On (a, d)-vertex-antimagic total labeling of Harary graphs

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1 On (a, d)-ertex-antimagic total labeling of Harary graphs M. Hussain 1, Kashif Ali 1, M. T. Rahim, Edy Tri Baskoro 3 1 COMSATS Institute of Information Technology, Lahore Campus, Pakistan {muhammad.hussain,kashif.ali}@ciitlahore.edu.pk FAST (National Uniersity) 160-Industrial Estate, hayatabad Peshawar, Pakistan. tariqsms@gmail.com 3 Combinatorial Mathematics Research Diision, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung Jl. Ganesa 10 Bandung 4013, Indonesia ebaskoro@math.itb.ac.id Abstract. Let G = (V, E) be a graph with ertices and e edges. An (a, d)-ertex-antimagic total labeling is a bijection λ from V (G) E(G) to the set of consecutie integers 1,,..., + e, such that the weights of the ertices form an arithmetic progression with the initial term a and common difference d. If λ(v (G)) = {1,,..., } then we call the labeling a super (a, d)-ertex-antimagic total. In this paper we construct (a, d)-ertex-antimagic total labeling on Harary graphs as well as for the disjoint union of k identical copies of Harary graphs. Keywords : (a, d)-ertex-antimagic total labeling, Harary graph. 1 Introduction All graphs in this paper are finite, simple and undirected. The graph G has the ertex-set V (G) and edge-set E(G). A general reference for graph-theoretic ideas can be seen in [13]. A labeling λ of a graph G is a mapping that assigns elements of a graph to the set of numbers (usually to positie or non-negatie integers). If the domain of mapping is the set of ertices (respectiely, the set of edges) then we call the labeling ertex labeling (respectiely, edge labeling). If the domain is V E then we call the labeling a total labeling. For a further explanation of ertex, edge and total labelings, see [7], [1].

2 M. Hussain, Kashif Ali, M. T. Rahim, Edy Tri Baskoro The ertex-weight wt(x) of a ertex x V, under a labeling λ : V E {1,,..., + e}, is the sum of alues λ(xy) assigned to all edges incident to a gien ertex x together with the alue assigned to x itself. A bijection λ : V E {1,,..., +e} is called an (a, d)-ertexantimagic total( in short, (a, d)-vat ) labeling of G if the set of ertex-weights of all ertices in G is {a, a+d, a+d,..., a+( 1)d}, where a > 0 and d 0 are two fixed nonnegatie integers. If d = 0 then we call λ a ertex-magic total labeling. An (a, d)-ertex-antimagic total labeling λ is called a super (a, d)- ertex-antimagic total (in short, super (a, d)-vat )) labeling if λ(v ) = {1,,..., } and λ(e) = { + 1, +,..., + e}. Super (a, 0)-ertex-antimagic total labeling on Harary graphs was studied by C. Balbuena et. al [5]. One of their result is rewritten as follows: Theorem A. For any odd p 5 and t, G = Cp t admits a super ( 17p+5, 0)- ertex-antimagic total labeling. In this paper we study super (a, d)-ertex-antimagic total labeling of the Harary graph. We also construct an (a, d)-ertex-antimagic total labeling on the Harary graph as well as for disjoint union of k identical copies of Harary graphs. For t and p 4, a Harary graph C t p is a graph constructed from a cycle C p by joining any two ertices at distance t in C p. Figure 1 shows the Harary graph C 7. Main Results In this section, we study super (a, d)-ertex-antimagic total labeling of the Harary graph C t p. We also construct an (a, d)-ertex-antimagic total labeling on the Harary graph C t p as well as for disjoint union of k identical copies of Harary graphs. Before proing these results let us consider the following fact. Lemma 1. Let t and p 5. If Harary graph G = C t p is (a, d)- ertex-antimagic total then d < 15 for p t and d < 10 for p = t. Proof. Let p = V (G) and q = E(G). Assume that there exists a bijection

3 On (a,d)-ertex-antimagic total labelings of Harary graph Fig. 1. Harary graph C 7. λ : V (G) E(G) {1,,..., p + q} which is (a, d)-ertex-antimagic total and W = {λ(u) + λ(u) : u E(G)} = {a, a + d,..., a + (p 1)d} is the set of ertex-weights. If p t then the minimum possible ertex-weight in (a, d)-ertexantimagic total labeling is = 15. and maximum ertex-weight is no more than 3p + (3p 1) + (3p ) + (3p 3) + (3p 4) = 15p 10. Thus, we hae and a + (p 1)d 15p 10, d a p 1

4 4 M. Hussain, Kashif Ali, M. T. Rahim, Edy Tri Baskoro If p = t then minimum ertex-weight in (a, d)-ertex-antimagic total labeling is = 10. and maximum ertex-weight is no more than 5p + (5p 1) + (5p ) + (5p Thus, we hae 3) = 10p 6. and a + (p 1)d 10p 6, d 10p 16 p 1 < 10. Theorem 1. For any een p 4 and t, G = C t p does not admit a super (a, d)-ertex-antimagic total labeling proided that d is een and p t. Proof. Suppose on the contrary that G admits a super (a, d)-ertex-antimagic total labeling. Let S p be the sum of all ertex labels and let S q be the sum of all edge labels. The sum of all ertex label and all the edge labels used to calculate the ertex-weights is S p + S q = p(p + 1) + pq + q(q + 1). The sum of all ertex-weights is x V p(p 1) wt(x) = ap + d Combining these two equations, we obtain a = 1 q(q + 1) (p + 1 (p 1)d) + q + p For een d, a will be integer only if p is odd, which is a contradiction. Hence, for een p 4 and een d there does not exist super (a, d)- VAT for G. In Theorem we construct the (a, d)-ertex-antimagic total labeling on Harary graph for d = 3.

5 On (a,d)-ertex-antimagic total labelings of Harary graph 5 Theorem. For any odd p 5 and t, G = Cp t admits a ( 13p+9, 3)-ertex-antimagic total labeling proided that p t. Proof. Let p = V (G) and q = E(G), then we denote the ertex and edge sets of G as follows: V = { i : 1 i p}, E = { i i+1 : 1 i p} { i i+t where all indices are taken in mod p. : 1 i p}. Now, we define the labeling λ : V E {1,,..., p + q} as follows: λ( i ) = i 1 1 i p. λ( i i+t ) = p (i 1) 1 i p. λ( i i+1 ) = 1(5p + t i + 1), for i = t, t +, t + 4,..., p 1. 3p + 1 (t i + 1), for i = t + 1, t + 3, t + 5,..., p. For i = 1,, 3,..., t 1, we consider the following two cases: Case 1. If t is een λ( i i+1 ) = p + 1 (t i + 1), for i = 1, 3, 5,..., t 1. 1(5p + t i + 1), for i =, 4, 6,..., t. Case. If t is odd λ( i i+1 ) = p + 1 (t i + 1), for i =, 4, 6,..., t 1. 1(5p + t i + 1), for i = 1, 3, 5,..., t. We hae ertices { i : 1 i p} and we can see that the ertex t has the weight 13p+9 and set of labels of ertices are consecutie

6 6 M. Hussain, Kashif Ali, M. T. Rahim, Edy Tri Baskoro integers 13p+9, 13p+15,..., 19p+3 with a = 13p+9 and d = 3. Thus, λ is a ( 13p+9, 3)-ertex-antimagic total labeling. In Theorem 3 and 4 we construct the (a, d)-ertex-antimagic total labeling on the Harary graph and disjoint union of k identical copies of the Harary graphs respectiely. Theorem 3. For p 5 and t, G = C t p admits a (7p + 3, 1)- ertex-antimagic total labeling, proided that p t. Proof. Let p = V (G) and q = E(G), then we denote the ertex and edge sets of G as follows: V = { i : 1 i p}, E = { i i+1 : 1 i p} { i i+t where all indices are taken mod p. : 1 i p}. Now, we define the labeling λ : V E {1,,..., p + q} as follows: λ( i ) = p + i 1 i p. λ( i i+t ) = p + 1 i 1 i p. 3p t + i, for 1 i t. λ( i i+1 ) = p t + i, for t + 1 i p. We hae ertices { i : 1 i p} and we can see that the ertex t+ has the weight 7p+3 and set of labels of ertices are consecutie integers 7p + 3, 7p + 4,..., 8p + with a = 7p + 3 and d = 1. Thus, λ is a (7p + 3, 1)-ertex-antimagic total labeling. Theorem 4. For n 5, k and t, G = kcn t admits a (7nk + 3, 1)-ertex-antimagic total labeling, proided that n t. Proof. Let p= V (G) and q= E(G) then p = nk, q = nk. We denote the ertex and edge sets of G as follows:

7 On (a,d)-ertex-antimagic total labelings of Harary graph 7 V = { j i : 1 i n, 1 j k}, E = { j i j i+1 : 1 i n, 1 j k} {j i j i+t where all indices are taken mod n. : 1 i n, 1 j k}, Now, we define labeling λ : V E {1,,..., p + q} as follows: λ( j i ) = p + j + k(i 1) 1 i n, 1 j k. λ( i i+t ) = p (j 1) k(i 1) 1 i n, 1 j k. 3p + j k(t + 1 i), if 1 i t, 1 j k. λ( j i j i+1 ) = p + j + k(i (t + 1)), if t + 1 i n, 1 j k. We hae ertex set { j i : 1 i n, 1 j k} and we can see that the ertex t+ 1 has the weight 7nk + 3 and set of labels of ertices is the set of consecutie integers 7nk + 3, 7nk + 4,..., 8nk + with a = 7nk+3 and d = 1. Thus, λ is a (7nk+3, 1)-ertex-antimagic total labeling. 3 Acknowledgement We are indebted to an anonymous referee for many useful remarks which improed the first ersion of this paper. References 1. M. Baca, F. Bertault, J. MacDougall, M. Miller, R. Simanjuntak and Slamin, Vertex-antimagic total labeling of graphs, Discuss. Math. Graph Theory, 3(003), M. Baca, J. Jengrol, M. Miller and J. Ryan, Antimagic labeling of generalized Petersen graphs that are plane, Ars Combin., 73(004), M. Baca, Y. Lin, M. Miller and R. Simanjuntak, New constructions of magic and antimagic graph labelings, Utilitas Math., 60 (001), M. Baca, M. Miller and Slamin, Eery generalized Petersen graph has a ertexmagic total labeling, Int. J. Comput. Math., 79(00), C. Balbuena, Barker, Das, Lin, Miller, Ryan, Slamin, Sugeng, and Tkad, On the degrees of a strongly ertex-magic graph, Discrete Mathematics, 306 (006), D. Froncek, P. Koar and T. Koaroa, Vertex-magic total labeling of product of cycles, Australas. J. Combin., 33(005),

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