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Lucky Edge Labeling for Some Graphs The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 LUCKY EDGE LABELING FOR SOME GRAPHS M.Mohana priya1, S.M.Santhiya2 1 Assistant Professor, Department of Mathematics, PSGR Krishnammal College for Women, Bharathidasan University, Coimbatore, Tamilnadu, India. 2 Student in Mathematics, PSGR Krishnammal College for Women, Bharathidasan University, Coimbatore, Tamilnadu, India. [email protected] [email protected] ABSTRACT: Let G be a simple graph with vertex set V(G) and set edge set E(G) respectively. Vertex set V(G) are labeled arbitrarily be 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 called lucky labeling and the graph which admits lucky labeling are called lucky labeled graph. If the Edge set E(G) is called proper coloring of G if it satisfies the condition E(e1)≠E(e2) where e1 and e2 adjacent edges. The least number k for which the graph G has a lucky edge labeling from the set{1,2,………k} is the lucky number of G is denoted by η(G). In this paper we compute lucky edge labeling for complete graph, regular graph, duber graph, connected graph and butterfly graph. And also the general formulas are discussed. KEYWORDS: Labeling, Colouring, Proper colouring, Lucky edge labeling. I. INTRODUCTION All the graphs we used are undirected graphs. The labeling of graphs was introduced by Rosa. A graph labeling is nothing but the assignment of integers or colors to the vertices or edges or both subject to certain terms and conditions. Defintion1.1 An undirected graph G = (V,E) consists of a set V of elements called vertices, and a multiset E (repetition of elements is allowed) of pairs of vertices called edges. A subgraph of a graph G = (V,E) is a graph H(U,F) such that U proper subset of V and F proper subset of E; G is a super graph for H. Fig 1.1 Volume XII, Issue IX, September/2020 Page No:1224 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 Defintion 1.2 A graph G is called labeled if its p points are distinguished from one another by names such as v1,v2,v3,……vn. Fig 1.2 Definition 1.3 Lucky labeling is coloring the vertices arbitrarily such that the sum of labels of all adjacent vertices of a vertex is not equal to the sum of labels of all adjacent vertices of any vertex which is adjacent it. Definition 1.4 A Proper labeling of graph is assigning to its vertices or edges with integers such that no two adjacent vertices or no two adjacent edges have the same labeling subject to some conditions. Definition 1.5 Let G be a simple graph with vertex set V(G) and edge set E(G) respectively. The labeling is said to be lucky edge labeling if the edge set E(G) is a proper coloring of G, i.e., if we have E(e1) E(e2) whenever e1 and e2 are adjacent edges. Definition 1.6 Let G be a simple graph with vertex set VG) and edge set EG) respectively. The vertex set VG) is labeled arbitary by positive integers and E(e) denotes the edge labeled such that it is the sum of labels of vertices incident with edge e. A lucky edge neighborhood labeling of G is an assignment of positive integers to the vertices of G so that each edge neighborhood labels are distinct. The least integer for which a graph G has a lucky edge neighborhood labeling from the set {1,2,…..k} is the lucky edge neighborhood number and is denoted by ηN(G). The graph which admits a lucky edge neighbrhood labeling is called a Lucky edge neighborhood labeled graph. Definition 1.7 + A graph obtained by joining each ui of the path Pn to a vertex vi is called comb and denoted by Pn . Volume XII, Issue IX, September/2020 Page No:1225 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 Definition 1.8 + Cn is graph obtained from Cn by attaching a pendent vertex from each vertex of the graph Cn is called crown. II. MAIN RESULTS LUCKY EDGE LABELING OF CERTAIN GRAPHS Theorem 2.1 The Lucky number of the Bistar graph Bm,n is given by 푚 + 2 푚 ≥ 푛 η(Bm,n)= { ; 푛 + 2 푛 ≥ 푚 Proof Let the vertices K2 be v1 and v2. Let the vertices joined to v1 by the pendent edges be u1,u2,…….um and let the vertices joined to v2 by the pendent edges be w1,w2,………wn. Label the vertices v1 and v2 with 1. Label the vertices u1 by the labels i+1 (i+1 to m) and label the vertices wj by the labels j+1 (j+1 to n). This induces a lucky edge labeling for Bm,n. Fig 2.1 η(Bm,n) = 5 Theorem 2.2 The lucky number of wheel graph Wn is η(Wn) = 2n-1. Proof Let the single vertices and the centre be v1 and the other vertices of (n-1) cycle that are joined to the vertices v1 be v2,v3,…….vn. Label the vertices vi by the label i(i = 1 to n). This induces a lucky edge labeling of Wn and η(Wn) = 2n – 1 . Volume XII, Issue IX, September/2020 Page No:1226 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 Fig 2.2 η(W5) = 9 Theorem 2.3 The lucky number of Fan graph Fn is η(Fn) = 4n+1 Proof Let the middle vertex to the join of all the C3 be v1 and let the remaining vertices be v2,v3,……vn. Label the vertices vi with labels i ( i = 1 to n ). This induces a lucky edge labeling of the Fan graph Fn with lucky number given by η(Fn) = 4n+1. Fig 2.3 η(F3) = 13 Theorem 2.4 The regular graph admits lucky edge labeling with the lucky number 9 Proof Let us label the vertices by v1,v2,…..v6. Label the vertices vi (i=1 to n). This induces the lucky edge labeling of the regular graph η(R5) = 9 Volume XII, Issue IX, September/2020 Page No:1227 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 Fig 2.4 η(R5) = 9 Theorem 2.5 The lucky number of the connected graph cn is (cm,n ) = 5. m 2 : m n (cm,n ) n 2 : n m proof Let the vertices be w1 and w2 . Let the vertices joined to w2 by v1,v2,……. And let the vertices of w1 are joined by u1,u2,……Label the vertices w1 and w2 with 1. Label the vertices ui, by the labels i+1 (i=1 to m) and label the vertices vi by the labels j+1 (j=1 to n). This induces a lucky edge labeling for Cm,n. Fig 2.5 η(C3,3) = 7 Volume XII, Issue IX, September/2020 Page No:1228 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 Theorem 2.6 Duber graph is a lucky graph with lucky number 2n-1. Proof Let the vertices in the inside be named as v1,v2,…..v6 and the cycle which is present outside be v7, v8,…..v12. Label the vertices vi by the label (i=1 to n). This induces a lucky edge labeling of Dn and η(Dn) = 2n-1 Fig : 2.6 η(Dn)=2n Theorem 2.7 Butterfly graph admits the lucky labeling with the lucky number 9 (i.e.,)η(Bn) = 9 Proof Let us label the single vertex as V1 and the other (n-1) cycles that are joined to the vertex V1 be V2, V3,……..Vn. Label the vertices Vi by the label (i=1 to n). This induces a lucky edge labeling of Bn and η(Bn) = 9. Fig: 2.7 η(Bn) = 9 Volume XII, Issue IX, September/2020 Page No:1229 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 Theorem 3.8 Lucky number of the diamond graph is η(Dn) = 7 Proof Let us label the vertices of n-1 cycles by V1,V2,…..Vn. Label the vertices Vi by the label i(i= 1 to n). This induces a lucky edge labeling of diamond graph Dn and η(Dn) = 7 Fig 2.8 η(Dn) = 7 CONCLUSION In this paper lucky numbers are found for connected graph, regular graph diamond graph, duber graph, etc., and we can extend this paper by found the lucky numbers some graphs too. From this we know that the graphs which admits lucky labelling must undergoes proper colouring. REFERENCES 1. Aishwarya. A “Lucky Edge Labeling of Different Graphs”, International Journal of Innovative Research in Science, Engineering and Technology, Volume 4, Issue 2, February 2015. 2. Bondy.J.A and Murthy.U.S.R, “Graph Theory with applications” ,Newyork Macmillan Ltd Press, 1976. 3. Douglas B. “West, Introduction to Graph Theory” ̵ Second edition, Published by Prentice Hall, 2001. 4. Maria Irudhaya Aspin Chithra. R and Nellai Murugan. A, “Lucky edge labeling of some special graphs”, International Journal of Recent Research Aspects ISSN: 2349 ̵ 7688, Special issue: Conscientious Computing Technologies, April 2018. 5. Nellai Murugan .A, Maria and Irudhaya Aspin Chithra. R “Lucky edge labeling of Triangular Graphs”, International Journal of Mathematics Trends and Technology (IJMTT), Volume 36 Number 2 ̵ August 2016.
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