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(IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No.7, July 2011 Even Harmonious Graphs with Applications P.B.SARASIJA R .BINTHIYA Department of Mathematics, Department of Mathematics, Noorul Islam Centre for Higher Education, Noorul Islam Centre for Higher Education, Kumaracoil,TamilNadu,India. Kumaracoil,TamilNadu,India. Abstract— In this paper we plan to contribute an even harmonious labeling of some well known graphs. Let G(V,E) be A function f is called the even harmonious labeling of a graph with p vertices and q edges. A graph G(p,q) is said to be a graph G(V, E) if f : V → {0, 1, 2, ……., 2q} is injective and even harmonious if there exists an injection f : V → { 0,1,…,2q} the induced function f * : E → {0, 2,…,2(q– 1)} is defined as such that the induced mapping f*(uv) = (f(u)+f(v)) (mod 2q) is a bijection from E onto { 0,2,4,…,2(q-1)}. f*(uv) = (f(u) + f(v)) (mod 2q ) is bijective, the resulting edge labels should be distinct. Keywords- Harmonious labeling, Path, Cycle, Complete A graph which admits even harmonious labeling is bipartite graph, Bistar. called an even harmonious graph. Theorem 1.1. If a graph G (p,q) is even harmonious graph I.INTRODUCTION then 2q and 0 are the maximal label and the minimal label of vertices respectively . Throughout this paper, by a graph we mean a finite, undirected, simple graph. For notations and terminology we Theorem 1.2. In even harmonious labeling two follow Bondy and Murthy [ 1 ]. Graph labeling where the consecutive integers cannot be the labels of any two vertices vertices are assigned values subject to certain conditions have in G. been motivated by practical problems. Labeled graphs serves as models in a wide range of applications such as Coding II. MAIN RESULTS theory, the x-ray crystallography , to design a communication network addressing system etc. Theorem 2.1. A path Pn (n ≥ 2 ) is even harmonious. We denoted the path on n vertices by Pn, the cycle on Proof. n vertices by Cn, the complete bipartite graph by Km,n and the star graph by K1,n. In [ 3 ] Graham and Sloane have introduced Case (i): The number of vertices is even. the harmonious labeling of a graph. Zhi – He Liang, Zhan – Let m = 2n LiBai [ 4 ] have introduced the odd harmonious labeling of a graph. For a detailed survey on graph labeling we refer to Let Pm be the path with vertices vi, 1 ≤ i ≤ m Gallian [2 ]. We refer [ 5] also. Define f(vi) = 2( i – 1 ), 1 ≤ i ≤ m . Let G(V,E) be a graph with p vertices and q edges. Then f is an even harmonious labeling of Pm. A graph G with q edges is harmonious if there is an injection ‘f’ from the vertices of G to the group of integers Case (ii): The number of vertices is odd. modulo q such that when each edge xy is assigned the label Let m = 2n + 1 (f(x) + f(y)) (mod q), the resulting edge labels should be distinct. Let Pm be the path with vertices vi, 1 ≤ i ≤ m Define f(v2i - 1) = 2( i – 1 ), i = 1,2,…, m/2. A graph G(p,q) is said to be odd harmonious if there exists an injection f : V → {0, 1, 2, …., 2q – 1 } such that f(v2i) = m+ ( 2i – 1 ), i = 1,2,…, m/2 induced mapping f*(uv) = (f(u) + f(v)) is bijection from E on Then f is an even harmonious labeling of Pm. to {1, 3, 5, …., 2q - 1}. Then f is said to become an odd harmonious labeling of G. Theorem 2.2. The complete bipartite graph Km,n is even harmonious. The floor function assigns to the real number x the largest integer that is less than or equal x. The value of the Proof. Let the vertices sets U and V be the bipartition of Km,n, floor function at x is denoted by x . where U = { uj, 1 ≤ j ≤ m} and V = { vi, 1 ≤ i ≤ n}. The ceiling function assigns to the real number x the Define f(uj) = 2( j – 1 ), 1 ≤ j ≤ m and smallest integer that is greater than or equal x. The value of f(vi) = 2mi, 1 ≤ i ≤ n. the ceiling function at x is denoted by x . Then f is an even harmonious labeling of Km,n. 161 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 7, July 2011 Corollary 2.3. The star graph K1,n is even harmonious. 10 0 Proof. 10 Replaced m by 1 in Theorem 2.2 the result followed. 14 6 12 Theorem 2.4. Any cycle of odd length is even harmonious. 2 8 4 Proof. 16 0 Let C2n+1 ( n ≥ 1) be the cycle of odd length with vertices v1,v2, …,v2n+1. Define f(vi) = 2( i – 1 ), 1 ≤ i ≤ 2n+ 1. 2 4 6 8 Then f is an even harmonious labeling of odd cycles. Figure 2 : K2 + K4 c Definition 2.5. The Bistar graph Bm,n is the graph obtained k from K2 by joining m pendant edges to one end of K2 and n Definition 2.8. Pn , the kth power of Pn, is the graph pendant edges to the other end of K2. obtained from the path Pn by adding edges that join all vertices u and v with distance (u,v) = k. Theorem 2.6. The Bistar graph Bm,n is an even harmonious. 2 Theorem 2.9. The graph Pn is an even harmonious. Proof . Let u, v be the vertices of K2 in Bistar graph Bm,n and U = { uj, 1 ≤ j ≤ m} and V = { v i, 1 ≤ i ≤ n} be the vertices Proof. 2 adjacent to u and v respectively. Let v1,v2,…,vn be the vertices of Pn . Define f(u) = 0, f(v) = 2q, f(uj) = 2n+2j, 1 ≤ j ≤ m and Define f(vi) = 2( i – 1), 1 ≤ i ≤ n. f(vi) = 2i, 1 ≤ i ≤ n. Then f is even harmonious. Then f becomes an even harmonious labeling of Bistar graph 2 For example, Figure 3 is an even harmonious graph of Pn . For example, even harmonious labeling of the Bistar graph Definition 2.10. [3] The friendship graph Fn ( n ≥ 1 ) B3,4 is shown in Figure 1. consists of n triangles with a common vertex. Theorem 2.7. The graph K2 + Knc is even harmonious. Theorem 2.11. The friendship graph F2n +1 is even harmonious. Proof. Let V(K2) = {u1,u2} and V(Knc) = { v1, v2,…, vn} . Proof. Define f(u1) = 2(n + 1) , f(u2) = 0 and f(vi) = 2i, 1 ≤ i ≤ n The friendship graph F 2n+1 consists of 2n + 1 triangle Then f becomes an even harmonious labeling of K2 + Knc. with 4n + 3 vertices and 6n + 3 edges. Let v 1 be the common For example, even harmonious labeling of the graph K2 + Knc. vertex and v1,vi,vi+1 , i= 2,4,…,2(2n+1) be the vertices of each is shown in Figure 2. triangle. Define f(v1) = 0, f( v2i +2) = 2(3i +1), i = 0, 1,…, 2n and f(v2j+1) = 2(3j -1), j = 1, 2,…, 2n + 1. Then f becomes an even harmonious graph. 2 14 4 8 12 14 2 4 4 12 12 0 2 6 0 10 6 10 16 0 6 0 2 4 6 8 Figure 1:B3,4 8 2 10 8 Figure 3 : Pn 162 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 7, July 2011 REFERENCES [1] J.A.Bondy and U.S.R.Murthy, Graph Theory with Applications, Macmillan, London, 1976. [2] J.A.Gallian, A dynamic survey of graph labeling, The electronics J. of Combinatorics, 16, 2009. [3] R.L.,Graham and N.J.A.,Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Discrete Meth., 1,1980, pp. 382-404. [4] Z-H Liang and Z-L Bai , On the odd harmonious graphs with applications, J. Appl. Math. Comput., 29, 2009, pp. 105-116. [5] B.Liu and X.Zhang, On harmonious labeling of graphs, Ars. Combin.,36 ,1993, pp. 315 – 326. 163 http://sites.google.com/site/ijcsis/ ISSN 1947-5500