ON A WEAKER VERSION OF SUM LABELING OF GRAPHS

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1 ON A WEAKER VERSION OF SUM LABELING OF GRAPHS IMRAN JAVAID, FARIHA KHALID, ALI AHMAD and M. IMRAN Communicated by the former editorial board In this paper, we introduce super weak sum labeling and weak sum labeling of a graph G with vertex set V and edge set E, defined as follows: A super weak sum (briefly sw-sum) labeling is a bijection L : V {1, 2,..., V } such that for every edge (u, v) in G, there is a vertex w in G with L(u) + L(v) = L(w). A graph that can be sw-sum labeled is called an sw-sum graph. It is obvious that an sw-sum graph cannot be connected. There must be at least one isolated vertex, namely the vertex with the largest label. The sw-sum number, ω(h), of a connected graph H is the least number r of isolated vertices K r such that G = H K r is an sw-sum graph. If the set {1, 2,..., V } is replaced by some subset S of Z + in the definition of sw-sum labeling, then such a labeling will be referred to as weak sum (briefly w-sum) labeling and the minimum number of isolates in such a labeling as w-sum number. We show that a lower bound for the sw-sum number is the minimum degree δ of a vertex in the graph. Graphs achieving this bound will be referred to as δ-optimal sw-summable. We provide labeling schemes for different families of graphs showing that they are δ-optimal sw-summable. We show that not all the graphs are δ-optimal sw-summable and conjecture that all the graphs are δ-optimal w-summable. AMS 2010 Subject Classification: 05C78, 05C62. Key words: sum graph, super weak sum labeling, weak sum labeling. 1. INTRODUCTION All the graphs considered in this paper are simple, finite and undirected. If a graph G has p vertices and q edges, then G will be referred to as (p, q)- graph and by [p], we mean the set {1, 2,..., p}. For a vertex v, the set of vertices adjacent with v are referred to as the neighborhood of v, denoted by N(v) and N(v) is the degree of v. A graph G is called a sum graph if there exists a labeling of the vertices of G by distinct positive integers such that the vertices u and v are adjacent if and only if there exists a vertex whose label is equal to the sum of labels of u and v. The sum number, σ(h), of a graph H is the least number r of isolated vertices needed so that G = H K r is a sum graph [1]. All sum graphs are MATH. REPORTS 16(66), 3 (2014),

2 414 Imran Javaid, Fariha Khalid, Ali Ahmad and M. Imran 2 necessarily disconnected. There must be at least one isolated vertex, namely the vertex with the largest label, so that the sum number r of a connected graph is always more than or equal to one. Sum labelings have important applications in graph storage. Many variations of sum labelings have been studied, for example integer sum labelings [2], mod sum labeling [2], exclusive sum labelings [4], sum* labeling [5] and mod sum* labeling [5]. In this paper, we are introducing a weaker version of sum labeling of a (p, q)-graph G, namely super weak sum (briefly sw-sum) labeling using integers from the set [p] in the following way: A labeling L : V [p] is called super weak sum labeling if for any (u, v) E(G), there exists a vertex w in G such that L(u) + L(v) = L(w). sw-sum graphs are necessarily disconnected so in order to sw-sum label a connected graph H, it becomes necessary to add a set of isolated vertices known as isolates as a disjoint union and the labeling scheme that requires the fewest isolates is termed as optimal. By this method, any graph can be embedded in an sw-sum graph by adding sufficient isolates. The smallest number of isolates required for a graph H to support an sw-sum labeling is known as the sw-sum number of a graph, denoted by ω(h). It is evident that ω(h) q. A lower bound for the sw-sum number of a graph is the minimum degree δ of a vertex in the graph. We prove this in the following lemma: Lemma 1.1. A lower bound for the sw-sum number ω(h) of a graph H is the minimum degree δ of a vertex in the graph. Proof. Let v V (H) be a vertex with maximum label. Then it has at least δ neighbors v 1, v 2,..., v δ. Since sum of labels of v and v i ; i = 1, 2,..., δ must be a label of another vertex, so we must have δ isolates to sw-sum label this graph. Hence δ ω(h). An sw-sum graph is termed as δ-optimal sw-summable if it needs δ isolates to sw-sum label a graph. If the set [p] is replaced by some subset S of Z + in the definition of swsum labeling, then such a labeling is referred to as weak sum (briefly w-sum) labeling. Since w-sum graphs are generalization of sw-sum graphs, so all the terminology mentioned above for sw-sum graphs holds for w-sum graphs as well. Note that, if all the labels x [p] of vertices in a graph G are replaced by kx for some k Z +, then this graph receives the labels from k[p] and the sum of labels of every two distinct vertices is a label of another vertex in G. Hence, we have the following lemma:

3 3 On a weaker version of sum labeling of graphs 415 Lemma 1.2. Every (p, q)-graph which is sw-summable is w-summable. Observation 1.3. In a δ-optimal sw-summable graph, if the degree of a vertex v receiving the largest label is d, then vertices in N(v) receive labels from the set [d]. If a (p, q)-graph is w-summable, then it may not be δ-optimal sw-summable. In order to show this first we define Cayley graph: Let X be a group and S X\{1}, an inverse closed subset. The Cayley graph Cay(X, S) is a graph with the vertex set X and two vertices x, y X adjacent whenever xy 1 S. Consider the w-sum labeling of Cay(Z 8, {±1, ±2}) K 4 in Figure 1. However, this graph does not support sw-sum labeling. By Observation 1.3, suppose that the vertex v 0 receives label 8, then the vertices v 1, v 2, v 6 and v 7 will receive labels from the set [4] and the vertices v 3, v 4 and v 5 will receive labels from the set {5, 6, 7}. It can be seen that there exists an edge say (x, y) with one vertex say x having label 7 such that L(x) + L(y) / [12]. Fig. 1 Weak sum labeling of Cay(Z 8, {±1, ±2}) K 4. Let L and L be two optimal w-sum labelings of a graph G. Labeling L is said to be smaller than L if the largest label under L is less than the largest label under L. From this definition, it follows that sw-sum labeling is the smallest w-sum labeling. In the next section, we provide labeling schemes showing that paths are 1- optimal sw-summable, cycles are 2-optimal sw-summable, wheels are 3-optimal sw-summable, complete graphs are (n 1)-optimal sw-summable and complete multipartite graphs K n1,n 2,...,n q are t-optimal sw-summable, where t is the minimum degree of a vertex in K n1,n 2,...,n q. Throughout the paper, the vertices are identified by their labels under L.

4 416 Imran Javaid, Fariha Khalid, Ali Ahmad and M. Imran 4 2. SUPER WEAK SUM LABELING Let P n and C n be path and cycle on n vertices. We show that paths are 1-optimal summable and cycles are 2-optimal summable by providing sw-sum labeling schemes of P n K 1 and C n K 2. Labeling scheme for P n K 1 : The vertex set of P n K 1 is given as {v 1, v 2,..., v n } {s 1 }. Let n = 2k or n = 2k + 1 depending upon whether n is even or odd. Define v 2i = i for i [k] and v 2i+1 = n i for i from 0 to k 1 or k depending upon whether n is even or odd, respectively, and s 1 = n + 1. Labeling scheme for C n K 2 : Let V (C n K 2 ) = {w 1, w 2,..., w n } {t 1, t 2 }. Let n = 2k or n = 2k + 1 depending upon whether n is even or odd. Define w 2i = n i + 1 for i [k] and w 2i+1 = i + 1 for i from 0 to k 1 or k depending upon whether n is even or odd, respectively, and t j = n + j for j = 1, 2. It is easy to see that P n K 1 and C n K 2 are sw-sum graphs. Hence, we have the following result: Theorem 2.1. ω(p n ) = 1 for all n 2 and ω(c n ) = 2 for all n 3. For every integer n 3, a wheel W n = (V, E) is a graph with V = {c, v 0, v 1,..., v n 1 }, E = {(c, v i ), (v i, v i+1 ) i = 0, 1,..., n 1} where indices of the vertices are considered modulo n. The vertex c is called the center of the wheel, each edge (c, v i ), for i = 0, 1, 2,..., n 1, is called a spoke, the vertices v 0, v 1,..., v n 1 are referred to as rim vertices and each edge (v i, v i+1 ) for i = 0, 1,..., n 1 is called a rim edge. Now, we show that wheels are 3-optimal sw-summable. Theorem 2.2. ω(w n ) = 3 for all n 3. Proof. By Lemma 1.1, ω(w n ) 3. Following is the labeling scheme for the wheels with three isolates: Let n = 2k or n = 2k + 1 depending upon whether n is even or odd. Label the central vertex by c = 1. Set d = n + 1. Assign labels to the vertices as v 2i = i + 2 and v 2i+1 = d i, where i [k 1] {0} for n even. For n odd we define v 2i = i + 2, where i [k] {0} and v 2i+1 = d i, where i [k 1] {0}. After labeling all the vertices of the graph W n, d is the maximum label on the graph. Note that v 1 = d, so v 1 + c = d + 1. v 0 + v 1 = d + 2 and v 1 + v 2 = d + 3 are larger than the maximum label in the graph. Hence, there must be three isolates with labels d + 1, d + 2 and d + 3. It is easy to see that the sum of the labels of every spoke and rim edge is a label of another vertex. Hence, the wheel W n is 3-optimal sw-summable. Let K n be a complete graph with vertex set {v 1, v 2,..., v n }. Now, we show that ω(k n ) = n 1 for all n 2.

5 5 On a weaker version of sum labeling of graphs 417 Theorem 2.3. For all n 2, ω(k n ) = n 1. Proof. Following is the sw-sum labeling scheme for complete graph K n with n 1 isolates: Let n = 2k or n = 2k +1 depending upon whether n is even or odd. Starting from any vertex, label the vertices as: v i = i, i = 1, 2,..., n. Since v i is adjacent with n 1 vertices, so there must be n 1 isolates. Consider any vertex v j then the edges incident at v j are (v i, v j ) with i j and i [n] \ {j}, j [n] \ {i}. Note that if i + j < n, then there is a vertex v k with k = i+j such that, v k = v i +v j and if i+j > n, then v i +v j [2n 1]\[n], which means (v i, v j ) is labeled by an isolate. Hence (v i, v j ) is an edge. We see that the sum of the labels of every edge on the graph K n is a label of another vertex. By Lemma 1.1, we conclude that ω(k n ) n 1. This gives that ω(k n ) = n 1 for all n. A complete multipartite graph is a graph whose vertex set can be partitioned into q subsets V 1, V 2,..., V q such that every (u, v) is an edge if and only if u and v belong to different partite sets. If V i = n i, 1 i q, then we denote complete multipartite graph as K n1,n 2,...,n q. For labeling purpose, we arrange the q-partitions in such a way that n 1 n 2... n q where n i s are the number of vertices in V i -class. We name the vertices from the classes V 1, V 2,..., V q as v 1, v 2,..., v n1, v n1 +1,..., v n1 +n 2,..., v s where s = n 1 +n n q. Now, since V q is the class having maximum number of vertices n q, so they attain the minimum degree. Let t denotes the minimum degree of a vertex in K n1,n 2,...,n q then t = s n q. In the following theorem, we shall prove that ω(k n1,n 2,...,n q ) = s n q. Theorem 2.4. K n1,n 2,...,n q is δ-optimal super weak summable. Proof. Let V = {V 1, V 2,..., V q } be the vertex set of K n1,n 2,...,n q and s = n 1 + n n q be the total number of vertices in the graph. Now, assign labels to the vertices as: v i = i, i [s]. The maximum label is s which is the label of the vertex v s. Now, the sum of v s + v i = s + i, i [t]. Since v s has maximum label on the graph, so the labels v s + v i are greater than the maximum label s, so they must be the isolates. Labels of the vertices of K n1,n 2,...,n q form the sequence {1, 2,..., s, s + 1,..., s + t}, so v s j + v i = s j + i {1, 2,..., s, s + 1,..., s + t}, i [s 1], j [s] and i s j. This shows that the sum of labels of every vertex is the label of another vertex on the graph. Since s j + i < s + i and they results in minimum isolates. This shows that K n1,n 2,...,n q can be sw-sum labeled using t isolates, so ω(k n1,n 2,...,n q ) t. Hence, by Lemma 1.1, ω(k n1,n 2,...,n q ) = t for all n i, i = 1, 2,..., q.

6 418 Imran Javaid, Fariha Khalid, Ali Ahmad and M. Imran 6 Corollary 2.5. Let K m,n be a bipartite graph then ω(k m,n ) = m if m n. In particular, ω(s n ) = ω(k 1,n ) = 1, where S n is a star. 3. WEAK SUM LABELING OF Cay(Z n, {±1, ±2}) We note that paths, cycles, wheels, complete graphs, complete multipartite graphs and stars are all δ-optimal sw-summable graphs. Earlier it was shown that Cay(Z 8, {±1, ±2}) K 4 is not sw-summable but w-summable. In this section, we show that Cay(Z n, {±1, ±2}) K 4 is w-summable for all n 5. Theorem 3.1. For all n 5, Cay(Z n, {±1, ±2}) is 4-optimal w-summable. Proof. Let v 0, v 1,..., v n 1 be the vertices of Cay(Z n, {±1, ±2}), where n = 3k + r, k Z + and r = 0, 1, 2. To show that Cay(Z n, {±1, ±2}) is 4- optimal w-summable, we define the labeling as follows: { i + 1(0 i k 1), r = 0, v 3i = i + 1(0 i k), r = 1, 2, v 3i+1 = { n i(0 i k 1), r = 0, 1, 2, k i(0 i k 2), r = 0, v 3i+2 = k i(0 i k 1), r = 1, k i(0 i k 1), r = 2, for r = 0, v 3k 1 = k + 1 and for r = 2, v 3k+1 = k + 2. Note that v 1 = n is the largest label of a vertex in the graph and v 0 +v 1 = n + 1, v 1 + v 3 = n + 2 for r = 0, 1, 2, v 1 + v 2 = v 1 + v n 1 = 4k + 2, r = 0, 4k + 3, r = 1, 4k + 5, r = 2, 4k + 1, r = 0, 4k + 2, r = 1, 4k + 4, r = 2. Now, it remains to show that v i + v j {v 1, v 2,..., v n 1 } {v 0 + v 1, v 1 + v 2, v 1 + v 3, v 1 + v n 1 } whenever (v i, v j ) is an edge. Note that { n + 2(1 i k 1), r = 0, v 3i + v 3i 2 = n + 2(1 i k), r = 1, 2, { k + 2i + 2(0 i k 1), r = 0, 1, v 3i + v 3i 1 = k + 2i + 3(0 i k 1), r = 3, v 3i + v 3i+1 = { n + 1(0 i k 1), r = 0, 1, 2,

7 7 On a weaker version of sum labeling of graphs 419 k i(0 i k 2), r = 0, v 3i + v 3i+2 = k i(0 i k 1), r = 1, k i(0 i k 1), r = 2, { n + k + 1(0 i k 1), r = 0, 1, v 3i 1 + v 3i+1 = n + k + 2(0 i k 1), r = 3, n + k + 2(0 i k 2), r = 0, v 3i+1 + v 3i+2 = n + k + 2(0 i k 1), r = 1, n + k + 3(0 i k 1), r = 2, { 2k + 2, r = 0, 1, v 0 + v n 2 = k + 2, r = 2, { 3k + 2, r = 0, 1, v n 2 + v n 1 = 2k + 3, r = 2, for r = 0, v 3k 3 + v 3k 1 = 2k + 1 and for r = 2, v n 3 + v n 2 = 3(k + 1), v n 3 + v n 1 = 3k + 4. We see that the sum of the labels of every edge is a label of another vertex. We conclude that Cay(Z n, {±1, ±2}) with n 5 can be weak sum labeled using only four isolates. Hence, by Lemma 1.1, Cay(Z n, {±1, ±2}) is 4-optimal w-summable. Concluding Remarks: In this paper, we have introduced a weaker version of sum labeling of a (p, q) graph using labels from the set [p]. We have seen that wheels, complete graphs, complete bipartite graphs can be super weak sum labeled using δ isolates. Also, note that a b [p] for any a, b [p]. Hence, if we define super difference labeling by replacing L(u) + L(v) with L(u) L(v) in the definition of sw-sum labeling, then all the classes of graphs mentioned above are super difference graphs. We note that not all the graphs are sw-summable and give w-sum labeling of Cay(Z n, {±1, ±2}). We have the following conjecture and open question for further work on this paper. Conjecture 3.2. All graphs are δ-optimal w-summable. Open Problem 3.3. Does there exist a graph which is an sw-sum graph but not δ-optimal sw-summable? Acknowledgments. The authors are grateful to the anonymous referee whose careful reading and valuable suggestions resulted in producing an improved paper. REFERENCES [1] F. Harary, Sum graphs and difference graphs. Congr. Numer. 72 (1990), [2] F. Harary, Sum graphs over all the integers. Discrete Math. 124 (1994), [3] K.M. Koh, M. Miller, W.F. Smyth and Y. Wang, On optimal summable graphs. Submitted for publication.

8 420 Imran Javaid, Fariha Khalid, Ali Ahmad and M. Imran 8 [4] M. Miller, J.F. Ryan, Slamin, K. Sugeng and M. Tuga, Exclusive sum graph labelings. Preprint. [5] M. Sutton, Summable graph labelings and their applications. Ph.D. Thesis, October Received 31 May 2012 Bahauddin Zakariya University Multan, Center for Advanced Studies in Pure and Applied Mathematics, Pakistan ijavaidbzu@gmail.com National University of Computer and Emerging Sciences, FAST, Lahore, Pakistan L125505@nu.edu.pk Jazan University, College of Computer and Information System, Jazan, KSA, ahmadsms@gmail.com National University of Sciences and Technology, Center for Advanced Mathematics and Physics, Islamabad, Pakistan, imrandhab@gmail.com