EVEN VERTEX EQUITABLE EVEN LABELING FOR CYCLE RELATED GRAPHS

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1 Kragujevac Joural of Matheatics Volue 43(3) (019), Pages EVEN VERTEX EQUITABLE EVEN LABELING FOR CYCLE RELATED GRAPHS A. LOURDUSAMY 1 AND F. PATRICK 1 Abstract. Let G be a graph with p vertices ad q edges ad A = {0,, 4,..., q+1} if q is odd or A = {0,, 4,..., q} if q is eve. A graph G is said to be a eve vertex equitable eve labelig if there exists a vertex labelig f : V (G) A that iduces a edge labelig f defied by f (uv) = f(u)+f(v) for all edges uv such that for all a ad b i A, v f (a) v f (b) 1 ad the iduced edge labels are, 4,..., q, where v f (a) be the uber of vertices v with f(v) = a for a A. A graph that adits eve vertex equitable eve labelig is called a eve vertex equitable eve graph. [ I this ] paper, we prove that the graphs C P, C (Q ) if 0, 3 (od 4), P ; C () if 0 (od 4), C e C ad the graph obtaied by duplicatig a arbitrary vertex ad edge of a cycle C adit a eve vertex equitable eve labelig. 1. Itroductio All graphs cosidered here are siple, fiite, coected ad udirected. The vertex set ad the edge set of a graph are deoted by V (G) ad E(G) respectively. We follow the basic otatios ad teriology of graph theory as i [3]. A labelig of a graph is a ap that carries the graph eleets to the set of ubers, usually to the set of o-egative or positive itegers. If the doai is the set of vertices the labelig is called vertex labelig. If the doai is the set of edges, the we speak about edge labelig. If the labels are assiged to both vertices ad edges the the labelig is called total labelig. For a dyaic survey of various graph labelig, we refer to Gallia []. Lourdusay et al. itroduced the cocept of vertex equitable labelig i [13]. Let G be a graph with p vertices ad q edges ad let A = { 0, 1,,..., } q. A vertex Key words ad phrases. Vertex equitable labelig, eve vertex equitable eve labelig. 010 Matheatics Subject Classificatio. Priary: 05C78. Received: Deceber 0, 016. Accepted: Jauary 10,

2 48 A. LOURDUSAMY AND F. PATRICK labelig f : V (G) A iduces a edge labelig f defied by f (uv) = f(u) + f(v) for all edges uv. For a A, let v f (a) be the uber of vertices v with f(v) = a. A graph G is said to be vertex equitable if there exists a vertex labelig f such that for all a ad b i A, v f (a) v f (b) 1 ad the iduced edge labels are 1,, 3,..., q. I [13], Lourdusay et al. proved that graphs like path, bistar, cob graph, cycle C for 0, 3 (od 4), K,, C (t) 3 for t, quadrilateral sake, K +K 1, K 1, K1,+k for 1 k 3, ladder ad arbitrary super divisio of ay path ad cycle C with 0, 3 (od 4) adit vertex equitable labelig. Also, they proved K 1, for 4, ay Euleria graph with edges where 1, (od 4), the wheel W, the coplete graph K if > 3 ad triagular cactus with q 0, 6, 9 (od 1) are ot vertex equitable graphs. I additio, they proved that if G is a graph with p vertices ad q edges, q is eve ad p < q + the G is ot vertex equitable graph. I [4 11], Jeyathi et al. proved that T p -tree, T K, where T is a T p -tree with eve uber of vertices, B(, + 1), the caterpillar S(x 1, x..., x ), crow, P, T ôp, T ôp, T ôc with 0, 3 (od 4) ad T õc with 0, 3 (od 4), tadpoles, C C, ared crows, [P ; C], P ôk 1,, jewel graph J, jelly fish graph (JF ), balaced lobster graph BL(,, ), L K, L ôk 1,, DA(T ) K 1, DA(T ) K 1, DA(Q ) K 1, DA(Q ) K 1, S (P K 1 ), S (B(, )), S (P P ), S (Q ), S(D(T )), S(D(Q )), S(DA(T )), S(DA(Q )), DA(Q ) K 1, DA(T ) K 1, KY (, ), P (.QS ), P (.QS ), C(.QS ), NQ() ad K 1, P adit vertex equitable labelig. Motivated by the cocept of vertex equitable labelig, Lourdusay et al. itroduced the cocept of eve vertex equitable eve labelig i [1]. I [1], they proved that path, cob, coplete bipartite, cycle, K + K 1, bistar, ladder, S(L ), S(B, ) ad L K 1 adit a eve vertex equitable eve labelig. I this paper, we prove that the ] if 0 (od 4), C e C graphs C P, C (Q ) if 0, 3 (od 4), [ P ; C () ad the graph obtaied by duplicatig a arbitrary vertex ad edge of a cycle C adit a eve vertex equitable eve labelig. Defiitio 1.1. [13] Let G be a graph with p vertices ad q edges ad let A = {0, 1,,..., q }. A vertex labelig f : V (G) A iduces a edge labelig f defied by f (uv) = f(u) + f(v) for all edges uv. For a A, let v f (a) be the uber of vertices v with f(v) = a. A graph G is said to be vertex equitable if there exists a vertex labelig f such that for all a ad b i A, v f (a) v f (b) 1 ad the iduced edge labels are 1,, 3,..., q. Defiitio 1.. [1] Let G be a graph with p vertices ad q edges ad A = {0,, 4,..., q + 1} if q is odd or A = {0,, 4,..., q} if q is eve. A graph G is said to be a eve vertex equitable eve labelig if there exists a vertex labelig f : V (G) A that iduces a edge labelig f defied by f (uv) = f(u) + f(v) for all edges uv such that for all a ad b i A, v f (a) v f (b) 1 ad the iduced edge labels are, 4,..., q,

3 EVEN VERTEX EQUITABLE EVEN LABELING 49 where v f (a) be the uber of vertices v with f(v) = a for a A. A graph that adits eve vertex equitable eve labelig is called a eve vertex equitable eve graph. Defiitio 1.3. [6] A ared crow is a cycle attached with paths of equal legths at each vertex of the cycle. It is deoted by C P, where P is a path of legth 1. Defiitio 1.4. [14] The graph C (Q ) is defied as isoorphic quadrilateral sake attached with each vertex of cycle C. is the uber of C 4 attached i the quadrilateral sake. Defiitio 1.5. [1] Let G be a graph with fixed vertex v ad let [P ; G] be the graph obtaied fro copies of G by joiig v ij ad v i+1j by eas of a edge, for soe j ad 1 i 1. Defiitio 1.6. [6] Let G 1 ad G be two graphs. The idetificatio of G 1 ad G is a graph obtaied by idetifyig a edge of G 1, with a edge of G ad is deoted by G 1 e G. Defiitio 1.7. [6] Let G be a graph ad v be ay vertex of G. A ew vertex v is said to be duplicatio of v if all the vertices which are adjacet to v are adjacet to v. The graph obtaied by duplicatio v is deoted by D(G, v ). Defiitio 1.8. [6] Let G be a graph ad e be ay edge of G. A ew edge e is said to be duplicatio of a edge e if all the edges which are icidet to e i G are icidet to e. The graph obtaied by duplicatio e is deoted by D(G, e ).. Mai Results Theore.1. The graph C P is a eve vertex equitable eve graph. Proof. Let u 11, u 1,..., u 1 be the vertices of the cycle C ad u 11 u 1... u 1, u 1 u u 3... u,..., u 1 u... u be the vertices of the path P attached with u i1 by idetifyig u ij with u i1 for 1 i, 1 j. The C P is of order ad size. 0,,..., + 1, if is odd, Defie f : V (C P ) A = 0,,...,, if is eve, as follows. Case 1. 0, 3 (od 4). For 1 i (i 1) + j 1, if i is odd, j is odd ad 1 j, (i 1) + j, if i is odd, j is eve ad 1 j, i j + 1, if i is eve, j is odd ad 1 j, i j, if i is eve, j is eve ad 1 j.

4 430 A. LOURDUSAMY AND F. PATRICK For + 1 i, (i 1) + j + 1, if i is odd, j is odd ad 1 j, (i 1) + j, if i is odd, j is eve ad 1 j, i j + 1, if i is eve, j is odd ad 1 j, i j +, if i is eve, j is eve ad 1 j. Case. 1, (od 4) ad is odd. For 1 i i j, if i is odd, j is odd ad 1 j, i j + 1, if i is odd, j is eve ad 1 j, (i 1) + j, if i is eve, j is odd ad 1 j, (i 1) + j 1, if i is eve, j is eve ad 1 j. For i = + 1, (i 1) + j +, if j is odd ad 1 j, (i 1) + j 1, if j is eve ad 1 j. For i = +, i j +, if j is odd ad 1 j 1, i j + 1, if j is eve ad 1 j 1, i j, if j =. For + 3 i, (i 1) + j, if i is eve, j is odd ad 1 j, (i 1) + j + 1, if i is eve, j is eve ad 1 j, i j +, if i is odd, j is odd ad 1 j, i j + 1, if i is odd, j is eve ad 1 j. Case 3. (od 4) ad is eve. For 1 i, (i 1) + j 1, if i is odd, j is odd ad 1 j, (i 1) + j, if i is odd, j is eve ad 1 j, i j + 1, if i is eve, j is odd ad 1 j, i j, if i is eve, j is eve ad 1 j.

5 EVEN VERTEX EQUITABLE EVEN LABELING 431 For i = + 1, i + j, if j = 1, i j +, if j is eve ad j, i j + 1, if j is odd ad j. For i = +, (i 1) + j + 1, if j is odd ad 1 j 1, (i 1) + j +, if j is eve ad 1 j 1,, if j =. For + 3 i, i j + 1, if i is eve, j is odd ad 1 j, i j +, if i is eve, j is eve ad 1 j, (i 1) + j + 1, if i is odd, j is odd ad 1 j, (i 1) + j, if i is odd, j is eve ad 1 j. Case 4. 1 (od 4) ad is eve. For 1 i i j + 1, if i is odd, j is odd ad 1 j, i j, if i is odd, j is eve ad 1 j, (i 1) + j 1, if i is eve, j is odd ad 1 j, (i 1) + j, if i is eve, j is eve ad 1 j. For + 1 i, (i 1) + j + 1, if i is eve, j is odd ad 1 j, (i 1) + j, if i is eve, j is eve ad 1 j, i j + 1, if i is odd, j is odd ad 1 j, i j +, if i is odd, j is eve ad 1 j. I above four cases, it ca be verified that the iduced edge labels of C P are, 4,..., ad v f (a) v f (b) 1 for all a, b A. Thus, C P is a eve vertex equitable eve graph. Theore.. The graph C (Q ) is a eve vertex equitable eve graph if 0, 3 (od 4). Proof. Let G = C (Q ). Let V (G) = {u i : 1 i } {u ij, v ij, w ij : 1 i ; 1 j }

6 43 A. LOURDUSAMY AND F. PATRICK ad E(G) = {u i u i+1 : 1 i 1} {u 1 u } {u i v i1, u i w i1 : 1 i } {v ij u ij, w ij u ij : 1 i ; 1 j } {u ij v ij+1, u ij w ij+1 : 1 i ; 1 j 1}. The G is of order 3 + ad size 4 +. Case 1. 0 (od 4). Defie f : V (G) A = {0,,..., 4 + } as follows: (4 + 1)(i 1), if i is odd ad 1 i, f(u i ) = (4 + 1)(i 1) +, if i is odd ad + 1 i, (4 + 1)i, if i is eve ad 1 i. For 1 j, (4 + 1)(i 1) + 4j, if i is odd ad 1 i, (4 + 1)(i 1) + 4j +, if i is odd ad + 1 i, (4 + 1)i 4j, if i is eve ad 1 i, (4 + 1)(i 1) + 4(j 1) +, if i is odd ad 1 i, f(v ij ) = (4 + 1)i 4j, if i is eve ad 1 i, (4 + 1)i 4j +, if i is eve ad + 1 i, (4 + 1)(i 1) + 4j, if i is odd ad 1 i, f(w ij ) = (4 + 1)i 4j +, if i is eve ad 1 i, (4 + 1)i 4j + 4, if i is eve ad + 1 i. Case. 3 (od 4). Defie f : V (G) A = {0,,..., } as follows: (4 + 1)i 1, if i is odd ad 1 i f(u i ) = (4 + 1)i + 1, if i is odd ad i, (4 + 1)(i 1) + 1, if i is eve ad 1 i. For 1 j, (4 + 1)i 4j 1, if i is odd ad 1 i (4 + 1)i 4j + 1, if i is odd ad i, (4 + 1)(i 1) + 4j + 1 if i is eve ad 1 i, (4 + 1)i 4(j 1) 1, if i is odd ad 1 i, f(v ij ) = (4 + 1)(i 1) + 4j 1, if i is eve ad 1 i, (4 + 1)i 4j + 1, if i is odd ad 1 i, f(w ij ) = (4 + 1)(i 1) + 4(j 1) + 1, if i is eve ad 1 i (4 + 1)(i 1) + 4j + 1, if i is eve ad + 1 i.

7 EVEN VERTEX EQUITABLE EVEN LABELING 433 I both the cases, it ca be verified that the iduced edge labels of C (Q ) are, 4,..., 8 + ad v f (a) v f (b) 1 for all a, b A. Thus, C (Q ) is a eve vertex equitable eve graph for 0, 3 (od 4). Theore.3. The graph [ ] P ; C () is a eve vertex equitable eve graph if 0 (od 4). Proof. Let u 1, u..., u be the vertices of a path P ad the vertex u i (1 i ) is attached with the ceter vertex of the i th copy of C (). Let u i = v i1 (ceter vertex i the i th copy of C). Let v ij ad v ij, for 1 i, j, be the reaiig vertices i the i th copy of C (). The [ ] P ; C () is of order ad size ,,..., +, if + 1 is odd, Defie f : V (G) A = 0,,..., + 1 if + 1 is eve, as follows: ( + 1)i ( + 1), if i is odd ad 1 i, f(v i1 ) = ( + 1)i, if i is eve ad 1 i, For 1 i ad i is odd, ( + 1)i ( + 1) j + 1, if j is odd ad j, f(v ij ) = ( + 1)i ( + 1) j +, if j is eve ad j, ( + 1)i ( + 1) j, if j is eve ad + 1 j, ( + 1)i ( + 1) + j, if j is eve ad j, f(v ij) = ( + 1)i ( + 1) + j 1, if j is odd ad j, ( + 1)i ( + 1) + j + 1, if j is odd ad + 1 j. For 1 i ad i is eve, ( + 1)i ( + 1) j + 1, if j is eve ad j, f(v ij ) = ( + 1)i ( + 1) j +, if j is odd ad j, ( + 1)i ( + 1) j, if j is odd ad + 1 j, ( + 1)i ( + 1) + j, if j is odd ad j, f(v ij) = ( + 1)i ( + 1) + j 1, if j is eve ad j, ( + 1)i ( + 1) + j + 1, if j is eve ad + 1 j. It ca be verified that the iduced edge labels of [ ] P ; C () are, 4,..., 4 + ad v f (a) v f (b) 1 for all a, b A. Thus, [ ] P ; C () is a eve vertex equitable eve graph for 0 (od 4). Theore.4. If G = C e C is a graph obtaied by idetifyig a edge of C with a edge of C the G is a eve vertex equitable eve graph.

8 434 A. LOURDUSAMY AND F. PATRICK Proof. Let v 1, v, v be the vertices of C ad u 1, u..., u be the vertices of C. Let G be a graph obtaied by idetifyig a edge v 1 v of C with a edge u u 1 of C. The G is of order + ad size ,,..., +, if + 1 is odd, Defie f : V (G) A = 0,,..., + 1, if + 1 is eve, as follows. Case 1. 0 (od 4). Subcase (od 4). f(v 1 ) =, i, if i is eve ad i, f(v i ) = i, if i is eve ad + 1 i, i 1, if i is odd ad i, ( + ) i +, if i is eve ad i, f(u i ) = ( + ) i, if i is eve ad + 1 i, ( + ) i + 1, if i is odd ad i. Subcase (od 4). Subcase = 3. ( + ) i + 1, if i is eve ad i, f(v i ) = ( + ) i +, if i is odd ad i + 1, ( + ) i, if i is odd ad + i, f(u 1 ) =4, f(u ) =0, f(u 3 ) =. Subcase 1... > 3. f(v 1 ) =, f(v ) =, i 3, if i is odd ad 3 i + 1, f(v i ) = i 1, if i is odd ad + i, i, if i is eve ad 3 i, ( + ) i + 1, if i is eve ad i, f(u i ) = ( + ) i +, if i is odd ad i + 1, ( + ) i, if i is odd ad + i. Subcase 1.3. (od 4). f(v 1 ) =,

9 EVEN VERTEX EQUITABLE EVEN LABELING 435 f(v ) =, i 3, if i is odd ad 3 i + 1, f(v i ) = i 1, if i is odd ad + i, i, if i is eve ad 3 i, ( + ) i +, if i is eve ad i + 1, f(u i ) = ( + ) i, if i is eve ad + i, ( + ) i + 1, if i is odd ad i. Subcase (od 4). f(v 1 ) =, i, if i is eve ad i, f(v i ) = i, if i is eve ad + 1 i, i 1 if i is odd ad i, ( + ) i + 1, if i is eve ad i, f(u i ) = ( + ) i +, if i is odd ad i ( + ) i, if i is odd ad + 1 i. Case. 3 (od 4). Subcase.1. 3 (od 4). f(v 1 ) = + 1, i, if i is eve ad i f(v i ) = i, if i is eve ad + 1 i, i 1, if i is odd ad i, ( + ) i +, if i is eve ad i f(u i ) = ( + ) i, if i is eve ad + 1 i, ( + ) i + 1, if i is odd ad i. Subcase.. (od 4) ad > 3. f(v 1 ) = 1, f(v ) = + 1, i 3, if i is odd ad 3 i + 1, f(v i ) = i 1, if i is odd ad + i, i, if i is eve ad 3 i,

10 436 A. LOURDUSAMY AND F. PATRICK ( + ) i +, if i is odd ad i, f(u i ) = ( + ) i + 1, if i is eve ad i + 1, ( + ) i 1, if i is eve ad + i. Subcase.3. 1 (od 4) ad > 3. f(v 1 ) = 1, f(v ) = + 1, i 3, if i is odd ad 3 i + 1, f(v i ) = i 1, if i is odd ad + i, i, if i is eve ad 3 i, ( + ) i +, if i is eve ad i, f(u i ) = ( + ) i + 1, if i is odd ad i ( + ) i 1, if i is odd ad + 1 i. Case 3. (od 4). Subcase (od 4). f(v 1 ) = +, i 1, if i is odd ad i, f(v i ) = i, if i is eve ad i + 1, i, if i is eve ad + i, + i + 1, if i is odd ad i 1, f(u i ) = + i, if i is eve ad i + i, if i is eve ad + 1 i 1. Subcase 3.. (od 4). Subcase =. f(v 1 ) = +, i, if i is eve ad i, f(v i ) = i 1, if i is odd ad i, + i, if i is eve ad i 1, f(u i ) = + i, if i is eve ad i, + i 1, if i is odd ad i. Subcase 3... > ad = 6. f(v 1 ) = +,

11 EVEN VERTEX EQUITABLE EVEN LABELING 437 i, if i is eve ad i, f(v i ) = i 1, if i is odd ad i, + i, if i is eve ad i 1, + i, if i is eve ad f(u i ) = i, + i 3, if i is odd ad i, + i 1, if i is odd ad 1 i. Subcase > ad > 6. f(v 1 ) = +, i, if i is eve ad i, f(v i ) = i 1, if i is odd ad i, f(u 3 ) =, + i, if i is eve ad i 1, f(u i ) = + i, if i is eve ad i, + i 1 if i is odd ad 5 i. Case 4. 1 (od 4), 1 (od 4) ad, > 5. f(v 1 ) = 1, f(v ) = 1, f(v 3 ) = + 3, i 3, if i is odd ad 4 i, f(v i ) = i 4, if i is eve ad 4 i + 4, i, if i is eve ad + 5 i, ( + 1) + i, if i is eve ad i 1, f(u i ) = + i, if i is odd ad i + i, if i is odd ad + 1 i 1. I above all cases, it ca be verified that the iduced edge labels of G are, 4,..., + ad v f (a) v f (b) 1 for all a, b A. Hece, the graph G is a eve vertex equitable eve graph. Theore.5. The graph obtaied by duplicatig a arbitrary vertex of a cycle C is a eve vertex equitable eve graph. Proof. Let u 1, u..., u be the vertices of cycle C. Let G = D(C, u ) be the graph obtaied by duplicatig a arbitrary vertex u of C. Without loss of geerality take u = u 1 ad the duplicatio of u 1 be u 1. The G is of order + 1 ad size +.

12 438 A. LOURDUSAMY AND F. PATRICK 0,,..., + 3, if + is odd, Defie f : V (G) A = 0,,..., +, if +, is eve, as follows. Case 1. 0 (od 4) ad > 4. f(u 1) = +, f(u ) = +, i +, if i is eve ad 1 i 4, f(u i ) = i, if i is eve ad i 1, i 1, if i is odd ad 1 i 1. Case. 1 (od 4) ad > 5. f(u 1) = + 1, f(u 1 ) = 1, f(u ) = + 3, i + 1, if i is eve ad i 5, f(u i ) = i 1, if i is eve ad i 1, i if i is odd ad i 1. Case 3. (od 4). f(u 1) =, f(u 1 ) = +, Case 4. 3 (od 4). f(u ) = +, i, if i is eve ad i 1, f(u i ) = i + 1, if i is odd ad i 4, i 1, if i is odd ad i 1. f(u 1) = + 3, f(u ) = + 1, i, if i is odd ad 1 i 1, f(u i ) = i + 1, if i is eve ad 1 i 3, i 1, if i is eve ad i 1. I above four cases, it ca be verified that the iduced edge labels of G are, 4,..., +4 ad v f (a) v f (b) 1 for all a, b A. Hece, the graph obtaied by duplicatig a arbitrary vertex of a cycle C is a eve vertex equitable eve graph.

13 EVEN VERTEX EQUITABLE EVEN LABELING 439 Theore.6. The graph obtaied by duplicatig a arbitrary edge of a cycle C is a eve vertex equitable eve graph. Proof. Let u 1, u..., u be the vertices of cycle C. Let G = D(C, e ) be the graph obtaied by duplicatig a arbitrary edge e of C. Without loss of geerality take e = u 1 u ad the duplicatio of e be e = u 1u. The G is of order + ad size ,,..., + 4, if + 3 is odd, Defie f : V (G) A = 0,,..., + 3, if + 3, is eve, as follows. Case 1. 0 (od 4). Subcase 1.1. For = 4, Subcase 1.. For > 4, f(u 1) =, f(u ) =, f(u 1 ) = +, f(u 1 ) =, f(u 1) =8, f(u ) =6, f(u 1 ) =0, f(u ) =, f(u 3 ) =f(u 4 ) = 4. f(u ) = + 4, i 1, if i is odd ad i, f(u i ) = i, if i is eve ad i, i, if i is eve ad + 1 i. Case. 1 (od 4) ad > 5. f(u 1) = 3, f(u ) =f(u 1 ) = + 1, f(u 1 ) =f(u ) = + 3, i + 1, if i is eve ad i 3, f(u i ) = i 1, if i is eve ad i, i, if i is odd ad i. Case 3. (od 4). f(u 1) = + 4,

14 440 A. LOURDUSAMY AND F. PATRICK f(u ) =, f(u 1 ) =, f(u 1 ) =, f(u ) = +, i, if i is eve ad i, f(u i ) = i, if i is eve ad + 1 i, i 1, if i is odd ad i. Case 4. 3 (od 4) ad > 7. f(u 1) =f(u ) = 1, f(u 1 ) =f(u ) = + 3, f(u 1 ) = + 1, i 1, if i is eve ad i, f(u i ) = i, if i is odd ad i 3, i, if i is odd ad i. I above four cases, it ca be verified that the iduced edge labels of G are, 4,..., +6 ad v f (a) v f (b) 1 for all a, b A. Hece, the graph obtaied by duplicatig a arbitrary edge of a cycle C is a eve vertex equitable eve graph. Refereces [1] S. Avadayappa ad R. Vasuki, New failies of ea graphs, Iteratioal Joural of Matheatical Cobiatorics (010), [] J. A. Gallia, A dyaic survey of graph labelig, Electro. J. Cobi. 5 (1998), DS6. [3] F. Harary, Graph Theory, Addiso-Wesley Professioal, Readig, MA, 197. [4] P. Jeyathi ad A. Maheswari, Soe results o vertex equitable labelig, Ope Joural of Discrete Matheatics (01), [5] P. Jeyathi ad A. Maheswari, Vertex equitable labelig of trasfored trees, Joural of Algoriths ad Coputatio 44 (013), 9 0. [6] P. Jeyathi ad A. Maheswari, Vertex equitable labelig of cycle ad path related graphs, Util. Math. 98 (015), [7] P. Jeyathi, A. Maheswari ad M. Vijayalaksi, Vertex equitable labelig of cycle ad star related graphs, Joural of Scietific Research 7 (015), [8] P. Jeyathi, A. Maheswari ad M. Vijayalaksi, Vertex equitable labelig of double alterate sake graphs, Joural of Algoriths ad Coputatio 46 (015), [9] P. Jeyathi, A. Maheswari ad M. Vijayalaksi, Vertex equitable labelig of super subdivisio graphs, Scietific Iteratioal 7 (015), [10] P. Jeyathi, A. Maheswari ad M. Vijayalaksi, New results o vertex equitable labelig, J. Algebra Cob. Discrete Struct. Appl. 3 (016), [11] P. Jeyathi, A. Maheswari ad M. Vijayalaksi, Vertex equitable labelig of uio of cycle sake related graphs, Proyeccioes 35 (016), [1] A. Lourdusay, J. S. Mary ad F. Patrick, Eve vertex equitable eve labelig, Sciecia Acta Xaveriaa 7 (016),

15 EVEN VERTEX EQUITABLE EVEN LABELING 441 [13] A. Lourdusay ad M. Seeivasa, Vertex equitable labelig of graphs, J. Discrete Math. Sci. Cryptogr. 11 (008), [14] L. Tailselvi ad P. Selvaraju, O ea labelig: quadrilateral sake attachet of path: P M (QS N ) ad cycle: C M (QS N ), Iteratioal Joural of Cobiatorial Graph Theory ad Applicatios 4 (011), Departet of Matheatics, St. Xavier s College, Maoaia Sudaraar Uiversity, Tiruelveli, Idia Eail address: lourdusay15@gail.co Eail address: patrick881990@gail.co