- Email: [email protected]
Some results on edge pair sum labeling
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Electronic Notes in Discrete Mathematics 48 (2015) 169–173 www.elsevier.com/locate/endm
Some results on edge pair sum labeling P. Jeyanthi Research Centre, Department of Mathematics Govindammal Aditanar College for Women Tiruchendur- 628 215, Tamil Nadu, India.
T. Saratha Devi Department of Mathematics G.Venkataswamy Naidu College Kovilpatti-628502, Tamil Nadu, India.
Cee-Choon Lau Faculty of Computer and Mathematical Sciences University Teknologi MARA (segamat Campus) 8500, Johore, Malaysia.
Abstract An injective map f : E(G) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling if the induced vertex function f ∗ : V (G) → Z − {0} defined by f ∗ (v) = f (e) is one-one where Ev denotes eEv
∗ the set of edges in G that areincident with a vertex v and f (V (G)) is either of p or ±k1 , ±k2 , · · · , ±k p−1 k p+1 according as the form ±k1 , ±k2 , · · · , ±k 2 2 2 p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that triangular snake, bistars, Cn Cn and K1,n K1,m are edge pair sum graphs.
Keywords: Edge pair sum labeling, edge pair sum graph,triangular snake,bistar http://dx.doi.org/10.1016/j.endm.2015.05.025 1571-0653/© 2015 Elsevier B.V. All rights reserved.
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Introduction
We consider only finite, simple and undirected graphs. A graph G(p, q) has vertex set V (G) and edge set E(G). A vertex labeling f of a graph G is an assignment of labels to the vertices of G that induces a label for each edge depending on the vertex labels. An edge labeling f of a graph G is an assignment of labels to the edges of G that induces a label for each vertex v depending on the labels of the edges incident to it. Terms used are in the sense of Harary [1]. Ponraj et al. introduced the concept of pair sum labeling in [7]. Analogous to pair sum labeling, we defined a new labeling called edge pair sum labeling in [2] and further studied in [3,4,5,6]. An injective map f : E(G) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling if the induced vertex function f ∗ : V (G) → Z −{0} defined by f ∗ (v) = f (e) is one-one where Ev denotes eEv
∗ the set ofedges in G that are incident with a vertex v andf(V (G))is either of the form ±k1 , ±k2 , · · · , ±k p2 or ±k1 , ±k2 , · · · , ±k p−1 k p+1 according 2 2 as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that triangular snake, bistars, Cn Cn and K1,n K1,m are edge pair sum graphs.
2
Main results
Theorem 2.1 Every graph is a subgraph of a connected edge pair sum graph. Proof. Let G be a graph with n vertices. Take two copies of the complete graph Kn . Let ui , vi (1 ≤ i ≤ n) be the vertices of the first and second copy of n Kn .Let ei , ei be the edges of the first and second copy of Kn . Form = 2 − 2n, V (K1,m ) = {w, wi , 1 ≤ i ≤ m}and E(K1,m ) = ei , 1 ≤ i ≤ m . The graph G∗ is obtained from Kn Kn K1,m by joining the vertex w to u1 and v1 . Obviously G is a subgraph of G∗ . Define f (u1 w) = 3 and f (v1 w) = −3. Label the edges ei by 4,6,8,...,(n2 − n + 2), ei by -4,-6,-8,...,-(n2 − n + 2), e1 by -1, 1 2 3
Email: [email protected] Email: [email protected] Email: [email protected]
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e2 by 2, e3 , e4 ,.., e m +1 by 5,7,9,11,...,m+1 and e m +2 ,e m +3 ,...,em by -5,-7,-9,2 2 2 11,...,−(m + 1). Clearly, this edge labeling is an edge pair sum labeling of G∗ . 2 The graph G∗ obtained in Theorem 2.1 does not contain G as an induced subgraph. Hence the following problem arises naturally. Problem 2.2 Is it possible to embed a given graph as an induced subgraph of a connected edge pair sum graph? Theorem 2.3 For n ≥ 2 , all triangular snakes Tn are edge pair sum graphs. Proof. Let V (Tn ) ={ui , vj , 1 ≤ i ≤ (n + 1) and 1 ≤ j ≤ n} and E(Tn )={ei = ui ui+1 ; ei = ui vi ; ei = ui+1 vi , 1 ≤ i ≤ n}. We consider the following two cases. Case(i) n is odd. , let f (ei ) = f (ei )−2 = f (ei )−1 = (3i−1) and f (e n+1+2i ) For 1 ≤ i ≤ n−1 2
2
) + 1, f (e n+1 ) = 1 = −f (e n+1 ) = f (e n+1+2i ) + 2 = f (e n+1+2i ) + 1 = −3( n+1−2i 2 2
2
2
2
and f (e n+1 ) = (4 − 3n). It can be easily verified that f ∗ (V (Tn )) = {±7, ±13, 2 ±19, ..., ±(3n−2), ±(3n−3), ±5, ±17, ±29, ..., ±(6n−13) (−6n+8)}. Hence, f is an edge pair sum labeling of V (Tn ). Case(ii) n is even. For 1 ≤ i ≤ n, let f (ei ) = −i, for 0 ≤ i ≤ (n − 1), let f (e1+i ) = (3n − i) and for 1 ≤ i ≤ n2 f (e2i−1 ) = −(3n + 2 − 4i) = f (e2i ) − 1. Then f ∗ (V (Tn )) = {±(n + 1), ±(n + 3), ..., ±(3n − 3), ±(3n − 1), n}. Hence, f is an edge pair sum labeling of V (Tn ). 2 Theorem 2.4 For n ≥ 3 , the graph Cn Cn is an edge pair sum graph. Proof. Let V (Cn Cn ) = {ui , vi , 1 ≤ i ≤ n} and E(Cn Cn ) = {ei = ui ui+1 , ei = vi vi+1 : 1 ≤ i ≤ (n − 1), en = un u1 , en = vn v1 }. We consider the following two cases. Case(i) n is odd. For 1 ≤ i ≤ n, let f (ei ) = i = −f (ei ). Then f ∗ (V (Cn Cn )) = {±3, ±5, ..., ±(2n − 1), ±(n + 1)}. Hence, f is an edge pair sum labeling. Case(ii) n is even. For 1 ≤ i ≤ (n − 1), let f (ei ) = i = −f (ei ) and f (en ) = (n + 1) = −f (en ). Then f ∗ (V (Cn Cn )) = {±3, ±5, ..., ±(2n − 3), ±2n, ±(n + 2)}. 2 Hence, Cn Cn is an edge pair sum graph.
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Theorem 2.5 The graph K1,n ∪ K1,m is an edge pair sum graph. Proof. Let V (K1,n ) = {v, vi , 1 ≤ i ≤ n}, V (K1,m ) = {u, ui , 1 ≤ i ≤ m} and E(K1,n ) = {ei = vvi , 1 ≤ i ≤ n}, E(K1,m ) = {ei = uui , 1 ≤ i ≤ m}. Case (i) n = m. For 1 ≤ i ≤ n f (ei ) = i and for 1 ≤ i ≤ m, f (ei ) = −i. Then f ∗ (V (K1,n ∪ K1,m )) = {±1, ±2, . . . , ±n, ±n(n + 1)/2}. Hence, f is an edge pair sum labeling. Case (ii) n < m, n and m are even or odd. Sub case (a) n ≡ 3(mod 4) or n ≡ 2(mod 4). For 1 ≤ i ≤ n, let f (ei ) = i = −f (ei ) and for 1 ≤ i ≤ (m − n)/2, let f (en+i ) = (n+2i) = −f (e(m+n)/2+i ). Then f ∗ (V (K1,n ∪K1,m )) = {±1, ±2, . . . , ±n, ±(n + 2), (n + 4), . . . , ±m, ±n(n + 1)/2}. Hence, f is an edge pair sum labeling. Sub case (b) n ≡ 0(mod 4) or n ≡ 1(mod 4). For 1 ≤ i ≤ n, let f (ei ) = i = −f (ei ), for 1 ≤ i ≤ (m − n − 2)/2, let f (en+2+i ) = (n + 1 + 2i) = −f (e(m+n+2)/2+i ) and f (en+1 ) = (n + 1) = −f (en+2 ). Then f ∗ (V (K1,n ∪K1,m )) = {±1, ±2, . . . , ±n, ±(n+1), ±(n+3), . . . , ±(m − 1), ±n(n + 1)/2}. Hence, f is an edge pair sum labeling. Case (iii) n < m, n is even and m is odd. For 1 ≤ i ≤ (n+2)/2, let f (ei ) = i, for 1 ≤ i ≤ (n−2)/2, let f (e(n+2)/2+i ) = −i, for 1 ≤ i ≤ 2 f (ei ) = −((n − 2)/2 + i), f (e3 ) = −(n + 1), and for 1 ≤ i ≤ (m − 3)/2, let f (e3+i ) = (n + 1 + 2i) = −f (e(m+3)/2+i ). Then f ∗ (V (K1,n ∪ K1,m )) = {±1, ±2, . . . , ±((n + 2)/2), ±(n + 1), ±(n + 3), . . . , ±(m + n − 2)} ∪ {−2(n + 1)}. Hence, f is an edge pair sum labeling. Case (iv) n < m, n is odd and m is even. For 1 ≤ i ≤ (n+3)/2, let f (ei ) = i, for 1 ≤ i ≤ (n−3)/2, let f (e(n+3)/2+i ) = −i, for 1 ≤ i ≤ 3, let f (ei ) = −((n − 3)/2 + i), f (e4 ) = −3((n + 1)/2) and for 1 ≤ i ≤ (m − 4)/2, let f (e4+i ) = (n + 2i) = −f (e(m+4)/2+i ). Then f ∗ (V (K1,n ∪K1,m )) = {±1, ±2, . . . , ±((n+3)/2), ±3((n+1)/2), ±(n+2), ±(n+ 4), . . . , ±(m + n − 4)} ∪ {−3(n + 1)}. Hence, K1,n ∪ K1,m is an edge pair sum graph. 2 Theorem 2.6 The bistar Bm,n is an edge pair sum graph. Proof. Let V (Bm,n ) = {u, ui , 1 ≤ i ≤ m and v, vi , 1 ≤ i ≤ n} and E(Bm,n ) = {e = uv; ei = uui , 1 ≤ i ≤ m and ei = vvi , 1 ≤ i ≤ n}. Case (i) m and n are even. We define f : E(Bm,n ) → {±1, ±2, . . . , ±(m + n + 1)} as follows:
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Let f (e1 ) = −2, f (e2 ) = 5, f (e) = −1, f (e1 ) = 3, f (e2 ) = −5, for 3 ≤ i ≤ (m + 2)/2, let f (ei ) = (2i + 1) = −f (e(m−2)/2+i ), and for 3 ≤ i ≤ (n + 2)/2, let f (ei ) = (2i + m − 1) = −f (e(n−2)/2+i ). Then f ∗ (V (Bm,n )) = {±2, ±3, ±5, ±7, . . . , ±(m + 3), ±(m + 5), . . . , ±(m + n + 1)}. Hence, f is an edge pair sum labeling. Case (ii) m and n are odd. Let f (e1 ) = −2, f (e) = −1, f (e1 ) = 3, for 2 ≤ i ≤ (m + 1)/2, let f (ei ) = (2i + 1) = −f (e(m−1)/2+i ), for 2 ≤ i ≤ (n + 1)/2, let f (ei ) = (2i + m) = −f (e(n−1)/2+i ). Then f ∗ (V (Bm,n )) = {±2, ±3, ±5, . . . , ±(m + 2), ±(m + 4), . . . , ±(m + n + 1)}. Hence, f is an edge pair sum labeling. Case (iii) Either m is odd or n is even. Let f (e1 ) = −2, f (e) = 1, for 2 ≤ i ≤ (m + 1)/2, let f (ei ) = (2i − 1) = −f (e(m−1)/2+i ) and for 1 ≤ i ≤ n/2, let f (ei ) = (m + 2i) = −f (en/2+i ). Then f ∗ (V (Bm,n )) = {±1, ±3, . . . , ±m, ±(m + 2), ±(m + 4), . . . , ±(m + n)} ∪ {−2}. Hence, f is an edge pair sum labeling of Bm,n . 2
References [1] F. Harary, Graph Theory, Narosa Publishing House, New Delhi, 1998. [2] P. Jeyanthi and T. Saratha Devi, Edge pair sum labeling, Journal of Scientific Research, 5(3) (2013), 457–467. [3] P. Jeyanthi and T. Saratha Devi, On edge pair sum labeling of graphs, International Journal of Mathematics Trends and Technology, 7(2) (2014), 106–113. [4] P. Jeyanthi and T. Saratha Devi, Some edge pair sum graphs, Journal of Discrete Mathematical Science and Cryptography, (To appear). [5] P. Jeyanthi and T. Saratha Devi, Edge pair sum labeling of spider graphs, Journal of Algorithm and Computation, (To appear). [6] P. Jeyanthi and T. Saratha Devi, New results on edge pair sum graphs, preprint. [7] R. Ponraj and J.V.X. Parthipan, Pair Sum Labeling of Graphs, The Journal of Indian Academy of Mathematics, 32(2) (2010), 587–595.
Electronic Notes in Discrete Mathematics 48 (2015) 169–173 www.elsevier.com/locate/endm
Some results on edge pair sum labeling P. Jeyanthi Research Centre, Department of Mathematics Govindammal Aditanar College for Women Tiruchendur- 628 215, Tamil Nadu, India.
T. Saratha Devi Department of Mathematics G.Venkataswamy Naidu College Kovilpatti-628502, Tamil Nadu, India.
Cee-Choon Lau Faculty of Computer and Mathematical Sciences University Teknologi MARA (segamat Campus) 8500, Johore, Malaysia.
Abstract An injective map f : E(G) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling if the induced vertex function f ∗ : V (G) → Z − {0} defined by f ∗ (v) = f (e) is one-one where Ev denotes eEv
∗ the set of edges in G that areincident with a vertex v and f (V (G)) is either of p or ±k1 , ±k2 , · · · , ±k p−1 k p+1 according as the form ±k1 , ±k2 , · · · , ±k 2 2 2 p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that triangular snake, bistars, Cn Cn and K1,n K1,m are edge pair sum graphs.
Keywords: Edge pair sum labeling, edge pair sum graph,triangular snake,bistar http://dx.doi.org/10.1016/j.endm.2015.05.025 1571-0653/© 2015 Elsevier B.V. All rights reserved.
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P. Jeyanthi et al. / Electronic Notes in Discrete Mathematics 48 (2015) 169–173
Introduction
We consider only finite, simple and undirected graphs. A graph G(p, q) has vertex set V (G) and edge set E(G). A vertex labeling f of a graph G is an assignment of labels to the vertices of G that induces a label for each edge depending on the vertex labels. An edge labeling f of a graph G is an assignment of labels to the edges of G that induces a label for each vertex v depending on the labels of the edges incident to it. Terms used are in the sense of Harary [1]. Ponraj et al. introduced the concept of pair sum labeling in [7]. Analogous to pair sum labeling, we defined a new labeling called edge pair sum labeling in [2] and further studied in [3,4,5,6]. An injective map f : E(G) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling if the induced vertex function f ∗ : V (G) → Z −{0} defined by f ∗ (v) = f (e) is one-one where Ev denotes eEv
∗ the set ofedges in G that are incident with a vertex v andf(V (G))is either of the form ±k1 , ±k2 , · · · , ±k p2 or ±k1 , ±k2 , · · · , ±k p−1 k p+1 according 2 2 as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that triangular snake, bistars, Cn Cn and K1,n K1,m are edge pair sum graphs.
2
Main results
Theorem 2.1 Every graph is a subgraph of a connected edge pair sum graph. Proof. Let G be a graph with n vertices. Take two copies of the complete graph Kn . Let ui , vi (1 ≤ i ≤ n) be the vertices of the first and second copy of n Kn .Let ei , ei be the edges of the first and second copy of Kn . Form = 2 − 2n, V (K1,m ) = {w, wi , 1 ≤ i ≤ m}and E(K1,m ) = ei , 1 ≤ i ≤ m . The graph G∗ is obtained from Kn Kn K1,m by joining the vertex w to u1 and v1 . Obviously G is a subgraph of G∗ . Define f (u1 w) = 3 and f (v1 w) = −3. Label the edges ei by 4,6,8,...,(n2 − n + 2), ei by -4,-6,-8,...,-(n2 − n + 2), e1 by -1, 1 2 3
Email: [email protected] Email: [email protected] Email: [email protected]
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P. Jeyanthi et al. / Electronic Notes in Discrete Mathematics 48 (2015) 169–173
e2 by 2, e3 , e4 ,.., e m +1 by 5,7,9,11,...,m+1 and e m +2 ,e m +3 ,...,em by -5,-7,-9,2 2 2 11,...,−(m + 1). Clearly, this edge labeling is an edge pair sum labeling of G∗ . 2 The graph G∗ obtained in Theorem 2.1 does not contain G as an induced subgraph. Hence the following problem arises naturally. Problem 2.2 Is it possible to embed a given graph as an induced subgraph of a connected edge pair sum graph? Theorem 2.3 For n ≥ 2 , all triangular snakes Tn are edge pair sum graphs. Proof. Let V (Tn ) ={ui , vj , 1 ≤ i ≤ (n + 1) and 1 ≤ j ≤ n} and E(Tn )={ei = ui ui+1 ; ei = ui vi ; ei = ui+1 vi , 1 ≤ i ≤ n}. We consider the following two cases. Case(i) n is odd. , let f (ei ) = f (ei )−2 = f (ei )−1 = (3i−1) and f (e n+1+2i ) For 1 ≤ i ≤ n−1 2
2
) + 1, f (e n+1 ) = 1 = −f (e n+1 ) = f (e n+1+2i ) + 2 = f (e n+1+2i ) + 1 = −3( n+1−2i 2 2
2
2
2
and f (e n+1 ) = (4 − 3n). It can be easily verified that f ∗ (V (Tn )) = {±7, ±13, 2 ±19, ..., ±(3n−2), ±(3n−3), ±5, ±17, ±29, ..., ±(6n−13) (−6n+8)}. Hence, f is an edge pair sum labeling of V (Tn ). Case(ii) n is even. For 1 ≤ i ≤ n, let f (ei ) = −i, for 0 ≤ i ≤ (n − 1), let f (e1+i ) = (3n − i) and for 1 ≤ i ≤ n2 f (e2i−1 ) = −(3n + 2 − 4i) = f (e2i ) − 1. Then f ∗ (V (Tn )) = {±(n + 1), ±(n + 3), ..., ±(3n − 3), ±(3n − 1), n}. Hence, f is an edge pair sum labeling of V (Tn ). 2 Theorem 2.4 For n ≥ 3 , the graph Cn Cn is an edge pair sum graph. Proof. Let V (Cn Cn ) = {ui , vi , 1 ≤ i ≤ n} and E(Cn Cn ) = {ei = ui ui+1 , ei = vi vi+1 : 1 ≤ i ≤ (n − 1), en = un u1 , en = vn v1 }. We consider the following two cases. Case(i) n is odd. For 1 ≤ i ≤ n, let f (ei ) = i = −f (ei ). Then f ∗ (V (Cn Cn )) = {±3, ±5, ..., ±(2n − 1), ±(n + 1)}. Hence, f is an edge pair sum labeling. Case(ii) n is even. For 1 ≤ i ≤ (n − 1), let f (ei ) = i = −f (ei ) and f (en ) = (n + 1) = −f (en ). Then f ∗ (V (Cn Cn )) = {±3, ±5, ..., ±(2n − 3), ±2n, ±(n + 2)}. 2 Hence, Cn Cn is an edge pair sum graph.
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Theorem 2.5 The graph K1,n ∪ K1,m is an edge pair sum graph. Proof. Let V (K1,n ) = {v, vi , 1 ≤ i ≤ n}, V (K1,m ) = {u, ui , 1 ≤ i ≤ m} and E(K1,n ) = {ei = vvi , 1 ≤ i ≤ n}, E(K1,m ) = {ei = uui , 1 ≤ i ≤ m}. Case (i) n = m. For 1 ≤ i ≤ n f (ei ) = i and for 1 ≤ i ≤ m, f (ei ) = −i. Then f ∗ (V (K1,n ∪ K1,m )) = {±1, ±2, . . . , ±n, ±n(n + 1)/2}. Hence, f is an edge pair sum labeling. Case (ii) n < m, n and m are even or odd. Sub case (a) n ≡ 3(mod 4) or n ≡ 2(mod 4). For 1 ≤ i ≤ n, let f (ei ) = i = −f (ei ) and for 1 ≤ i ≤ (m − n)/2, let f (en+i ) = (n+2i) = −f (e(m+n)/2+i ). Then f ∗ (V (K1,n ∪K1,m )) = {±1, ±2, . . . , ±n, ±(n + 2), (n + 4), . . . , ±m, ±n(n + 1)/2}. Hence, f is an edge pair sum labeling. Sub case (b) n ≡ 0(mod 4) or n ≡ 1(mod 4). For 1 ≤ i ≤ n, let f (ei ) = i = −f (ei ), for 1 ≤ i ≤ (m − n − 2)/2, let f (en+2+i ) = (n + 1 + 2i) = −f (e(m+n+2)/2+i ) and f (en+1 ) = (n + 1) = −f (en+2 ). Then f ∗ (V (K1,n ∪K1,m )) = {±1, ±2, . . . , ±n, ±(n+1), ±(n+3), . . . , ±(m − 1), ±n(n + 1)/2}. Hence, f is an edge pair sum labeling. Case (iii) n < m, n is even and m is odd. For 1 ≤ i ≤ (n+2)/2, let f (ei ) = i, for 1 ≤ i ≤ (n−2)/2, let f (e(n+2)/2+i ) = −i, for 1 ≤ i ≤ 2 f (ei ) = −((n − 2)/2 + i), f (e3 ) = −(n + 1), and for 1 ≤ i ≤ (m − 3)/2, let f (e3+i ) = (n + 1 + 2i) = −f (e(m+3)/2+i ). Then f ∗ (V (K1,n ∪ K1,m )) = {±1, ±2, . . . , ±((n + 2)/2), ±(n + 1), ±(n + 3), . . . , ±(m + n − 2)} ∪ {−2(n + 1)}. Hence, f is an edge pair sum labeling. Case (iv) n < m, n is odd and m is even. For 1 ≤ i ≤ (n+3)/2, let f (ei ) = i, for 1 ≤ i ≤ (n−3)/2, let f (e(n+3)/2+i ) = −i, for 1 ≤ i ≤ 3, let f (ei ) = −((n − 3)/2 + i), f (e4 ) = −3((n + 1)/2) and for 1 ≤ i ≤ (m − 4)/2, let f (e4+i ) = (n + 2i) = −f (e(m+4)/2+i ). Then f ∗ (V (K1,n ∪K1,m )) = {±1, ±2, . . . , ±((n+3)/2), ±3((n+1)/2), ±(n+2), ±(n+ 4), . . . , ±(m + n − 4)} ∪ {−3(n + 1)}. Hence, K1,n ∪ K1,m is an edge pair sum graph. 2 Theorem 2.6 The bistar Bm,n is an edge pair sum graph. Proof. Let V (Bm,n ) = {u, ui , 1 ≤ i ≤ m and v, vi , 1 ≤ i ≤ n} and E(Bm,n ) = {e = uv; ei = uui , 1 ≤ i ≤ m and ei = vvi , 1 ≤ i ≤ n}. Case (i) m and n are even. We define f : E(Bm,n ) → {±1, ±2, . . . , ±(m + n + 1)} as follows:
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Let f (e1 ) = −2, f (e2 ) = 5, f (e) = −1, f (e1 ) = 3, f (e2 ) = −5, for 3 ≤ i ≤ (m + 2)/2, let f (ei ) = (2i + 1) = −f (e(m−2)/2+i ), and for 3 ≤ i ≤ (n + 2)/2, let f (ei ) = (2i + m − 1) = −f (e(n−2)/2+i ). Then f ∗ (V (Bm,n )) = {±2, ±3, ±5, ±7, . . . , ±(m + 3), ±(m + 5), . . . , ±(m + n + 1)}. Hence, f is an edge pair sum labeling. Case (ii) m and n are odd. Let f (e1 ) = −2, f (e) = −1, f (e1 ) = 3, for 2 ≤ i ≤ (m + 1)/2, let f (ei ) = (2i + 1) = −f (e(m−1)/2+i ), for 2 ≤ i ≤ (n + 1)/2, let f (ei ) = (2i + m) = −f (e(n−1)/2+i ). Then f ∗ (V (Bm,n )) = {±2, ±3, ±5, . . . , ±(m + 2), ±(m + 4), . . . , ±(m + n + 1)}. Hence, f is an edge pair sum labeling. Case (iii) Either m is odd or n is even. Let f (e1 ) = −2, f (e) = 1, for 2 ≤ i ≤ (m + 1)/2, let f (ei ) = (2i − 1) = −f (e(m−1)/2+i ) and for 1 ≤ i ≤ n/2, let f (ei ) = (m + 2i) = −f (en/2+i ). Then f ∗ (V (Bm,n )) = {±1, ±3, . . . , ±m, ±(m + 2), ±(m + 4), . . . , ±(m + n)} ∪ {−2}. Hence, f is an edge pair sum labeling of Bm,n . 2
References [1] F. Harary, Graph Theory, Narosa Publishing House, New Delhi, 1998. [2] P. Jeyanthi and T. Saratha Devi, Edge pair sum labeling, Journal of Scientific Research, 5(3) (2013), 457–467. [3] P. Jeyanthi and T. Saratha Devi, On edge pair sum labeling of graphs, International Journal of Mathematics Trends and Technology, 7(2) (2014), 106–113. [4] P. Jeyanthi and T. Saratha Devi, Some edge pair sum graphs, Journal of Discrete Mathematical Science and Cryptography, (To appear). [5] P. Jeyanthi and T. Saratha Devi, Edge pair sum labeling of spider graphs, Journal of Algorithm and Computation, (To appear). [6] P. Jeyanthi and T. Saratha Devi, New results on edge pair sum graphs, preprint. [7] R. Ponraj and J.V.X. Parthipan, Pair Sum Labeling of Graphs, The Journal of Indian Academy of Mathematics, 32(2) (2010), 587–595.