Vol 6, No 12 (2015)

(1)

Available online throug

ISSN 2229 – 5046

E AND V –MAGIC STRENGTH OF H-MULTIPLE LABELING OF GRAPHS

D. S. T. RAMESH

1

_{, A. SUBRAMANIAN}

2

**_{AND K. SUNITHA*}**

3

1

_{Associate Professor, Department of Mathematics,}

Nazareth Margoschis College, Nasarath, India.

2

Associate Professor, Department of Mathematics,

M. D. T. Hindu College, Tirunelveli, India.

3

_{Assistant Professor, Department of Mathematics,}

Women’s Christian College, Nagercoil, India.

(Received On: 21-11-15; Revised & Accepted On: 17-12-15)

ABSTRACT

A

one-to-one map

f

from

V

∪

E

of a graph G onto the integers

{

h h h

, 2 , 3 ,..., (

m

+

n h

)

}

is a E-Magic h-multiple labeling if there exist a constant k such that

( )

( ) ( )

( ).

v V e E

k f

d v f v

f e

ε

∈ ∈

=

∑

+

∑

A one-to-one map

f

from V

∪

E

of

a graph G onto the integers

{

h h h

, 2 , 3 ,..., (

m

+

n h

)

}

is a V-Magic h-multiple labeling if there exist a constant k such that

( )

v N u

f u

f u v

k

∈

+

∑

=

where h is any positive integer. In this Paper, we obtain the E-magic strength of

h-multiple labeling of the path

P

_n on n-vertices, the n-bi star

B

_{n n}_, obtained from two disjoint copies of

K

_1,_n by joining the center vertices by an edge and the tree

<

K

_1,_n

: 2

>

obtained from

B

_{n n}_, by subdividing the middle edge with a new vertex.Also we obtain the V-magic strength of h-multiple labeling of the path

P

_n on n-vertices and 4-connected graph

4,n

H

on n-vertices

Keywords: Magic labeling, E-Magic Strength, E-Magic h-multiple labelling, graph

nP

₂, V-Magic h-multiple labeling

and structure of

H

_{m n}_,

1. INTRODUCTION

A labeling of a graph G(V,E) is a mapping from the set of vertices ,edges or both vertices and edges to the set of labels. Based on the domain we distinguish vertex labelings, edge labelings and total labelings. In most applications the labels are positive (or nonnegative) integers, though in general real numbers could be used. Various labelings are obtained based on the requirements put on the mapping. Magic labelings were introduced by sedlacek in 1963[6]. In general for a magic type labeling we require the sum of labels related to a vertex (a vertex magic labeling) or to an edge (an edge magic labeling) to be constant all over the graph.

Definition 1.1: For any magic labeling f of G, there is a constant k such that

f x

( )

+

f y

( )

+

f xy

(

)

=

k

,

( )

xy

∈

E G

.

Corresponding Author: K. Sunitha*

3

_{Assistant Professor, Department of mathematics,}

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2. E-MAGIC STRENGTH OF H-MULTIPLE LABELING OF GRAPHS

Definition 2.1 [3]: The E-magic strength of G, Em(G),is defined as the minimum of all k where the minimum is taken over all magic labelings f of G. That is Em(G)=min{k:f is a magic labeling of G}.Let f be a magic labeling of G with the constant k. Then adding all constants obtained at each edge we get

( )

( ) ( )

( )

v V e E

k f

d v f v

f e

ε

∈ ∈

=

∑

+

∑

Definition 2.2: Let h be any positive integer.A one-to-one map

f

from

V

∪

E

of a graph G onto the integers

{

h h h

, 2 , 3 ,..., (

m

+

n h

)

}

is a E-Magic h-multiple labeling if there exist a constant k such that

( )

( ) ( )

( )

v V e E

k f

d v f v

f e

ε

∈ ∈

=

∑

+

∑

A graph which admits E-Magic h-multiple labeling then it is called E-Magic h-multiple graph.

Theorem2.1:

Em P

(

₂n

)

=

h n

(5

+

1)

and

Em P

(

2n+1

)

=

h n

(5

+

3)

Proof: We prove this theorem by assigning a magic labeling to

P

₂_n and

P

₂_n₊₁

Let

v v

₁

,

₂

,...,

v

₂n be the consecutive vertices and

e e

1

,

2

,...,

e

2n−1 be the consecutive edges of

P

2n

That is ,

e

_i

=

v v

_{i i}₊₁

for

1 ≤ ≤

i

2 n

−

1

Define

f V

:

∪ →

E

{ , 2 ,..., (2

h

n

−

1) }

h

by

2 1

2

( ) 1

( ) ( ) 1

( ) (4 ) 1 2 1

i

f v ih for i n

f v h n i for i n

f e h n i for i n

− = ≤ ≤

= + ≤ ≤

= − ≤ ≤ −

Thus

Em P

(

₂_n

)

≤

h n

(5

+

1)

Since

ε

(

P

₂_n

)

=

2 n

−

1

, and hence

ε

+ =

v

4 n

−

1

If f is a magic labeling of

P

₂_n with constant

k f

( )

then

( )

( ) ( )

( )

v V e E

k f

d v f v

f e

ε

∈ ∈

=

∑

+

∑

That is ,

2 1 2 1

1 2

2 1

(2

1) ( )

2 ( )

( )

(

)

( )

n n

i n i

i i

n

k f

h v

h

f e

− −

= =

−

=

∑

+

∑

=

2 2 1 2 1

1 1 2

( )

n n n

i i i

h

f v

h

f e

h

f v

− − = = =

+

∑

= 2 1 2

(

2 ... (4

1) )

( )

n i i

h

n

h

f v

− =

+

+ +

−

+

∑

= 2 1 2

(4

1)(2 )

( )

n i i

h n

n

h

f v

− =

−

+

∑

2 1 2

( )

2 (4

1)

( )

2

1

2

1

n i i

f v

h n n

k f

h

n

− =

−

∴

=

+

−

∑

−

(3)

Hence

k f

( )

≥

(5

n

+

1)

h

which implies that

Em P

(

₂_n

)

≥

h n

(5

+

1)

∴

Em P

(

2n

)

=

h n

(5

+

1)

Similarly, we can prove that

Em P

(

₂_n₊₁

)

=

h n

(5

+

3)

Example 2.1: Fig. 2.1 Shows that

Em P

(

₉

)

=

46

Fig. 2.1

Theorem 2.2:

Em B

(

_{n n}_,

)

=

h n

(5

+

6)

Proof: Let

V B

(

_{n n}_,

) { , ; ,

=

u v u u

₁ ₂

,...,

u v v

_n

; ,

₁ ₂

,..., }

v

_n and

E B

(

_{n n}_,

) {

=

uu vv uv

_i

,

_i

,

:1

≤ ≤

i

n

}

Define a function

f V

:

∪ →

E

{ , 2 ,..., (2

h

n

+

1) }

h

by

( )

_i

f u

=

ih

for

1 ≤ ≤

i

n f u

, ( )

=

h n

(

+

2), ( )

f v

=

h n

(

+

1), ( )

f u v

=

h n

(3

+

3), ( )

f u u

i

=

h n

(4

+ −

4 i

)

for

1 ≤ ≤

i

n

and

f vv

(

_i

)

=

h n

(2

+ −

3 i

)

for

1 ≤ ≤

i

n

Clearly f is a E-magic h-multiple labeling of

B

_{n n}_, with

k f

( )

=

h n

(5

+

6)

and hence

Em B

(

_{n n}_,

)

≤

h n

(5

+

6)

Let f be a E-magic h-multiple labeling of

B

_{n n}_, with

k f

( )

Then

( )

( ) ( )

( )

v V e E

k f

d v f v

f e

ε

∈ ∈

=

∑

+

∑

1 1

( )

(

1) ( )

(

1) ( )

( )

n n i i

i i e E

h

f v

h

f u

h n

f u

h n

f v

h

f e

= = ∈

=

∑

+

∑

+

∑

(2

1) ( )

( )

v V e E

n

k f

h

f v

h

f e

nhf u

nhf v

∈ ∈

+

=

∑

+

∑

+

=

h

+

2 h

+ +

... (4

n

+

3)

h

+

nhf u

( )

+

nhf v

( )

=

(4

3)(4

4)

( ( )

( ))

2 h n

n

nh f u

f v

+

( )

(4

5)

( ( ( )

( )) 1)

2

1 h

k f

h n

n f u

f v

n

∴

=

+ +

+

Since

k f

( )

is an integer,

( ( )

( )) 1

2

1 n f u

f v

h

n

+







₊







is also an integer.

That is

n f u

( ( )

+

f v

( )) 1

+ ≡

0 mod(2

n

+

1)

And hence

n f u

( ( )

+

f v

( ))

≡

2 mod(2

n

+

1)

which implies that

f u

( )

+

f v

( )

≡

2 mod(2

n

+

1)

But

f u

( )

+

f v

( )

≥

3

and thus

f u

( )

+

f v

( )

≥

2 n

+

3 ( )

(4

5) [ (2

3) 1] / 2

1 k f

h

n

h

n

∴

≥

+ +

+

=

h n

(5

+

6)

(4)

Hence

Em B

(

_{n n}_,

)

=

h n

(5

+

6)

Example2.2: Fig. 2.2 Shows that

Em B

(

_5,5

)

=

155

Fig. 2.2

Theorem2.3:

Em

(

<

K

_1,_n

: 2 )

> =

h n

(4

+

9)

Proof: Define the labeling f on

<

K

_1,_n

: 2

>

as follows

( )

, ( )

2 , ( )

(

3)

f u

=

h f v

=

h f w

=

h n

+

( )

i

(

2)

f u

=

h i

+

for

1 ≤ ≤

i

n

,

( )

_i

(

3 )

f v

=

h n

+ +

i

for

1 ≤ ≤

i

n

,

(

)

(3

5), (

)

(3

4), (

_i

)

(4

6 )

f uw

=

h n

+

f vw

=

h n

+

f uu

=

h n

+ −

i

for

1 ≤ ≤

i

n

(

_i

)

(3

4 )

f vv

=

h n

+ −

i

for

1 ≤ ≤

i

n

One can check that the above labeling f is a E-magic h-multiple labeling of

<

K

_1,_n

: 2

>

with

k f

( )

=

h

(4

n

+

9)

Thus

Em

(

<

K

_1,_n

: 2 )

> ≤

h n

(4

+

9)

Now we show that

Em

(

<

K

_1,_n

: 2 )

> ≥

h n

(4

+

9)

Let f be a magic labeling of

<

K

_1,_n

: 2

>

with constant

k f

( )

Then

( )

( ) ( )

( )

v V e E

k f

d v f v

f e

ε

∈ ∈

=

∑

+

∑

That is,

1 1

(2

2) ( )

( )

(

1) ( )

(

1) ( ) 2 ( )

( )

n n i i

i i e E

n

k f

h u

h

f v

h n

f u

h n

f v

h w

h

f e

= = ∈

+

=

∑

+

∑

+

∑

(2

2 )

( )

v V e E

n

f

h

f v

h

f e

nhf u

nhf v

hf w

∈ ∈

+

=

∑

+

∑

+

=

h

+

2 h

+ +

... (4

n

+

5)

h

+

nhf u

( )

+

nhf v

( )

+

hf w

( )

=

(4

5)(4

6)

( ( )

( ))

( )

2 h n

n

nh f u

f v

hf w

+

₊

(4

5)(4

6)

( ( ( )

( ))

( )) / (2

2)

2(2

2)

h

n

h n f u

f v

f w

n

+

=

+

( )

(4

7)

( ( ( )

( ))

( ) 1)

2

2 h

k f

h n

n f u

f v

f w

n

∴

=

+ +

+

(5)

Since

k f

( )

is an integer and

f u

( )

+

f v

( )

≥

3

,

Em

(

<

K

_1,_n

: 2 )

> ≥

h n

(4

+

9)

Hence

Em

(

<

K

_1,_n

: 2 )

> =

h n

(4

+

9)

Em

(

<

K

_1,4

: 2 )

> =

50

Fig. 2.3

Definition 2.2: The graph

nP

₂ is defined as the disjoint union of n copies of

P

₂.

Theorem2.4:

Em

((2

n

+

1)

P

₂

)

=

h n

(9

+

6)

for any

n

≥

0,

n

is odd

Proof: Let the vertices of

(2

n

+

1)

P

₂ be

u u

₁

,

₂

,...,

u

₂_n₊₁

; ,

v v

₁ ₂

,...,

v

₂_n₊₁ and let the edge set be

{

u v

_{i i}

:1

≤ ≤

i

2 n

+

1}

Define

f V

:

∪ →

E

{ , 2 , 3 ,..., (6

h

h h

h

n

+

3)}

in such a way that

f u

( )

_i

=

ih

for

1 ≤ ≤

i

2 n

+

1 ( )

_i

(6

4 2 )

f v

=

h n

+ −

i

for

1 ≤ ≤

i

n

,

(

_{n i}

)

(6

5 2 )

f v

₊

=

h n

+ −

i

for

1 ≤ ≤ +

i

n

1

,

(

_{i i}

)

(3

2 )

f u v

=

h n

+ +

i

for

1 ≤ ≤

i

n

,

(

_{n i n i}

)

(2

1 )

f u v

₊ ₊

=

h n

+ +

i

for

1 ≤ ≤ +

i

n

1

It is easy to check that f is a E- magic h-multible labeling of

(2

n

+

1)

P

₂ with

k f

( )

=

h n

(9

+

6)

(2

n

+

1)

P

₂ with constant

k f

( )

Then

( )

( ) ( )

( )

v V e E

k f

d v f v

f e

ε

∈ ∈

=

∑

+

∑

(2

1) ( )

( )

v V e E

n

k f

h

f v

h

f e

∈ ∈

+

=

∑

+

∑

=

h

+

2 h

+ +

... (6

n

+

3)

h

=

(6

3)(6

4)

2 h n

+

n

+

(6

3)(6

4)

( )

(2

1)2

h n

n

k f

n

+

∴

=

+

=

h n

(9

+

6)

(6)

Em

(11 )

P

₂

=

255

Fig. 2.4

Theorem2.5:

Em C

(

₂_n₊₁

)

=

h n

(5

+

4)

C

₂_n₊₁ be

v v

_{1 2}

...

v

₂_n₊_{1 1}

v

Define

f V

:

∪ →

E

{ , 2 , 3 ,..., (5

h

h h

h n

+

4)}

in such a way that

f v

(

₂_i₊₁

)

=

h i

(

+

1)

for

0 ≤ ≤

i

n

2

(

_i

)

(

1 )

f v

=

h n

+ +

i

for

1 ≤ ≤

i

n

, 1

(

_{i i}

)

(4

2 )

f v v

₊

=

h n

+ −

i

for

1 ≤ ≤

i

2 n

, 2 1 1

(

_n

)

(4

2)

f v

₊

v

=

h n

+

It is easy to check that f is a E- magic h-multiple labeling of

C

₂_n₊₁ with

k f

( )

=

h n

(5

+

4)

Thus,

Em C

(

₂_n₊₁

)

≤

h n

(5

+

4)

C

₂_n₊₁ with constant

k f

( )

Then

( )

( ) ( )

( )

v V e E

k f

d v f v

f e

ε

∈ ∈

=

∑

+

∑

(2

1) ( )

2 ( )

( )

v V e E

n

k f

h

f v

h

f e

∈ ∈

+

=

∑

+

∑

( )

v V e E v V

h

f v

h

f e

h

f v

∈ ∈ ∈

=

∑

+

∑

+

∑

=

2 ... (4

2)

( )

v V

h

n

h

f v

∈

+

+ +

+

∑

(4

2)(4

3)

2 ... (2

1)

2 h n

n

h

n

h

+

≥

+ +

+

( )

(5

4)

k f

h n

∴

≥

+

2 1

(

_n

)

(5

4)

Em C

₊

≥

h n

+

(7)

Em C

(

₉

)

=

48

Fig 2.5

Theorem 2.6:

Em P

(

_n2

)

=

3 nh

P

_n2be

v v

₁

,

₂

,...,

v

_n

Define

f V

:

∪ →

E

{ , 2 , 3 ,..., 3

h

h h

nh

}

in such a way that

( )

_i

f v

=

ih

for

1 ≤ ≤

i

n

, 1

(

_{i i}

)

(3

(2

1))

f v v

₊

=

h n

−

i

+

for

1 ≤ ≤ −

i

n

1

, 2

(

_{i i}

)

(3

(2

2))

f v v

₊

=

h n

−

i

+

1 ≤ ≤ −

i

n

2

Thus,

Em P

(

_n2

)

≤

3 nh

But since

ε

(

P

_n2

)

=

(3

n

−

3) ,

h

we have

Em P

(

_n2

)

≥

3 nh

Hence

Em P

(

_n2

)

=

3 nh

Em P

(

₆2

)

=

90

Fig. 2.6

3. V- MAGIC STRENGTH OF H-MULTIPLE LABELING OF GRAPHS

Definition 3.1 [4]: A one-to-one map

f V

:

∪ →

E

{1, 2, 3,...,

m

+

n

}

is a vertex –magic total labeling of G if there is a constant k so that for every vertex u,

( )

k

v N u

w u

f u

f u v

k

∈

=

+

∑

=

. So the magic requirement is the

associated weight

w u

_k

( )

=

k

for all u. The fixed integer k is called the magic constant of

f

.

Definition 3.2 [4]: Let h be any positive integer. A one-to-one map

f

from

V

∪

E

of a graph G onto the integers

{

h h h

, 2 , 3 ,..., (

m

+

n h

)

}

is a V-Magic h-multiple labeling if there exist a constant k such that

( )

v N u

f u

f u v

k

∈

+

∑

=

. A graph which admits V-Magic h-multiple labeling then it is called V-Magic h-multiple

(8)

Lemma 3.1: [1] If a non-trivial graph G is vertex magic, then the magic constant k is given by

1 (

1)

2 p

q q

k

q

p

+

= +

+

Theorem3.1:

(

)

(5

3)

3

2

n

h n

k P

=

−

if

n

is

odd

and

n

≥

Proof: Let

{ ,

v v

₁ ₂

,..., }

v

n be the consecutive vertices and

{ ,

e e

1 2

,...,

e

n−1

}

be the consecutive edges of

P

n.

That is

E P

(

_n

)

=

{

e

_i

=

v v

_{i i}₊₁

/ 1

≤ ≤ −

i

n

1}

Define

f V

:

∪ →

E

{ , 2 , 3 ,..., (2

h

h h

h

n

−

1)}

as follows 1

( )

(2

1)

( )

(

2 )

2 (

)

2 ( )

(

)

2

i i

f v

h n

f v

h n

i

for

i

n

h n i

if

i

is

odd

and f e

i

h n

if

i

is

even

=

−

=

− +

≤ ≤

−





= 



₋



Let f be an V –Magic h-multiple labeling of a graph G with the magic number k. Then

f E

( )

=

{ , 2 , 3 ,...,

h

h h

hm

}

and

( )

(

)

v N u

f u

f uv

k for

all

u

V

∈

+

∑

=

∈

That is

( )

(

_n

)

( )

(

)

u V u V v N u

nk P

f u

f uv for

all

u

V

∈ ∈ ∈

=

∑

+

∑ ∑

∈

( ) 2

( )

(

1)

(

1)

(

1)

2

u V e E

f u

f e

hn n

∈ ∈

=

+

=

− +

+

−

∑

(

1)

(

)

(

1)

(

1)

2

n

h n

k P

h n

+

h n

∴

=

− +

+

−

(

1)

2 (

1)

2 (5

3)

3

2 h n

h n

if

n

is odd

and

n

+

=

− +

−

=

≥

Example 3.1: Fig. 3.1 Shows that

k P

(

₅

)

=

55

Fig 3.1

Remark 3.1: In [5], Bondy and Murty constructed an m-connected graph

H

_{m n}_, on n-vertices that has exactly {mn/2} edges. The structure of

H

_{m n}_, depends on the parities of m and n; there are three cases.

(9)

Case-2: m odd, n even. Let m=2r+1.Then

H

₂_r₊_1,_n is constructed by first drawing

H

_{2 ,}_{r n}and then adding edges joining vertex i to vertex i+(n/2) for

1 ≤ ≤

i

n

/ 2

Case-3: m odd, n odd. Let m=2r+1.Then

H

₂_r₊_1,_n is constructed by first drawing

H

_{2 ,}_{r n} and then adding edges joining vertex 0 to vertices(n-1)/2 and (n+1)/2 and vertex i to vertex i+(n+1)/2 for

1 ≤ ≤

i

(

n

−

1) / 2

Theorem 3.2:

(

_4,

)

(13

5)

2

n

h

n

k H

=

+

if

n

is

odd

Proof: Let the vertex set of

H

_4,_n

be

V

=

{ ,

v v

1 2

,..., }

v

n and the edge set of

H

4,n be

1 1 2 1 1 2

{

_{i i}

/ 1

1}

{

_n

}

{

_{i i}

/ 1

2}

{

_n

}

{

_n

}

E

=

v v

₊

≤ ≤ − ∪

i

n

v v

∪

v v

₊

≤ ≤ − ∪

i

n

v v

₋

∪

v v

Define

f V

:

∪ →

E

{ , 2 , 3 ,..., 3

h

h h

nh

}

as follows

(6

1 )

1(mod 2)

2 ( )

(5

1 )

0(mod 2)

2

i

h

n

i

for

i

f v

h

n

i

for

i



_{+ −}

_≡



= 



_{+ −}

_≡



1

(

2 )

1(mod 2),

2 (

)

(

2)

0(mod 2)

2

i i

h

n

i

for

i

n

f v v

h

i

for

i

+



_{+ +}

_≡

_≠



= 



₊

_≡



1

(

_n

)

f v v

=

h

2

1 2 (

)

1, 2,...,

2 (

)

1

3 (3

2 )

,

,...,

2

i i

n

h n i

for

i

f v v

n

h n

i

for

i

n

+

−



₋

₌



= 

₊



₋

₌

₋



1 1

(

_n

)

(

2)

f v v

₋

=

h n

+

2

(

_n

)

2 f v v

=

hn

Suppose there exists an vertex –magic h-multiple total labeling f of a graph G with the magic number k. Then by lemma [3.1],

(

1)

(2 )(2

1)

(2 )

2

1 [2

4 2]

2 (13

5)

2 h n

h n

n

k

h n

n

h n

n

(10)

k H

(

_4,7

)

=

96

Fig. 3.2

REFERENCES

1. G.Marimuthu and M.Balakrishnan, E-Super vertex magic labelings of graphs, Discrete Applied Mathematics 160(2012) 1766-1774.

2. D.G.Akka and Nanda S.Warad, Super Magic Strength of a graph, Indian J. Pure Appl. math., 41(4):557-568, August 2010.

3. Selvam Avadayappan, R.Vasuki and P.Jeyanthi, Magic Strength of a graph, Indian J. Pure appl.math., 31(7): 873-883, July 2000.

4. C.Y.Ponnappan, C.T.Nagaraj, C.Ranjitham, Vertex-Magic h-Multiple Labeling, International Journal of IT, Engineering and Applied Sciences Research (IJIEASR) Volume 3, No.11, November 2014.

5. J.A.Bondy and U.S.R.Murty, Graph Theory with applications, Macmillan, London, 1976.

6. J.Sedlacek, problem 27.in: Theory of graphs and its Applications.Proc.Symposium, 1963; pp.163-167.

Source of support: Nil, Conflict of interest: None Declared

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