Pelabelan selimut cycle-anti ajaib pada graf double cones, graf friendship dan graf grid Pn × P3 ngartikelM0109063

(1)

commit to user

Surya Aji Nugroho, Mania Roswitha, and Titin Sri Martini

Department of Mathematics

Faculty of Mathematics and Natural Sciences Sebelas Maret University

Abstract. A simple graph G = (V(G), E(G)) admits an H-covering if every edge in

E(G) belongs to a subgraph ofGthat is isomorphic toH and there is a bijective function

ξ:V(G)∪E(G)→ {1,2, . . . ,|V(G)|+|E(G)|}such that for all subgraphsH′ _isomorphic to H, the H-weights w(H′_{) =} ∑

v∈V₍H′₎ξ(v) +

∑

e∈E₍H′₎ξ(e) constitute an arithmetic

progressiona, a+d, a+ 2d, . . . , a+ (t−1)dwhereaanddare positive integers andtis the number of subgraphs ofGisomorphic toH. The labelingξis called a super (a, d)-H-anti magic total labeling, ifξ(V(G)) ={1,2, . . . ,|V(G)|}. The aim of this research is to study (a, d)-H-anti magic covering on double cones, friendship, and grid Pn×P3 with cycle.

Keywords: (a, d)-H-anti magic covering, double cones, friendship, grid

1.

Introduction

A labeling of a graph is a map that carries graph elements to positive or

non-negative integers (Wallis [8]). Dozens graph labelings techniques are studied recently.

Some of them are magic and anti magic labeling. Sedl´aˇcek introduced magic graphs

in 1964 (Gallian [2]). In 1970 Kotzig and Rosa [6] defined an edge-magic total

labeling of a graph

G

(

V, E

) as a bijection

f

from

V

∪

E

to

{

1

,

2

, . . . ,

|

V

∪

E

|}

such

that for all edges

xy

,

f

(

x

) +

f

(

y

) +

f

(

xy

) is constant. Guti´errez and Llad´o [3]

then developed the research into

H

-supermagic covering in 2005. An edge-covering

of

G

is a family of different subgraphs

H

1

, H

2

, . . . , H

k

such that any edge of

E

belongs to at least one of the subgraphs

H

i

,

1

≤

i

≤

k.

If every

H

i

is isomorphic

to a given graph

H

, then

G

admits an

H

-covering. Suppose that

G

admits an

H

-covering. A bijective function

f

:

V

(

G

)

∪

E

(

G

)

→ {

1

,

2

, . . . ,

|

V

(

G

)

|

+

|

E

(

G

)

|}

is

called an

H

-magic labeling of

G

if there exists a positive integer

c

such that for each

subgraph

H

′

isomorphic to

H

satisfies

f

(

H

′

) =

∑

v∈V(H′₎

f

(

v

) +

∑

e∈E(H′₎

f

(

e

) =

c

.

When

f

(

V

(

G

)) =

{

1

,

2

, . . . ,

|

V

(

G

)

|}

, it is said that

G

is

H

-supermagic. Llad´o and

Moragas [7] proved that a wheel

W

n

, a prism

C

n

×

K

2

, a book

K

1,n

×

K

2

, and

windmill

W

(

r, k

) are

C

h

-magic.

Hartsfield and Ringel [4] introduced anti magic labeling in 1990, followed by

Bodendiek and Walther [1] which defined (

a, d

)-anti magic labeling as follows. A

connected graph

G

(

V, E

) is said to be (

a, d

)-anti magic if there exist positive integers

a

,

d

, and a bijection

f

:

E

→ {

1

,

2

, . . . ,

|

E

|}

such that

g

_f

(

V

) =

{

a, a

+

d, . . . , a

+(

|

V

|−

1)

d

}

with

g

f

(

v

) =

∑

{

f

(

uv

)

|

uv

∈

E

(

G

)

}

. In [1] Bodendiek and Walther proved the

Herschel graph is not (

a, d

)-anti magic. In 2009, Inayah

et al.

[5] developed magic

coverings into a new labeling, namely (

a, d

)-

H

- anti magic total labeling. An (

a, d

)-H

-anti magic total labeling of a graph

G

is a bijective function

ξ

:

V

(

G

)

∪

E

(

G

)

→

{

1

,

2

, . . . ,

|

V

(

G

)

|

+

|

E

(

G

)

|}

such that for all subgraphs

H

′

isomorphic to

H

, the

H

-weights

w

(

H

′

) =

∑

v∈V(H′₎

ξ

(

v

) +

∑

e∈E(H′₎

ξ

(

e

) constitute an arithmetic progression

(2)

commit to user

admits (

a, d

)-cycle

C

n

-anti magic covering for some

d

. In Gallian [2], Susilowati

et

al.

proved that ladders

P

n

×

P

2

admits (

a, d

)-cycle-anti magic covering for some

d

.

This research aims to find (

a, d

)-

H

-anti magic covering on double cones

DC

n

,

friendship

D

n

k

, and grid

P

n

×

P

3

.

2.

Technique of Partitioning A Multiset

2.1.

k

-balance multiset

Let

k

∈

N

and

Y

be a multiset that contains positive integers.

Y

is said to

be

k

-balanced if there exists

k

subsets of

Y

, say

Y

₁

, Y

₂

, . . . , Y

k

, such that for every

i

∈

[1

, k

],

|

Y

i

|

=

|Y_k|

,

∑

Y

i

=

∑

Y

k

∈

N

, and

⊎

k

i=1

Y

i

=

Y

. If this is the case for every

i

∈

[1

, k

] then

Y

i

is called a balanced subset of

Y

.

Lemma 2.1.

Let

x

,

y

,

z

, and

r

be positive integers and

k

≥

3

is odd. Then the

multiset

Y

= [

x, x

+

k

−

1]

⊎

[

x

+ 1

, x

+

k

]

⊎

[

y

+ 1

, y

+

k

]

⊎

[

y, y

+

k

−

1]

⊎

[

z, z

+

k

−

1]

⊎

[

z

+ 1

, z

+

k

]

⊎

[

r, r

+

k

−

1]

is

k

-balanced.

Proof.

Let

x

,

y

,

z

, and

r

k

≥

3 is odd. For every

i

∈

[1

, k

],

define

Y

i

=

{

a

i

, b

i

, c

i

, d

i

, e

i

, f

i

, g

i

}

where

a

i

=

{

x

+

i−₂1

for

i

odd;

x

+

k−₂1

+

i

2

for

i

even;

e

i

=

{

z

+

i−₂1

for

i

odd;

z

+

k−₂1

+

i

2

for

i

even;

b

i

=

{

x

+

k+1

2

+

i−1

2

for

i

odd;

x

+

i

2

for

i

even;

f

i

=

{

z

+

k−1

2

+

i+1

2

for

i

odd;

z

+

i

2

for

i

even;

c

i

=

{

y

+

k

−

i−₂1

for

i

odd;

y

+

k+1

2

−

i

2

for

i

even;

g

i

=

r

+

k

−

i.

d

i

=

{

y

+

k−1

2

−

i−1

2

for

i

odd;

y

+

k

−

i

2

for

i

even;

Further, we define the sets

A

=

{

a

i

|

1

≤

i

≤

k

}

= [

x, x

+

k

−

1];

E

=

{

e

i

|

1

≤

i

≤

k

}

= [

z, z

+

k

−

1];

B

=

{

b

i

|

1

≤

i

≤

k

}

= [

x

+ 1

, x

+

k

];

F

=

{

f

i

|

1

≤

i

≤

k

}

= [

z

+ 1

, z

+

k

];

C

=

{

c

i

|

1

≤

i

≤

k

}

= [

y

+ 1

, y

+

k

];

G

=

{

g

i

|

1

≤

i

≤

k

}

= [

r, r

+

k

−

1]

.

D

=

{

d

i

|

1

≤

i

≤

k

}

= [

y, y

+

k

−

1];

Since

A

⊎

B

⊎

C

⊎

D

⊎

E

⊎

F

⊎

G

=

Y

, we have

⊎

k

i=1

Y

i

=

Y

. Since

|

Y

i

|

= 7 and

∑

Y

i

=

−

1₂

+

7₂k

+

r

+ 2

x

+ 2

y

+ 2

z

is constant, for every

i

∈

[1

, k

], we conclude that

Y

is

k

-balanced.

Lemma 2.2.

Let

x

and

y

k

≥

5

is odd. Then the multiset

(3)

commit to user

Proof.

Let

x

and

y

k

≥

5 is odd. For every

i

∈

[1

, k

], define

Y

i

=

{

a

i

, b

i

, c

i

}

where

a

i

=

x

+

i,

1

≤

i

≤

k

;

c

i

=

{

y

−

2

i

+

k,

1

≤

i

≤

k−1

2

;

y

−

2

i

+ 2

k,

k+1

2

≤

i

≤

k

.

b

i

=

{

x

+

i

+

k+1

2

,

1

≤

i

≤

k−1

2

;

x

+ 1 +

i

−

k+1

2

,

k+1

2

≤

i

≤

k

;

Further, we define the sets

A

=

{

a

i

|

1

≤

i

≤

k

}

= [

x, x

+

k

−

1];

C

=

{

c

i

|

1

≤

i

≤

k

}

= [

y

+ 1

, y

+

k

]

.

B

=

{

b

i

|

1

≤

i

≤

k

}

= [

x

+ 1

, x

+

k

];

Since

A

⊎

B

⊎

C

=

Y

, we have

⊎

k

i=1

Y

i

=

Y

. Since

|

Y

i

|

= 3 and

∑

Y

i

=

1₂

+

3₂k

+2

x

+

y

is constant, for every

i

∈

[1

, k

], we conclude that

Y

is

k

-balanced.

2.2. (

k, δ

)

-anti balance multiset

Let

k

∈

N

and let

X

be a multiset containing positive integers. Then

X

is

said to be (

k, δ

)-anti balanced if there exists

k

subset of

X

, say

X

1

, X

2

, X

3

, . . . , X

k

,

such that for every

i

∈

[1

, k

],

|

X

i

|

=

|X|_k

,

⊎

k_i₌₁

X

i

=

X

and for

i

∈

[1

, k

−

1],

∑

(

X

i+1

)

−

∑

(

X

i

) =

δ

is satisfied.

Lemma 2.3.

Let

k

≥

2

be an integer. If

X

=











[1

,

8

k

+ 5]

⊎

[2

, k

]

⊎

[

k

+ 3

,

2

k

+ 1]

⊎

[2

k

+ 4

,

3

k

+ 2]

⊎

[3

k

+ 5

,

4

k

+ 3]

⊎

[7

k

+ 5

,

8

k

+ 5]

⊎

[7

k

+ 6

,

8

k

+ 4]

⊎

[7

k

+ 6

,

8

k

+ 4]

⊎ {

4

k

+ 5

,

4

k

+ 8

, . . . ,

4

k

+ 5 + 3(

k

−

1)

}

⊎{

4

k

+ 7

,

4

k

+ 10

, . . . ,

4

k

+ 7 + 3(

k

−

1)

}

,

if

j

= 0

;

[1

,

3

k

]

⊎

[4

,

3

k

+ 3]

⊎

[3

k

+ 4

,

6

k

+ 3]

⊎

[6

k

+ 4

,

8

k

+ 3]

⊎

[6

k

+ 6

,

8

k

+ 5]

⊎ {

2

,

5

,

8

, . . . ,

3

k

+ 2

} ⊎ {

2

,

5

,

8

, . . . ,

3

k

+ 2

}

,

if

j

= 1

.

then

X

is

(2

k,

2

j

+ 2)

-anti balanced for

j

= 0

,

1

.

Proof.

For

i

∈

[1

, k

] define

X

_ij

=

{

6

m

+

i,

6

m

+

i

+1

,

4

k

−

6

m

−

i

+5

,

4

k

−

6

m

−

i

+4

,

4

k

+

2

m

+3

i

+2

,

8

k

−

i

+6

,

8

k

−

i

+5

,

4

k

+3

i

+3

}

, with

m

= 0

,

1 for

j

= 0 and

X

_ij

=

{

2

m

+

3

i

−

2

,

2

m

+3

i

+1

,

8

k

−

m

−

2

i

+7

,

8

k

−

m

−

2

i

+5

,

3

i

−

1

,

3

i

+2

,

6

k

−

3

i

+5

,

3

k

+2

m

+3

i

+1

}

,

with

m

= 0

,

1 for

j

= 1. It is easy to verify that each

i

∈

[1

, k

],

|

X

_ij

|

= 8,

X

i

⊂

X

,

and

⊎

k

i=1

X

i

=

X

. Since

∑

(

X

_ij

) = 2

m

+ 32

k

+ 4

i

+ 26 +

j

(2

m

−

7

k

+ 4

i

−

8) for every

i

∈

[1

, k

] and

∑

(

X

i+1

)

−

∑

(

X

i

) = 2

j

+ 2 for every

i

∈

[1

, k

],

X

is (2

k,

2

j

+ 2)-anti

balanced for

j

= 0

,

1.

Lemma 2.4.

Let

x

,

t

, and

k

≥

2

t

is odd. If

X

= [

x

+

1

, x

+

tk

]

then

X

is

(

k, t

)

-anti balanced.

Proof.

For

i

∈

[1

, k

] define

X

i

=

{

x

+

i

+

jk

}

with

j

= 0

,

1

, . . . ,

(

t

−

1). It is easy

to verify that for every

i

∈

[1

, k

],

|

X

i

|

=

t

,

X

i

⊂

X

, and

⊎

k

(4)

commit to user

∑

(

X

i

) =

nt 2

2

−

nt

2

+

tx

+

ti

for every

i

∈

[1

, k

] and

∑

(

X

i+1

)

−

∑

(

X

i

) =

t

for every

i

∈

[1

, k

],

X

is (

k, t

)-anti balanced.

Lemma 2.5.

Let

x

,

j

,

t

, and

k

≥

2

t

is odd. If

X

=

[

x

+ 1

, x

+

tk

]

then

X

is

(

k, j

2

−

j

+

t

)

-anti balanced.

Proof.

For

i

∈

[1

, k

] define

X

i

=

{

x

+

j

(

i

−

1) + 1

, x

+

j

(

i

−

1) + 2

, x

+

j

(

i

−

1) +

3

, . . . , x

+

j

(

i

−

1) +

j, x

+

i

+

jk, x

+

i

+ (

j

+ 1)

k, . . . , x

+

i

+ (

t

−

1)

k

}

. It is easy

to verify that for every

i

∈

[1

, k

],

|

X

i

|

=

t

,

X

i

⊂

X

, and

⊎

k_i₌₁

X

i

=

X

. Since

∑

(

X

i

) =

₂j

−

ij

−

j 2

2

+

ij

2

+

jk

2

−

j2_k

2

+

it

−

kt

2

+

kt2

2

+

tx

for every

i

∈

[1

, k

] and

∑

(

X

i+1

)

−

∑

(

X

i

) =

j

2

−

j

+

t

for every

i

∈

[1

, k

],

X

is (

k, j

2

−

j

+

t

)-anti balanced.

Lemma 2.6.

Let

x

,

t

, and

k

≥

2

t

is odd. If

X

= [

x

+

1

, x

+

tk

]

then

X

is

(

k, t

2

)

-anti balanced.

Proof.

For

i

∈

[1

, k

] define

X

i

=

{

x

+ (

i

−

1)

t

+ 1

, x

+ (

i

−

1)

t

+ 2

, x

+ (

i

−

1)

t

+

3

, . . . , x

+ (

i

−

1)

t

+

t

}

. It is easy to verify that for every

i

∈

[1

, k

],

|

X

i

|

=

t

,

X

_i

⊂

X

, and

⊎

k

i=1

X

i

=

X

. Since

∑

(

X

_i

) =

t

2

−

t2

2

+

t

2

i

+

xt

for every

i

∈

[1

, k

] and

∑

(

X

i+1

)

−

∑

(

X

i

) =

t

2

for every

i

∈

[1

, k

],

X

is (

k, t

2

)-anti balanced.

Lemma 2.7.

Let

n

≥

5

and

k

be positive integers,with

k

= 2

n

and

n

is odd. If

X

=

⊎

n

1

{

1

,

2

} ⊎

[

k

+ 3

,

2

k

+ 2]

⊎

[

k

+ 3

,

2

k

+ 2]

then

X

is

(

k,

1)

-anti balanced.

Proof.

For

i

∈

[1

, k

] define

X

i

=











{

1

2

(3 +

i

+ 6

n

)

,

1

2

(5 +

i

+ 4

n

)

,

2

}

,

for 1

≡

i

(mod 4);

{

1₂

(4 +

i

+ 6

n

)

,

1₂

(6 +

i

+ 4

n

)

,

1

}

,

for 2

≡

i

(mod 4);

{

1₂

(3 +

i

+ 4

n

)

,

1₂

(5 +

i

+ 6

n

)

,

2

}

,

for 3

≡

i

(mod 4);

{

1

2

(4 +

i

+ 4

n

)

,

1

2

(6 +

i

+ 6

n

)

,

1

}

,

for 0

≡

i

(mod 4).

It is easy to verify that each

i

∈

[1

, k

],

|

X

i

|

= 3,

X

i

⊂

X

, and

⊎

k_i₌₁

X

i

=

X

,with

k

=

2

n

,

n

≥

5 is odd. Since

∑

(

X

i

) = 6+

i

+5

n

for every

i

∈

[1

, k

],

∑

(

X

i

+1)

−

∑

(

X

i

) = 1

for every

i

∈

[1

, k

],

X

is (

k,

1)-anti balanced.

3.

Main results

3.1. (

a, d

)

-

C

₃

-anti magic coverings on double cones

Double cones is defined by

DC

n

=

C

n

+

K

2

, for

n

≥

3.

G

∼

=

DC

n

has

|

V

(

DC

n

)

|

=

n

+ 2 and

|

E

(

DC

n

)

|

= 3

n

. We derive an upper bound of the difference

d

for

DC

n

to be (

a, d

) -

C

3

-anti magic covering.

Theorem 3.1.

If

G

is

(

a, d

)

-

H

-anti magic then

d

≤

24n−24

2n−1

.

Proof.

Let

t

be a number of subgraphs of

DC

n

isomorphic to

C

3

, say

H

i′

, with

t

= 2

n

.

(5)

commit to user

1)+(4

n

+2

−

6+2)+(4

n

+2

−

6+3)+(4

n

+2

−

6+4)+(4

n

+2

−

6+5)+(4

n

+2) = 24

n

−

3

and the least possible

H

_i′

-weight is 1 + 2 + 3 + 4 + 5 + 6 = 21.

(2

n

−

1)

d

≤

(24

n

−

3)

−

21

d

≤

24

n

−

24

2

n

−

1

Theorem 3.2.

Let

n

≥

5

and

n

be positive integers. Graph

DC

n

is

(14 + 7

n

+

(n+1)

2

,

1)

-

C

3

-anti magic.

Proof.

We define a bijective function

ξ

:

V

(

DC

n

)

∪

E

(

DC

n

)

→ {

1

,

2

, . . . ,

4

n

+ 2

}

.

Let

P

be the set of label used to label vertices and edges of subgraph of

DC

n

which

isomorphic to

C

_n

as follows. The vertices are labeled using integers on interval

[

x

+ 1

, x

+

n

] and the edges are labeled using integers on interval [

y, y

+

n

−

1].

According to Lemma 2.2 with

n

=

k

,

x

= 2 and

y

=

n

+ 3,

P

is

k

-balanced. Vertices

u

₁

and

u

₂

are labeled using 1 and 2. Let

Q

be the set of label of the rest. Edges

u

₁

vi

and

u

2

vi

are labeled using integers on interval [2

n

+ 3

,

4

n

+ 2] such that every

H

i′

-weight satisfies Lemma 2.7.

Q

is (2

n,

1)-anti balanced. It is easy to verify that

ξ

is

a bijective function from

V

(

DC

n

)

∪

E

(

DC

n

) to

{

1

,

2

,

3

, . . . ,

4

n

+ 2

}

. For 1

≤

i

≤

2

n

,

w

(

H

_i′

)











13 +

i

+ 7

n

+

(n+1)₂

,

for 1

≡

i

(mod 4);

13 +

i

+ 7

n

+

(n+1)₂

,

for 2

≡

i

(mod 4);

11 +

(1−i₂ )

+

(3−i₂ )

+ 2

i

+

(1−n₂ )

+ 8

n,

for 3

≡

i

(mod 4);

10 +

(2−i₂ )

+

(4−i₂ )

+ 2

i

+

(1−n₂ )

+ 8

n,

for 0

≡

i

(mod 4).

Since 1

≤

i

≤

2

n

,

w

(

H

′

i+1

)

−

w

(

H

i′

) = 1 and

w

(

H

1′

) = 14 + 7

n

+

(n+1)

2

is the least

weight, then

DC

n

is (14 + 7

n

+

( n+1)

2

,

1)-

C

3

-anti magic.

3.2. (

a, d

)

-

C

k

-anti magic coverings on friendship

Friendship is a graph consisting of

n

cycles with a common vertex. Let

H

be

C

k

with

|

V

(

C

k

)

|

=

k

and

|

E

(

C

k

)

|

=

k

.

G

∼

=

D

nk

has

|

V

(

D

n

k

)

|

= 1 +

n

(

k

−

1) and

|

E

(

D

n

k

)

|

=

nk

. We derive an upper bound of the difference

d

for

D

nk

to be (

a, d

)

-C

k

Theorem 3.3.

If

G

is

(

a, d

)

-

H

-anti magic then

d

≤

2

k

(2

k

−

1)

.

Proof.

Let

t

D

n

k

isomorphic to

C

k

, say

H

i′

, with

t

=

n

.

Since

D

n

k

is (

a, d

)-

C

k

-anti magic, the maximum possible

H

i′

-weight is (2

kn

−

n

+ 1

−

k

−

k

+ 1) + (2

kn

−

n

+ 1

−

k

−

k

+ 2) + (2

kn

−

n

+ 1

−

k

−

k

+ 3) +

. . .

+ (2

kn

−

(6)

commit to user

is 1 + 2 +

. . .

+ 2

k

=

k

(2

k

+ 1).

(

n

−

1)

d

≤

k

(3

−

2

k

−

2

n

+ 4

kn

)

−

k

(2

k

+ 1)

d

≤

2

k

(2

k

−

1)(

n

−

1)

n

−

1

= 2

k

(2

k

−

1)

Theorem 3.4.

Let

n

≥

2

,

j

and

k

be positive integers. Graph

D

n

k

is

(

a, d

)

-

C

k

-anti

magic.

Proof.

We define a bijective function

ξ

:

V

(

D

n_k

)

∪

E

(

D

_kn

)

→ {

1

,

2

, . . . ,

1 +

n

(2

k

−

1)

}

.

Then we define 3 cases of labelings.

1. Case

d

= 2

k

−

1

The center vertex is labeled with 1. Use the set

X

= [2

, n

(2

k

−

1) + 1] to label the

rest of the vertices and edges such that every

H

_i′

-weight satisfies Lemma 2.4 with

k

=

n

,

x

= 1, and

t

= 2

k

−

1.

X

is (

n,

2

k

−

1)-anti balanced. It is easy to verify

that

ξ

is a bijective function from

V

(

D

n

k

)

∪

E

(

D

kn

) to

{

1

,

2

,

3

, . . . ,

1 +

n

(2

k

−

1)

}

. For

1

≤

i

≤

n

,

w

(

H

_i′

) = 1+(2

k

−

1)(1+

i

+(

k

−

1)

n

). Since

w

(

H

_i′₊₁

)

−

w

(

H

_i′

) = 2

k

−

1 and

w

(

H

₁′

) = (2

k

−

1)(2 + (

k

−

1)

n

), then

D

n

k

is (1 + (2

k

−

1)(2 + (

k

−

1)

n

)

,

2

k

−

1)-

C

k

-anti

magic.

2. Case

d

=

j

2

−

j

+ 2

k

−

1

We label the center vertex with 1. The set

X

= [2

, n

(2

k

−

1) + 1] is used to label

the rest of the vertices and edges such that every

H

_i′

-weight satisfies Lemma 2.5

with

k

=

n

,

x

= 1,

t

= 2

k

−

1, and 1

≤

j

≤

2

k

−

1.

X

is (

n, j

2

−

j

+ 2

k

−

1)-anti

balanced. It is easy to check that

ξ

is a bijective function from

V

(

D

n

k

)

∪

E

(

D

nk

) to

{

1

,

2

,

3

, . . . ,

1 +

n

(2

k

−

1)

}

. For 1

≤

i

≤

n

,

w

(

H

_i′

) =

1

2

(

j

−

j

2

−

2 + 2

i

(+

j

2

−

j

−

1) +

k

(4 + 4

i

−

6

n

) + 2

n

+ 4

k

2

n

+

jn

−

j

2

n

) + 1. Since

w

(

H

′

i+1

)

−

w

(

H

i′

) =

j

2

−

j

+ 2

k

−

1

and

w

(

H

₁′

) =

₂1

(

j

2

−

j

+ 8

k

−

4 + (2

k

−

j

−

1)(2

k

+

j

)

n

−

2) + 1, then

D

n_k

is

(

1

2

(

j

2

−

j

+ 8

k

−

4 + (2

k

−

j

−

1)(2

k

+

j

)

n

−

2) + 1

, j

2

−

j

+ 2

k

−

1)-

C

k

-anti magic.

3. Case

d

= (1

−

2

k

)

2

Put 1 as the label of the center vertex. Use the set

X

= [2

, n

(2

k

−

1) + 1] to label the

rest of the vertices and edges such that every

H

_i′

-weight satisfies Lemma 2.6 with

k

=

n

,

x

= 1, and

t

= 2

k

−

1.

X

is (

n,

(2

k

−

1)

2

)-anti balanced. It is easy to verify

that

ξ

is a bijective function from

V

(

D

n

k

)

∪

E

(

D

kn

) to

{

1

,

2

,

3

, . . . ,

1 +

n

(2

k

−

1)

}

. For

1

≤

i

≤

n

,

w

(

H

_i′

) = (2

k

−

1)(2

−

i

+

k

(2

i

−

1))+1. Since

w

(

H

_i′₊₁

)

−

w

(

H

_i′

) = (1

−

2

k

)

2

and

w

(

H

₁′

) = 2

k

2

+

k

, then

D

n

k

is (2

k

2

+

k,

(1

−

2

k

)

2

)-

C

k

-anti magic.

3.3. (

a, d

)

-

C

4

-anti magic coverings on grid

(7)

commit to user

set is

V

(

G

)

×

V

(

H

) and whose edge set is the set of all pairs (

u

₁

v

₁

)(

u

₂

v

₂

) such that

either

u

1

u

2

∈

E

(

G

) and

v

1

=

v

2

, or

v

1

v

2

∈

E

(

H

) and

u

1

=

u

2

. A graph grid is

defined by

P

n

×

P

m

. Let

m

= 3. Let

H

be

C

4

with

|

V

(

C

4

)

|

= 4 and

|

E

(

C

4

)

|

= 4.

G

∼

=

P

n

×

P

3

has

|

V

(

P

n

×

P

3

)

|

= 3

n

and

|

E

(

P

n

×

P

3

)

|

= 5

n

−

3. We derive an upper

bound of the difference

d

for

P

n

×

P

3

to be (

a, d

)-

C

4

Theorem 3.5.

If

G

is

(

a, d

)

-

H

-anti magic then

d

≤

64₂n−_n−88₂

.

Proof.

Let

t

P

n

×

P

3

isomorphic to

C

4

, say

H

i′

, with

t

= 2(

n

−

1). Since

P

n

×

P

3

is (

a, d

)-

C

4

-anti magic, the maximum possible

H

i′

-weight

is (8

n

−

3

−

8 + 1) + (8

n

−

3

−

8 + 2) + (8

n

−

3

−

8 + 3) + (8

n

−

3

−

8 + 4) + (8

n

−

3

−

8 + 5) + (8

n

−

3

−

8 + 6) + (8

n

−

3

−

8 + 7) + (8

n

−

3) = 64

n

−

52 and the least

possible

H

_i′

-weight is 1 + 2 +

. . .

+ 8 = 36.

(2(

n

−

1)

−

1)

d

≤

64

n

−

52

−

36

d

≤

64

n

−

88

2

n

−

3

Theorem 3.6.

Let

n

≥

4

be positive integer and

n

even. Graph

P

n

×

P

3

is

(3+

57₂n

,

1)

-C

₄

-anti magic.

Proof.

ξ

:

V

(

P

n

×

P

3

)

∪

E

(

P

n

×

P

3

)

→ {

1

,

2

, . . . ,

8

n

−

3

}

.

Let

R

be the set of labels with

R

= [1

,

8

n

−

3]. Partition

R

into 7 sets,

A

= [1

, n

],

B

= [

n

+1

,

2

n

],

C

= [2

n

+1

,

3

n

],

D

= [3

n

+1

,

4

n

],

E

= [4

n

+1

,

5

n

],

F

= [5

n

+1

,

6

n

−

1],

and

G

= [6

n,

8

n

−

3]. The set

A

is used to label

u

i

,

B

for

w

i

,

C

for

u

i

v

i

,

D

for

w

i

v

i

,

E

for

v

i

, and

F

for

v

i

v

i+1

such that every

H

i′

-weight satisfies the Lemma 2.1 with

k

=

n

−

1,

z

= 4

n

+ 1, and

r

= 5

n

+ 1. For

H

_i′

which contains vertices

u

i

,

x

= 1,

y

= 3

n

+ 1 and for

H

_i′

which contains vertices

w

i

,

x

=

n

+ 1,

y

= 2

n

+ 1. Then

G

is

used to label

u

i

u

i+1

and

w

i

w

i+1

.

R

is (2(

n

−

1)

,

that

ξ

V

(

P

n

×

P

3

)

∪

E

(

P

n

×

P

3

) to

{

1

,

2

,

3

, . . . ,

8

n

−

3

}

.

Hence, we have

w

(

H

_i′

) = 2+

i

+

57n

2

. Since

w

(

H

′

i+1

)

−

w

(

H

i′

) = 1 and

w

(

H

1′

) = 3+

572n

,

then

P

n

×

P

3

is (3 +

57₂n

,

1)-

C

4

-anti magic.

Theorem 3.7.

Let

n

≥

4

be positive integer and

n

even. Graph

P

n

×

P

3

is

(30 +

32(

n

−

1)

,

2)

-

C

₄

-anti magic.

Proof.

ξ

:

V

(

P

n

×

P

3

)

∪

E

(

P

n

×

P

3

)

→ {

1

,

2

, . . . ,

8

n

−

3

}

.

Let

L

i

=

H

i′

. Let

R

be the set of label with

R

= [1

,

8

n

−

3]. Partition

R

into 6 sets,

A

= [1

, n

],

B

= [

n

+1

,

2

n

],

C

= [2

n

+1

,

3

n

],

D

= [3

n

+1

,

4

n

],

E

= [4

n

+1

,

7

n

−

3], and

F

= [7

n

−

2

,

8

n

−

3]. The set

A

is used to label

u

i

,

B

for

w

i

,

C

for

v

i

w

i

,

D

for

u

i

v

i

,

E

for

u

i

u

i+1

,

v

i

v

i+1

,

w

i

w

i+1

, and

F

for

v

i

such that every

L

i

-weight satisfies the Lemma

(8)

commit to user

which contain vertices

w

i

,

m

= 1.

R

is (2(

n

−

1)

,

that

ξ

V

(

P

n

×

P

3

)

∪

E

(

P

n

×

P

3

) to

{

1

,

2

,

3

, . . . ,

8

n

−

3

}

.

Hence, we have

w

(

L

m

i

) = 2(13 + 2

i

+ 16(

n

−

1) +

m

). Since

w

(

L

0i+1

)

−

w

(

L

1i

) = 2

and

w

(

L

1

i

)

−

w

(

L

0i

) = 2 constitute an arithmatic progression

L

10

, L

11

, L

02

, . . . , L

1n−1

,

and

w

(

L

0₁

) = 30 + 32(

n

−

1) then

P

n

×

P

3

is (30 + 32(

n

−

1)

,

2)-

C

4

-anti magic.

Theorem 3.8.

Let

n

≥

4

be positive integer and

n

even. Graf

P

n

×

P

3

is

(26 +

25(

n

−

1)

,

4)

-

C

₄

-anti magic.

Proof.

ξ

:

V

(

P

n

×

P

3

)

∪

E

(

P

n

×

P

3

)

→ {

1

,

2

, . . . ,

8

n

−

3

}

.

Let

L

i

=

H

i′

. Let

R

be the set of label with

R

= [1

,

8

n

−

3]. Partition

R

into 3 sets,

A

= [1

,

3

n

],

B

= [3

n

+ 1

,

6

n

−

3], and

C

= [6

n

−

2

,

8

n

−

3]. The set

A

is used to

label all the vertices of grid

P

n

×

P

3

, the set

B

is used to label

u

i

u

i+1

,

v

i

v

i+1

, and

w

i

w

i+1

,

C

for

u

i

v

i

and

v

i

w

i

such that every

L

i

-weight satisfies the Lemma 2.3 with

k

=

n

−

1 and

j

= 1. For

L

i

which contains vertices

u

i

,

m

= 0 and for

L

i

which

contains vertices

w

i

,

m

= 1.

R

is (2(

n

−

1)

,

that

ξ

V

(

P

n

×

P

3

)

∪

E

(

P

n

×

P

3

) to

{

1

,

2

,

3

, . . . ,

8

n

−

3

}

.

Hence, we have

w

(

L

m

i

) = 18 + 8

i

+ 25(

n

−

1) + 4

m

. Since

w

(

L

0i+1

)

−

w

(

L

1i

) = 4 and

w

(

L

1_i

)

−

w

(

L

0_i

) = 4 constitute an arithmatic progression

L

0₁

, L

1₁

, L

0₂

, . . . , L

1_n−₁

, and

w

(

L

0

1

) = 26 + 25(

n

−

1) then

P

n

×

P

3

is (26 + 25(

n

−

1)

,

4)-

C

4

-anti magic.

4.

Conclusion

In this section, we conclude that a double cones is (

a, d

)-

C

3

-anti magic with

d

= 1

for

n

≥

5, a friendship is (

a, d

)-

C

k

-anti magic with

d

=

{

2

k

−

1

, j

2

−

j

+ 2

k

−

1

,

(1

−

2

k

)

2

}

for

n

≥

2 and

k

are positive integers, and a grid

P

n

×

P

3

is (

a, d

)-

C

4

-anti magic

with

d

=

{

1

,

2

,

4

}

for

n

≥

4 is positive integers.

References

[1] Bodendiek, R. and Walther, G. , Arithmetisch Antimagische Graphen, Graphentheorie III,

Mannhein, 1993.

[2] Gallian, J. A. , A Dynamics Survey of Graph Labeling, The Electronic Journal Combinatorics

16_{(2013), #DS6.}

[3] Guti´errez, and Llad´o,Magic Coverings, J. Combin. Math. Combin. Comput55_{(2005), 43-46.}

[4] Hartsfield, N. and Ringel, G. , Pearls in Graph Theory, Academy Press, Boston, San Diego,

New York, London, 1990.

[5] Inayah, A. N. M., Salman, and R. Simanjuntak, On (a, d)-H-Antimagic Coverings of Graphs,

J. Combin Math. Combin. Comput71_{(2009), 1662-1680.}

[6] Kotzig, A. and A. Rosa, Magic Valuations of Finite Graphs, Canad. Math. Bull 13 _(1970),

451-461.

[7] Llad´o, A. and J. Moragas,Cycle Magic Graph, Discrete Mathematics307_{(2007), 2925-2933.}