CUBO A Mathematical Journal
Vol.20, No¯ 02, (13–21). June 2018

http: // dx. doi. org/ 10. 4067/ S0719-06462018000200013

Odd Vertex Equitable Even Labeling of Cycle Related
Graphs

P. Jeyanthi
1
and A. Maheswari

2

1 Research Centre, Department of Mathematics,

Govindammal Aditanar College for Women,

Tiruchendur-628215, Tamilnadu, India.
2 Department of Mathematics,

Kamaraj College of Engineering and Technology,

Virudhunagar, Tamil Nadu, India.

jeyajeyanthi@rediffmail.com, bala nithin@yahoo.co.in

ABSTRACT

Let G be a graph with p vertices and q edges and A = {1, 3, ..., q} if q is odd or

A = {1, 3, ..., q + 1} if q is even. A graph G is said to admit an odd vertex equitable

even labeling 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 2, 4, ..., 2q where vf(a)

be the number of vertices v with f(v) = a for a ∈ A. A graph that admits an odd

vertex equitable even labeling is called an odd vertex equitable even graph. Here, we

prove that the subdivision of double triangular snake (S(D(Tn))), subdivision of double

quadrilateral snake (S(D(Qn))), DA(Qm) ⊙ nK1 and DA(Tm) ⊙ nK1 are odd vertex

equitable even graphs.

http://dx.doi.org/10.4067/S0719-06462018000200013


14 P. Jeyanthi and A. Maheswari CUBO
20, 2 (2018)

RESUMEN

Sea G un grafo con p vértices y q aristas, y A = {1, 3, ..., q} si q es impar o A =

{1, 3, ..., q + 1} si q es par. Se dice que un grafo G admite un etiquetado par equitativo

de vértices impares si existe un etiquetado de vértices f : V(G) → A que induce un

etiquetado de ejes f∗ definido por f∗(uv) = f(u) + f(v) para todos los ejes uv tales

que para todo a y b en A, |vf(a) − vf(b)| ≤ 1 y las etiquetas de ejes inducidas son

2, 4, ..., 2q donde vf(a) es el número de vértices v con f(v) = a para a ∈ A. Un grafo que

admite un etiquetado par equitativo de vértices impares se dice grafo par equitativo de

vértices impares. Aqúı demostramos que la subdivisión de serpientes triangulares dobles

(S(D(Tn))), la subdivisión de serpientes cuadriláteras dobles (S(D(Qn))), DA(Qm) ⊙

nK1 y DA(Tm) ⊙ nK1 son grafos pares equitativos de vértices impares.

Keywords and Phrases: Odd vertex equitable even labeling, odd vertex equitable even graph,

double triangular snake, subdivision of double quadrilateral snake, double alternate triangular

snake, double alternate quadrilateral snake, subdivision graph.

2010 AMS Mathematics Subject Classification: 05C78.



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Odd Vertex Equitable Even Labeling of Cycle Related Graphs 15

1 Introduction:

All graphs considered here are simple, finite, connected and undirected. Let G(V, E) be a graph

with p vertices and q edges. We follow the basic notations and terminology of graph theory as

in [2]. The vertex set and the edge set of a graph are denoted by V(G) and E(G) respectively.

A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain

conditions and a detailed survey of graph labeling can be found in [1]. The concept of vertex

equitable labeling was due to Lourdusamy and Seenivasan [6]. 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) be 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 vertex

equitable labeling. Motivated by the concept of vertex equitable labeling [6], Jeyanthi, Maheswari

and Vijayalakshmi extend this concept and introduced a new labeling namely odd vertex equitable

even (OVEE) labeling in [3]. A graph G with p vertices and q edges and A = {1, 3, ..., q} if q is

odd or A = {1, 3, ..., q + 1} if q is even. A graph G is said to admit an odd vertex equitable even

labeling 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 2, 4, ..., 2q where vf(a) be the number of vertices v with f(v) = a for

a ∈ A. A graph that admits an odd vertex equitable even (OVEE) labeling then G is called an odd

vertex equitable even (OVEE) graph. In [3], [4] and [5] the same authors proved that nC4-snake,

CS(n1, n2, ..., nk, ni ≡ 0(mod4), ni ≥ 4, be a generalized kCn -snake, TÔQSn and TÕQSn are

odd vertex equitable even graphs. They also proved that the graphs path, Pn ⊙ Pm(n, m ≥ 1),

K1,n ∪ K1,n−2(n ≥ 3), K2,n, Tp-tree, cycle Cn (n≡ 0 or 1 (mod4)), quadrilateral snake Qn,

ladder Ln, Ln ⊙ K1, arbitrary super subdivision of any path Pn, S(Ln), LmÔPn, Ln ⊙ Km and〈
LnÔK1,m

〉
are odd vertex equitable even graphs. Also they proved that the graphs K1,n is an

odd vertex equitable even graph iff n ≤ 2 and the graph G = K1,n+k ∪ K1,n is an odd vertex

equitable even graph if and only if k = 1, 2 and cycle Cn is an odd vertex equitable even graph

if and only if n≡ 0 or 1(mod4). Let G be a graph with p vertices and q edges and p ≤
⌈
q
2

⌉
+ 1,

then G is not an odd vertex equitable even graph. In addition they proved that if every edge of a

graph G is an edge of a triangle, then G is not an odd vertex equitable even graph.

We use the following definitions in the subsequent section.

Definition 1.1. 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 1.2. 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 1.3. 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



16 P. Jeyanthi and A. Maheswari CUBO
20, 2 (2018)

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 1.4. A double alternate quadrilateral snake DA(Qn) consists of two alternate quadri-

lateral 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 viwi and xiyi for i = 1, 2, ..., n − 1.

Definition 1.5. Let G be a graph. The subdivision graph S(G) is obtained from G by subdividing

each edge of G with a vertex.

Definition 1.6. 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 i
th vertex of G1

to every vertex of the ith copy of G2.

2 Main Results

In this section, we prove that S(D(Tn)), S(D(Qn)), DA(Qm) ⊙ nK1 and

DA(Tm) ⊙ nK1 are odd vertex equitable even graphs.

Theorem 2.1. Let G1(p1, q1), G2(p2, q2),...,Gm(pm, qm) be an odd vertex equitable even graphs

with each qi is even for i = 1, 2, ..., m − 1, qm is even or odd and let ui, vi be the vertices of

Gi(1 ≤ i ≤ m) labeled by 1, qi if qi is odd or qi + 1 if qi is even. 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 an odd vertex equitable even graph.

Proof. The graph G has p1 + p2 + ... + pm − (m − 1) vertices and
∑m

i=1
qi edges and fi be an odd

vertex equitable even labeling of Gi(1 ≤ i ≤ m).

Let A =

{
1, 3, 5, ...,

∑m
i=1

qi, if
∑m

i=1
qi is odd

1, 3, 5, ...,
∑m

i=1 qi + 1, if
∑m

i=1 qi is even

}

.

Define a vertex labeling f : V(G) → A as follows: f(x) = f1(x) if x ∈ V(G1), f(x) = fi(x)+
∑i−1

k=1
qk

if x ∈ V(Gi) for 2 ≤ i ≤ m. The edge labels of the graph G1 will remain fixed, the edge labels of

the graph Gi(2 ≤ i ≤ m) are 2q1 +2, 2q1 +4, ..., 2(q1 +q2);2(q1 +q2)+2, 2(q1 +q2)+4, ..., 2(q1 +

q2 + q3);...,2
∑m−1

i=1
qi + 2, 2

∑m−1
i=1

qi + 4, ..., 2
∑m

i=1
qi. Hence the edge labels of G are distinct

and is
{
2, 4, 6, ..., 2

∑m
i=1

qi
}
. Also |vf(a) − vf(b)| ≤ 1 for all a, b ∈ A. Hence G is an odd vertex

equitable even graph.

Theorem 2.2. The graph S(D(Tn)) is an odd vertex equitable even graph.

Proof. Let Gi = S(D(T2)) 1 ≤ i ≤ n − 1 and ui, vi be the vertices with labels 1 and q + 1

respectively. By Theorem 2.1, S(D(T2)) admits an odd vertex equitable even labeling. An odd

vertex equitable even labeling of Gi = S(D(T2)) is given in Figure 1.



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Odd Vertex Equitable Even Labeling of Cycle Related Graphs 17

r rr

❅
❅
❅
❅
❅

r

r

r

r r

r

1 11
3

1

7

9

5

5

7

Figure 1.

Theorem 2.3. The graph S(D(Qn)) is an odd vertex equitable even graph.

Proof. Let Gi = S(D(Q2)) 1 ≤ i ≤ n − 1 and ui, vi be the vertices with labels 1 and q + 1

respectively. By Theorem 2.1, S(D(Q2)) admits an odd vertex equitable even labeling. An odd

vertex equitable even labeling of Gi = S(D(Q2)) is given in Figure 2.

r rr

r r r

rr r

r r

r r

1
1

15

3

9 9 13

11

13

773

5

Figure 2.

Theorem 2.4. The double quadrilateral graph D(Q2n) is an odd vertex equitable even graph.

Proof. Let Gi = D(Q4) 1 ≤ i ≤ n−1 and ui, vi be the vertices with labels 1 and q+1 respectively.

By Theorem 2.1, D(Q4) admits an odd vertex equitable even labeling. An odd vertex equitable

even labeling of Gi = D(Q4) is given in Figure 3.



18 P. Jeyanthi and A. Maheswari CUBO
20, 2 (2018)

r r r

r r r r

r r r r❇
❇
❇
❇❇

1
7

15

1 5 9 11

3 7 11 13

Figure 3.

Theorem 2.5. Let G1(p1, q), G2(p2, q), ..., Gm(pm, q) be an odd vertex equitable even graphs with

q odd and ui,vi be vertices of Gi(1 ≤ i ≤ m) labeled by 1 and q. 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 an odd vertex equitable even graph.

Proof. The graph G has p1 + p2 + ... + pm vertices and mq + (m − 1) edges.

Let fi be the odd vertex equitable even labeling of Gi(1 ≤ i ≤ m) and

let A = {1, 3, ..., mq + (m − 1)}.

Define a vertex labeling f : V(G) → A as

f(x) = fi(x) + (i − 1)(q + 1) if x ∈ Gi for 1 ≤ i ≤ m.

The edge labels of Gi are incresed by 2(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 2i(q + 1), 1 ≤ i ≤ m − 1.

Hence the edge labels of G are distinct and is {2, 4, ..., 2(mq + m − 1)}.

Also |vf(a) − vf(b)| ≤ 1 for all a, b ∈ A.

Then the graph G is an odd vertex equitable even graph.

Theorem 2.6. The graph DA(T2) ⊙ nK1 is an odd vertex equitable even graph for n ≥ 1.

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 = {1, 3, ..., 4n + 5}.

Define a vertex labeling f : V(G) → A as follows:

For 1 ≤ i ≤ n f(u1) = 1, f(u2) = 4n + 5, f(v) = 2n + 1, f(w) = 2n + 5, f(u1i) = 2i − 1,

f(u2i) = 4n + 5 − 2(i − 1),

f(vi) =

{
3 if i=1

2i + 3 if 2 ≤ i ≤ n,

f(wi) =

{
2(n + i) + 1 if 1 ≤ i ≤ n − 1

4n + 3 if i=n.



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Odd Vertex Equitable Even Labeling of Cycle Related Graphs 19

It can be verified that the induced edge labels of DA(T2)⊙nK1 are 2, 4, ..., 8n+10 and |vf(a) − vf(b)| ≤

1 for all a, b ∈ A.

Hence f is an odd vertex equitable even labeling DA(T2) ⊙ nK1.

An odd vertex equitable even labeling of DA(T2) ⊙ 3K1 is shown in Figure 4.

s

s

s

s

s

s

s

s

s
s

s

s

s

s
s

s

1

11

17

7

15 11 9

17

15

13

9
7

3

5

3

1

Figure 4.

Theorem 2.7. The graph DA(Q2) ⊙ nK1 is an odd vertex equitable even graph for n ≥ 1.

Proof. Let G = DA(Q2) ⊙ nK1. Let V(G) = {u1, u2, v, w, x, y} ∪ {vi, wi, xi, yi : 1 ≤ i ≤ n} ∪ {uij :

1 ≤ i ≤ 2, 1 ≤ j ≤ n} and

E(G) = {u1u2, u1v, vw, wu2, u1x, xy, yu2} ∪ {vvi, wwi, xxi, yyi : 1 ≤ i ≤ n} ∪ {uiuij : 1 ≤ i ≤

2, 1 ≤ j ≤ n}.

Here |V(G)| = 6(n + 1) and |E(G)| = 6n + 7.

Let A = {1, 3, ..., 6n + 7}.

Define a vertex labeling f : V(G) → A as follows:

For 1 ≤ i ≤ n f(u1) = 1, f(u2) = 6n + 7, f(u1i) = 2i − 1, f(u2i) = 6n − 2i + 9, f(v) = 2n + 1,

f(w) = 2n + 3, f(x) = 4n + 5, f(y) = 4n + 7, f(vi) = 2i + 1, f(wi) = f(xi) = 2n + 2i + 3,

f(yi) = 4n + 2i + 5.

It can be verified that the induced edge labels of DA(Q2)⊙nK1 are 2, 4, ..., 12n+14 and |vf(a) − vf(b)| ≤

1 for all a, b ∈ A.

Hence f is an odd vertex equitable even labeling of DA(Q2) ⊙ nK1.



20 P. Jeyanthi and A. Maheswari CUBO
20, 2 (2018)

An odd vertex equitable even labeling of DA(Q2) ⊙ 4K1 is shown in Figure 5.

s

s

s s

s

s

s

s

s
s

s
s

s

s

s

s

s

s

s

s
sss

s

s

s

s

s

s

s

1

21 23

31

119

13

15

17
19 23 25

27

29

25

27

29

31

19

17
15139

7
5

3

1

3

5

7

Figure 5.

Theorem 2.8. The graph DA(Qm) ⊙ nK1 is an odd vertex equitable even graph for m, n ≥ 1.

Proof. By Theorem 2.7, DA(Q2)⊙nK1 is an odd vertex equitable even graph. Let Gi = DA(Q2)⊙

nK1 for 1 ≤ i ≤ m − 1. Since each Gi has 6n+7 edges, by Theorem 2.5, DA(Qm) ⊙ nK1 admits

odd vertex equitable even labeling.

An odd vertex equitable even labeling of DA(Q4) ⊙ 4K1 is shown in Figure 6.

s s s s

s s s s

s s s s✑
✑
✑✑

✡
✡
✡✡

❚
❚
❚

✭✭✭✭

❜
❜
❜

✔
✔
✔✔

s

s

s

s
s

s
s

s

s

s

s

s

s

s

s s

s

s

s

ss
s

s

s

s

s

s

s

s

s s
s s

s

s

s

s

s

s

s

s

s

s
s

s

s

s

s
1

21 23

31

119

33

53 55

63

4341

1

3

5

7

13

15

17

19
23 25 27

29

31
29

27

25

19

17

15
139

7
5

3

39

37
35 33

35
37 39 41 45

47

49

51

63

61

59

57

61

59

57
5551

49

47

45

Figure 6.



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Odd Vertex Equitable Even Labeling of Cycle Related Graphs 21

Theorem 2.9. The graph DA(Tm) ⊙ nK1 is an odd vertex equitable even graph for m, n ≥ 1.

Proof. By Theorem 2.6, DA(T2)⊙ nK1 is an odd vertex equitable even graph. Let Gi = DA(T2)⊙

nK1 for 1 ≤ i ≤ m − 1. Since each Gi has 4n+5 edges, by Theorem 2.5, DA(Tm) ⊙ nK1 admits

odd vertex equitable even labeling.

An odd vertex equitable even labeling of DA(T4) ⊙ 3K1 is shown in Figure 7.

r r r r

r r

r r❏
❏

❏
❏

❏
◗

◗
◗◗

✑
✑

✑r

r

r

r
r

r

r
r

r

r
r

r

r

r r

r
r r

r

r

r

r
r

r

1

11

17

7

15 11 9

17
15

13

973

5

3

1

19

21 23

33 29 27

35

33

31

272511

19

29

35

25

Figure 7.

References

[1] A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics,

19(2017), #DS6.

[2] F. Harary, Graph theory, Addison Wesley, Massachusetts, 1972.

[3] P. Jeyanthi, A. Maheswari and M. Vijaya Lakshmi, Odd Vertex Equitable Even labeling,

Proyecciones Journal of Mathematics, Vol.36(1)(2017), 1-11.

[4] P. Jeyanthi, A. Maheswari and M. Vijaya Lakshmi, Odd Vertex Equitable Even labeling of

cyclic snake related graphs, Proyecciones Journal of Mathematics, Vol.37(4)(2018), 613-625.

[5] P. Jeyanthi, A. Maheswari and M. Vijaya Lakshmi, Odd Vertex Equitable Even labeling of

Ladder Graphs, Jordon Journal of Mathematics and Statistics, to appear.

[6] A. Lourdusamy and M. Seenivasan, Vertex equitable labeling of graphs, Journal of Discrete

Mathematical Sciences and Cryptography, Vol.11,(6)(2008), 727-735.


	Introduction:
	Main Results