How to construct new super edge-magic graphs from some old ones

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1 How to construct new super edge-magic graphs from some old ones E.T. Baskoro 1, I W. Sudarsana 2 and Y.M. Cholily 1 1 Department of Mathematics Institut Teknologi Bandung (ITB), Jalan Ganesa 10 Bandung 40132, Indonesia {ebaskoro, yus}@dns.math.itb.ac.id 2 Department of Mathematics, Tadulako University, Jalan Sukarno-Hatta Palu, Indonesia isudarsana203@yahoo.com Abstract. In this paper, we study the property of super edge-magic total graphs. We give some further necessary conditions for such graphs. Based on this condition we provide some algorithms to contruct new super edge-magic total graphs from some old ones. Keywords : super, edge-magic total labeling 1 Introduction All graphs, in this paper, are finite and simple. A general reference for graph-theoretic ideas can be seen in [4]. For a graph G with vertex-set V (G) and edge-set E(G) an edge-magic total labeling is a bijection λ: V (G) E(G) {1, 2,, V (G) E(G) } with satisfying the property that there exists an integer k such that λ(x) + λ(xy) + λ(y) = k, for any edge xy in G. We call λ(x)+λ(xy)+λ(y) the edge sum of xy, and k the magic constant of graph G. In particular, if λ(v (G)) = {1, 2,, V (G) } then λ is called super edge-magic total labeling. A graph is called (super) edge-magic total if it admits any (super) edge-magic total labeling. The notion of edge-magic total graphs was introduced and studied by Kotzig and Rosa [5] with a different name as graphs with magic valuations, while the term of super edge-magic total graphs was firstly introduced by Enomoto et al. [1]. They showed that a Supported by Hibah Bersaing XII DP3M-DIKTI Indonesia, 2004, DIP Number: 004/XXIII/1/ /2004. Supported by Hibah Pekerti DP3M-DIKTI Indonesia, 2004.

2 2 E.T. Baskoro, I W. Sudarsana and Y.M. Cholily star S n+1 = K 1,n is the only complete bipartite graph which is super edge-magic total. They also showed that any odd cycle is super edge-magic total, but any wheel is not. Since then, a number of papers have studied super edge-magic property in graphs. For instances, Figueroa-Centeno et al. [3] and [2] derived a necessary and sufficient condition for a graph to be super edge-magic total and they also showed several class of graphs, such as fans f n = Pn +K 1 with n 6, ladders L n = Pn P 2 for odd n, and the generalized prism G = C m P n for odd m and n 2, are super edge-magic. They also studied the relationships between super edgemagic labeling with other labelings. However, a conjecture Every tree is super edge-magic total proposed by Enomoto et al. [1] still remains open. In this paper, we study super edge-magic total labelings. We derive more necessary conditions to be able to know more deeply the property of such labelings. Based on this condition we give some algorithms to contruct a new super edge-magic labeling from some old ones. By using these algorithms we can provide more evidence to support the correctness of the conjecture proposed by Enomoto et al. 2 Some further necessary conditions Several necessary conditions for a graph to be super edge-magic have been derived by several authors. Enomoto et al. [1] showed that if a nontrivial graph G is super edge-magic then E(G) 2 V (G) 3. Furthermore, Figueroa-Centeno et al. [3] provide a neccessary and sufficient condition for a graph being super edge-magic as in the following lemma. Lemma 1. A (p, q)-graph G is super edge-magic if and only if there exists a bijective function f : V (G) {1, 2,, p} such that the set S = {f(u) + f(v) : uv E(G)} consists of q consecutive integers. In such a case, f extends to a super edge-magic labeling of G with the magic constant k = p+q+s, where s = min(s) and S = {f(u) + f(v) : uv E(G)} = {k (p + 1), k (p + 2),, k (p + q)}.

3 How to construct new super edge-magic graphs 3 Further, in order to know what possible values of k s for graph G to be super, we add the following neccessary conditions. Lemma 2. Let a (p, q)-graph G be super edge-magic total. Then, the magic constant k of G satisfies p + q + 3 k 3p. Proof. Since G is super edge-magic total then the vertices of G receive labels 1, 2,, p and the edges receive p + 1, p + 2,, p + q so that by Lemma 1 S = {f(u) + f(v) : uv E(G)} consists of consecutive integers a, a + 1,, a + q 1 for some positive integer a. The smallest possible magic constant of G obtained if a = 3. In this case the vertices of G with labels 1 and 2 are adjacent and the magic constant for this case must be k = (a + q 1) + (p + 1) = p + q + 3. If the vertices of labels p 1 and p are adjacent in G then we obtain the biggest possible magic constant of G, namely k = (p 1)+p+(p+1) = 3p. Therefore we obtain p+q +3 k 3p. The lower and upper bounds in Lemma 2 are tight, since the super edge-magic labelings λ 1 and λ 2 on a star S n of n vertices shown in Fig. 1 have the magic constant 2n + 2 and 3n, respectively. 2 3 n 2n-1 2n-2 n+1 1 λ n-1 2n-1 2n-2 n+1 n λ 2 Fig. 1. A star graph achieves the lower and upper bounds of k. Corollary 1 If k is the magic constant of a tree with p vertices then 2p + 2 k 3p. Furthermore, the magic constant of a (p, q)-graph G with c components ranges between 2p c + 3 to 3p.

4 4 E.T. Baskoro, I W. Sudarsana and Y.M. Cholily Proof. The first statement holds, since in any tree the number of edges is one less than the number of vertices. If a (p, q)-graph G has c components then in each component G i (i = 1, 2,, c) we have E(G i ) V (G i ) 1. Thus, E(G) V (G) c and by Lemma 2 it implies that 2p c + 3 k 3p. 3 Duality in super edge-magic labeling Given any edge-magic total labeling λ on a (p, q)-graph G, Wallis et al. [6] define the dual labeling λ of labeling λ as follows. λ (v i ) = M λ(v i ), v i V (G), and λ (x) = M λ(x), x E(G), where M = p + q + 1. It is easy to see that if λ is edge-magic total with the magic constant k then λ is edge-magic total with the magic constant k = 3M k. It is also easy to see that if λ is super edge-magic total then λ is no longer super edge-magic total. In the next theorem, we introduce another dual property which preserve the superness of edge-magic total labelings. Theorem 1. Let a (p, q)-graph G be super edge-magic total. Let λ be a super edge-magic total labeling of G with the magic constant k. Then, the labeling λ defined: λ (v i ) = p + 1 λ(v i ), v i V (G), and λ (x) = 2p + q + 1 λ(x), x E(G) is a super edge-magic total labeling with the magic constant k = 4p + q + 3 k. Proof. Let uv E(G). Then, λ (u) + λ (uv) + λ (v) = (p + 1 λ(u)) + (2p + q + 1 λ(uv)) + (p + 1 λ(v)) = 4p + q + 3 (λ(u) + λ(uv) + λ(v)) = 4p + q + 3 k a constant. Therefore, λ is a super edge-magic total labeling of G with magic constant k = 4p + q + 3 k. The labeling λ in Theorem 1 is called the dual super labeling of λ on G.

5 How to construct new super edge-magic graphs 5 4 Construction of new labelings In this section, we give algorithms to construct new super edge-magic graphs by extending the old ones. Theorem 2. From any super edge-magic (p, q)-graph G with the magic constant k, we can construct a new super edge-magic total graph from G by adding one pendant incident to vertex x of G whose label k 2p 1. The magic constant of the new graph is k = k + 2. Proof. In the new graph, define a labeling in the following. Preserve all vertices labels of G. Increase the labels of all edges (except the new one) by 2. Label the new vertex and edge by p + 1 and p + 2 respectively. It can be verified that the resulting labeling on the new graph is super edge-magic total labeling with magic constant k = k + 2. Since 2p + 2 k 3p (by Lemma 2), the proposition holds for any value of k. Theorem 3. Let a (p, q)-graph G be super edge-magic total with the magic constant k and k 2p + 3. Then, a new graph formed from G by adding exactly two pendants incident to two distinct vertices x and y of G whose labels k 2p and k 2p 2 respectively is super edge-magic total with the magic constant k = k + 4. Proof. In the new graph denote by u and v the new vertices adjacent to x and y, respectively. Then, define a labeling in the new graph as follows. Preserve labels of all the vertices of G. Add all edge labels (except the new ones) by 4. Label vertices x and y by p+1 and p+2, respectively and label two new edges xu and yv by p + 3 and p + 4, respectively. For the new edges, clearly we have the edge sum of each is k + 4. Since each label of old edge increased by 4 then we get the edge sum of each old edge is also k + 4. Therefore, the new graph is super edge-magic total labeling. This process works only if k 2p 2 1. This implies that k 2p + 3. Theorem 4. Let a (p, q)-graph G be super edge-magic total with the magic constant k and k 2p + 3. Then, a new graph formed from G by adding exactly three pendants incident to three distinct vertices x, y and z of G whose labels k 2p, k 2p 1 and k 2p 2 respectively is super edge-magic total with the magic constant k = k + 6. Proof. In the new graph, define a labeling as follows. Preserve all vertex labels of G in the new graph. Increase all edge labels (except

6 6 E.T. Baskoro, I W. Sudarsana and Y.M. Cholily the new ones) by 6 in the new graph. Label the three new vertices u which adjacent to x, y and z by using the second row of either matrix A or B. Label the corresponding new edge e by using the third row from A or B. x y z x y z A = u : p + 1 p + 3 p + 2, B = u : p + 2 p + 1 p + 3 e : p + 5 p + 4 p + 6 e : p + 4 p + 6 p + 5 For the new edges, clearly we have the edge sum of each is k + 6 (from the above matrix). Since each label of old edge increased by 6 then we get the edge sum of each old edge in the new graph is k + 6. Therefore, the new graph is super edge-magic total. Note that this process works only if k 2p 2 1. This implies that k 2p+3. Alternatively, we have the following theorem for adding three pendants. Theorem 5. Let a (p, q)-graph G be super edge-magic total with the magic constant k and k 2p + 4. Then, a new graph formed from G by adding exactly three pendants incident to three distinct vertices x, y and z of G whose labels k 2p + 1, k 2p 1 and k 2p 3 respectively is super edge-magic total with the magic constant k = k + 6. Proof. The proof is similar with the one of Theorem 4 by using the following matrix C. x y z C = u : p + 1 p + 2 p + 3 e : p + 4 p + 5 p + 6 Theorem 6. Let p be an odd integer. Let a (p, q)-graph G be super edge-magic total with the magic constant k = (5p + 3)/2. Then, a new graph formed from G by adding exactly p pendants incident to all vertices of G is also super edge-magic total with the magic constant k = (9p + 3)/2. Proof. In the new graph, define a labeling as follows. Preserve all vertex labels of G in the new graph. Increase all edge labels (except the new ones) by 2p in the new graph. Label each new vertex u which adjacent to the old vertex v by using the second row of the following matrix. Label the corresponding new edge e = vu by using the third row.

7 How to construct new super edge-magic graphs 7 p 1 p+1 p+3 v : 1 2 p 1 p p+3 3p+5 3p 1 3p+1 u : 2p p + 1 p e : 3p 1 3p 3 2p + 2 3p 3p 2 2p + 3 2p + 1 For the new edges, clearly we have the edge sum of each is 9p+3 2 (from the above matrix). Since each label of old edge increased by 2p then we get the edge sum of each old edge in the new graph is k + 2p = 9p+3 2. Therefore, the new graph is super edge-magic total. Note that the (extension) construction method in Theorem 6 only works for a super edge-magic total graph with the magic constant k = (5p + 3)/2. There are several graphs of p vertices known to have the magic constant (5p + 3)/2, such as odd cycles and paths with odd number of vertices. Let C(n, s) be a graph contructed from path P n of n vertices by adding s pendants to each vertex of P n. Let us call C(n, s) by a caterpillar with s legs. We know that C(n, s) is super edge-magic total [5]. The following corollary shows one way how to label this graph so that super edge-magic total. Corollary 2 For odd n and s 1, the graph C(n, s) is super edgemagic total. Proof. Take a super edge-magic total labeling for path P n, for odd n, with the magic constant (5n + 3)/2, namely label the vertices in the odd positions of P n from left to right consecutively by 1, 2,, (n + 1)/2; And then label the even positions from left to right consecutively by (n + 3)/2, (n + 5)/2,, n; Next, it is easy to label all the edges of P n so that we have a super edge-magic total labeling. Apply Theorem 6 to P n. Denote the resulting graph by C(n, 1). For i = 1, 2,, s 1, apply Theorem 6 repeatedly to graph C(n, i) and denote the resulting graph by C(n, i + 1). In the final result, we have a super edge-magic labeling for C(n, s). Let T p be a tree of p vertices, for p 3. For h 1 we denote by T n + A h a graph which is obtained by adding h pendants to one vertex of tree T p. Then, we have: Theorem 7. From any super edge-magic total tree T p with the magic constant k = 2p s, for some s {1, 2,, p}, we can construct a new super edge-magic total tree T p + A h.

8 8 E.T. Baskoro, I W. Sudarsana and Y.M. Cholily Proof. Apply Theorem 2 to T p h times by attaching a new pendant each to vertex whose label k 2p 1. By using Theorem 7 we can obtain a class of trees which is super edge-magic total from just one super edge-magic total tree. For instances, the tree in Fig. 2(b) is obtained by applying Theorem 7 to the tree in (a). This theorem provides more facts to support the correctness of the Conjecture proposed by Enomoto et al. [1] (a) 3 2h+11 2h h+12 2h (b) 2h+8 3 2h+13 h h+7 h+7 Fig. 2. The tree in (b) is formed from the tree in (a) by using Theorem 7. References 1. H. Enomoto, A.S. Llado, T. Nakamigawa and G. Ringel: Super edge-magic graphs, SUT J. Math. 2 (1998), R.M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, On super edgemagic graphs, Ars Combin., 64 (2002) R.M. Figueroa-Centeno, R. Ichishima and F.A. Muntaner-Batle, The place of super edge-magic labelings among other classes of labelings, Discrete Math., 231 (2001) N. Hartsfield and G. Ringel: Pearls in Graph Theory (Academic Press, 1990). 5. A. Kotzig and A. Rosa: Magic valuations of finite graphs, Canad. Math. Bull. 13 (1970), W. D. Wallis, E. T. Baskoro, M. Miller and Slamin: Edge-magic total labelings, Australasian Journal of Combinatorics 22 (2000)