EDGE TRIMAGIC TOTAL LABELING OF GRAPHS

1

EDGE TRIMAGIC TOTAL LABELING OF GRAPHS

C. Jayasekaran1 and M. Regees2

1 Department of Mathematics, Pioneer Kumaraswamy College, Nagercoil-629003, Tamilnadu, India.

2 Department of Mathematics, Malankara Catholic College, Mariagiri, Kaliakavilai-629153, Tamilnadu, India.

E-Mail: 1. [email protected], 2. [email protected]

Abstract

An edge magic total labeling of a (p, q) graph is a bijection f: V(G) E(G) {1, 2, …, p+q}

such that for each edge uvE(G), the value of f(u)+f(uv)+f(v) is a constant k. If there exists two

constants k1 and k2 such that f(u)+f(uv)+f(v) is either k1or k2, it is said to be an edge bimagic total

labeling. An edge trimagic total labeling of a (p, q) graph is a bijection f: V(G) E(G) {1, 2, …,

p+q} such that for each edge uvE(G), the value of f(u)+f(uv)+f(v) is either k1or k2 or k3. In this

paper we prove the graphs Pn K2, Pn 2, Cn 2, Pn2, (Pn; S1) and triangular snake graph TSn are

edge trimagic total and super edge trimagic total.

Keywords: Function, Bijection, Magic labeling, Trimagic labeling.

AMS Subject Classification: 05C78

1. Introduction

We begin with simple, finite and undirected graph G = (V, E). A graph labeling is an

assignment of integers to elements of a graph, the vertices or edges or both subject to certain

Int Jr. of Mathematical Sciences & Applications Vol.3, No.1, January-June 2013Copyright Mind Reader PublicationsISSN No: 2230-9888 www.journalshub.com

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conditions. The concept of graph labeling was introduced by Rosa in 1967. In 1970 Kotzig and

Rosa[6] defined, magic labeling of graph G is a bijection f: VE{1, 2, …, p+q} such that, for

each edge uvE(G), f(u)+f(uv)+f(v) is a magic constant. In 1996, Ringel and Llado called this

labeling as edge magic. In 2001, Wallis introduced this as edge magic total labeling. In 2004,

J.Baskar Babujee [1] introduced the bimagic labeling of graphs.

In 2013, C. Jayasekaran, M. Regees and C. Davidraj[4] introduced the edge trimagic total

labeling of graphs. An edge trimagic total labeling of a (p, q) graph G is a bijection

f: V(G)E(G) {1, 2, …, p+q} such that for each edge uvE, the value of f(u)+f(uv)+f(v) is

equal to any of the distinct constant k1or k2 or k3. A graph G is said to be edge trimagic total if it

admits an edge trimagic total labeling. An edge trimagic total labeling is called super edge

trimagic total labeling if G has the additional property that the vertices are labeled with smallest

positive integers. A simple graph in which there exists an edge between every pair of vertices is

called a complete graph. The complete graph with n vertices is denoted by Kn. If G is of order n,

the Corona of G with H, G H is the graph obtained by taking one copy of G and n copies of H

and joining the ith vertex of G with an edge to every vertex in the ith copy of H. Square of a graph

G denoted by G2 has the same vertex set as of G and two vertices are adjacent in G2 if they are at

a distance 1 or 2 apart in G. A star graph Sm is the complete graph K1, m. If v1(i), v2

(i), …, v

(i)m+1

and u1, u2, …, un be the vertices of the star graph Sm and the path Pn, then the graph (Pn; Sm) is

obtained from n copies of Sm and the path Pn by joining ui with the central vertex v1(i) of the ith

copy of Sm by means of an edge for 1 ≤ i ≤ n [2]. A triangular cactus is a connected graph all of

whose blocks are triangles. A triangular snake is a triangular cactus whose block-cut point-graph

is a path. A triangular snake is obtained from a path v1, v2, …, vn by joining vi and vi+1 to a new

vertex wi for i = 1, 2, …, n-1[5]. The triangular snake graph is denoted by TSn.

C. Jayasekaran and M. Regees

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For further references, we use dynamic survey of graph labeling by J.A.Gallian[5]. We

follow the notations and terminology of [3]. In [4], we introduced the concept edge trimagic and

super edge trimagic total labeling and proved that the pyramid graph Py(n), K4 snake graph,

wheel snake nW4 and a fan graph Fn are edge trimagic total and super edge trimagic total graphs.

In this paper, we prove the corona graphs Pn K2, Pn 2, Cn 2, square graph Pn2, (Pn; S1) and

triangular snake graph TSn are edge trimagic total and super edge trimagic total.

2. Trimagic Labeling of the Corona Graphs Pn K2, Pn 2 and Cn 2

In this section we prove the corona graphs Pn K2, Pn 2 and Cn 2 are edge trimagic

total and super edge trimagic total. And give examples for edge trimagic total labeling for each

of the above graphs.

Theorem 2.1. The graph Pn K2 has an edge trimagic total labeling for all positive integer n.

Proof: Let V = {u1, u2, …, un}{v1, v2, …, vn}{w1, w2, …, wn} be the vertex set and

E = {uivi / 1≤ i ≤ n}{ uiwi /1≤ i ≤ n}{ uiui+1 / 1≤ i ≤ n–1}}{ viwi /1≤ i ≤ n} be the edge set of

the graph Pn K2 . Then Pn K2 has 3n vertices and 4n-1 edges.

Define a bijection f: V E {1, 2, …, 7n–1} such that

f(ui) = i, 1≤ i ≤ n; f(vi) = n+i, 1≤ i ≤ n; f(wi) = 2n+i, 1≤ i ≤ n;

f(uiui+1) = 7n –2i, 1≤ i ≤ n–1; (uivi) = 7n–2i+1, 1≤ i ≤ n; f(uiwi) = 5n–2i+2, 1≤ i ≤ n and


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f(viwi) = 5n–2i+1, 1≤ i ≤ n.

We prove the above labeling is edge trimagic total.

For the edges uiui+1, 1≤ i≤ n–1;

f ui +f uiui+1 +f ui+1 = i+7n–2i+i+1 = 7n+1 = λ1(say).

For the edges uivi, 1≤ i ≤ n;

f ui +f uivi +f vi = i+7n–2i +1+n+i = 8n+1 = λ2(say).

For the edges uiwi, 1≤ i ≤ n;

f ui +f uiwi +f wi = i+ n–2i +2+2n+i = 7n+2 = λ3(say).

For the edges viwi, 1≤ i ≤ n;

f vi +f viwi +f wi = n+i +5n–2i+1+2n+i = 8n+1 = λ2.

Hence for each edge uvE, f(u)+f(uv)+f(v) yields any one of the magic constants

λ1 = 7n+1, λ2 = 8n+1 and λ3 = 7n+2.

Therefore, the graph Pn K2 admits an edge trimagic total labeling for all positive integer n.

Theorem 2.2. The graph Pn K2 has a super edge trimagic total labeling.

Proof: We proved that the graph Pn K2 admits an edge trimagic total labeling. The labeling

given in the proof of Theorem 2.1, the vertices get labels 1, 2, …, 3n. Since the graph Pn K2 has


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3n vertices and the 3n vertices have labels 1, 2, …, 3n, the graph Pn K2 is a super edge trimagic

total.

Example 2.3. A super edge trimagic total labeling of the graph P5 K2 is given in figure1.

Figure 1: P5 K2 with λ1 = 36, λ2 = 41 and λ3 = 37.

Theorem 2.4. The graph Pn 2 admits an edge trimagic total labeling.


E = {uivi /1≤ i ≤ n}{ uiwi /1≤ i ≤ n}{ uiui+1 /1≤ i ≤ n–1}be the edge set of Pn 2. Then

Pn 2 has 3n vertices and 3n–1 edges.

Case 1. n is odd.


f ui = 2n+

i+1

2, 1≤ i ≤ n and i is odd

2n+n+i+1

2, 1≤ i ≤ n and i is even

f vi =

i+1


n+1

2+i


4

u1

v1 w1

1 2 3 5

6 7 8 9 10 11 12 13 14 15 16 18 20 22 24

17 19 21 23 25 26 28 30 32 34

33 31 29 27


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f wi = n+

i+1


n+n+i+1


f uivi = n–i, 1≤ i ≤ n and i is odd

4n–i, 1≤ i ≤ n and i is even

f uiwi = n–i+1, 1≤ i ≤ n and i is odd

4n–i+1, 1≤ i ≤ n and i is even

and f(uiui+1) = 6n–i, 1≤ i ≤ n-1.

Now we prove the above labeling is an edge trimagic total.

Consider the edges uivi, 1≤ i ≤ n.

For odd i, f ui +f uivi +f vi = 2n+i+1

2+ n–i+

i+1

2 = 7n+1 = λ1(say).

For even i, f ui +f uivi +f vi = 2n+n+i+1

2+4n–i+

n+1

2

i

2 = 7n+1 = λ1.

Consider the edges uiwi, 1≤ i≤ n.

For odd i, f ui +f uiwi +f wi = 2n+i+1

2+ n–i+1+n+

i+1

2 = 8n+2 = λ2(say).

For even i, f ui +f uiwi +f wi = 2n+n+i+1

2+4n–i+1+n+

n+i+1

2 = 8n+2 = λ2.

Consider the edges uiui+1, 1≤ i ≤ n–1.

For odd i, f ui +f uiui+1 +f ui+1 = 2n+i+1

2+6n–i+2n+

n+i+1+1

2 =

21n+3

2 = λ3(say).


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For even i, f ui +f uiui+1 +f ui+1 = 2n+n+i+1

2+6n–i+2n+

i+1+1

2 =

21n+3

2 = λ3.

Hence for each edge uvE, f(u)+f(uv)+f(v) yields any one of the constants λ1 = 7n+1,

λ2 = 8n+2 and λ3 = 21n+3

2

Therefore, the graph Pn 2 admits an edge trimagic total labeling for odd n.

Case 2. n is even.


f ui = 2n+

i+1


2n+n+i


f vi =

i+1


n+i


f wi = n+

i+1


n+n+i


f uivi = 6n – i, 1≤ i ≤ n and i is odd

n – i+1, 1≤ i ≤ n and i is even

f uiwi = 6n – i –1, 1≤ i ≤ n and i is odd

n – i, 1≤ i ≤ n and i is even

and f(uiui+1) = 4n– i, 1≤ i ≤ n–1.

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For odd i, f ui +f uivi +f vi = 2n+i+1

2+6n–i+

i+1

2 = 8n+1 = λ1(say).

For even i, f ui +f uivi +f vi = 2n+n+i

2+ n–i+1+

n+i

2 = 8n+1 = λ1.

Consider the edges uiwi, 1≤ i ≤ n.

For odd i, f ui +f uiwi +f wi = 2n+i+1

2+6n–i–1+n+

i+1

2 = 9n = λ2 (say).

For even i, f ui +f uiwi +f wi = 2n+n+i

2+ n–i+n+

n+i

2 = 9n = λ2.


For odd i, f ui +f uiui+1 +f ui+1 = 2n+i+1

2+4n–i+2n+

n+i+1

2 =

17n+2

2 = λ3(say).

For even i, f ui +f uiui+1 +f ui+1 = 2n+n+i

2+4n–i+2n+

i+1+1

2 =

17n+2

2 = λ3.

Hence for each edge uvE, f(u)+f(uv)+f(v) yields any one of the constants λ1= 8n+1,

λ2 = 9n and λ3 = 17n+2

2 Therefore, the graph Pn 2 admits an edge trimagic total labeling for

even n.

The theorem follows from case 1 and case 2.


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Theorem 2.5. The graph Pn 2 admits a super edge trimagic total labeling.

Proof: We proved that the graph Pn 2 admits an edge trimagic total labeling. The labeling

given in the proof of Theorem 2.4, the vertices get labels 1, 2, …, 3n. Since the graph Pn 2 has

3n vertices and the 3n vertices have labels 1, 2, …, 3n for odd and even n, Pn 2 is a super edge

trimagic total.

Example 2.6. A super edge trimagic total labeling of P7 2 and P6 2 are given in figure 2

and figure 3, respectively.

Figure 2: P7 2 with λ1 = 50, λ2 = 8 and λ3 = 75.

Figure 3: P6 2 with λ1 = 49, λ2 = 54 and λ3 = 52.

Theorem 2.7. The graph Cn 2 admits an edge trimagic total labeling.


E = {uivi / 1≤ i ≤ n}{ uiwi /1≤ i ≤ n}{ uiui+1 / 1≤ i ≤ n–1}{u1un} be the edge set of Cn 2.

Then Cn 2 has 3n vertices and 3n edges.

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40 41

29 28

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22 23 19 20 21 13 15 17 14 16 18

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Case1. n is odd.

Define a bijection f: V E {1, 2, …, 6n} such that

f ui =

i+1


n+i+1


f vi = n+

i+1


n+n+i+1


f wi = 2n+

i+1


2n+n+i+1


f(uivi) = 5n–i+1, 1≤ i ≤ n and f(uiwi) = 4n–i+1, 1≤ i ≤ n;

f(uiui+1) = 6n–i, 1≤ i ≤ n–1 and f(u1un) = 6n.



For odd i, f ui +f uivi +f vi = i+1

2+ n–i+1+n+

i+1

2 = 6n+2 = λ1(say).

For even i, f ui +f uivi +f vi = n+i+1

2+ n–i+1+n+

n+i+1

2 = 7n+2 = λ2(say).


For odd i, f ui +f uiwi +f wi = i+1

2+4n–i+1+2n+

i+1

2 = 6n+2 = λ1.


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For even i, f ui +f uiwi +f wi = n+i+1

2+4n–i+1+2n+

n+i+1

2 = 7n+2 = λ2.


For odd i, f ui +f uiui+1 +f ui+1 = i+1

2+6n–i+

n+i+1+1

2 =

13n+3

2 = λ3(say).

For even i, f ui +f uiui+1 +f ui+1 = n+i+1

2+6n–i+

i+1+1

2 =

13n+3

2 = λ3.

For the edges u1un, f u1 +f u1un +f un = 1+1

2+6n+

n+1

2 =

13n+3

2 = λ3.


λ2 = 7n+2 and λ3 = 13n+3

2 Therefore, the graph Cn 2 admits an edge trimagic total labeling for

odd n.

Case 2. n is even.

Define a bijection f: V E {1, 2, …, 6n} such that

f ui =

i+1


n+i


f vi = n+

i+1


n+n+i


f wi = 2n+

i+1


2n+n+i



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f(uivi) = 5n–i+1, 1≤ i ≤ n; f(uiwi) = 4n–i+1, 1≤ i ≤ n;

f(uiui+1) = 6n–i, 1≤ i ≤ n–1 and f(u1un) = 6n.




2+ n–i+1+n+

i+1

2 = 6n+2 = λ1(say).

For even i, f ui +f uivi +f vi = n+i

2+ n–i+1+n+

n+i

2 = 7n+1 = λ2(say).


For odd i, f ui +f uiwi +f wi = i+1

2+4n–i+1+2n+

i+1

2 = 6n+2 = λ1.

For even i, f ui +f uiwi +f wi = n+i

2+4n–i+1+2n+

n+i

2 = 7n+1 = λ2.



2+6n–i+

n+i+1

2 =

13n+2

2 = λ3(say).

For even i, f ui +f uiui+1 +f ui+1 = n+i

2+6n–i+

i+1+1

2 =

13n+2

2 = λ3.

For the edge u1un, f u1 +f u1un +f un = 1+1

2+6n+

n+n

2 = 7n+1 = λ2.


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λ2 = 7n+2 and λ3 = 13n+3

2 Therefore, the graph Cn 2 admits an edge trimagic total labeling for

even n. Thus, the graph Cn 2 admits an edge trimagic total labeling for even n.


Theorem 2.8. The graph Cn 2 admits a super edge trimagic total labeling.

Proof: We proved that the graph Cn 2 has an edge trimagic total labeling. The labeling given

in the proof of Theorem 2.7, the vertices get labels 1, 2, …, 3n. Since the graph Cn 2 has 3n

vertices and the 3n vertices have labels 1, 2, …, 3n for odd and even n, the graph Cn 2 is a

super edge trimagic total.

Example 2.9. A super edge trimagic total labeling of C8 2 and C5 2 are given in figure 4

and figure 5, respectively.

Figure 4: Figure 5:

C8 2 with λ1 = 0, λ2 = 57 and λ3 = 53. C5 2 with λ1 = 32, λ2 = 37 and λ3 = 34.

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3. Trimgic Labeling for the Graphs Pn2, (Pn; S1) and TSn

In this section we prove the square graph Pn2, (Pn; S1) and the triangular snake TSn are

edge trimagic total and super edge trimagic total. And give examples for edge trimagic total

labeling for each of the above graphs.

Theorem 3.1. The square graph Pn2 admits an edge trimagic total labeling.

Proof: Let V = {v1, v2, …,vn} be the vertex set and E = {vivi+1 /1≤ i ≤ n–1}{ vivi+2 /1≤ i ≤ n–2}

be the edge set of the square graph Pn2. Then the square graph Pn

2 has n vertices and 2n–3 edges.

Case1. n is odd.


f vi =

i+1


n+i+1


f(vivi+1) = 2n–i, 1≤ i ≤ n–1 and f(vivi+2) = 3n–i–2, 1≤ i ≤ n–2.


Consider the edges vivi+1, 1≤ i ≤ n–1.

For odd i, f vi +f vivi+1 +f vi+1 = i+1

2+2n–i+

n+i+1+1

2 =

n+3

2 = λ1(say).

For even i, f vi +f vivi+1 +f vi+1 = n+i+1

2+2n–i+

i+1+1

2 =

n+3

2 = λ1.


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2+3n–i–2+

i+2+1

2 = 3n = λ2(say.

For even i, f vi +f vivi+2 +f vi+2 = n+i+1

2+3n–i–2+

n+i+2+1

2 = 4n = λ3(say).

Hence for each edge uvE, f(u)+f(uv)+f(v) yields any one of the constants λ1 = n+3

2,

λ2 = 3n and λ3 = 4n. Therefore, the square graph Pn2 admits an edge trimagic total labeling for

odd n.

Case 2. n is even.


f vi =

i+1


n+i


f(vivi+1) = 2n–i, 1≤ i ≤ n–1 and f(vivi+2) = 3n–i–2, 1≤ i ≤ n–2,




2+2n–i+

n+i+1

2 =

n+2

2 = λ1(say).

For even i, f vi +f vivi+1 +f vi+1 = n+i

2+2n–i+

i+1+1

2 =

n+2

2 = λ1.



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2+3n–i–2+

i+2+1

2 = 3n = λ2(say.

For even i, f vi +f vivi+2 +f vi+2 = n+i

2+3n–i–2+

n+i+2

2 = 4n–1 = λ3(say).

Hence for each edge uvE, f(u)+f(uv)+f(v) yields any one of the constants λ1 = n+2

2,

λ2 = 3n and λ3 = 4n 1 Therefore, the square graph Pn2 admits an edge trimagic total labeling for

even n.


Theorem 3.2. The graph Pn2 admits a super edge trimagic total labeling.

Proof: We proved the square graph Pn2 admits an edge trimagic total labeling. The labeling given

in the proof of Theorem 3.1, the vertices get labels 1, 2, …, n. Since the square graph Pn2 has n

vertices and the n vertices have labels 1, 2, …, n for odd and even n, the square graph Pn2 is a


Example 3.3. A super edge trimagic total labeling of P92 and P8

2 are given in figure 6 and figure

7, respectively.

Figure 6: P92 with λ1 = 24, λ2 = 27 and λ3 = 36.

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Figure 7: P82 with λ1 = 21, λ2 = 24 and λ3 = 31.

Theorem 3.4. The graph (Pn; S1) admits an edge trimagic total labeling.


E = {uivi /1≤ i ≤ n}{viwi /1≤ i ≤ n}{uiui+1 / 1≤ i ≤ n–1} be the edge set. Then (Pn; S1) has 3n

vertices and 3n–1 edges.

Case1. n is odd.

Define a bijection f: V E {1, 2, …, 6n-1} such that

f ui =

i+1


n+i+1


f vi = n+

i+1


n+n+i+1


f wi = 2n+

i+1


2n+n+i+1


f uivi = n–i, 1≤ i ≤ n and i is odd

4n–i, 1≤ i ≤ n and i is even

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3


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f viwi = n– i+1, 1≤ i ≤ n and i is odd

4n– i+1, 1≤ i ≤ n and i is even

and f(uiui+1) = 6n–i, 1≤ i ≤ n–1.




2+ n– i+n+

i+1

2 = 6n+1 = λ1(say).


2+4n– i+n+

n+i+1

2 = 6n+1 = λ1.

Consider the edges viwi, 1≤ i ≤ n.

For odd i, f vi +f viwi +f wi = n+i+1

2+ n– i+1+2n+

i+1

2 = 8n+2 = λ2(say).

For even i, f vi +f viwi +f wi = n+n+i+1

2+4n–i+1+2n+

n+i+1

2 = 8n+2 = λ2.


For odd i, f ui +f uiui+1 +f u i+1 = i+1

2+6n–i+

n+i+1+1

2 =

13n+3

2 = λ3(say).


2+6n–i+

i+1+1

2 =

13n+3

2 = λ3.

Hence for each edge uvE, f(u)+f(uv)+f(v) yields any one of the constant λ1 = 6n+1,

λ2 = 8n+2 and λ3 = 13n+3

2


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Therefore, the graph (Pn; S1) admits an edge trimagic total labeling for odd n.

Case 2. n is even.

Define a bijection f: VE {1, 2, …, 6n–1} such that

f ui =

i+1


n+i


f vi = n+

i+1


n+n+i


f wi = 2n+

i+1


2n+n+i


f uivi = n– i+1, 1≤ i ≤ n and i is odd


f viwi = n – i, 1 ≤ i ≤ n and i is odd


and f(uiui+1) = 6n– i, 1≤ i ≤ n–1.




2+ n–i+1+n+

i+1

2 = 6n+2 = λ1(say).


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2+4n–i+2+n+

n+i

2 = 6n+2 = λ1.

Consider the edges viwi, 1≤ i ≤ n.

For odd i, f(vi)+f viwi +f wi = n+i+1

2+ n–i+2n+

i+1

2 = 8n+1 = λ2(say).

For even i, f vi +f viwi +f wi = n+n+i

2+4n–i+1+2n+

n+i

2 = 8n+1 = λ2.



2+6n–i+

n+i+1

2 =

13n+2

2 = λ3(say).


2+6n–i+

i+1+1

2 =

13n+2

2 = λ3.


λ2 = 8n+1 and λ3 = 13n+2

2 Therefore, the graph ( Pn; S1) admits an edge trimagic total labeling for

even n.


Theorem 3.5. The graph (Pn; S1) admits a super edge trimagic total labeling.

Proof: We proved that the graph ( Pn; S1) admits an edge trimagic total labeling. The labeling

given in the proof of Theorem 3.4, the vertices get labels 1, 2, …, 3n. Since the graph ( Pn; S1)

has 3n vertices and the 3n vertices have labels 1, 2, …, 3n for both odd and even n, ( Pn; S1) is a



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Example 3.6. An edge trimagic total labeling of (P7; S1) and (P6; S1) are given in figure 8 and

figure 9, respectively.

.

Figure 8: (P7; S1) with λ1 = 47, λ2 = 43 and λ3 = 58.

Figure 9: (P6; S1) with λ1 = 40, λ2 = 38 and λ3 = 49.

Theorem 3.7. The triangular snake graph TSn admits an edge trimagic total labeling.

Proof: Let V = {u1, u2, …,un}{v1, v2, …, vn} be the vertex set and E = {uivi /1≤ i ≤ n–1}

{ui+1vi /1≤ i ≤ n–1}{uiui+1 / 1≤ i ≤ n–1} be the edge set of the triangular snake graph TSn. Then

the triangular snake graph TSn has 2n–1 vertices and 3n-3 edges.

Case1. n is odd.

Define a bijection f: V E {1, 2, …, n–4} such that

w1

v1

u1

19 15 17 20 16

21 18

5 1 6 2 3 7 4

8 12 13 9 10 14 11

35 33 27 25 31 23 29

26 34 30 32 24 28 22

41 39 40 37 38 36

w7

v7

u7

29

11

30

13

7 10

27 23 21 25

24 26 28 22

16 15 17 14

8 9

19

20

35 33 34 31 32

w1

v1

u1

18

4 1 5 2 3 6

12

w6

v6

u6


315

22

f ui =

i+1


n+i+1


f vi = n+

i+1

2, 1≤ i ≤ n–1 and i is odd

n+n+i–1

2, 1≤ i ≤ n–1 and i is even

f(uivi ) = 4n–i–2, 1≤ i ≤ n–1 and i is odd

3n–i–1, 1≤ i ≤ n–1 and i is even

f(ui+1vi) = 4n–i–3, 1≤ i ≤ n–1 and i is odd

3n–i, 1≤ i ≤ n–1 and i is even

and f(uiui+1) = 5n–i–3, 1≤ i ≤ n–1.


Consider the edges uivi, 1≤ i ≤ n–1.


2+4n–i–2+ n+

i+1

2 = 5n–1 = λ1(say).


2+3n–i–1+ n+

n+i–1

2 = 5n–1 = λ1.

Consider the edges ui+1vi, 1≤ i ≤ n–1.

For odd i, f ui+1 +f ui+1vi +f vi = n+i+1+1

2 4n i 3 n

i+1

2 =

11n–3

2. = λ2(say).

For even i, f ui+1 +f ui+1vi +f vi = 1+i+1

2 3n i n

n+i–1

2 = n+1

2 = λ3(say).


316

23



2+ n–i–3+

n+i+1+1

2 =

11n– 3

2 = λ2.


2+ n–i–3+

i+1+1

2 =

11n–3

2 = λ2.

Hence for each edge uvE, f(u)+f(uv)+f(v) yields any one of the constants λ1 = 5n–2,

λ2 = 11n –3

2 and λ3 =

n+1

2 Therefore, the triangular snake graph TSn admits an edge trimagic total

labeling for odd n.

Case 2. n is even.

Define a bijection f: V E {1, 2, …, n–4} such that

f ui =

i+1


n+i


f vi = n+

i+1

2, 1≤ i ≤ n–1 and i is odd

n+n+i

2, 1≤ i ≤ n–1 and i is even

f(uivi ) = 4n–i–2, 1≤ i ≤ n–1 and i is odd

3n–i–1, 1≤ i ≤ n–1 and i is even

f(ui+1vi) = 4n–i–3, 1≤ i ≤ n–1 and i is odd

3n–i, 1≤ i ≤ n–1 and i is even

and f(uiui+1) = 5n–i–3, 1≤ i ≤ n–1.


317

24


Consider the edges uivi, 1≤ i ≤ n–1.


2+4n–i–2+ n+

i+1

2 = 5n–1 = λ1(say).


2+3n–i–1+ n+

n+i

2 = 5n–1 = λ1.

Consider the edges ui+1vi, 1≤ i ≤ n–1.

For odd i, f ui+1 +f ui+1vi +f vi = n+i+1

2 4n i 3 n

i+1

2 =

11n– 4

2 = λ2(say).

For even i, f ui+1 +f ui+1vi +f vi = i+1+1

2 3n i 2 n

n+i

2 =

n–2

2 = λ3(say).



2+ n–i–3+

n+i+1

2 =

11n–4

2 = λ2.


2+ n–i–3+

i+1+1

2 =

11n– 4

2 = λ2.

Hence for each edge uvE, f(u)+f(uv)+f(v) yields any one of the constants λ1 = 5n–1,

λ2 = 11n– 4

2 and λ3 =

n–2

2 Therefore, the triangular snake graph TSn admits an edge trimagic total

labeling for odd n.



318

25

Theorem 3.8. The triangular snake graph TSn admits a super edge trimagic total labeling.

Proof: We proved that the triangular snake graph TSn admits an edge trimagic total labeling. The

labeling given in the proof of Theorem 3.7, the vertices get labels 1, 2, …, 2n–1. Since the

triangular snake graph has 2n–1 vertices and the 2n–1 vertices have labels 1, 2, …, 2n–1 for odd

and even n, the triangular snake graph TSn is a super edge trimagic total.

Example 3.9. An edge trimagic total labeling of triangular graphs TS9 and TS8 are given in

figure 10 and figure 11, respectively.

Figure 10: TS9 with λ1 = 44, λ2 = 48 and λ3 = 41.

Figure 11: TS8 with λ1 = 3 , λ2 = 42 and λ3 = 35.

Conclusion

In this paper we proved the the corona graphs Pn K2, Pn 2, Cn 2, the square graph

Pn2, (Pn; S1) and triangular snake TSn are edge trimagic total and super edge trimagic total. There

may be many interesting trimagic graphs can be constructed in future.

41 1

V1

u1

6 40 2 39 7 38 3 37 8 36 4 35 9 5 34

10 14 11 15 12 16 13 17

33 32 24 25 31 30

22 23

29 28

20 21

27 26

18 19

u9

v8

5 1

u1

v1

36 2 6 3 7 4 8 35 34 33 32 31 30

9 13 10 14 11 15 12

29 28

21 20

27

26 19

18 24

25 17

16 23 22

u8

v7


319

26

REFERENCES

[1] J. Baskar Babujee,“On Edge Bimagic Labeling”, Journal of Combinatorics Information &

System Sciences, Vol.28-29, Nos. 1-4, pages. 239- 244 (2004).

[2] A. Durai Basker, S. Arockiaraj, B. Rajendran “F- Geometric Mean Labeling of Some Chain

Graphs and Thorn Graphs”, KAAGUJEVAC Journal of Mathematics, Vol. 37(1), pages 163-186

(2013).

[3] N.Hartsfield and G.Ringel, “Pearls in Graph Theory”, Academic press, Cambridge (1990).

[4] C. Jayasekaran, M. Regees and C. Davidraj, “Edge Trimagic Labeling of Some Graphs”,

Accepted for publication, International Journal for Combinatorial Graph theory and applications

(2013).

[5] Joseph A. Gallian, “A Dynamic Survey of Graph Labeling”, The Electronic Journal of

Combinatorics, 19 (2012), #DS6.

[6] A. otzig and A. Rosa,“Magic Valuations of finite graphs”, Canad. Math. Bull., Vol. 13, pages.

415- 416 (1970).


320

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EDGE TRIMAGIC TOTAL LABELING OF GRAPHS

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