TOTAL MEAN CORDIAL LABELING OF SOME CYCLE RELATED GRAPHS

Journal of Applied Mathematics & Informatics.
2015.
Jan,
33(1_2):
101-110

- Received : July 17, 2014
- Accepted : October 10, 2014
- Published : January 30, 2015

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A Total Mean Cordial labeling of a graph
G
= (
V,E
) is a function
f
:
V
(
G
) → {0, 1, 2} such that
where
x, y
∈
V
(
G
),
xy
∈
E
(
G
), and the total number of 0, 1 and 2 are balanced. That is |
ev_{f}
(
i
) −
ev_{f}
(
j
)| ≤ 1,
i, j
∈ {0, 1, 2} where
ev_{f}
(
x
) denotes the total number of vertices and edges labeled with
x
(
x
= 0, 1, 2). If there is a total mean cordial labeling on a graph
G
, then we will call
G
is Total Mean Cordial. Here, We investigate the Total Mean Cordial labeling behaviour of prism, gear, helms.
AMS Mathematics Subject Classification : 05C78.
G
= (
V,E
) we mean a finite, undirected graph with neither loops nor mul-tiple edges. The product graph
G
_{1}
×
G
_{2}
is defined as follows: Consider any two points
u
= (
u
_{1}
,
u
_{2}
) and
v
= (
v
_{1}
,
v
_{2}
) in
V
=
V
_{1}
×
V
_{2}
. Then
u
and
v
are adjacent in
G
_{1}
×
G
_{2}
whenever [
u
_{1}
=
v
_{1}
and
u
_{2}
adj
v
_{2}
] or [
u
_{2}
=
v
_{2}
and
u
_{1}
adj
v
_{1}
]. The join of two graphs
G
_{1}
and
G
_{2}
is denoted by
G
_{1}
+
G
_{2}
and whose vertex set is
V
(
G
_{1}
+
G
_{2}
) =
V
(
G
_{1}
) ∪
V
(
G
_{2}
) and edge set
E
(
G
_{1}
+
G
_{2}
) =
E
(
G
_{1}
) ∪
E
(
G
_{2}
) ∪ {
uv
:
u
∈
V
(
G
_{1}
),
v
∈
V
(
G
_{2}
)}. The order and size of
G
are denoted by
p
and
q
respectively. Ponraj, Ramasamy and Sathish Narayanan
[3]
introduced the concept of Total Mean Cordial labeling of graphs and studied about their be-havior on Path, Cycle, Wheel and some more standard graphs. In
[4]
, Ponraj and Sathish Narayanan proved that
is Total Mean Cordial if and only if
n
= 1 or 2 or 4 or 6 or 8. Also in
[5]
, Ponraj, Ramasamy and Sathish Narayanan studied about the Total Mean Cordiality of Lotus inside a circle, bistar, flower graph,
K
_{2,}
_{n}
, Olive tree,
In this paper, we investigate the Total Mean Cordiality of some cycle related graphs. Let
x
be any real number. Then the symbol ⌈
x
⌉ stands for the smallest integer greater than or equal to
x
.
Definition 2.1.
A Total Mean Cordial labeling of a graph
G
= (
V,E
) is a function
f
:
V
(
G
) → {0, 1, 2} such that
where
x, y
∈
V
(
G
),
xy
∈
E
(
G
), and the total number of 0, 1 and 2 are balanced. That is |
ev_{f}
(
i
) -
ev_{f}
(
j
)| ≤ 1,
i, j
∈ {0, 1, 2} where
ev_{f}
(
x
) denotes the total number of vertices and edges labeled with
x
(
x
= 0, 1, 2). If there exists a total mean cordial labeling on a graph
G
, we will call
G
is Total Mean Cordial.
Prisms are graphs of the form
C_{m}
×
P_{n}
. We now look into the graph prism
C_{n}
×
P
_{2}
.
Theorem 2.2.
Prisms are Total Mean Cordial.
Proof
. It is clear that
p
+
q
= 5
n
. Let
V
(
C_{n}
×
P
_{2}
) = {
u_{i}
,
v_{i}
: 1 ≤
i
≤
n
} and
E
(
C_{n}
×
P
_{2}
) = {
u
_{1}
u_{n}
,
v
_{1}
v_{n}
}∪{
u_{i}v_{i}
: 1 ≤
i
≤
n
}∪{
u_{i}
u
_{i}
_{+1}
,
v_{i}
v
_{i}
_{+1}
: 1 ≤
i
≤
n
−1}.
Case 1.
n
≡ 0 (mod 6).
Let
n
= 6
t
and
t
> 0. Define a map
f
:
V
(
C_{n}
×
P
_{2}
) → {0, 1, 2} by
f
(
u
_{2}
_{t}
_{+1}
) = 0,
f
(
v
_{2}
_{t}
_{+1}
) =
f
(
v
_{6}
_{t}
) = 2,
f
(
u
_{6}
_{t}
) = 1. In this case
ev_{f}
(0) =
ev_{f}
(1) =
ev_{f}
(2) = 10
t
.
Case 2.
n
≡ 1 (mod 6).
Let
n
= 6
t
+ 1 and
t
> 0. Define a map
f
:
V
(
C_{n}
×
P
_{2}
) → {0, 1, 2} by
f
(
v
_{2}
_{t}
_{+1}
) =
f
(
v
_{2}
_{t}
_{+2}
) = 1. Here
ev_{f}
(0) =
ev_{f}
(2) = 10
t
+ 2,
ev_{f}
(2) = 10
t
+ 1.
Case 3.
n
≡ 2 (mod 6).
Let
n
= 6
t
+ 2 and
t
> 0. Define a map
f
:
V
(
C_{n}
×
P
_{2}
) → {0, 1, 2} by
f
(
u
_{6}
_{t}
_{+2}
) = 1,
f
(
v
_{6}
_{t}
_{+2}
) = 2. In this case
ev_{f}
(0) =
ev_{f}
(2) = 10
t
+ 3,
ev_{f}
(1) = 10
t
+ 4.
Case 4.
n
≡ 3 (mod 6).
Let
n
= 6
t
+ 3 and
t
≥ 0. A Total Mean Cordial labeling of
C
_{3}
×
P
_{2}
is given in
figure 1
.
Assume
t
> 0. Define a map
f
:
V
(
C_{n}
×
P
_{2}
) → {0, 1, 2} by
f
(
u
_{2}
_{t}
_{+2}
) = 0,
f
(
u
_{6}
_{t}
_{+3}
) = 1,
f
(
v
_{2}
_{t}
_{+2}
) =
f
(
v
_{6}
_{t}
_{+3}
) = 2. In this case
ev_{f}
(0) =
ev_{f}
(1) =
ev_{f}
(2) = 10
t
+ 5.
Case 5.
n
≡ 4 (mod 6).
Let
n
= 6
t
+ 4 and
t
≥ 0. A Total Mean Cordial labeling of
C
_{4}
×
P
_{2}
is given in
figure 2
.
Assume
t
> 0. Define a map
f
:
V
(
C_{n}
×
P
_{2}
) → {0, 1, 2} by
In this case
ev_{f}
(0) =
ev_{f}
(1) = 10
t
+ 7,
ev_{f}
(2) = 10
t
+ 6.
Case 6.
n
≡ 5 (mod 6).
Let
n
= 6
t
- 1 and
t
> 0. Define a map
f
:
V
(
C_{n}
×
P
_{2}
) → {0, 1, 2} by
f
(
u
_{6}
_{t}
_{-1}
) = 1,
f
(
v
_{6}
_{t}
_{-1}
) = 2. In this case
ev_{f}
(0) =
ev_{f}
(2) = 10
t
- 2,
ev_{f}
(1) = 10
t
- 1. □
The gear graph
G_{n}
is obtained from the wheel
W_{n}
=
C_{n}
+
K
_{1}
where
C_{n}
is the cycle
u
_{1}
u
_{2}
. . .
u_{n}u
_{1}
and
V
(
K
_{1}
) = {
u
} by adding a vertex between every pair of adjacent vertices of the cycle
C_{n}
.
Theorem 2.3.
The gear graph G_{n} is Total Mean cordial.
Proof
. Let
V
(
G_{n}
) =
V
(
W_{n}
) ∪ {
v_{i}
: 1 ≤
i
≤
n
} and
E
(
G_{n}
) =
E
(
W_{n}
) ∪ {
u_{i}v_{i}
,
v_{j}
u
_{j}
_{+1}
: 1 ≤
i
≤
n
, 1 ≤
j
≤
n
} −
E
(
C_{n}
). Clearly
p
+
q
= 5
n
+ 1.
Case 1.
n
≡ 0 (mod 12).
Let
n
= 12
t
and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
_{+1}
) = 0,
f
(
v
_{12}
_{t}
) = 1.
Case 2.
n
≡ 1 (mod 12).
Let
n
= 12
t
+ 1 and
t
> 0. Assign the label to the vertices
u_{i}
(1 ≤
i
≤ 12
t
),
v_{i}
(1 ≤
i
≤ 12
t
−1) as in case 1. Then put the labels 0, 1, 2 to the vertices
v
_{12}
_{t}
,
u
_{12}
_{t}
_{+1}
,
v
_{12}
_{t}
_{+1}
, respectively.
Case 3.
n
≡ 2 (mod 12).
Let
n
= 12
t
+ 2 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{9}
_{t}
_{+2}
) =
f
(
v
_{12}
_{t}
_{+2}
) = 2,
f
(
u
_{12}
_{t}
_{+2}
) = 0,
f
(
u
_{12}
_{t}
_{+1}
) = 1.
Case 4.
n
≡ 3 (mod 12).
Let
n
= 12
t
- 9 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
_{-3}
) = 0,
f
(
v
_{12}
_{t}
_{-9}
) = 1.
Case 5.
n
≡ 4 (mod 12).
Let
n
= 12
t
− 8 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{12}
_{t}
_{-8}
) = 1,
f
(
v
_{12}
_{t}
_{-8}
) = 2.
Case 6.
n
≡ 5 (mod 12).
Let
n
= 12
t
− 7 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
_{-2}
) = 0,
f
(
v
_{5}
_{t}
_{-2}
) = 1,
f
(
u
_{9}
_{t}
_{-4}
) = 2,
f
(
v
_{12}
_{t}
_{-7}
) = 1,
Case 7.
n
≡ 6 (mod 12).
Let
n
= 12
t
− 6 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{9}
_{t}
_{-4}
) = 2,
f
(
v
_{12}
_{t}
_{-6}
) = 2.
Case 8.
n
≡ 7 (mod 12).
Let
n
= 12
t
− 5 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{12}
_{t}
_{-6}
) =
f
(
u
_{12}
_{t}
_{-5}
) = 1,
f
(
v
_{12}
_{t}
_{-6}
) = 0,
f
(
v
_{12}
_{t}
_{-5}
) = 2.
Case 9.
n
≡ 8 (mod 12
Let
n
= 12
t
− 4 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
_{-1}
) = 0,
f
(
v
_{5}
_{t}
_{-1}
) =
f
(
v
_{12}
_{t}
_{-4}
) = 1,
f
(
u
_{9}
_{t}
_{-2}
) = 2.
Case 10.
n
≡ 9 (mod 12).
Let
n
= 12
t
− 3 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{9}
_{t}
_{-2}
) = 2,
f
(
v
_{12}
_{t}
_{-3}
) = 2.
Case 11.
n
≡ 10 (mod 12).
Let
n
= 12
t
− 2 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
) = 0,
f
(
v
_{12}
_{t}
_{-2}
) = 2.
Case 12.
n
≡ 11 (mod 12).
Let
n
= 12
t
− 1 and
t
> 0. Define a map
f
:
V
(
G_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{9}
_{t}
) = 0,
f
(
v
_{12}
_{t}
_{-1}
) = 1.
The following
table 1
shows that
G_{n}
is a Total Mean Cordial graph. □
The helm
H_{n}
is the graph obtained from a wheel by attaching a pendant edge at each vertex of the
n
-cycle.
Theorem 2.4.
Helms H_{n} are Total Mean Cordial.
Proof
. Let
V
(
H_{n}
) = {
u
,
u_{i}
,
v_{i}
: 1 ≤
i
≤
n
} and
E
(
H_{n}
) = {
u_{i}
u
_{i}
_{+1}
: 1 ≤
i
≤
n
− 1} ∪ {
u_{n}u
_{1}
} ∪ {
uu_{i}
,
u_{i}v_{i}
: 1 ≤
i
≤
n
}. Clearly the order and size of
H_{n}
are 2
n
+ 1 and 3
n
respectively.
Case 1.
n
≡ 0 (mod 12).
Let
n
= 12
t
and
t
> 0. Construct a vertex labeling
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
and
f
(
v
_{5}
_{t}
_{+1}
) = 1.
Case 2.
n
≡ 1 (mod 12).
Let
n
= 12
t
+ 1 and
t
> 0. Define a map
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
_{+1}
) = 0,
f
(
v
_{5}
_{t}
_{+1}
) = 1,
f
(
u
_{12}
_{t}
) =
f
(
u
_{12}
_{t}
_{+1}
) = 1,
f
(
v
_{12}
_{t}
) = 2 and
f
(
v
_{12}
_{t}
_{+1}
) = 0.
Case 3.
n
≡ 2 (mod 12).
Let
n
= 12
t
+ 2 and
t
> 0. Define a map
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{12}
_{t}
_{+1}
) =
f
(
u
_{12}
_{t}
_{+2}
) = 1 and
f
(
v
_{12}
_{t}
_{+1}
) =
f
(
v
_{12}
_{t}
_{+2}
) = 2.
Case 4.
n
≡ 3 (mod 12).
The Total Mean Cordial labeling of
H
_{3}
is given in
figure 3
.
Let
n
= 12
t
+ 3 and
t
> 0. Define a map
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
_{+2}
) = 0,
f
(
v
_{5}
_{t}
_{+2}
) = 1.
Case 5.
n
≡ 4 (mod 12).
Let
n
= 12
t
- 8 and
t
> 0. Define a map
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{12}
_{t}
_{-8}
) = 1 and
f
(
v
_{12}
_{t}
_{-8}
) = 2.
Case 6.
n
≡ 5 (mod 12).
The Total Mean Cordial labeling of
H
_{3}
is given in
figure 4
.
Let
n
= 12
t
+ 5 and
t
> 0. Define a function
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
_{+3}
) = 0,
f
(
v
_{5}
_{t}
_{+3}
) = 1,
f
(
u
_{12}
_{t}
_{+4}
) =
f
(
u
_{12}
_{t}
_{+5}
) = 1 and
f
(
v
_{12}
_{t}
_{+4}
) =
f
(
v
_{12}
_{t}
_{+5}
) = 2.
Case 7.
n
≡ 6 (mod 12).
Let
n
= 12
t
- 6 and
t
> 0. Define a function
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
Case 8.
n
≡ 7 (mod 12).
Let
n
= 12
t
- 5 and
t
> 0. Define a function
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{12}
_{t}
_{-6}
) =
f
(
u
_{12}
_{t}
_{-5}
) = 1,
f
(
v
_{12}
_{t}
_{-6}
) = 2 and
f
(
v
_{12}
_{t}
_{-5}
) = 0.
Case 9.
n
≡ 8 (mod 12).
Let
n
= 12
t
- 4 and
t
> 0. Construct a vertex labeling
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
_{-1}
) = 0,
f
(
v
_{5}
_{t}
_{-1}
) = 1,
f
(
u
_{12}
_{t}
_{-5}
) =
f
(
u
_{12}
_{t}
_{-4}
) = 1 and
f
(
v
_{12}
_{t}
_{-5}
) =
f
(
v
_{12}
_{t}
_{-4}
) = 2.
Case 10.
n
≡ 9 (mod 12).
Let
n
= 12
t
- 3 and
t
> 0. Define
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
Case 11.
n
≡ 10 (mod 12).
Let
n
= 12
t
- 2 and
t
> 0. Define a function
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{5}
_{t}
) = 0,
f
(
v
_{5}
_{t}
) = 1,
f
(
u
_{12}
_{t}
_{-2}
) = 1 and
f
(
v
_{12}
_{t}
_{-2}
) = 2.
Case 12.
n
≡ 11 (mod 12).
Let
n
= 12
t
- 1 and
t
> 0. Define a function
f
:
V
(
H_{n}
) → {0, 1, 2} by
f
(
u
) = 1,
f
(
u
_{12}
_{t}
_{-2}
) =
f
(
u
_{12}
_{t}
_{-1}
) = 1 and
f
(
v
_{12}
_{t}
_{-2}
) =
f
(
v
_{12}
_{t}
_{-1}
) = 2.
The following
table 2
shows that
H_{n}
is a Total Mean Cordial graph. □
Dr. R. Ponraj received his Ph.D in Manonmaniam Sundaranar University, Tirunelveli. He is currently an Assistant Professor at Sri Paramakalyani College, Alwarkurichi, India. His Research interest is in Discrete Mathematics.
Department of Mathematics, Sri Paramakalyani College, Alwarkurichi, Tamil Nadu, India-627412.
e-mail: ponrajmaths@gmail.com
Mr. S. Sathish Narayanan did his M.Phil in St. Johns College, Palayamkottai. He is persuing doctoral research work. His Research interest is in Graph labeling.
Department of Mathematics, Sri Paramakalyani College, Alwarkurichi, Tamil Nadu 627 412, India.
e-mail: sathishrvss@gmail.com

1. Introduction

Terminology and notations in graph theory we refer Harary
[2]
. New terms and notations shall, however, be specifically defined whenever necessary. By a graph
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2. Main results

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BIO

Gallian J.A.
(2013)
A Dynamic survey of graph labeling
The Electronic Journal of Combinatorics
# Ds6.
16

Harary F.
2001
Graph theory
Narosa Publishing house
New Delhi

Ponraj R.
,
Ramasamy A.M.S.
,
Sathish Narayanan S.
Total Mean Cordial labeling of graphs (communicated).

Ponraj R.
,
Sathish Narayanan S.
Total Mean Cordiality ofKcn+2K2(communicated).

Ponraj R.
,
Ramasamy A.M.S.
,
Sathish Narayanan S.
Total Mean Cordial labeling of some graphs (communicated).

Citing 'TOTAL MEAN CORDIAL LABELING OF SOME CYCLE RELATED GRAPHS
'

@article{ E1MCA9_2015_v33n1_2_101}
,title={TOTAL MEAN CORDIAL LABELING OF SOME CYCLE RELATED GRAPHS}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2015.101}, DOI={10.14317/jami.2015.101}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={PONRAJ, R.
and
NARAYANAN, S. SATHISH}
, year={2015}
, month={Jan}