + Hence,P n hasluky edge labeling graph. For example, lucky edge labeling of P + and P + 6 are given in the figure a and figure b and η(p + n ) =. u u u u u v v v v v u u Figure a and η(p n + ) = u u u u 6 v Figure b and η(p + n ) = Theorem.: C n : n,, (mod ) has {a, b, c} lucky edge labeling and C n : n 0 (mod ) has {a, b} lucky edge labeling for any a, b, c. v v v v v 6 International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 7

4 Dr. A. Nellai Murugan & R. Maria Irudhaya Aspin Chitra Proof: Let V[C n ] = { u i : i n }and E[C n ] = {{(u i u i+ ): i n-} {(u n u )}}. Case : Let C n be the graph whenn 0 (mod ). Let f: V[C n ] {, } defined by f(u i ) =, i. f(u i- ) =, i. when i (, f * (u i u i+ ) =, f * (u n u ) =. when i, f * (u i- u i ) =. It is clear that the lucky edge labeling of C n : n 0 (mod ) is {,, }. For example, Lucky edge labeling ofc is given in the figure a and η(c n ) =. Case : Let C n be the graph when n (mod ). Let f: V[C n ] {,, } defined by f(u i ) =, i. f(u i- ) =, i. f(u i )=, i (mod ) and i = n. when i, f * (u i u i+ ) =. when i, f * (u i- u i ) =. when i=n, f * (u i u ) = and f * (u i- u i ) =. It is clear that the lucky edge labeling of C n : n (mod ) is clearly {,,, }. For example, Lucky edge labeling of C is shown in the figure b and η(c n ) =. u u u u u u u u u Figure a and η(c n ) = Figure b and η(c n ) = International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 7

5 Lucky Edge Labeling of P n, C n and Corona of P n, C n Case : Let C n be the graph when n (mod ). Let f: V[C n ] {,, } defined by f(u i ) =, i ( f(u i- ) =, i ( f(u i )=, and i = n, n-. u u u u 7 u u 6 u u 6 u 6 u u u u Figure c and η(c n ) = 6 Figure d and η(c n ) = when i (, f * (u i u i+ ) =. when i ( f * (u i- u i ) =. when i= n- and n-, f * (u i u i+ ) =. f * (u n u ) =. It is clear that the lucky edge labeling of C n : n (mod ) is clearly {,,,, 6}. For example, Lucky edge labeling of C 6 is shown in the figure c and η(c n ) = 6. Case : Let C n be the graph when n (mod ). Let f: V[C n ] {,, } defined by f(u i ) =, i f(u i- ) =, i and i= n- f(u i )=, i (mod ). Then the induced edge coloring is when i, f * (u i u i+ ) =. when i, f * (u i- u i ) =. f * (u i- u i ) = and f * (u n u ) =. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 7

6 Dr. A. Nellai Murugan & R. Maria Irudhaya Aspin Chitra It is clear that the lucky edge labeling of C n : n (mod )is clearly {,,, }. For example, Lucky edge labeling of C 7 is shown in the figure d and η(c n ) =. Hence,C n has lucky edge labeling graph. Theorem.: C + n : n 0,,, (mod ) has {a, b, c} lucky edge labeling for any a, b, c. Proof: Let V[C + n ] = {u i : i n} and {v i : i n} and E[C + n ] = {{(u i u i+ ): i n-} {(u n u )} {(u i v i ): i n}}. Case : Let C + n be the graph whenn 0 (mod ). Let f: V[C n ] {,, } defined by f(u i ) =, i. f(u i- ) =, i. f(v i ) =, i n. when i (, f * (u i u i+ ) =. f * (u n u ) =. when i, when i n, f * (u i- u i ) =. It is clear that the lucky edge labeling of C n + : n 0(mod ) is {,,, }. For example, Lucky edge labeling ofc + is shown in the figure a and η(c n + ) =. Case : Let C n + be the graph when n (mod ). Let f: V[C n + ] {,, } defined by f(u i ) =, i. f(u i- ) =, i. f(u i )=, i (mod ) and i = n. f(v ) =, f(v n- ) = f(v i ) =, i n. when i, f * (u i u i+ ) =. when i, when i=n, f * (u i- u i ) =. f * (u i u ) = and f * (u i- u i ) =. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 7

7 Lucky Edge Labeling of P n, C n and Corona of P n, C n whenn- i n, f * (u v ) =. when i n-, It is clear that the lucky edge labeling of C n + :n (mod ) is {,,,, 6}. For example, Lucky edge labeling of C + is shown in the figure b and η(c n + ) = 6. Case : Let C n + be the graph when n (mod ). Let f: V[C n + ] {,, } defined by f(u i ) =, i ( f(u i- ) =, i ( f(u i )=, and i = n, n-. f(v ) =, i = n-, n- and f(v i ) =, i =, n. f(v i ) =, i n-. when i (, f * (u i u i+ ) = when i (, f * (u i- u i ) =. when i= n-, n-, f * (u i u i+ ) =. f * (u n u ) =. when i =, n-, when i n-, f * (u i v i ) =, i = n. It is clear that the lucky edge labeling of C n + :n (mod ) is {,,,, 6}. For example, Lucky edge labeling of C 6 + is shown in the figure c and η(c n + ) = 6. Case : Let C n + be the graph when n (mod ). Let f: V[C n + ] {,, } defined by f(u i ) =, i f(u i- ) =, i and i= n- f(u i )=, i (mod ). f(v i ) =,n- i n. f(v i ) =, i n-. International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 76

8 Dr. A. Nellai Murugan & R. Maria Irudhaya Aspin Chitra Figure a and η(c n + ) = Figure b and η(c n + ) = 6 Figure c and η(c + n ) = 6 Figure d and η(c + n ) = 6 when i, f * (u i u i+ ) = when i, f * (u i- u i ) =. f * (u i- u i ) =. f * (u n u ) =. whenn- i n, when i n-, International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 77

9 Lucky Edge Labeling of P n, C n and Corona of P n, C n It is clear that the lucky edge labeling of C n + :n (mod ) is {,,,, 6}. For example, Lucky edge labeling of C 7 + is shown in the figure d and η(c n + ) = 6. Hence,C n + has lucky edge labeling graph.. CONCLUSION Among all labelling lucky edge labelling has a special importance because it is incorporated with coloring of graphs REFERENCES [] GALLIAN.J.A,A Dynamical survey of graphs Labeling, The Electronic Journal of combinatorics. 6(00) #DS6. [] HARRARY.F, Graph Theory, Adadison-Wesley Publishing Company Inc, USA, 969. [] NELLAIMURUGAN.A- STUDIES IN GRAPH THEORY SOME LABELING PROBLEMS IN GRAPHS AND RELATED TOPICS,Ph.D, Thesis September 0... AUTHORS BIOGRAPHY Dr. A. Nellai Murugan is working as a Associate Professor in thedepartment of Mathematics, V.O. ChidambaramCollege, Tuticorin. He has 0 years of teachingexperience and 0 years of research experience. Hehas participated in number of conferences/seminar atnational and international level. He has publishedmore than 0 research article in the reputed research journals. He is guiding 6 Ph.D research scholars. R.Maria Irudhaya Aspin Chitra is a full time Ph.D Research Scholar working in Graph theory at Department of Mathematics, V.O. ChidambaramCollege, Tuticorin. She has communicated more research articles International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page 78