Cordial Labelings in the Context of Triplication

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 04 Issue: 07 | July -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2303
Cordial labelings in the context of triplication
S. Gurupriya1, S.Bala2
1B.E(Final year), Department of Computer Science
Sri Sairam Engineering College, West Tambaram, Chennai, Tamilnadu, India
2Assistant professor, Department of Mathematics
S.I.V.E.T.College, Gowrivakkam, Chennai, Tamilnadu, India
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Abstract - In this paper, we introduce the extended triplicate
graph of a ladder and investigate the existence of cordial
labeling, total cordial labeling, product cordial labeling, total
product cordial labeling and prime cordial labeling for the
extended triplicate graph of a ladder graph by presenting
algorithms.
Key Words: Ladder graph, Triplicate graph, Graph
labelings.
1.INTRODUCTION
Graph theory has various applications in the fieldof
computer programming and networking, marketing and
communications, business administration and so on. Some
major research topics in graph theory are Graph coloring,
Spanning trees, Planargraphs,Networksand Graphlabeling.
Graph labeling has been observedandidentifiedforitsusage
towards communication networks. That is, the concept of
graph labeling can be applied to network security, network
addressing, channel assignmentprocessandsocial networks
[3].
In 1967, Rosa introduced the concept of graph
labeling [4]. A graph labeling is an assignment of integers to
the vertices or edges or both subject to certain condition(s).
If the domain of mapping is the set of vertices (edges) then
the labeling is called a vertex(an edge) labeling.
In 1987, Cahit introduced the notion of cordial
labeling [2]. A graph G is said to admit a cordial labeling if
there exists a function f : V → {0, 1} such that the induced
function f* : E → {0, 1} defined as f*(vivj) = | f(vi) - (f(vj) | or
(f(vi) + f(vj)) (mod 2) satisfies the property that the number
of vertices labeled ‘0’ and the number of vertices labeled ‘1’
differ by atmost one and the number of edges labeled ‘0’ and
the number of edges labeled ‘1’ differ by atmost one. A
graph G is said to admit a total cordial labeling if thereexists
a function f : V → {0, 1} such that the induced function
f* : E → {0, 1} defined as f*(vivj) = | f(vi) - (f(vj) | or
(f(vi) + f(vj)) (mod 2) satisfies the property that the number
of vertices and edges labeled with ‘0’ and the number of
vertices and edges labeled with ‘1’ differ by atmost one.
In 2004, Sundaram, Ponraj and Somasundaram have
introduced the concept of product cordial labeling [ 5,6 ]. A
graph G is said to admit product cordial labeling if there
exists a function f : V → {0, 1} such that the induced function
f* : E → {0, 1} defined as
f*(vivj) = {(f(vi) × f(vj) | vivj ∈ E} satisfies the property that
the number of vertices labeled ‘0’ and thenumberofvertices
labeled ‘1’ differ by atmost 1 and number of edges labeled ‘0’
and the number of edges labeled ‘1’ differ by atmost 1. A
graph that admits product cordial labeling is called product
cordial graph.
A graph is called total product cordial graph if
there exists a function f : V → {0, 1} such that the induced
function f* : E → {0, 1} defined as
f*(vivj) ={(f(vi) × f(vj) | vivj ∈ E} satisfies the property that
the number of 0’s on the vertices and edges taken together
differ by atmost one with the number of 1’s on the vertices
and edges taken together.
In 2011, Bala and Thirusangu introduced the
concept of the extended triplicate graph of a path Pn
((ETG(Pn)) and proved many results on this newly defined
concept [1]. Let V = { v1, v2,…,vn+1} and E = { e1, e2 , ….
, en} be the vertex and Edge set of a path Pn. For every vi ∈ V,
construct an ordered triple {vi , vi
′ , vi
″} where 1≤ i ≤
n+1 and for every edge vivj ∈ E, construct four edges
vivj
′, vj
′ vi
″ , vjvi
′ and vi
′ vj
″ where j = i +1, then the graph with
this vertex set and edge set is called a Triplicate Graph of a
path Pn. It is dentoted by TG(Pn). Clearly the Triplicate graph
TG(Pn) is disconnected. Let V1 = {v1, v2 …,v3n+1} and
E1 = { e1, e2,…., e4n}be the vertex and edge set of TG(Pn). Ifnis
odd, include a new edge (vn+1 , v1) and if n is even, include a
new edge (vn ,v1) in the edge set of TG(Pn). This graph is
called the Extended Triplicate of the pathPn anditisdenoted
by ETG(Pn).
In 2014 , Thirusangu et.al proved some results on
Duplicate Graph of Ladder Graph [7].
A ladder graph Ln is a planar undirected graph with 2n
vertices and 3n– 2 edges. It is obtained as the cartesian
product of two path graphs, one of which has only one edge:
Ln,1 = Pn × P1, where n is the number of rungs in the ladder.
Motivated by the study, the present work is aimed to
provide label for the extended triplicate graph of a
ladder graph and prove the existence of cordial
labeling, total cordial labeling, product cordiallabeling

and total product cordial labeling for the extended
triplicate graph of a ladder graph.
Throughout this work, graph G = (V, E), we mean a
simple, finite, connected and undirected graph with p
vertices and q edges.
K.Thirusangu and E.Bala (2011) introduced the concept of
triplicate graph and proved many results on this newly
defined concepts.
Motivated by thestudy,thepresent work isaimedto
provide label for the extended triplicate graph of a ladder
graph and prove the existence of cordial labeling, total
cordial labeling, product cordial labeling and total product
cordial labeling for the extended triplicate graph of a ladder
graph.
Throughout this work, graph G = (V, E), we mean a simple,
finite, connected and undirectedgraph withpverticesandq
edges.
2. STRUCTURE OF THE EXTENDED TRIPLICATE GRAPH
OF LADDER
In this section we discuss about the structure of the
extended triplicate graph of ladder by presenting algorithm.
Algorithm 2.1:
Input ladder graph Ln
procedure triplicate of graph Ln
for i = 1 to n do
V {
end for
for i = 1 to n-1 do
E1 ∪ (
end for
for i = 2 to n do
E2 ∪ ∪ ∪
end for
for i=1 to n do
E3 ∪ ∪ ∪
end for
E E1∪ E2∪E3
end procedure
output : Triplicate graph of ladder Ln
From the above algorithm2.1,thetriplicategraphof
a ladder denoted by TG(Ln) is a disconnected graph with 6n
vertices and 12n - 8 edges. To make it as a connected graph,
for convenience, we include an edge to the edge set E
as defined in the above algorithm. Thus the graph so
obtained is called an extended triplicate graph of ladder Ln
and is denoted by ETG(Ln). By the construction, it is clear
that, the graph ETG(Ln) has 6n vertices and 12n - 7 edges.
Illustration 2.1:
The structure of extended triplicate graph of ladder ETG(L4)
is given in figure 1.
Fig-1 : ETG(L4)
3 CORDIAL AND TOTAL CORDIAL LABELING
In this section, we present an algorithm and prove the
existence of cordial and total cordial labelingoftheextended
triplicate graph of ladder (ETG(Ln)).
Algorithm 3.1
procedure (cordial labeling for ETG(Ln))
for i = 1 to n do
V {
end for
for i = 1 to n do
← 0
←
←
end for
end procedure
output labeled vertices of ETG(Ln)
Theorem 3.1
The extended triplicate graph of a ladder graph
admits cordial labeling.
Proof:
We know that, the extended triplicate graph of a
ladder has 6n vertices and 12n – 7 edges. Consider the

arbitrary vertex vi ∈ V. To label the vertices,usingalgorithm
3.1, define a map f: V → {0,1} . Clearly the number of
vertices labeled with ‘0’ is 3n and ‘1’ is 3n.
Thus the number of vertices labeled with ‘0’ and the number
of vertices labeled with ‘1’ differ by atmost one.
In order to get the labels for the edges, define the induced
map f* : E → {0,1}such that for any vivj ∈ E, f*(vivj) = (f(vi ) +
f(vj)) (mod 2). Thus,
(i) For 1≤ i ≤ n - 1, the edges receives the following
labels:
= =
(ii) For 2 ≤ i ≤ n , the edges receives the labels as
follows:
= = =
(iii) For 1 ≤ i ≤ n , the edges receives the labels as
follows:
= =
=
(iv) = 1
Clearly the number of edges labeled with ‘0’ is 6n – 4 and ‘1’
is 6n – 3.
Thus, the number of edges labeled with ‘0’ and ‘1’ differ by
atmost one.
Hence ETG(Ln) admits cordial labeling.
Theorem 3.2
Extended triplicate graph of ladder admits total cordial
labeling.
Proof:
By theorem 3.1 , using the map f on V and there by
the induced map f* on E, the total number of vertices and
edges labeled together with ‘0’ and ‘1’ is 9n-4 and 9n-3
respectively.
Thus for all n, the number of zeroes on the vertices and
edges taken together differ by atmost 1 with the number of
one’s on vertices and edges taken together.
Hence the extended triplicate graph of ladder admits total
cordial labeling.
Illustration 3.1
ETG(L4) with its cordial labeling is given below in
figure 2.
Fig-2: ETG(L4) and its cordial labeling
4 PRODUCT CORDIAL AND TOTAL PRODUCT CORDIAL
LABELING
In this section we present an algorithm and prove the
existence of product cordial and total product cordial
labelings for the extended triplicate graph of ladder
(ETG(Ln)).
Algorithm 4.1
procedure (product cordial labeling for ETG(Ln))
for i = 1 to n do
V {
end for
for i = 1 to n do
← ←
←
end for
end procedure
output labeled vertices of ETG(Ln)
Theorem 4.1
The extended triplicate graph of a ladder admits
product cordial labeling.
Proof:
The extendedtriplicategraphoftwighas6nvertices
and 12n - 7 edges. Using algorithm 4.1, define the function f:
V → {0,1} to label the vertices. Thus the number of vertices
labeled with ‘0’ is 3n and ‘1’ is 3n.

To obtain the edge labels, define theinducedfunctionf*:E →
{0,1}such that for any vivj ∈ E, f*(vivj) = (f(vi ) × f(vj))(mod2).
(i) For 1≤ i ≤ n - 1, the edges receives the following
labels:
= =
=
(ii) For 2 ≤ i ≤ n , the edges receives the labels as
follows:
=
= =
(iii) For 1 ≤ i ≤ n , the edges receives the labels as
follows:
=
=
(iv) = 0
Clearly the number of edges labeled with ‘0’ is 6n – 3 and ‘1’
is 6n – 4
Thus, for all n, the number of edges labeled with ‘0’ and ‘1’
differ by atmost one.
Hence ETG(Ln) admits product cordial labeling.
Illustration 4.1:
ETG(L4) with its product cordial labeling is given below in
figure 3 .
Fig-3: ETG(L4) and its product cordial labeling
Theorem 4.2
The extended triplicate graph of ladder admitstotal
Proof:
By theorem 4.1, using the map f on V and there by the
induced map f* on E, we have the total number of vertices
and edges labeled together with ‘0’ and ‘1’ is 9n-3 and 9n-4
respectively.
Thus for all n, the number of zeroes on the vertices
and edges taken together differ by atmost1withthenumber
of one’s on vertices and edges taken together.
Hence the extended triplicate graph of ladder admits total
5 CONCLUSION
In this paper, we have introduced and proved the existence
of cordial labeling, total cordial labeling, product cordial
labeling and total product cordial labeling for the extended
triplicate graph of ladder by presenting algorithms.
REFERENCES
[1]. Bala.E, Thirusangu.K, Some graph labelings in
extended triplicate graph of a path Pn,
International Review of Applied Engineering
Research, (IRAER), Vol 1, No.1 (2011), 81-92.
[2]. Cahit.I, Cordial graphs; a weaker version of graceful
and harmonious graphs, Ars Combin, 23(1987),
201-207.
[3]. Gallian.J.A, A dynamic survey of graph labeling, The
Electronic Journal of Combinatorics 18 (2011),
#DS6.
[4]. Rosa.A, On certain valuations of the vertices of a
graph, Theory of graphs (Internat.Symposium,
Rome, July 1996), Gordon and Breach, N.Y. and
Dunod Paris (1967), 349-355.
[5]. Sundaram.M, Ponraj.R and Somasundaram.S,
Product cordial graphs, Bull. Pure and Applied
Sciences (Mathematics and Statistics), Vol. 23E
(2004), 155-163.
[6]. Sundaram.M, Ponraj.R and Somasundaram.S, Total
product cordial labeling of graphs, Bull. Pure and
Appl. Sci. Sect. E Math. Stat., Vol. 25 (2006), 199-
203.
[7].Thirusangu. K, Ulaganathan.P.P. and
Vijayakumar. P, Some Cordial Labeling of
Duplicate Graph of Ladder Graph, Annals of Pure
and Applied Mathematics Vol. 8, No. 2, (2014), 43-
50.

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