      - Cordial labeling of Some Cycle Related Graphs

American Journal of Engineering Research (AJER) 2016 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-5, Issue-10, pp-90-95 www.ajer.org Research Paper Open Access

 p  - Cordial labeling of Some Cycle Related Graphs  2  D. S. T. Ramesh1, K. Jenita Devanesam2 1

Department of Mathematics, Margoschis College, Nazareth – 628 617, Tamil Nadu, INDIA 2 Department of Mathematics, Pope’s College, Sawyerpuram-628251, Tamil Nadu, India.

 p  - cordial labeling of a graph G with p vertices is a bijection f: V(G)  {1, 2, 3,…, p}  2  1 if f (u)  f(v)   p    2  and e (0)  e (1)  1 . If a graph has  p  - cordial defined by f (e  uv)    2  0 otherwise  p  - cordial where  p  represents the nearest integer less than or equal to p labeling then it is called  2   2  2 ABSTRACT: A

f

f

 p  2  - co rd ial g raph for n ≥ 3 , excep t fo r n = 4 and the graph  p obtained by duplication of an arbitrary vertex by a new edge in cycle C n is  2  - co rd ia l and the graph  p obtained by duplication of an arbitrary edge by a new vertex in cycle Cn is  2  -cordial. By a graph we mean In th is p aper we p rove C n is

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 v k 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

 p  2  - 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.

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n  i if i is even and i   2   n  i if i is odd and i  n  2 Define f(ui) =  n  1  i if i is even and n  i  n  2  n  i  1 if i is odd in  2  1 if i  n  Then the labels of the vertices u1, u2, u3, …, un/2 are respectively n-1, 2, n-3, 4,…n-

n n +1, , 2 2

Hence f (ui )  f (ui 1 ) , i = 1, 2, 3, …,

n -1, are respectively 2

n-3, n-5, n-7, …, 7, 5, 3, 1.

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

And the labels of vertices u n , u n 2

1

2

2

n n -1 times, 1, 1, 1, … times.

4 4

, …, un-1, un are respectively

n n n n +2, -1, +4, -3, …, n, 1 2 2 2 2 n n n n And f (ui )  f (ui 1 ) , i = , +1, +2, +3, …, n-3, n-2, n-1, are respectively 2, 3, 5, 7, …, n-5, n-3, n-1.

2 2 2 2 n n Thus the corresponding edge labels are 1,1,1, … times, 0, 0, 0, … times.

4 4 And f (un )  f (u1 )  n  2 The label of the corresponding edge is 0.

Hence e f (0)  e f (1) 

n 2

Case(ii): n = 4m+1, m = 1, 2, 3, …

Let the vertices be u1, u2, …, un.

 i if i     n  i if  Define f(ui) =   n  1  i if   i  1 if    1 if Here ef(1) =

n i  2 n and i    2

is even and i is odd i is odd

and

n  in 2

n i is even    i  n 2 in

n 1 n 1 , ef(0) = 2 2

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

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Let the vertices be u1, u2, …, un.

 i if    n 1 i  Define f(ui) =  n  1  i if    i   1  Hence e f (0)  e f (1) 

n 1 2 n if i is odd and i  2 n i is even and   i  n 2 n if i is odd in 2 if i  n i

i is even and

n 2

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

  i if    ni Define f(ui) =    n  1  i   i 1   1 

e f (0) 

n i    1 2 n if i is odd and i    2 n if i is odd and    i  n 2 n if i is even    i  n 2 if in i is even and

n 1 n 1 , e f (1)  2 2 u8 = 1 u7 = 8

0

0

0

u1 = 7

0 u2 = 2

u6 = 3

1

1

u5 = 6

u3 = 5

1

1 u4 = 4

The

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

 p  2  –

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

i if i is even and i  2m   4m  3  i if i is odd and i  2m  Define f(ui) = 4m  1  i if i is even and m  i  n  i  2 if i is odd 2m  i  n  1 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

i if i is even and i  2m  4m  4  i if i is odd and i  2m  Define f(ui) =  i  2 if i is even and m  i  n  4m  2  i if i is odd 2m  i  n  1 if i  n  ef(0) = 2m+2, ef(1) = 2m+2 v" = 5

v' = 4

v5 = 1

v1 = 7

v4 = 6

v3 = 3

v2 = 2

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

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i if i is even and i  2m  2   4m  5  i if i is odd and i  2m  Define f(ui) =  4m  3  i if i is even and m  2  i  n  i  2 if i is odd 2m  1  i  n  1 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

i if i is even and i  2m   4m  6  i if i is odd and i  2m  1  Define f(ui) =  4m  4  i if i is odd and m  1  i  n  i  2 if i is even 2m  i  n  1 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

 p  2  -

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.

V(G)  n  1 and E (G)  n  2 .

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

Then

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

i if i is even and i  2m   4m  2  i if i is odd and i  2m  Define f (ui )  4m  1  i if i is even and m  i  n  i  1 if i is odd 2m  i  n  1 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

i if i is even and i  2m   4m  3  i if i is odd and i  2m  Define f (ui )  4m  2  i if i is odd and m  i  n  i  1 if i is even 2m  i  n  1 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

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i if i is even and i  2m   4m  3  i if i is odd and i  2m  Define f (ui )  4m  3  i if i is even and m  i  n  i  2 if i is odd 2m  i  n  1 if i  n  ef(0) = 2m+2, ef(1) = 2m+2 v' = 4 u6 = 1

u1 = 6

u5 = 7

u2 = 2

u4 = 3

u3 = 5

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

i if i is even and i  2m  4m  4  i if i is odd  Define f (ui )    i  2 if i is even and 2m  i  n  1 if i  n ef(0) = 2m+2, ef(1) = 2m+3.

REFERENCES [1].

[2].

[3].

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

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

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