ISSN 2148-838Xhttp://dx.doi.org/10.13069/jacodesmath.59822 J. Algebra Comb. Discrete Appl. 3(2) • 97–104 Received: 14 April 2015 Accepted: 16 Feburary 2016 Journal of Algebra Combinatorics Discrete Structures and Applications New results on vertex equitable labeling Research Article Pon Jeyanthi, Anthony Maheswari, Mani Vijayalakshmi Abstract: The concept of vertex equitable labeling was introduced in [9]. A graph G is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the induced edge labels are 1, 2, 3, · · · , q. A graph G is said to be a vertex equitable if it admits a vertex equitable labeling. In this paper, we prove that the graphs, subdivision of double triangular snake S(D(Tn)), subdivision of double quadrilateral snake S(D(Qn)), subdivision of double alternate triangular snake S(DA(Tn)), subdivision of double alternate quadrilateral snake S(DA(Qn)), DA(Qm) ⊙ nK1 and DA(Tm) ⊙ nK1 admit vertex equitable labeling. 2010 MSC: 05C78, 05C38 Keywords: Vertex equitable labeling, Vertex equitable graph, Double triangular snake graph, Double alternate triangular snake graph, Double alternate quadrilateral snake graph 1. Introduction All graphs considered here are simple, finite, connected and undirected. We follow the basic notation and terminology of graph theory as in [2]. A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. There are several types of labeling and a detailed survey of graph labeling can be found in [1]. The concept of vertex equitable labeling was due to Lourdusamy and Seenivasan [9]. Let G be a graph with p vertices and q edges and A = { 0, 1, 2, · · · , ⌈ q 2 ⌉} . A graph G is said to be vertex equitable if there exists a vertex labeling f : V (G) → A that induces an edge labeling f∗ defined by f∗(uv) = f(u)+f(v) for all edges uv such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the induced edge labels are 1, 2, 3, · · · , q, where vf (a) is the number of vertices v with f(v) = a for a ∈ A. The vertex labeling f is known as vertex equitable labeling. A graph G is said to be a vertex equitable if it admits a vertex equitable labeling. In [9] they proved that the graphs like paths, bistars B(n, n), combs, Pon Jeyanthi (Corresponding Author); Research Centre, Department of Mathematics, Govindammal Aditanar College for Women, Tiruchendur - 628 215, Tamilnadu, India (email: jeyajeyanthi@rediffmail.com). Anthony Maheswari; Department of Mathematics, Kamaraj College of Engg. and Technology, Virudhunagar, Tamilnadu, India (email: bala_nithin@yahoo.co.in). Mani Vijayalakshmi; Department of Mathematics, Department of Mathematics, Dr. G. U. Pope College of Engineering, Sawyerpuram, Thoothukudi District, Tamil Nadu, India (email: viji_mac@rediffmail.com). 97 P. Jeyanthi et al. / J. Algebra Comb. Discrete Appl. 3(2) (2016) 97–104 cycles Cn if n ≡ 0 or 3 (mod 4), K2,n, C (t) 3 for t ≥ 2, quadrilateral snakes, K2 + mK1, K1,n ∪ K1,n+k if and only if 1 ≤ k ≤ 3, ladders, arbitrary super division of any path and cycle Cn with n ≡ 0 or 3 (mod 4) are vertex equitable. Also they proved that the graphs K1,n if n ≥ 4, any Eulerian graph with n edges where n ≡ 1 or 2 (mod 4), the wheel Wn, the complete graph Kn if n > 3 and triangular cactus with q ≡ 0 or 6 or 9 (mod 12) are not vertex equitable. In addition, they proved that if G is a graph with p vertices and q edges, q is even and p < ⌈ q 2 ⌉ + 2 then G is not vertex equitable. Motivated by these results, we [3]-[6] proved that Tp-trees, T ⊙ Kn where T is a Tp-trees with even number of vertices, T ◦̂Pn, T ◦̂2Pn, T ◦̂Cn(n ≡ 0 , 3 (mod 4)), T ◦̃Cn(n ≡ 0 , 3 (mod 4)), bistar B(n, n + 1), square graph of Bn,nand splitting graph of Bn,n, the caterpillar S(x1, x2, · · · , xn) and Cn ⊙ K1, P 2n, tadpoles, Cm ⊕ Cn, armed crowns , [ Pm; C 2 n ] , ⟨Pm◦̂K1,n⟩, kC4-snakes for all k ≥ 1, generalized kCn-snakes if n ≡ 0 (mod 4), n ≥ 4 and the graphs obtained by duplicating an arbitrary vertex and an arbitrary edge of a cycle Cn, total graph of Pn, splitting graph of Pn and fusion of two edges of a cycle Cn are vertex equitable graphs. In this paper, we prove that S(D(Tn)), S(D(Qn)), S(DA(Tn)), S(DA(Qn)), DA(Qm) ⊙ nK1 and DA(Tm) ⊙ nK1 are vertex equitable graphs. We use the following definitions in the subsequent section. Definition 1. The double triangular snake D(Tn) is a graph obtained from a path Pn with vertices v1, v2, · · · , vn by joining vi and vi+1 to the new vertices wi and ui for i = 1, 2, · · · , n − 1. Definition 2. The double quadrilateral snake D(Qn) is a graph obtained from a path Pn with vertices u1, u2, · · · , un by joining ui and ui+1 to the new vertices vi, xi and wi, yi respectively and then joining vi, wi and xi, yi for i = 1, 2, · · · , n − 1. Definition 3. A double alternate triangular snake DA(Tn) consists of two alternate triangular snakes that have a common path. That is, a double alternate triangular snake is obtained from a path u1, u2, · · · , un by joining ui and ui+1(alternatively) to the two new vertices vi and wi for i = 1, 2, · · · , n − 1. Definition 4. A double alternate quadrilateral snake DA(Qn) consists of two alternate quadrilateral snakes that have a common path. That is, a double alternate quadrilateral snake is obtained from a path u1, u2, · · · , un by joining ui and ui+1(alternatively) to the two new vertices vi, xi and wi, yi respectively and adding the edges viwiand xiyi for i = 1, 2, · · · , n − 1. Definition 5. Let G be a graph. The subdivision graph S(G) is obtained from G by subdividing each edge of G with a vertex. Definition 6. The corona G1 ⊙ G2 of the graphs G1 and G2 is defined as the graph obtained by taking one copy of G1 (with p vertices) and p copies of G2 and then joining the ith vertex of G1 to every vertex of the ith copy of G2. 2. Main results Theorem 2.1. Let G1(p1, q1), G2(p2, q2), · · · , Gm(pm, qm) be vertex equitable graphs with qi even (i = 1, 2, · · · , m) and ui, vi be the vertices of Gi (1 ≤ i ≤ m) labeled by 0 and qi2 . Then the graph G obtained by identifying v1 with u2 and v2 with u3 and v3 with u4 and so on until we identify vm−1 with um is also a vertex equitable graph. Proof. First we assign the label i∑ j=1 qj 2 , 1 ≤ i ≤ m−1 to the common vertices between the two graphs Gi, Gi+1. Then we add the number i∑ j=1 qj 2 to all the remaining vertex labels of the graph Gi+1, 1 ≤ i ≤ m−1. Hence the edge labels are 1, 2, · · · , q1; q1 + 1, q1 + 2, · · · , q1 + q2, q1 + q2 + 1, q1 + q2 + 2, · · · , q1 + q2 + q3; · · · ; m−1∑ j=1 qj + 1, m−1∑ j=1 qj + 2, · · · , m∑ j=1 qj. Theorem 2.2. The graph S(D(Tn)) is a vertex equitable graph. 98 P. Jeyanthi et al. / J. Algebra Comb. Discrete Appl. 3(2) (2016) 97–104 b bb bb b b b b 3 2 0 5 1 5 3 4 2 Figure 1. b b b b b b bb b b b b b 21 3 4 1 6 3 0 7 7 54 6 Figure 2. Proof. The vertex equitable labeling shown in Figure 1 together with Theorem 2.1 proves the result. Theorem 2.3. The graph S(D(Qn)) is a vertex equitable graph. Proof. The vertex equitable labeling shown in Figure 2 together with Theorem 2.1 proves the result. Theorem 2.4. The graph S(DA(Tn)) is a vertex equitable graph. Proof. Let G = S(DA(Tn)). Let u1, u2, · · · , un be the vertices of path Pn. Case i. The triangle starts from u1. We construct DA(Tn) by joining u2i−1 and u2i to the new vertices vi, wi for 1 ≤ i ≤ ⌊ n 2 ⌋ . Let V (G) = V (DA(Tn)) ∪ {u′i|1 ≤ i ≤ n − 1} ∪ { xi, yi, x ′ i, y ′ i|1 ≤ i ≤ ⌊ n 2 ⌋} and E(G) = E(DA(Tn)) ∪ {uiu′i|1 ≤ i ≤ n} ∪ {u ′ iui+1|1 ≤ i ≤ n − 1} ∪ {xivi, x ′ iwi, viyi, wiy ′ i|1 ≤ i ≤ ⌊ n 2 ⌋ } ∪ {u2i−1xi, u2i−1x′i, yiu2i, y ′ iu2i|1 ≤ i ≤ ⌊ n 2 ⌋ }. We consider the following two sub cases: Subcase i. n is even. Here |V (G)| = 5n − 1 and |E(G)| = 6n − 2. Let A = {0, 1, 2, · · · , 3n − 1}. Define a vertex labeling f : V (G) → A as follows: For 1 ≤ i ≤ n 2 , f(u2i−1) = 6(i − 1), f(u′2i−1) = 6i − 5, f(u2i) = f(yi) = 6i − 1, f(xi) = f(y ′ i) = 6i − 3, f(wi) = f(x ′ i) = 6i − 4, f(vi) = 6i − 2 and f(u ′ 2i) = 6i if 1 ≤ i ≤ n−2 2 . It can be verified that the induced edge labels of S(DA(Tn)) are 1, 2, · · · , 6n − 2 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ A. Hence f is a vertex equitable labeling of S(DA(Tn)) . Subcase ii. n is odd. 99 P. Jeyanthi et al. / J. Algebra Comb. Discrete Appl. 3(2) (2016) 97–104 b b b b b bbbb b b b b b b b b b b b b b b b b b b b b b 2 8 14 4 10 16 2 8 14 3 9 15 3 9 15 5 11 17 0 17 5 6 11 12 1 6 7 121 13 Figure 3. Here |V (G)| = 5n − 4 and |E(G)| = 6n − 6. Let A = {0, 1, 2, · · · , 3n − 3}. Define a vertex labeling f : V (G) → A as follows: We label the vertices u2i−1 ( 1 ≤ i ≤ ⌈ n 2 ⌉) and u2i, u′2i−1, u ′ 2i, vi, v ′ i, wi, w ′ i( 1 ≤ i ≤ n−1 2 ) as in sub case (i). It can be verified that the induced edge labels of S(DA(Tn)) are 1, 2, · · · , 6n − 6 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ A. Hence f is a vertex equitable labeling of S(DA(Tn)). Case ii. The triangle starts from u2. We construct DA(Tn) by joining u2i and u2i+1 to the new vertices vi, wi for 1 ≤ i ≤ ⌈ n−2 2 ⌉ . Let V (G) = V (DA(Tn)) ∪ {u′i|1 ≤ i ≤ n − 1} ∪ { xi, yi, x ′ i, y ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉} and E(G) = E(DA(Tn)) ∪ {uiu′i|1 ≤ i ≤ n} ∪ {u ′ iui+1|1 ≤ i ≤ n − 1} ∪ {xivi, x ′ iwi, viyi, wiy ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉ } ∪ {u2ixi, u2ix′i, u2i+1yi, u2i+1y ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉ }. We consider the following two sub cases: Subcase i. n is odd. Here |V (G)| = 5n − 4 and |E(G)| = 6n − 6. Let A = {0, 1, 2, · · · , 3n − 3}. Define a vertex labeling f : V (G) → A as follows: f(u2i−1) = 6(i − 1) if 1 ≤ i ≤ ⌈ n 2 ⌉ , for 1 ≤ i ≤ ⌊ n 2 ⌋ , f(u2i) = f(u′2i−1) = 6i − 5, f(u′2i) = 6i − 4, f(wi) = f(x ′ i) = 6i − 3, f(xi) = f(y ′ i) = 6i − 2, f(yi) = 6i, f(vi) = 6i − 1. It can be verified that the induced edge labels of S(DA(Tn)) are 1, 2, · · · , 6n − 6 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ A. Hence f is a vertex equitable labeling of S(DA(Tn)). Subcase ii. n is even. Here |V (G)| = 5n − 7 and |E(G)| = 6n − 10. Let A = {0, 1, 2, · · · , 3n − 5}. Define a ver- tex labeling f : V (G) → A as follows: We label the vertices u2i−1, u′2i−1, u2i ( 1 ≤ i ≤ ⌈ n 2 ⌉) and vi, v ′ i, wi, w ′ i, xi, x ′ i, yi, y ′ i, u ′ 2i ( 1 ≤ i ≤ ⌈ n−2 2 ⌉) as in sub case (i). It can be verified that the induced edge labels of S(DA(Tn)) are 1, 2, · · · , 6n − 10 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ A. Hence f is a vertex equitable labeling of S(DA(Tn)). An example for the vertex equitable labeling of S(DA(T6)) where the two triangles start from u1 is shown in Figure 3. Theorem 2.5. The graph S(DA(Qn)) is a vertex equitable graph. Proof. Let G = S(DA(Qn)) . Let u1, u2, · · · , un be the vertices of path Pn. Case i. The quadrilateral starts from u1. We construct DA(Qn) by joining u2i−1 and u2i to the new vertices vi, wi and xi, yi respectively and then joining vi, xi and wi, yi for 1 ≤ i ≤ ⌊ n 2 ⌋ . Let V (G) = V (DA(Qn)) ∪ {u′i|1 ≤ i ≤ n − 1} ∪{ v′i, w ′ i, x ′ i, y ′ i, zi, z ′ i|1 ≤ i ≤ ⌊ n 2 ⌋} and E(G) = E(DA(Qn)) ∪ {uiu′i|1 ≤ i ≤ n} ∪ {u ′ iui+1|1 ≤ i ≤ n − 100 P. Jeyanthi et al. / J. Algebra Comb. Discrete Appl. 3(2) (2016) 97–104 1} ∪ {viv′i, vix ′ i, xizi, w ′ iwi, wiy ′ i, y ′ iyi, yiz ′ i|1 ≤ i ≤ ⌊ n 2 ⌋ } ∪ {u2i−1v′i, u2i−1w ′ i, u2izi, u2iz ′ i|1 ≤ i ≤ ⌊ n 2 ⌋ }. We consider the following two sub cases: Subcase i. n is even. Here |V (G)| = 7n − 1 and |E(G)| = 8n − 2. Let A = {0, 1, 2, · · · , 4n − 1}. Define a vertex labeling f : V (G) → A as follows: For 1 ≤ i ≤ n 2 , f(u2i−1) = 8(i − 1), f(u2i) = f(u′2i−1) = 8i − 1, f(xi) = f(zi) = 8i − 2, f(x′i) = 8i − 3, f(wi) = f(w ′ i) = 8i − 7, f(yi) = f(z ′ i) = 8i − 5, f(y ′ i) = 8i − 6 and f(u′2i) = 8i if 1 ≤ i ≤ n−2 2 . It can be verified that the induced edge labels of S(DA(Qn)) are 1, 2, · · · , 8n−2 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ A. Hence f is a vertex equitable labeling of S(DA(Qn)) . Subcase ii. n is odd. Here |V (G)| = 7n − 6 and |E(G)| = 8n − 8. Let A = {0, 1, 2, · · · , 4n − 4}. Define a vertex labeling f : V (G) → A as follows: We label the vertices u2i−1 ( 1 ≤ i ≤ ⌈ n 2 ⌉) and u2i, u′2i−1, u ′ 2i, vi, v ′ i, wi, w ′ i, xi, x′i, yi, y ′ i, zi, z ′ i ( 1 ≤ i ≤ n−1 2 ) as in sub case (i). It can be verified that the induced edge labels of S(DA(Qn)) are 1, 2, · · · , 8n − 8 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ A. Hence f is a vertex equitable labeling of S(DA(Qn)). Case ii. The quadrilateral starts from u2. We construct DA(Qn) by joining u2i and u2i+1 to the new vertices vi, wi and xi, yi respectively and then joining vi, xi and wi, yi for 1 ≤ i ≤ ⌈ n−2 2 ⌉ . Let V (G) = V (DA(Qn)) ∪ {u′i|1 ≤ i ≤ n − 1} ∪{ v′i, w ′ i, x ′ i, y ′ i, zi, z ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉} and E(G) = E(DA(Qn)) ∪ {uiu′i|1 ≤ i ≤ n − 1} ∪ {u ′ iui+1|1 ≤ i ≤ n − 1} ∪ {viv′i, vix ′ i, xizi, w ′ iwi, wiy ′ i, xix ′ i, y ′ iyi, yiz ′ i|1 ≤ i ≤ ⌈ n−2 2 ⌉ } ∪ {u2iv′i, u2iw ′ i, u2i+1zi, u2i+1z ′ i| 1 ≤ i ≤ ⌈ n−2 2 ⌉ }. We consider the following two sub cases: Subcase i. n is odd. Here |V (G)| = 7n − 6 and |E(G)| = 8n − 8. Let A = {0, 1, 2, · · · , 4n − 4}. Define a vertex labeling f : V (G) → A as follows: For 1 ≤ i ≤ ⌈ n 2 ⌉ , f(u2i−1) = 8(i − 1) , f(u′2i−1) = 8i − 7. For 1 ≤ i ≤ ⌊ n 2 ⌋ , f(u2i) = 8i − 7 , f(u′2i) = 8i, f(vi) = f(v ′ i) = 8i − 3, f(xi) = f(zi) = 8i − 1, f(x′i) = 8i − 2, f(wi) = f(w ′ i) = 8i − 6, f(yi) = f(z ′ i) = 8i − 4, f(y ′ i) = 8i − 5. It can be verified that the induced edge labels of S(DA(Qn)) are 1, 2, · · · , 8n − 8 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ A. Hence f is a vertex equitable labeling of S(DA(Qn)). Subcase ii. n is even. Let |V (G)| = 7n − 11 and |E(G)| = 8n − 14. Let A = {0, 1, 2, · · · , 4n − 7}. Define a ver- tex labeling f : V (G) → A as follows: We label the vertices u2i−1, u2i, u′2i−1 ( 1 ≤ i ≤ ⌈ n 2 ⌉) and u′2i, vi, v ′ i, wi, w ′ i, xi, x ′ i, yi, y ′ i, ( 1 ≤ i ≤ ⌈ n−2 2 ⌉) as in sub case (i). It can be verified that the induced edge labels of S(DA(Qn)) are 1, 2, · · · , 8n−14 and |vf (i) − vf (j)| ≤ 1 for all i, j ∈ A. Hence f is a vertex equitable labeling of S(DA(Qn)). An example for the vertex equitable labeling of S(DA(Q7)) where the two quadrilaterals start from u1 is shown in Figure 4. Theorem 2.6. Let G1(p1, q), G2(p2, q), · · · , Gm(pm, q) ) be vertex equitable graphs with q odd ui, vi be vertices of Gi (1 ≤ i ≤ m) labeled by 0 and ⌈ q 2 ⌉ . Then the graph G obtained by joining v1 with u2 and v2 with u3 and v3 with u4 and so on until joining vm−1 with um by an edge is also a vertex equitable graph. Proof. The graph G has p1 + p2 + · · · + pm vertices and mq + (m − 1) edges. Let fi be the vertex equitable labeling of Gi (1 ≤ i ≤ m) and let A = { 0, 1, 2, · · · , ⌈ mq+m−1 2 ⌉} . Define a vertex labeling f : V (G) → A as f(x) = fi(x) + (i−1)(q+1) 2 if x ∈ Gi for 1 ≤ i ≤ m . The edge labels of Gi are increased by (i − 1)(q + 1) for i = 1, 2, · · · , m under the new labeling f. The bridge between the two graphs Gi , Gi+1 will get the label i(q + 1), 1 ≤ i ≤ m − 1. Hence the edge labels of G are distinct and is {1, 2, · · · , mq + m − 1}. Also |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Then the graph G is a vertex equitable graph. 101 P. Jeyanthi et al. / J. Algebra Comb. Discrete Appl. 3(2) (2016) 97–104 b b b b b b b b b b b b b b bbb b bb b b b b b b b b bbb b bb b b b b bb b b b b b b 1 2 3 9 10 11 17 18 19 4 5 6 12 13 14 20 21 22 1 4 9 12 17 20 3 6 11 14 19 22 0 7 7 8 8 15 15 16 16 23 23 24 24 Figure 4. b b b b bb b b b b b b b b b b b b b b b b b b b b b b b b 6 7 8 11 2 3 4 5 10 9 87 13 14 151 1 2 3 4 15 14 13 11 6 12 105 0 16 Figure 5. Remark 2.7. [7] The graph DA(Qm)⊙nK1 and DA(Tm)⊙nK1 are vertex equitable graphs if m, n = 1, 2. Theorem 2.8. The graph DA(Q2) ⊙ nK1 is a vertex equitable graph for n ≥ 3 Proof. Let G = DA(Q2) ⊙ nK1. Let V (G) = {u1, u2, v, w, x, y} ∪ {uij|1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vi, wi, xi, yi|1 ≤ i ≤ n} and E(G) = {u1u2, u1v, vw, wu2, u1x, xy, yu2} ∪ {uiuij|1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vvi, wwi, xxi, yyi|1 ≤ i ≤ n}. Here |V (G)| = 6(n+1) and |E(G)| = 6n+7. Let A = { 0, 1, 2, · · · , ⌈ 6n+7 2 ⌉} . Define a vertex labeling f : V (G) → A as follows: For 1 ≤ i ≤ n, f(u1i) = i, f(vi) = i+1, f(yi) = n+2+i, f(ui) = 0, f(u2) = 3n+4, f(v) = n+1, f(w) = 2(n+1), f(x) = n+2, f(y) = 2(n+2), f(u2i) = 3n+4−i if 1 ≤ i ≤ n − 1 , f(u2n) = 2n + 3, f(w1) = 1, f(wi) = 3n + 5 − i if 2 ≤ i ≤ n, f(xi) = n + i + 1 if 1 ≤ i ≤ n − 1, f(xn) = 2n + 3. It can be verified that the induced edge labels of DA(Q2) ⊙ nK1 are 1, 2, · · · , 6n + 7 and |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Hence f is a vertex equitable labeling of DA(Q2) ⊙ nK1. An example for the vertex equitable labeling of DA(Q2) ⊙ 4K1 is shown in Figure 5. Theorem 2.9. The graph DA(Qm) ⊙ nK1 is a vertex equitable graph for m, n ≥ 3 . Proof. By Theorem 2.8, DA(Q2) ⊙ nK1 is a vertex equitable graph. Let Gi = DA(Q2) ⊙ nK1 for 1 ≤ i ≤ m − 1. Since each Gi has 6n + 7 edges, by Theorem 2.6, DA(Qm) ⊙ nK1 admits vertex equitable labeling. An example for the vertex equitable labeling of DA(Q6) ⊙ 4K1 is shown in Figure 6. 102 P. Jeyanthi et al. / J. Algebra Comb. Discrete Appl. 3(2) (2016) 97–104 b b b b bb b b b b b b b b b b b b bb b b bb b b b b b b b b b b b b b b b b b b b b b b b bb b b b bb b b bb b b 0 6 12 16 105 32 2822 16 21 26 7 8 9 10 2724 23 22 1 15 14 13 2120 19 18 3 4 5 2 3031 17 29 2524 23 26 7 8 11 6 1 2 4 3 31 30 27 29 15 14 11 13 17 18 20 19 Figure 6. b bb b b b b b b b b b b b b b 1 2 3 8 7 6 5 6 7 2 3 4 0 9 5 4 Figure 7. Theorem 2.10. The graph DA(T2) ⊙ nK1 is a vertex equitable graph for n ≥ 3 . Proof. Let G = DA(T2) ⊙ nK1. Let V (G) = {u1, u2, u, w} ∪ {uij|1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vi, wi|1 ≤ i ≤ n} and E(G) = {u1u2, u1v, vu2, u1w, wu2} ∪ {uiuij|1 ≤ i ≤ 2, 1 ≤ j ≤ n} ∪ {vvi, wwi|1 ≤ i ≤ n}. Here |V (G)| = 4(n + 1) and |E(G)| = 4n + 5. Let A = {0, 1, 2, · · · , ⌈ 4n+5 2 ⌉ }. Define a vertex labeling f : V (G) → A as follows. For 1 ≤ i ≤ n, f(u1i) = i, f(u2i) = 2n + 3 − i, f(vi) = i + 1, f(wi) = n + 1 + i, f(u1) = 0, f(u2) = 2n + 3, f(v) = n + 1, f(w) = n + 2. It can be verified that the induced edge labels of DA(T2) ⊙ nK1 are 1, 2, · · · , 4n + 5 and |vf (a) − vf (b)| ≤ 1 for all a, b ∈ A. Hence f is a vertex equitable labeling of DA(T2) ⊙ nK1 . An example for the vertex equitable labeling of DA(T2) ⊙ 3K1 is shown in Figure 7. Theorem 2.11. The graph DA(Tm) ⊙ nK1 is a vertex equitable graph for m, n ≥ 3 . Proof. By Theorem 2.10, DA(T2) ⊙ nK1 is a vertex equitable graph. Let Gi = DA(T2) ⊙ nK1 , 103 P. Jeyanthi et al. / J. Algebra Comb. Discrete Appl. 3(2) (2016) 97–104 b b b b b b b b b b b b b b b b bbb bbb b b b b b b b b b b 5 6 7 2 3 4 161514 131211 1 2 3 6 7 8 17 16 15 10 11 12 5 4 0 9 14 13 189 Figure 8. 1 ≤ i ≤ m−1. Since each Gi has 4n+5 edges, by Theorem 2.6, DA(Tm)⊙nK1 admits a vertex equitable labeling. An example for the vertex equitable labeling of DA(Tm) ⊙ 4K1 is shown in Figure 8. Acknowledgment: The authors would like to thank the referees for their valuable suggestions to improve the paper. References [1] J. A. Gallian, Graph labeling, Electron. J. Combin. (2015) (Dynamic Survey #DS6). [2] F. Harary, Graph theory, Addison-Wesley, Reading Mass, 1972. [3] P. Jeyanthi, A. Maheswari, Some results on vertex equitable labeling, Open J. Discrete Math. 2(2) (2012) 51–57. [4] P. Jeyanthi, A. Maheswari, Vertex equitable labeling of transformed trees, J. Algorithms Comput. 44(1) (2013) 9–20. [5] P. Jeyanthi, A. Maheswari, Vertex equitable labeling of cyclic snakes and bistar graphs, J. Sci. Res. 6(1) (2014) 79–85. [6] P. Jeyanthi, A. Maheswari, M. Vijayalaksmi, Vertex equitable labeling of cycle and star related graphs, J. Sci. Res. 7(3) (2015) 33–42. [7] P. Jeyanthi, A. Maheswari, Vertex equitable labeling of cycle and path related graphs, Util. Math. 98 (2015) 215–226. [8] P. Jeyanthi, A. Maheswari, M. Vijayalakshmi, Vertex equitable labeling of double alternate snake graphs, J. Algorithms Comput. 46 (2015) 27–34. [9] M. Seenivasan, A. Lourdusamy, Vertex equitable labeling of graphs, J. Discrete Math. Sci. Cryptogr. 11(6) (2008) 727–735. 104 http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6/pdf http://dx.doi.org/10.4236/ojdm.2012.22009 http://dx.doi.org/10.4236/ojdm.2012.22009 http://jac.ut.ac.ir/pdf_346_b7a385d110f5ec4c2d805c82bcf3079e.html http://jac.ut.ac.ir/pdf_346_b7a385d110f5ec4c2d805c82bcf3079e.html http://dx.doi.org/10.3329/jsr.v6i1.15044 http://dx.doi.org/10.3329/jsr.v6i1.15044 http://dx.doi.org/10.3329/jsr.v7i3.22810 http://dx.doi.org/10.3329/jsr.v7i3.22810 http://jac.ut.ac.ir/pdf_357_fff6f09f5a1ff8b27a45504304da01c5.html http://jac.ut.ac.ir/pdf_357_fff6f09f5a1ff8b27a45504304da01c5.html http://www.dx.doi.org/10.1080/09720529.2008.10698401 http://www.dx.doi.org/10.1080/09720529.2008.10698401 Introduction Main results References