Construction of Some (n,m) Petersen Graphs. with Vertex Magic Total Labeling

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1 Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 4, Construction of Some (n,m) Petersen Graphs with Vertex Magic Total Labeling R. Senthil Amutha Department of Mathematics Sree Saraswathi Thyagaraja College, Pollachi, India N. Murugesan Department of Mathematics, Government Arts College Coimbatore, India Abstract. A vertex magic total labeling on a graph with v vertices and e edges is a one to one map taking the vertices and edges onto the integers 1, 2, 3, v + e with the property that the sum of the label on the vertex and the labels of its incident edges is constant, independent of the choice of the vertex. In this paper we discuss algorithms for vertex magic total labeling of (n,m) Petersen graphs when n is even. Mathematics Subject Classification: 05C78 Keywords: Path, Cycle, Labeling, Vertex magic total labeling, (n,m)petersen graph I. INTRODUCTION A labeling of a graph is an assignment of labels, usually numbers to its vertices or its edges or sometimes to both of them. There are various attempts on labeling of graphs in the literature. J.A Gallian [1] in the year 1998 completely compiled the survey of graph labelings. J.A. MacDougall and others [3] introduced Vertexmagic total labelings of graphs, W. D. Wallis and others [6] introduced Edgemagic total labelings that generalize the idea of a magic square and can be referred for magic labelings. A vertex magic total labeling on a graph with v vertices and e edges is a one to one map taking the vertices and edges onto the integers 1, 2, 3, v + e with the property that the sum of the label on the vertex and the labels of its incident edges is constant k, independent of the choice of the vertex. In this paper, we construct the vertex magic total labeling of (n,m) Petersen graphs when n is even.

2 154 R. Senthil Amutha and N. Murugesan II. VERTEX MAGIC TOTAL LABELING A graph G(V,E) has vertex set V = V(G) and edge set E = E(G) and e = E and v = V. In general, by labeling a graph we mean an injective map defined from the set of vertices to the set of natural numbers, the same is extended to the set of edges. In this paper we define labeling from the set of vertices and edges to the set of natural numbers, such that the sum of labels of vertex and the edges incident to that vertex is a constant. Let G = {V, E, f} be a simple graph with v = V and e = E. A one to one and onto mapping f from V E to the finite subset {1,2, v+e} of natural numbers such that for every vertex x, f(x) + f(xy i ) = k, where y i s are vertices adjacent to x. For example, consider the path P 3.This is a graph with 3 vertices and 2 edges as given below Fig.1: Path P 3 In this graph the integers from 1 to 5 are used such that (i). At the vertex labeled 5, vertex label + incident edge label = = 6 (ii). At the vertex labeled 3, vertex label + sum of incident edges label = 3 + (1+2) = 6 (iii). At the vertex labeled 4, vertex label + incident edge label = = 6. This graph has vertex magic total labeling with constant k = 6. As another example consider the complete graph K Fig.2: Complete graph K 3 This is a graph with 3 vertices and 3 edges which are labeled from 1 to 6 such that (i). At the vertex labeled 6, vertex label + sum of incident edges label = 6 + (1 +2) = 9 (ii).at the vertex labeled 5, vertex label + sum of incident edges label = 5 + (1+3) = 9 (iii).at the vertex labeled 4, vertex label+ sum of incident edges label = 4 + (2 +3) = 9. This graph has vertex magic total labeling with constant k = Next let us consider the Path P Fig.3: Path P 2

3 Construction of some (n,m) Petersen graphs 155 This is a graph with 2 vertices and 1 edge which are labeled from 1 to 3 such that (i). At the vertex labeled 2, vertex label + sum of incident edges label = = 3 (ii). At the vertex labeled 3, vertex label + sum of incident edges label = = 4 This labeling is not vertex magic total labeling because the constant k differs from vertex to vertex. III. VERTEX MAGIC TOTAL LABELING FOR P(N,M) PETERSEN GRAPHS WHEN m IS ODD There are a number of standard graphs.one such graph is the Petersen graph. A generalized Petersen graph P(n,m), 1 m < n /2, consists of an outer n-cycle y 1,y 2,y 3,,y n, a set of spokes y i x i, 1 i n, the n inner edges x i x i+m, 1 i n, with indices taken modulo n. The standard Petersen graph is (5,2) is given below. Fig.4: (5,2) Petersen graph MirkaMiller and others [4] have discussed that P(n,m) is regular graph of degree 3 and has v = 2n vertices and e = 3n edges. Therefore there are totally 5n labeling required to label the Petersen graph P(n,m). It was also found in [4] that the feasible values for the constant k to define vertex magic total labeling on the Petersen graphs P(n,m) is 17n/ k 23n/ When m = 1, the P(n,1) Petersen graph is obtained. It is a graph which can be defined as a Cartesian product P 2 x C n of a path on two vertices with a cycle on n vertices. If n is even, n 4 then the generalized Petersen graph P (n,m) has a vertex magic total labeling with k = 17n/ as in [4]. The following is an algorithm for finding vertex magic total labeling (VMTL) for a P(n,1) Petersen graph when n is even and m is odd. Algorithm: In this algorithm anticlockwise and clockwise directions are represented as AC & C respectively. For all suffices l of x,when l > n, l = a mod n. Step 1: Start assigning any edge y i y i+1 in the outer cycle as 1. Step 2: Assign the edge y i+2 y i+3 as 2.In the same manner assign the edges in the outer cycle y i+4 y i+5, y i+6 y i+7 with consecutive numbers 3,4, in C. Step 3: Assign the spoke y i+1 x i+1 with the next natural number and continue assigning the spokes y i+(n-1) x i+(n-1), y i+n-3 x i+n-3, with the consecutive numbers in AC. Step 4: For the spoke y i+2 x i+2 assign the next number and start assigning the spokes y i+4 x i+4, y i+6 x i+6 up to y i x i with the consecutive numbers in C. Step 5: Assign the inner cycle edges x i x i+(n-m), x i+(n-2) x i+(n-m-2) upto x i+2 x i+(n-m+2) in AC. Next assign edges x i+2 x i+(m+2), x i+4 x i+(m+4) up to x i x i+m in C.

4 156 R. Senthil Amutha and N. Murugesan Step 6: Start assigning the remaining outer cycle edges as y i y i+(n-1 ), y i+(n-2) y i+ +(nthe next number at x i+1,x i+3,.up to x i+ (n-1) in C 3),, y i+2 y i+1 in AC. Now all the outer edges are labeled. Step 7: Label the inner cycle vertices x i, x i+(n- 2), x i+(n-4),...., x i+2 in AC. Assign Step 8: Label the outer cycle vertices y i+(n-2) ), y i+(n-4), y i in AC. Next assign y i+ +3,y i+5..., y i+ +1 in C. Step 9: Stop the process if all the vertices of edges are labeled. Let us use this algorithm for a P( (6,1) Petersen graph and(8,3) Petersen graph to get a vertex magic total labeling as in the figures below. Fig.5: VMTL of P(6,1) Petersen graph and P(8,3) Petersen graph IV V. VERTEX MAGIC TOTAL LABELING P(n,,m) PETERSEN GRAPHS WHEN m IS EVEN The following is an algorithm for the construction of the vertex magic total labeling (VMTL) of the P(n,m) Petersen graph when both n and m are even. Algorithm: As in the previous algorithm anticlockwise and clockwise directions are represented as AC & C respectively. For all suffices l of x, when l > n, l = a mod n. Step 1: Start assigning any edge y i y i+1 in the outer cycle as 1. Step 2: Assign the edgee y i+2 y i+3 as 2.In the same manner assign the edges in the outer cycle y i+4y i+5, y i+6 y i+7 with consecutive numbers 3,4,, in C. Step 3: Assign the spoke y i+1 x i+1 with the next natural number and continue assigning the spokes y i+( (n-1)x i+(n-1), y i+n-3 x i+n-3, with the consecutive numbers in AC. Step 4: For the spoke y i+2 x i+2 assign the next number and start assigning the spokes y i+4 x i+ +4, y i+6 x i+6 up to y i x i with the consecutive numbers in C. Step 5: Assign the inner edges x i x i+(n-m), x i+(n n-2)x i+(n-m-2) upto x i+2 x i+( (n-m+2) in AC. Next assign edges x i+(n-m m+3)x i+3, x i+( (n-m+5)x i+5 up to x i+(n-m+ 1) x i+1 in C. Step 6: Startt assigning the remaining outer cycle edges as y i y i+(n-1 1), y i+(n-2) y i+ +(n- number at x i, x i+2,.up to x i+(n-2) in C 3),, y i+2 y i+1 in AC. Now all the outer edges are labeled. Step 7: Label the inner vertices xi+(n-1), x i+(n- -3),..., x i+1 in AC. Assign the next Step 8: Label the outer cycle vertices y i+(n-2) ), y i+(n-4), y i in AC. Next assign y i+ +3,y i+5..., y i+ +1 in C. Step 9: Stop the process if all the vertices of edges are labeled.

5 Constructionn of some (n,m) Petersen graphs 157 Using this algorithm the P(6,2) Petersen graph and P(8,2) Petersen graph labeled as a vertex magic total labeling as in the figures given below. is Fig.6: VMTL of P(6,2) Petersen graph and P(8,2) Petersen graph V. PERFORMANCE OF THE ALGORITHMS For a given graph theory problem it is desirable to have an algorithm which guarantees a solution in a execution time proportional to some constant power of v or e.in other words the execution time t can be expressed as t α v k or t β e q Here, t is the execution time for the generation of alll such graphs, α and β are the multiplicative constants whichh shows the number of units of time taken to generate one such graph with respect to vertices and edges respectively, k and q are constants, such that lower these values better is the algorithm. Such an algorithm whose computation time is bounded by a polynomial in v or e is called a polynomial bounded algorithm. Different algorithms bounded by the same power of v may take different amounts of the actual computer time for the same graph because of their different multiplicative constants [5]. For the above two algorithms constructed, it is possible to generate v or e number of (n, m) Petersen graphs depending on the choice of the edge in the outer cycle. Therefore a polynomial bound algorithm can be found for these algorithms with k = 1and q = 1. If each such graph is generated in α or β units of time, the algorithm to generate all such (n, m) Petersen graphs would consume αv or βe units of time. VI. CONCLUSION In this paper we have discussed algorithms to find vertex magic total labeling of (n,m) Petersen graphs when n is even. A similar attempt can also be made for (n,m)petersenn graphs when n is odd. It may be interesting to develop such an algorithm for vertex magic total labeling of some other standard graphs.

6 158 R. Senthil Amutha and N. Murugesan REFERENCES [1] J. A. Gallian, A dynamic survey of graph labeling, Electronic J. Combinatorics 5 (1998), #DS6 [2] John Clark, Derek Allan Holton, A first look at graph theory, Allied publishers ltd.(1991). [3] J.A.MacDougall, Mirka Miller & Slamin & W.D.Wallis, Vertex-magic total labellings of graphs, Utilitas Math. 61 (2002) [4] Mirka Miller, Martin Baca, James A.Macdougall, Vertex magic total labeling of generalized and Petersen graphs and Polytopes, [5] Narsingh Deo, Graph theory with applications to Engineering and Computer Science 56 (2000). [6] W. D. Wallis, E. T. Baskoro, Mirka Miller & Slamin, Edge-magic total labellings, Australian J. Combin.22 (2000), [7] D. B. West, An Introduction to Graph Theory, Prentice-Hall (2001). Received: July, 2011