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Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 9, Issue 2 (December 2014), pp. 811-825
Difference Cordial Labeling of Graphs Obtained
from Triangular Snakes
R. Ponraj and S. Sathish Narayanan
Department of Mathematics
Sri Paramakalyani College
Alwarkurichi 627412, India
ponrajmaths@gmail.com; sathishrvss@gmail.com
Received: June 10, 2013; Accepted: June 24, 2014
Abstract
In this paper, we investigate the difference cordial labeling behavior of corona of triangular
snake with the graphs of order one and order two and also corona of alternative triangular snake
with the graphs of order one and order two.
Keywords: Corona; triangular snake; complete graph
MSC 2010 No.: 05C78; 05C38
1. Introduction:
(
) be
Throughout this paper we have considered only simple and undirected graph. Let
) graph. The cardinality of is called the order of and the cardinality of is called the
a(
size of . The corona of the graph with the graph ,
is the graph obtained by taking
th
one copy of and copies of and joining the i vertex of with an edge to every vertex in
the ith copy of . Graph labeling are used in several areas like communication network, radar,
astronomy, database management, see Gallian (2011). Rosa (1967) introduced graceful labeling
of graphs which was the foundation of the graph labeling. Consequently Graham (1980)
811
812
R. Ponraj and S. S. Narayanan
introduced harmonious labeling, Cahit (1987) initiated the concept of cordial labeling, and kproduct cordial labeling by Ponraj et al. (2012). Recently Ponraj et al. (2012) introduced k- Total
product cordial labeling of graphs. Ebrahim Salehi (2010) defined the notion of product cordial
set. On analogous of this, the notion of difference cordial labeling has been introduced by Ponraj
et al. (2013). Ponraj et al. (2013) studied the Difference cordial labeling behavior of quite a lot of
graphs like path, cycle, complete graph, complete bipartite graph, bistar, wheel, web and some
more standard graphs . In this paper we investigate the difference cordial labeling behavior of
( )
,
,
, ( )
, ( )
and
where
and
respectively denotes the triangular snake and complete graph. Let be any real number. Then
⌊ ⌋ stands for the largest integer less than or equal to and ⌈ ⌉ stands for the smallest integer
greater than or equal to . Terms and definitions not defined here are used in the sense of Harary
(2001).
2. Difference Cordial Labeling
Definition 2.1.
) graph. Let be a map from ( ) to *
+. For each edge , assign the
Let G be a (
label | ( )
( )|. is called difference cordial labeling if
and | ( )
( )|
where ( ) and ( ) denote the number of edges labeled with 1 and not labeled with 1
respectively. A graph with a difference cordial labeling is called a difference cordial graph.
The triangular snake is obtained from the path
triangle . Let be the path
. Let
( )
( )
*
by replacing each edge of the path by a
+
and
( )
( )
*
+.
We now investigate the difference cordiality of corona of triangular snake
.
Theorem 2.2.
is difference cordial.
with
,
and
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
813
Proof:
Clearly,
has
vertices and
(
)
( )
edges. Let
*
+
*
+
*
+
and
(
Case 1.
Define
)
( )
*
+.
is even
(
)
*
(
+ as follows:
)
(
⌊
⌋
)
(
)
(
)
(
)
(
)
(
(
⌈
⌉
⌊
⌋
⌈
⌉
⌊
⌋
⌊
⌋
⌈
⌉
⌈
⌉
⌊
⌋
)
⌈
⌉
)
⌊
⌋
(
)
(
)
(
)
(
(
and
(
)
,
)
)
.
814
Case 2.
R. Ponraj and S. S. Narayanan
is odd
Label the vertices
and
(
) as in case (i). Now, define,
(
Table 1 shows that
)
(
)
(
)
(
)
(
)
(
and
,
)
.
is a difference cordial labeling.
Table 1. The edge conditions of difference cordial labeling of
( )
( )
Nature of n
(
)
(
)
Example. A difference cordial labeling of
is given in Figure 1.
1
2
4
9
10
8
5
3
13
7
6
11
14
12
and
respectively. Let
Figure 1.
Theorem 2.3.
is difference cordial.
Proof:
Clearly, the order and size of
are
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
(
)
( )
815
*
+
*
+
*
+
and
(
)
( )
*
+.
to the set *
Define an injective map from the vertices of
+ as follows:
( )
(
)
(
)
.
( )
(
⌊
⌊
)
⌋
⌊
⌋
⌈ ⌉
( )
(
⌊
⌊
⌋
)
⌊
⌋
⌊
⌊
⌋
)
⌊
⌋
⌈ ⌉
( )
(
⌋
⌋
⌋
⌈ ⌉
Table 2. The conditions of difference cordial labeling of
( )
( )
Nature of n
(
)
(
)
Theorem 2.4.
is difference cordial.
Proof:
Clearly, the order and size of
are
and
respectively. Let
816
R. Ponraj and S. S. Narayanan
(
)
)
( )
( )
*
+
*
+
*
+
and
(
Case 1.
*
+.
is even.
to the set *
Define an injective map from the vertices of
(
)
(
⌊ ⌋
)
⌈
(
(
)
⌉
⌊ ⌋
)
⌈
(
(
+ as follows:
)
⌉
⌊ ⌋
)
⌈
⌉,
( )
( )
( )
(
Case 2.
)
, (
(
) and
)
and
(
)
.
is odd
Label the vertices
(
) as in case 1. Define,
( )
( )
( )
.
and
(
)
.
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
817
Table 3. The edge conditions of difference cordial labeling of
( )
( )
Nature of n
(
)
(
)
Example.
The graph
with a difference cordial labeling is shown in figure 2.
17 21 20 24 23 27 26
16
25
22
19
1
4
9
13
0
2 6 5
8 7 12 11 15 14
18
3
1
Figure 2.
An alternate triangular snake ( ) is obtained from a path
by joining
(alternatively) to new vertex . That is, every alternate edge of a path is replaced by
and
.
Theorem 2.5.
( )
is difference cordial.
Proof:
Case 1.
Let the first triangle start from
( ( )
and the last triangle ends with
)
( ( ))
*
. Here,
+
{
+
{
is even. Let
}
and
( ( )
)
( ( ))
*
In this case, the order and size of ( )
( ( )
)
*
+ as follows:
are
and
}.
, respectively. Define a map
818
R. Ponraj and S. S. Narayanan
( )
(
)
( )
(
)
(
)
⌊ ⌋
(
)
⌊ ⌋
(
⌊ ⌋
)
⌊ ⌋
⌈ ⌉
(
⌊ ⌋
)
⌊ ⌋
⌈ ⌉
Table 4. The conditions of difference cordial labeling of ( )
( )
( )
Nature of n
(
)
(
)
Case 2.
Let the first triangle be starts from
and the last triangle ends with
In this case, the order and size of
( )
vertices
(
) and
are
(
. Here, also
and
respectively. Label the
) and
(
⌊
case 1 and define,
(
(
⌊
⌋
⌊
⌋
is even.
)
⌊
⌋
⌈
⌉
)
⌊
⌋
⌈
⌉
⌋) as in
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
819
Table 5. The conditions of difference cordial labeling of ( )
( )
( )
Nature of n
(
)
(
)
Case 3.
Let the first triangle be starts from
and the last triangle ends with
case, the order and size of ( )
(
) and
are
and
(
. Here,
is odd. In this
, respectively. Label the vertices
) and
(
⌊
⌋) as in case (i)
and define,
(
(
⌊
⌋
⌊
⌋
)
⌊
⌋
⌊
⌋
)
⌊
⌋
⌊
⌋
Table 6. The conditions of difference cordial labeling of ( )
( )
( )
Nature of n
(
)
(
)
□
Theorem 2.6.
( )
is difference cordial.
Proof:
Case 1.
Let the first triangle be starts from
( ( )
and
)
and the last triangle ends with
( ( ))
*
+
{
. Here,
is even. Let
}
820
R. Ponraj and S. S. Narayanan
( ( )
)
( ( ))
*
+
In this case, the order and size of ( )
( ( )
)
{
are
{
and
}.
respectively. Define a map
} by
( )
( )
( )
( )
( )
Since
and
(
)
(
)
( )
,
is a difference cordial labeling of ( )
.
Case 2.
Let the first triangle be starts from
and the last triangle ends with
case, the order and size of ( )
map
from the vertices of ( )
(
to the set {
)
(
)
(
)
(
are
)
( )
( )
(
)
(
)
and
. Here
is even. In this
respectively. Define a one-one
} as follows:
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
Since
( )
( )
and
821
is a difference cordial labeling of ( )
,
.
Case 3.
Let the first triangle be starts from
and the last triangle ends with
case, the order and size of ( )
and
(
)
(
are
and
. Here,
is odd. In this
respectively. Label the vertices
(
) as in Case 2 and define
)
, (
)
,
,
( )
Since
( )
( )
(
)
(
)
,
is a difference cordial labeling of ( )
.
□
Theorem 2.7.
( )
is difference cordial.
Proof:
Case 1.
Let the first triangle be starts from
Let
( ( )
)
)
( ( ))
and the last triangle ends with
( ( ))
*
+
{
+
{
. In this case
is even.
}
and
( ( )
*
In this case, the order and size of
injective map
( )
from the vertices of ( )
are
and
to the set {
}.
respectively. Define an
} as follows:
822
R. Ponraj and S. S. Narayanan
( )
(
)
(
)
(
)
(
)
⌊ ⌋
(
)
⌊ ⌋
(
)
(
)
(
(
⌊ ⌋
)
⌊ ⌋
(
⌊ ⌋
)
⌊ ⌋
⌈ ⌉
(
⌊ ⌋
)
⌊ ⌋
⌈ ⌉
)
⌊ ⌋
⌈ ⌉
(
(
⌊ ⌋
⌊ ⌋
⌊ ⌋
⌊ ⌋
)
⌊ ⌋
⌈ ⌉
)
⌊ ⌋
⌈ ⌉
Table 7. The conditions of difference cordial labeling of ( )
( )
( )
Nature of n
(
)
(
)
Case 2.
Let the first triangle be starts from
and the last triangle ends with
this case, the order and size of ( )
(
as in case 1 and define
),
(
are
) and
and
. Here,
is even. In
respectively. Label the vertices
(
⌊
⌋)
AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
(
(
(
(
(
⌊
⌋
⌊
⌋
⌊
⌋
⌊
⌋
⌊
⌋
823
)
⌊
⌋
⌈
⌉
)
⌊
⌋
⌈
⌉
)
⌊
⌋
⌈
⌉
)
⌊
⌋
⌈
⌉
)
⌊
⌋
⌈
⌉
Table 8. The conditions of difference cordial labeling of ( )
( )
( )
Nature of n
(
)
(
)
Case 3.
Let the first triangle be starts from
case, the order and size of
(
and the last triangle ends with
( )
are
) and
and
. Here,
respectively. Label the vertices
(
⌊
⌋) as in case (i) and
define
(
(
(
(
(
⌊
⌋
⌊
⌋
⌊
⌋
⌊
⌋
⌊
⌋
is odd. In this
)
⌊
⌋
⌊
⌋
)
⌊
⌋
⌊
⌋
)
⌊
⌋
⌊
⌋
)
⌊
⌋
⌊
⌋
)
⌊
⌋
⌊
⌋
824
R. Ponraj and S. S. Narayanan
Table 9. The conditions of difference cordial labeling of ( )
( )
( )
Nature of n
(
)
(
)
Example.
A difference cordial labeling of ( )
triangle ends with
is given in Figure 3.
14
with the first triangle starts from
17
15
2
6
10
7
4
1
18
1
6
13
3
and the last
5 8
Figure 3.
9
( )
1
1
12
3. Conclusions
In this paper we have studied about difference cordial labeling behavior of
,
,
, ( )
, ( )
and ( )
. Investigation of difference cordiality of
join, union and composition of two graphs are the open problems for future research.
Acknowledgement
The authors thank to the referees for their valuable suggestions and commands.
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AAM: Intern. J., Vol. 9, Issue 2 (December 2014)
825
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