commit to user
Surya Aji Nugroho, Mania Roswitha, and Titin Sri Martini
Department of MathematicsFaculty 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
ksuch that any edge of
E
belongs to at least one of the subgraphs
H
i,
1
≤
i
≤
k.
If every
H
iis 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
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
2admits (
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
nk
, 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|
=
|Yk|,
∑
Y
i=
∑
Y
k
∈
N
, and
⊎
ki=1
Y
i=
Y
. If this is the case for every
i
∈
[1
, k
] then
Y
iis 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
be positive integers and
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−21for
i
odd;
x
+
k−21+
i2
for
i
even;
e
i=
{
z
+
i−21for
i
odd;
z
+
k−21+
i2
for
i
even;
b
i=
{
x
+
k+12
+
i−1
2
for
i
odd;
x
+
i2
for
i
even;
f
i=
{
z
+
k−12
+
i+1
2
for
i
odd;
z
+
i2
for
i
even;
c
i=
{
y
+
k
−
i−21for
i
odd;
y
+
k+12
−
i
2
for
i
even;
g
i=
r
+
k
−
i.
d
i=
{
y
+
k−12
−
i−1
2
for
i
odd;
y
+
k
−
i2
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
⊎
ki=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
be positive integers and
k
≥
5
is odd. Then the multiset
commit to user
Proof.
Let
x
and
y
be positive integers and
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−12
;
y
−
2
i
+ 2
k,
k+12
≤
i
≤
k
.
b
i=
{
x
+
i
+
k+12
,
1
≤
i
≤
k−1
2
;
x
+ 1 +
i
−
k+12
,
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
⊎
ki=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,
⊎
ki=1X
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
⊎
ki=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
be positive integers and
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
⊎
kcommit to user
∑
(
X
i) =
nt 22
−
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
be positive integers and
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
⊎
ki=1X
i=
X
. Since
∑
(
X
i) =
2j−
ij
−
j 22
+
ij
2
+
jk2
−
j2k
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
be positive integers and
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
⊎
ki=1
X
i=
X
. Since
∑
(
X
i) =
t2
−
t2
2
+
t
2i
+
xt
for every
i
∈
[1
, k
] and
∑
(
X
i+1)
−
∑
(
X
i) =
t
2for 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
=
⊎
n1
{
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=
{
12
(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);
{
12
(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
⊎
ki=1X
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
nhas
|
V
(
DC
n)
|
=
n
+ 2 and
|
E
(
DC
n)
|
= 3
n
. We derive an upper bound of the difference
d
for
DC
nto be (
a, d
) -
C
3-anti magic covering.
Theorem 3.1.
If
G
is
(
a, d
)
-
H
-anti magic then
d
≤
24n−242n−1
.
Proof.
Let
t
be a number of subgraphs of
DC
nisomorphic to
C
3, say
H
i′, with
t
= 2
n
.
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
nis
(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
nwhich
isomorphic to
C
nas 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
1and
u
2are labeled using 1 and 2. Let
Q
be the set of label of the rest. Edges
u
1vi
and
u
2vi
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
nis (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
kwith
|
V
(
C
k)
|
=
k
and
|
E
(
C
k)
|
=
k
.
G
∼
=
D
nkhas
|
V
(
D
n
k
)
|
= 1 +
n
(
k
−
1) and
|
E
(
D
nk
)
|
=
nk
. We derive an upper bound of the difference
d
for
D
nkto be (
a, d
)
-C
k-anti magic covering.
Theorem 3.3.
If
G
is
(
a, d
)
-
H
-anti magic then
d
≤
2
k
(2
k
−
1)
.
Proof.
Let
t
be a number of subgraphs of
D
nk
isomorphic to
C
k, say
H
i′, with
t
=
n
.
Since
D
nk
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
−
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
nk
is
(
a, d
)
-
C
k-anti
magic.
Proof.
We define a bijective function
ξ
:
V
(
D
nk)
∪
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
nk
)
∪
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
nk
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
nk
)
∪
E
(
D
nk) to
{
1
,
2
,
3
, . . . ,
1 +
n
(2
k
−
1)
}
. For 1
≤
i
≤
n
,
w
(
H
i′) =
12
(
j
−
j
2
−
2 + 2
i
(+
j
2−
j
−
1) +
k
(4 + 4
i
−
6
n
) + 2
n
+ 4
k
2n
+
jn
−
j
2n
) + 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
nkis
(
12
(
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
)
2Put 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
nk
)
∪
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
)
2and
w
(
H
1′) = 2
k
2+
k
, then
D
nk
is (2
k
2+
k,
(1
−
2
k
)
2)-
C
k-anti magic.
3.3. (
a, d
)
-
C
4-anti magic coverings on grid
commit to user
set is
V
(
G
)
×
V
(
H
) and whose edge set is the set of all pairs (
u
1v
1)(
u
2v
2) such that
either
u
1u
2∈
E
(
G
) and
v
1=
v
2, or
v
1v
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
4with
|
V
(
C
4)
|
= 4 and
|
E
(
C
4)
|
= 4.
G
∼
=
P
n×
P
3has
|
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
3to be (
a, d
)-
C
4-anti magic covering.
Theorem 3.5.
If
G
is
(
a, d
)
-
H
-anti magic then
d
≤
642n−n−882.
Proof.
Let
t
be a number of subgraphs of
P
n×
P
3isomorphic to
C
4, say
H
i′, with
t
= 2(
n
−
1). Since
P
n×
P
3is (
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
3is
(3+
572n,
1)
-C
4-anti magic.
Proof.
We define a bijective function
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
iv
i,
D
for
w
iv
i,
E
for
v
i, and
F
for
v
iv
i+1such 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
iu
i+1and
w
iw
i+1.
R
is (2(
n
−
1)
,
1)-anti balanced. It is easy to verify
that
ξ
is a bijective function from
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
+
57n2
. Since
w
(
H
′i+1
)
−
w
(
H
i′) = 1 and
w
(
H
1′) = 3+
572n,
then
P
n×
P
3is (3 +
572n,
1)-
C
4-anti magic.
Theorem 3.7.
Let
n
≥
4
be positive integer and
n
even. Graph
P
n×
P
3is
(30 +
32(
n
−
1)
,
2)
-
C
4-anti magic.
Proof.
We define a bijective function
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
iw
i,
D
for
u
iv
i,
E
for
u
iu
i+1,
v
iv
i+1,
w
iw
i+1, and
F
for
v
isuch that every
L
i-weight satisfies the Lemma
commit to user
which contain vertices
w
i,
m
= 1.
R
is (2(
n
−
1)
,
2)-anti balanced. It is easy to verify
that
ξ
is a bijective function from
V
(
P
n×
P
3)
∪
E
(
P
n×
P
3) to
{
1
,
2
,
3
, . . . ,
8
n
−
3
}
.
Hence, we have
w
(
L
mi
) = 2(13 + 2
i
+ 16(
n
−
1) +
m
). Since
w
(
L
0i+1)
−
w
(
L
1i) = 2
and
w
(
L
1i
)
−
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
3is (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
3is
(26 +
25(
n
−
1)
,
4)
-
C
4-anti magic.
Proof.
We define a bijective function
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
iu
i+1,
v
iv
i+1, and
w
iw
i+1,
C
for
u
iv
iand
v
iw
isuch that every
L
i-weight satisfies the Lemma 2.3 with
k
=
n
−
1 and
j
= 1. For
L
iwhich contains vertices
u
i,
m
= 0 and for
L
iwhich
contains vertices
w
i,
m
= 1.
R
is (2(
n
−
1)
,
4)-anti balanced. It is easy to verify
that
ξ
is a bijective function from
V
(
P
n×
P
3)
∪
E
(
P
n×
P
3) to
{
1
,
2
,
3
, . . . ,
8
n
−
3
}
.
Hence, we have
w
(
L
mi
) = 18 + 8
i
+ 25(
n
−
1) + 4
m
. Since
w
(
L
0i+1)
−
w
(
L
1i) = 4 and
w
(
L
1i)
−
w
(
L
0i) = 4 constitute an arithmatic progression
L
01, L
11, L
02, . . . , L
1n−1, and
w
(
L
01
) = 26 + 25(
n
−
1) then
P
n×
P
3is (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
3is (
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.
