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e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-5, Issue-10, pp-90-95 www.ajer.org Research Paper Open Access

w w w . a j e r . o r g Page 90

p 2

  

 - Cordial labeling of Some Cycle Related Graphs D. S. T. Ramesh1, K. Jenita Devanesam2

1Department of Mathematics, Margoschis College, Nazareth – 628 617, Tamil Nadu, INDIA

2Department of Mathematics, Pope’s College, Sawyerpuram-628251, Tamil Nadu, India.

ABSTRACT: A p 2

  

 - cordial labeling of a graph G with p vertices is a bijection f: V(G) {1, 2, 3,…, p}

defined by

1 (u) f(v)

(e uv) 2

0 if f p f

otherwise

 

  



ande (0)f e (1)f 1. If a graph has p 2

  

 - cordial labeling then it is called p

2

  

 - cordial where p 2

  

 represents the nearest integer less than or equal to

2 p

In th i s p ape r w e p r ov e Cn i s 2

 p

  - co rd ial g raph for n ≥ 3 , e xc ep t f o r n = 4 and the graph obtained by duplication of an arbitrary vertex by a new edge in cycle Cn i s

2

 p

  - co rd ia l and the graph obtained by duplication of an arbitrary edge by a new vertex in cycle Cn is

2

 p

  -cordial. By a graph we mean a finite, undirected graph without multiple edges or loops. For graph theoretic terminology, we refer to Harary [2] and Bondy [1].

Keywords: p 2

  

  - cordial, cycle.

Definition1.1: Duplication of a vertex vk by a new edge e = vꞌvꞌꞌ in a graph G produces a new graph Gꞌ such that N(vꞌ) = {vk, vꞌꞌ} and N(vꞌꞌ) = {vk,vꞌ}.

Definition1.2: Duplication of an edge e = uv by a new vertex w in a graph G produces a new graph Gꞌ such that N(w) = {u, v}.

Theorem1.1: Cn is 2

 p

  - cordial graph for n ≥ 3, except for n = 4.

Proof:

Case(i): n = 4m, m = 2, 3, 4, … Let the vertices be u1, u2, …, un.

(2)

w w w . a j e r . o r g Page 91

Define f(ui) =

2 2

1 2

1 2

1

i if i is even and i n n i if i is odd and i n

n i if i is even and n i n

i if i is odd n i n

if i n

    

 

Then the labels of the vertices u1, u2, u3, …, un/2 are respectively n-1, 2, n-3, 4,…n-

2 n+1,

2 n,

Hence

( )i ( i1)

f u f u , i = 1, 2, 3, …,

2

n-1, are respectively

n-3, n-5, n-7, …, 7, 5, 3, 1.

Thus the corresponding edge labels are 0, 0, 0, …

4

n-1 times, 1, 1, 1, …

4

n times.

And the labels of vertices u

2 1 n , u

2 2

n , …, un-1, un are respectively

2 n+2,

2 n-1,

2 n+4,

2

n-3, …, n, 1

And f u( )i f u( i1), i = 2 n,

2 n+1,

2 n+2,

2

n+3, …, n-3, n-2, n-1, are respectively 2, 3, 5, 7, …, n-5, n-3, n-1.

Thus the corresponding edge labels are 1,1,1, …

4

n times, 0, 0, 0, …

4 ntimes.

And f u( n)f u( )1  n 2

The label of the corresponding edge is 0.

Hence (0) (1)

f f 2

e e n

Case(ii): n = 4m+1, m = 1, 2, 3, … Let the vertices be u1, u2, …, un.

Define f(ui) =

2

2

1 2

1 2

1

i if i is even and i n

n i if i is odd and i n

n i if i is odd and n i n

i if i is even n i n

if i n

 

  

 

  

     

  

  

  

Here ef(1) = 1

2 n

, ef(0) = 1 2 n

Case (iii): n = 4m + 2, m = 1, 2, 3, …,

(3)

w w w . a j e r . o r g Page 92

1 2 n

Define f(ui) =

2 1

1 2

1 2

2 1

i if i is even and i n n i if i is odd and i n

n i if i is even and n i n

i if i is odd n i n

if i n

 

 

     

 

Hence (0) (1)

f f 2

e e n

Case (iv): n = 4m+3, m = 1, 2, 3, …, Let the vertices be u1, u2, …, un.

Define f(ui) =

2 1

2

1 2

1 2

1

i if i is even and i n

n i if i is odd and i n

n i if i is odd and n i n

i if i is even n i n

if i n

 

  

 

  

     

  

  

  

1 1

(0) , e (1)

2 2

f f

n n

e

1 7

u =

2 2

u =

3 5

u =

4 4

u =

5 6

u =

6 3

u =

7 8

u =

8 1

u =

0 0

0 0

1

1 1

1

The 2

 p

  - cordial labeling of c8

(4)

w w w . a j e r . o r g Page 93

Theorem1.2: The graph obtained by duplication of an arbitrary vertex by a new edge in cycle Cn is 2

 p

  

cordial.

Proof: Let u1, u2, …, un be vertices and e1, e2, e3, …, en be edges of cycle Cn

Without loss of generality we duplicate vn by an edge en+1with end vertices vꞌ and vꞌꞌ.

Let the graph so obtained be G.

Then |V(G)| = n+2 and |E(G)| = n+3.

To define f: V(G){1, 2, 3, …, n+2} we consider the following cases.

Case(i): n = 4m, m = 1, 2, 3, …, Let f(vꞌ) = 2m+1, f(vꞌꞌ) = 2m+2

Define f(ui) =

2

4 3 2

4 1

2 2

1

i if i is even and i m

m i if i is odd and i m

m i if i is even and m i n

i if i is odd m i n

if i n

 

     

 



Here ef(0) = 2m+1, ef(1) = 2m+2 Case(ii): n = 4m+1, m = 1, 2, 3, …, Let f(vꞌ) = 2m+2, f(vꞌꞌ) = 2m+3

Define f(ui) =

2

4 4 2

2

4 2 2

1

i if i is even and i m

m i if i is odd and i m

i if i is even and m i n

m i if i is odd m i n

if i n

 

   

   



ef(0) = 2m+2, ef(1) = 2m+2

5 1

v =

1 7

v =

4 6

v =

2 2

v =

3 3

v =

' 4

v = v =" 5

Case(iii): n = 4m+2, m = 1, 2, 3, …, Let f(vꞌ) = 2m+1, f(vꞌꞌ) = 2m+4

(5)

w w w . a j e r . o r g Page 94

Define f(ui) =

2 2

4 5 2

4 3 2

2 2 1

1

i if i is even and i m

m i if i is odd and i m

m i if i is even and m i n

i if i is odd m i n

if i n

 

      

  



ef(0) = 2m+2, ef(1) = 2m+3

Case(iv): n = 4m+3, m = 1, 2, 3, …, Let f(vꞌ) = 2m+2, f(vꞌꞌ) = 2m+3

Define f(ui) =

2

4 6 2 1

4 4 1

2 2

1

i if i is even and i m

m i if i is odd and i m

m i if i is odd and m i n

i if i is even m i n

if i n

 

      

 



ef(0) = 2m+3, ef(1) = 2m+3

Theorem1.3: The graph obtained by duplication of an arbitrary edge by a new vertex in cycle Cn is 2

 p

  -

cordial expect for n = 3.

Proof: Let u1, u2, …, unbe vertices and e1, e2, e3, …, en be edges of cycle Cn Without loss of generality we duplicate the edge unu1 by a vertex vꞌ.

ThenV(G)  n 1and E G( )  n 2.

To define f: V(G){1, 2, 3, …, n+1}we consider the following cases.

Case(i): n = 4m, m = 1, 2, 3, …, p = 4m+1, q = 4m+2

Let f(vꞌ) = 2m+1

2

4 2 2

( ) 4 1

1 2

1

i

i if i is even and i m

m i if i is odd and i m

Define f u m i if i is even and m i n

i if i is odd m i n

if i n

 

     

 



ef(0) = 2m+1, ef(1) = 2m+1

Case(ii): n = 4m+1, m = 1, 2, 3, …, p = 4m+2; q = 4m+3

2

4 3 2

( ) 4 2

1 2

1

i

i if i is even and i m

m i if i is odd and i m

Define f u m i if i is odd and m i n

i if i is even m i n

if i n

 

     

 



ef(0) = 2m+1, ef(1) = 2m+2

Case(iii): n = 4m+2, m = 1, 2, 3, …, p = 4m+3; q = 4m+4.

Let f(vꞌ) = 2m+2

(6)

w w w . a j e r . o r g Page 95 2

4 3 2

( ) 4 3

2 2

1

i

i if i is even and i m

m i if i is odd and i m

Define f u m i if i is even and m i n

i if i is odd m i n

if i n

 

     

 



ef(0) = 2m+2, ef(1) = 2m+2

' 4

v =

1 6

u =

2 2

u =

3 5

u =

4 3

u =

5 7

u =

6 1

u =

Case(iv): n = 4m+3, m = 1, 2, 3, …, p = 4m+4; q = 4m+5

Let f(vꞌ) = 2m+2

2

4 4

( ) 2 2

1

i

i if i is even and i m

m i if i is odd

Define f u

i if i is even and m i n

if i n

 

   

ef(0) = 2m+2, ef(1) = 2m+3.

REFERENCES

[1]. J. A. Bondy and U. S. R. Murty, Graph Theory with applications, Macmillan press, London (1976) [2]. J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of combinatorics (2014).

[3]. F. Harary, Graph Theory Addison – Wesley, Reading Mars., (1968)

References

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