Vertex Graceful Labeling of C j C k C l

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1 Applied Mathematical Sciences, Vol. 8, 01, no. 8, HIKARI Ltd, Vertex Graceful Labeling of C j C k C l P. Selvaraju 1, P. Balaganesan,5, J. Renuka 3 and M.L. Suresh 1 Department of Mathematics Vel Tech Multi Tech Dr. Rangarajan Dr. Sankanthula Engineering College Avadi, Chennai , India Research Scholar, Department of Mathematics Hindustan University Chennai , India 3 Departments of Mathematics Sri Sai Ram Engineering College Chennai , India 5 Department of Mathematics Saveetha School of Engineering Saveetha University Chennai , Tamil Nadu, India Copyright c 01 P. Selvaraju, P. Balaganesan J. Renuka, M.L. Suresh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The disjoint union C j C k C l is component of three cycles C j,c k and C l. In this paper, we prove that the disjoint union graph C j C k C l is vertex graceful for odd j + k + l with l j + k +5. Mathematics Subject Classification: 05C78 Keywords: Vertex graceful, disjoint of Union cycles, vertex labeling, edge labeling

2 08 P. Selvaraju, P. Balaganesan J. Renuka, M.L. Suresh 1 Introduction Graph labeling, where the vertices are assigned values subject to certain conditions have often been motivated by practical problems. Labeled graphs serves as useful mathematical models for a broad range of applications such as coding theory, including the design of good radar type codes, synch-set codes, missile guidance codes and convolution codes with optimal auto correlation properties. They facilitate the optimal nonstandard encoding of integers. All graphs in this paper are finite, simple graphs and undirected graph. The symbols V (G) and E(G) denote the vertex set and edge set of a graph G. The cardinality of the vertex set is called the order of G. The cardinality of the edge set is called the size of G. A graph with p vertices and q edges is called a G(p, q) graph. Most graph labeling methods trace their origin to one introduced by Rosa [1] called such a labeling a β-valuation and Golomb [] subsequently called graceful labeling, and one introduced by Graham and Sloane [3] called harmonious labeling. Several infinite families of graceful and harmonious graphs have been reported. Many illustrious works on graceful graphs brought a tide to different ways of labeling the elements of graph such as odd graceful.. A graph G with p vertices and q edges is said to be vertex graceful if there exists labeling f : V (G) 1,, 3,,p} such that the induced labeling f + : E(G) Z q defined by f + (u, v) =f(u)+f(v)(mod q) is a bisection. The concept of vertex graceful was introduced by Lee, Pan and Tsai in 005 []. In this paper, we show that the disjoint union of three cycles with odd order, namely C j C k C l is vertex graceful for odd j + k + l with l j + k +5. Main Results Theorem.1. The disjoint union C j C k C l of cycles C j,c k and C l is vertex graceful for odd for odd j + k + l with l j + k +5. Proof. Let G be the disjoint union on n vertices and q edges, then n = q = j + k + l. Let V (G) =u i :0 i j 1} v i :0 i k 1} w i : 0 }, E(G) =u i u (i+1)mod j :0 i j 1} v i v (i+1)mod k :0 i k 1} w i w (i+1)mod l :0}. θ = ( ) l j k 3 mod. By symmetry, we only need to consider three cases as below: Case (i): k, l are even and j is odd. We label the vertices as follows: i +1, 0 i ( ) j 1 f(u i )= (j i), ( ) j+1 i j 1

3 Vertex graceful labeling of C j C k C l 09 q i, 0 i ( k f(v i )= 1 q (k i)+1, ( k i k 1 j +1,i=0 j +i + θ, 1 i l j k 3 θ l j k+1 θ j +i ++θ, i l j+k 7 θ l j+k 3 θ j +i ++θ, i l+j+k 9 θ j +i +3+θ, i = l+j+k 5 θ l+j+k 1 θ j +i ++θ, i l f(w i )= 1 θ l j 1+(l i) θ, 1 i 3l j k 7 θ 3l j k 3 θ j 3+(l i) θ, i 3l+j k 9 θ j +(l i) θ, i= 3l+j k 5 θ j 1+(l i) θ, j +(l i) θ, i = 3l+j+k 1 θ j +1+(l i) θ, 3l+j k 1 θ i 3l+j+k 5 θ Let D be the label set of all edges, then we have D = D 1 D D 3, where D 1 = u i u (i+1)mod j :0 i j 1},D = v i v (i+1)mod k :0 i k 1},D 3 = w i w (i+1)mod l :0}. It is obvious that the labels of each edge are different. So g maps E onto 0, 1,,, (q 1)}. According to the definition of vertex graceful labeling, we can conclude that the disjoint union cycles C j C k C l is vertex graceful for k, l are even and j is odd and l j + k +5. u u 1 5 u v 0 18 v v 1 v 1 w 11 w 10 w 9 w 8 w 7 w w 0 17 w 1 1 w 6 w 3 8 w 10 w 5 Figure 1: Case (ii): j, k are even and l is odd. We label the vertices as follows:

4 050 P. Selvaraju, P. Balaganesan J. Renuka, M.L. Suresh f(u i )= i +1 (j i), ( j q i, 0 i ( j 1 i j 1, 0 i ( k 1 f(v i )= q (k i)+1, ( k i k 1 j +1,i=0 j +i + θ, 1 i l j k 3 θ l j k+1 θ j +i ++θ, i l j+k 7 θ l j+k 3 θ j +i ++θ, i l+j+k 11 θ l+j+k+1 θ j +i +6+θ, i l 7 l 5 j 5+(l i) θ, i 3l j k 13 θ f(w i )= j +(l i) θ, i= 3l j k 9 θ 3l j k 5 θ j 3+(l i) θ, i 3l+j k 9 θ j +(l i) θ, i= 3l+j+k 5 θ j 1+(l i) θ, j +(l i) θ, i = 3l+j+k 1 θ j +1+(l i) θ, 3l+j k 1 θ i 3l+j+k 5 θ By an argument similar to the one in case 1, we have this assignment provides a vertex graceful labeling for j, k are even and l is odd and l j+k+5. u u 3 v 0 v u 7 1 u v 1 v w 1 w 11 w 10 w 9 w 8 w w 0 w 1 w w 3 w w 5 w 6 Figure : Case (iii): j, k, l are all odd. We label the vertices as follows:

5 Vertex graceful labeling of C j C k C l 051 i +1, 0 i ( ) j 1 f(u i )= (j i), ( ) j+1 i j 1 q i, 0 i ( ) k 1 f(v i )= q (k i)+1, ( ) k+1 i k 1 j +1,i=0 j +i + θ, 1 i l j k 3 θ j +i ++θ, l j k+1 θ i l j+k 5 θ j +i +1+θ, i = l j+k 1 θ l j+k+3 θ j +i + θ, i l+j+k 3 θ l+j+k+1 θ j +i ++θ, i l 3 l 1 f(w i )= j +(l i) θ, i 3l j k 5 θ j +(l i) θ +1, i = 3l+j k 1 θ 3l j k+3 θ j +(l i) θ, i 3l+j k 3 θ 3l+j k+1 θ j 1+(l i) θ, i 3l+j+k 5 θ j +(l i) θ, i = 3l+j+k 1 θ j +1+(l i) θ, j +1+(l i) θ, Since the proof in this case is similar to the one in case 1, we omit it. Thus, we have that this assignment provides a vertex graceful labeling for C j C k C l, when j, k, l are all odd and l j + k + 5. Therefore, we can conclude that the graph of the disjoint union of C j C k C l is vertex graceful for odd j + k + l with l j + k +5. u 0 1 v u 1 u v 1 v w 10 w 9 w 8 w 7 w w 0 w 1 w w 3 w w 5 Figure 3:

6 05 P. Selvaraju, P. Balaganesan J. Renuka, M.L. Suresh Conjecture: The disjoint union of C j C k C l is vertex graceful for any j, k, l. References [1] J.A.Gallian, A Dynamic Survey of Graph Labeling, Electronic. J. Combinatorics. [] S.W. Golomb, (197 How to number a graph, in Graph Theory and Computing, R.C. Read, ed., Academic Press, New York, pp [3] R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J.Alg. Discrete Math., 1 (1980) [] F. Harary. (1969) Graph Theory, Addison-Wesley, Reading MA. [5] Jie Xi, Harmonious labeling of C j C k C l, Advanced Materials Research Vol. 30, 011, [6] A. Rosa, (1967) On certain valuations of the vertices of a graph, in Theory of Graphs, International Symposium, Rome, July 1966, Gordon and Breach, New York and Dunod, Paris, pp Received: May 9, 01