Pelabelan selimut cycle-anti ajaib pada graf double cones, graf friendship dan graf grid Pn × P3 ngartikelM0109063
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(a, d)-H-ANTI MAGIC COVERINGS ON SOME CLASSES
OF
GRAPHS
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 of G that is isomorphic to H and there 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 a, a + d, a + 2d, . . . , a + (t − 1)d where a and d are positive integers and t is the
number of subgraphs of G isomorphic to H. The labeling ξ is called a super (a, d)-H-anti
magic total labeling, if ξ(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 nonnegative integers (Wallis [8]). Dozens graph labelings techniques are studied recently.
Some of them are magic and anti magic labeling. Sedláček 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. Gutiérrez and Lladó [3]
then developed the research into H-supermagic covering in 2005. An edge-covering
of G is a family of different subgraphs H1 , H2 , . . . , Hk such that any edge of E
belongs to at least one of the subgraphs Hi , 1 ≤ i ≤ k. If every Hi is isomorphic
to a given graph H, then G admits an H-covering. Suppose that G admits an Hcovering. 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. Lladó and
Moragas [7] proved that a wheel Wn , a prism Cn × K2 , a book K1,n × K2 , and
windmill W (r, k) are Ch -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 gf (V ) = {a, a+d, . . . , a+(|V |−
∑
1)d} with gf (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) →
commit to user
{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
a, a + d, a + 2d, . . . , a + (k − 1)d where a and d are positive integers and k is the
number of subgraphs of G isomorphic to H. Inayah et. al [5] proved that fan Fn
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admits (a, d)-cycle Cn -anti magic covering for some d. In Gallian [2], Susilowati et
al. proved that ladders Pn × P2 admits (a, d)-cycle-anti magic covering for some d.
This research aims to find (a, d)-H-anti magic covering on double cones DCn ,
friendship Dkn , and grid Pn × P3 .
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 Y1 , Y2 , . . . , Yk , such that for every
∑
⊎
Yi = kY ∈ N , and ki=1 Yi = Y . If this is the case for every
i ∈ [1, k], |Yi | = |Yk | ,
i ∈ [1, k] then Yi 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 be positive integers and k ≥ 3 is odd. For every i ∈ [1, k],
define Yi = {ai , bi , ci , di , ei , fi , gi } where
ai =
{
bi =
{
x+
ci =
{
di =
{
for i odd;
y + k − i−1
2
i
k+1
y + 2 − 2 for i even;
y + k−1
− i−1
for i odd;
2
2
for i even;
y + k − 2i
x+
x+
x+
i−1
2
k−1
2
k+1
2
i
2
for i odd;
+
+
i
2
i−1
2
for i even;
for i odd;
for i even;
ei =
{
z+
fi =
{
z+
z+
z+
i−1
2
k−1
2
k−1
2
i
2
for i odd;
+
+
i
for i even;
2
i+1
for i odd;
2
for i even;
gi = r + k − i.
Further, we define the sets
A = {ai |1 ≤ i ≤ k} = [x, x + k − 1];
E = {ei |1 ≤ i ≤ k} = [z, z + k − 1];
B = {bi |1 ≤ i ≤ k} = [x + 1, x + k];
F = {fi |1 ≤ i ≤ k} = [z + 1, z + k];
C = {ci |1 ≤ i ≤ k} = [y + 1, y + k];
G = {gi |1 ≤ i ≤ k} = [r, r + k − 1].
D = {di |1 ≤ i ≤ k} = [y, y + k − 1];
⊎
Since A ⊎ B ⊎ C ⊎ D ⊎ E ⊎ F ⊎ G = Y , we have ki=1 Yi = Y . Since |Yi | = 7 and
∑
+ r + 2x + 2y + 2z is constant, for every i ∈ [1, k], we conclude that
Yi = − 21 + 7k
2
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Y is k-balanced.
Lemma 2.2. Let x and y be positive integers and k ≥ 5 is odd. Then the multiset
Y = [x + 1, x + k] ⊎ [x + 1, x + k] ⊎ [y, y + k − 1] is k-balanced.
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Proof. Let x and y be positive integers and k ≥ 5 is odd. For every i ∈ [1, k], define
Yi = {ai , bi , ci } where
ai = x + i, 1 ≤ i ≤ k;
bi =
{
x+i+
k+1
,
2
x+1+i−
ci =
1≤i≤
k+1
,
2
k+1
2
{
y − 2i + k,
1≤i≤
y − 2i + 2k,
k+1
2
k−1
;
2
≤ i ≤ k.
k−1
;
2
≤ i ≤ k;
Further, we define the sets
A = {ai |1 ≤ i ≤ k} = [x, x + k − 1];
C = {ci |1 ≤ i ≤ k} = [y + 1, y + k].
B = {bi |1 ≤ i ≤ k} = [x + 1, x + k];
⊎
∑
+2x+y
Since A⊎B ⊎C = Y , we have ki=1 Yi = Y . Since |Yi | = 3 and Yi = 21 + 3k
2
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 X1 , X2 , X3 , . . . , Xk ,
⊎
such that for every i ∈ [1, k], |Xi | = |X|
, ki=1 Xi = X and for i ∈ [1, k − 1],
k
∑
∑
(Xi+1 ) − (Xi ) = δ is satisfied.
Lemma
X=
2.3. Let k ≥ 2 be an integer. If
[1, 8k + 5] ⊎ [2, k] ⊎ [k + 3, 2k + 1] ⊎ [2k + 4, 3k + 2]
⊎[3k + 5, 4k + 3] ⊎ [7k + 5, 8k + 5] ⊎ [7k + 6, 8k + 4]
⊎[7k + 6, 8k + 4] ⊎ {4k + 5, 4k + 8, . . . , 4k + 5 + 3(k − 1)}
⊎{4k + 7, 4k + 10, . . . , 4k + 7 + 3(k − 1)},
if j = 0;
[1, 3k] ⊎ [4, 3k + 3] ⊎ [3k + 4, 6k + 3] ⊎ [6k + 4, 8k + 3]
⊎[6k + 6, 8k + 5] ⊎ {2, 5, 8, . . . , 3k + 2} ⊎ {2, 5, 8, . . . , 3k + 2}, if j = 1.
then X is (2k, 2j + 2)-anti balanced for j = 0, 1.
Proof. For i ∈ [1, k] define Xij = {6m+i, 6m+i+1, 4k−6m−i+5, 4k−6m−i+4, 4k+
2m+3i+2, 8k −i+6, 8k −i+5, 4k +3i+3}, with m = 0, 1 for j = 0 and Xij = {2m+
3i−2, 2m+3i+1, 8k−m−2i+7, 8k−m−2i+5, 3i−1, 3i+2, 6k−3i+5, 3k+2m+3i+1},
with m = 0, 1 for j = 1. It is easy to verify that each i ∈ [1, k], |Xij | = 8, Xi ⊂ X,
⊎
∑
and ki=1 Xi = X. Since (Xij ) = 2m + 32k + 4i + 26 + j(2m − 7k + 4i − 8) for every
∑
∑
i ∈ [1, k] and (Xi+1 ) − (Xi ) = 2j + 2 for every i ∈ [1, k], X is (2k, 2j + 2)-anti
balanced for j = 0, 1.
Lemma 2.4. Let x, t, and k ≥ 2 becommit
positive
integers and t is odd. If X = [x +
to user
1, x + tk] then X is (k, t)-anti balanced.
Proof. For i ∈ [1, k] define Xi = {x + i + jk} with j = 0, 1, . . . , (t − 1). It is easy
⊎
to verify that for every i ∈ [1, k], |Xi | = t, Xi ⊂ X, and ki=1 Xi = X. Since
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∑
(Xi ) =
nt2
2
−
nt
2
+ tx + ti for every i ∈ [1, k] and
i ∈ [1, k], X is (k, t)-anti balanced.
∑
(Xi+1 ) −
∑
(Xi ) = t for every
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 Xi = {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], |Xi | = t, Xi ⊂ X, and ki=1 Xi = X. Since
∑
2
2
2
− j 2k + it − kt2 + kt2 + tx for every i ∈ [1, k] and
(Xi ) = 2j − ij − j2 + ij 2 + jk
2
∑
∑
(Xi+1 )− (Xi ) = 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, t2 )-anti balanced.
Proof. For i ∈ [1, k] define Xi = {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], |Xi | = t,
⊎
∑
2
Xi ⊂ X, and ki=1 Xi = X. Since (Xi ) = 2t − t2 + t2 i + xt for every i ∈ [1, k] and
∑
∑
(Xi+1 ) − (Xi ) = t2 for every i ∈ [1, k], X is (k, t2 )-anti balanced.
Lemma 2.7. Let n ≥ 5 and k be positive integers,with k = 2n and n is odd. If
⊎
X = n1 {1, 2} ⊎ [k + 3, 2k + 2] ⊎ [k + 3, 2k + 2] then X is (k, 1)-anti balanced.
Proof. For i ∈ [1, k] define
{ 21 (3 + i + 6n), 21 (5 + i + 4n), 2},
{ 1 (4 + i + 6n), 1 (6 + i + 4n), 1},
2
2
Xi =
1
{ 2 (3 + i + 4n), 21 (5 + i + 6n), 2},
1
{ 2 (4 + i + 4n), 21 (6 + i + 6n), 1},
for 1 ≡ i (mod 4);
for 2 ≡ i (mod 4);
for 3 ≡ i (mod 4);
for 0 ≡ i (mod 4).
⊎
It is easy to verify that each i ∈ [1, k], |Xi | = 3, Xi ⊂ X, and ki=1 Xi = X,with k =
∑
∑
∑
2n, n ≥ 5 is odd. Since (Xi ) = 6+i+5n for every i ∈ [1, k], (Xi +1)− (Xi ) = 1
for every i ∈ [1, k], X is (k, 1)-anti balanced.
3. Main results
3.1. (a, d) - C3 -anti magic coverings on double cones
Double cones is defined by DCn = Cn +K2 , for n ≥ 3. G ∼
= DCn has |V (DCn )| =
n + 2 and |E(DCn )| = 3n. We derive an upper bound of the difference d for DCn
to be (a, d) - C3 -anti magic covering.
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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 DCn isomorphic to C3 , say Hi′ , with t = 2n.
Since DCn is (a, d)-C3 -anti magic, the maximum possible Hi′ -weight is (4n + 2 − 6 +
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1)+(4n+2−6+2)+(4n+2−6+3)+(4n+2−6+4)+(4n+2−6+5)+(4n+2) = 24n−3
and the least possible Hi′ -weight is 1 + 2 + 3 + 4 + 5 + 6 = 21.
(2n − 1)d ≤ (24n − 3) − 21
24n − 24
d ≤
2n − 1
Theorem 3.2. Let n ≥ 5 and n be positive integers. Graph DCn is (14 + 7n +
(n+1)
, 1)-C3 -anti
2
magic.
Proof. We define a bijective function ξ : V (DCn ) ∪ E(DCn ) → {1, 2, . . . , 4n + 2}.
Let P be the set of label used to label vertices and edges of subgraph of DCn which
isomorphic to Cn 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
u1 and u2 are labeled using 1 and 2. Let Q be the set of label of the rest. Edges u1 vi
and u2 vi are labeled using integers on interval [2n + 3, 4n + 2] such that every Hi′ weight satisfies Lemma 2.7. Q is (2n, 1)-anti balanced. It is easy to verify that ξ is
a bijective function from V (DCn ) ∪ E(DCn ) to {1, 2, 3, . . . , 4n + 2}. For 1 ≤ i ≤ 2n,
13 + i + 7n + (n+1)
,
for 1 ≡ i (mod 4);
2
13 + i + 7n + (n+1) ,
for 2 ≡ i (mod 4);
2
w(Hi′ )
11 + (1−i)
+ (3−i)
+ 2i + (1−n)
+ 8n, for 3 ≡ i (mod 4);
2
2
2
10 + (2−i) + (4−i) + 2i + (1−n) + 8n, for 0 ≡ i (mod 4).
2
2
2
′
Since 1 ≤ i ≤ 2n, w(Hi+1
) − w(Hi′ ) = 1 and w(H1′ ) = 14 + 7n +
weight, then DCn is (14 + 7n +
(n+1)
, 1)-C3 -anti
2
(n+1)
2
is the least
magic.
3.2. (a, d) - Ck -anti magic coverings on friendship
Friendship is a graph consisting of n cycles with a common vertex. Let H be
Ck with |V (Ck )| = k and |E(Ck )| = k. G ∼
= Dkn has |V (Dkn )| = 1 + n(k − 1) and
|E(Dkn )| = nk. We derive an upper bound of the difference d for Dkn to be (a, d) Ck -anti magic covering.
Theorem 3.3. If G is (a, d)-H-anti magic then d ≤ 2k(2k − 1).
commit
user
Proof. Let t be a number of subgraphs
of Dkntoisomorphic
to Ck , say Hi′ , with t = n.
Since Dkn is (a, d)-Ck -anti magic, the maximum possible Hi′ -weight is (2kn − n + 1 −
k − k + 1) + (2kn − n + 1 − k − k + 2) + (2kn − n + 1 − k − k + 3) + . . . + (2kn −
n + 1 − 1) + (2kn − n + 1) = k(3 − 2k − 2n + 4kn) and the least possible Hi′ -weight
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is 1 + 2 + . . . + 2k = k(2k + 1).
(n − 1)d ≤ k(3 − 2k − 2n + 4kn) − k(2k + 1)
2k(2k − 1)(n − 1)
d ≤
n−1
= 2k(2k − 1)
Theorem 3.4. Let n ≥ 2, j and k be positive integers. Graph Dkn is (a, d)-Ck -anti
magic.
Proof. We define a bijective function ξ : V (Dkn ) ∪ E(Dkn ) → {1, 2, . . . , 1 + n(2k − 1)}.
Then we define 3 cases of labelings.
1. Case d = 2k − 1
The center vertex is labeled with 1. Use the set X = [2, n(2k − 1) + 1] to label the
rest of the vertices and edges such that every Hi′ -weight satisfies Lemma 2.4 with
k = n, x = 1, and t = 2k − 1. X is (n, 2k − 1)-anti balanced. It is easy to verify
that ξ is a bijective function from V (Dkn ) ∪ E(Dkn ) to {1, 2, 3, . . . , 1 + n(2k − 1)}. For
′
1 ≤ i ≤ n, w(Hi′ ) = 1+(2k −1)(1+i+(k −1)n). Since w(Hi+1
)−w(Hi′ ) = 2k −1 and
w(H1′ ) = (2k − 1)(2 + (k − 1)n), then Dkn is (1 + (2k − 1)(2 + (k − 1)n), 2k − 1)-Ck -anti
magic.
2. Case d = j 2 − j + 2k − 1
We label the center vertex with 1. The set X = [2, n(2k − 1) + 1] is used to label
the rest of the vertices and edges such that every Hi′ -weight satisfies Lemma 2.5
with k = n, x = 1, t = 2k − 1, and 1 ≤ j ≤ 2k − 1. X is (n, j 2 − j + 2k − 1)-anti
balanced. It is easy to check that ξ is a bijective function from V (Dkn ) ∪ E(Dkn ) to
{1, 2, 3, . . . , 1 + n(2k − 1)}. For 1 ≤ i ≤ n, w(Hi′ ) = 21 (j − j 2 − 2 + 2i(+j 2 − j − 1) +
′
k(4 + 4i − 6n) + 2n + 4k 2 n + jn − j 2 n) + 1. Since w(Hi+1
) − w(Hi′ ) = j 2 − j + 2k − 1
and w(H1′ ) =
1 2
(j
2
− j + 8k − 4 + (2k − j − 1)(2k + j)n − 2) + 1, then Dkn is
( 21 (j 2 − j + 8k − 4 + (2k − j − 1)(2k + j)n − 2) + 1, j 2 − j + 2k − 1)-Ck -anti magic.
3. Case d = (1 − 2k)2
Put 1 as the label of the center vertex. Use the set X = [2, n(2k − 1) + 1] to label the
rest of the vertices and edges such that every Hi′ -weight satisfies Lemma 2.6 with
k = n, x = 1, and t = 2k − 1. X is (n, (2k − 1)2 )-anti balanced. It is easy to verify
that ξ is a bijective function from V (Dkn ) ∪ E(Dkn ) to {1, 2, 3, . . . , 1 + n(2k − 1)}. For
′
)−w(Hi′ ) = (1−2k)2
Since w(Hi+1
1 ≤ i ≤ n, w(Hi′ ) = (2k −1)(2−i+k(2i−1))+1.
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and w(H1′ ) = 2k 2 + k, then Dkn is (2k 2 + k, (1 − 2k)2 )-Ck -anti magic.
3.3. (a, d)-C4 -anti magic coverings on grid
The cartesian product of simple graphs G and H is the graph G×H whose vertex
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set is V (G) × V (H) and whose edge set is the set of all pairs (u1 v1 )(u2 v2 ) such that
either u1 u2 ∈ E(G) and v1 = v2 , or v1 v2 ∈ E(H) and u1 = u2 . A graph grid is
defined by Pn × Pm . Let m = 3. Let H be C4 with |V (C4 )| = 4 and |E(C4 )| = 4.
G∼
= Pn × P3 has |V (Pn × P3 )| = 3n and |E(Pn × P3 )| = 5n − 3. We derive an upper
bound of the difference d for Pn × P3 to be (a, d)-C4 -anti magic covering.
Theorem 3.5. If G is (a, d)-H-anti magic then d ≤
64n−88
.
2n−2
Proof. Let t be a number of subgraphs of Pn × P3 isomorphic to C4 , say Hi′ , with
t = 2(n − 1). Since Pn × P3 is (a, d)-C4 -anti magic, the maximum possible Hi′ -weight
is (8n − 3 − 8 + 1) + (8n − 3 − 8 + 2) + (8n − 3 − 8 + 3) + (8n − 3 − 8 + 4) + (8n −
3 − 8 + 5) + (8n − 3 − 8 + 6) + (8n − 3 − 8 + 7) + (8n − 3) = 64n − 52 and the least
possible Hi′ -weight is 1 + 2 + . . . + 8 = 36.
(2(n − 1) − 1)d ≤ 64n − 52 − 36
64n − 88
d ≤
2n − 3
Theorem 3.6. Let n ≥ 4 be positive integer and n even. Graph Pn ×P3 is (3+ 57n
, 1)2
C4 -anti magic.
Proof. We define a bijective function ξ : V (Pn ×P3 )∪E(Pn ×P3 ) → {1, 2, . . . , 8n−3}.
Let R be the set of labels with R = [1, 8n − 3]. Partition R into 7 sets, A = [1, n],
B = [n+1, 2n], C = [2n+1, 3n], D = [3n+1, 4n], E = [4n+1, 5n], F = [5n+1, 6n−1],
and G = [6n, 8n − 3]. The set A is used to label ui , B for wi , C for ui vi , D for wi vi ,
E for vi , and F for vi vi+1 such that every Hi′ -weight satisfies the Lemma 2.1 with
k = n − 1, z = 4n + 1, and r = 5n + 1. For Hi′ which contains vertices ui , x = 1,
y = 3n + 1 and for Hi′ which contains vertices wi , x = n + 1, y = 2n + 1. Then G is
used to label ui ui+1 and wi wi+1 . R is (2(n − 1), 1)-anti balanced. It is easy to verify
that ξ is a bijective function from V (Pn × P3 ) ∪ E(Pn × P3 ) to {1, 2, 3, . . . , 8n − 3}.
′
Hence, we have w(Hi′ ) = 2+i+ 57n
. Since w(Hi+1
)−w(Hi′ ) = 1 and w(H1′ ) = 3+ 57n
,
2
2
then Pn × P3 is (3 +
57n
, 1)-C4 -anti
2
magic.
Theorem 3.7. Let n ≥ 4 be positive integer and n even. Graph Pn × P3 is (30 +
32(n − 1), 2)-C4 -anti magic.
Proof. We define a bijective function ξ : V (Pn ×P3 )∪E(Pn ×P3 ) → {1, 2, . . . , 8n−3}.
Let Li = Hi′ . Let R be the set of label with R = [1, 8n − 3]. Partition R into 6 sets,
A = [1, n], B = [n+1, 2n], C = [2n+1,commit
3n], Dto=user
[3n+1, 4n], E = [4n+1, 7n−3], and
F = [7n − 2, 8n − 3]. The set A is used to label ui , B for wi , C for vi wi , D for ui vi , E
for ui ui+1 , vi vi+1 , wi wi+1 , and F for vi such that every Li -weight satisfies the Lemma
2.3 with k = n − 1 and j = 0. For Li which contains vertices ui , m = 0 and for Li
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which contain vertices wi , m = 1. R is (2(n−1), 2)-anti balanced. It is easy to verify
that ξ is a bijective function from V (Pn × P3 ) ∪ E(Pn × P3 ) to {1, 2, 3, . . . , 8n − 3}.
1
0
Hence, we have w(Lm
i ) = 2(13 + 2i + 16(n − 1) + m). Since w(Li+1 ) − w(Li ) = 2
and w(L1i ) − w(L0i ) = 2 constitute an arithmatic progression L01 , L11 , L02 , . . . , L1n−1 ,
and w(L01 ) = 30 + 32(n − 1) then Pn × P3 is (30 + 32(n − 1), 2)-C4 -anti magic.
Theorem 3.8. Let n ≥ 4 be positive integer and n even. Graf Pn × P3 is (26 +
25(n − 1), 4)-C4 -anti magic.
Proof. We define a bijective function ξ : V (Pn ×P3 )∪E(Pn ×P3 ) → {1, 2, . . . , 8n−3}.
Let Li = Hi′ . Let R be the set of label with R = [1, 8n − 3]. Partition R into 3 sets,
A = [1, 3n], B = [3n + 1, 6n − 3], and C = [6n − 2, 8n − 3]. The set A is used to
label all the vertices of grid Pn × P3 , the set B is used to label ui ui+1 , vi vi+1 , and
wi wi+1 , C for ui vi and vi wi such that every Li -weight satisfies the Lemma 2.3 with
k = n − 1 and j = 1. For Li which contains vertices ui , m = 0 and for Li which
contains vertices wi , m = 1. R is (2(n − 1), 4)-anti balanced. It is easy to verify
that ξ is a bijective function from V (Pn × P3 ) ∪ E(Pn × P3 ) to {1, 2, 3, . . . , 8n − 3}.
0
1
Hence, we have w(Lm
i ) = 18 + 8i + 25(n − 1) + 4m. Since w(Li+1 ) − w(Li ) = 4 and
w(L1i ) − w(L0i ) = 4 constitute an arithmatic progression L01 , L11 , L02 , . . . , L1n−1 , and
w(L01 ) = 26 + 25(n − 1) then Pn × P3 is (26 + 25(n − 1), 4)-C4 -anti magic.
4. Conclusion
In this section, we conclude that a double cones is (a, d)-C3 -anti magic with d = 1
for n ≥ 5, a friendship is (a, d)-Ck -anti magic with d = {2k − 1, j 2 − j + 2k − 1, (1 −
2k)2 } for n ≥ 2 and k are positive integers, and a grid Pn ×P3 is (a, d)-C4 -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] Gutiérrez, and Lladó, Magic Coverings, J. Combin. Math. Combin. Comput 55 (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. Comput 71 (2009), 1662-1680.
commit
to user
[6] Kotzig, A. and A. Rosa, Magic Valuations
of Finite
Graphs, Canad. Math. Bull 13 (1970),
451-461.
[7] Lladó, A. and J. Moragas, Cycle Magic Graph, Discrete Mathematics 307 (2007), 2925-2933.
[8] Wallis, W. D. , Magic Graphs, Birkhäuser, Boston, Basel, Berlin, 2001.
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(a, d)-H-ANTI MAGIC COVERINGS ON SOME CLASSES
OF
GRAPHS
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 of G that is isomorphic to H and there 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 a, a + d, a + 2d, . . . , a + (t − 1)d where a and d are positive integers and t is the
number of subgraphs of G isomorphic to H. The labeling ξ is called a super (a, d)-H-anti
magic total labeling, if ξ(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 nonnegative integers (Wallis [8]). Dozens graph labelings techniques are studied recently.
Some of them are magic and anti magic labeling. Sedláček 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. Gutiérrez and Lladó [3]
then developed the research into H-supermagic covering in 2005. An edge-covering
of G is a family of different subgraphs H1 , H2 , . . . , Hk such that any edge of E
belongs to at least one of the subgraphs Hi , 1 ≤ i ≤ k. If every Hi is isomorphic
to a given graph H, then G admits an H-covering. Suppose that G admits an Hcovering. 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. Lladó and
Moragas [7] proved that a wheel Wn , a prism Cn × K2 , a book K1,n × K2 , and
windmill W (r, k) are Ch -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 gf (V ) = {a, a+d, . . . , a+(|V |−
∑
1)d} with gf (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) →
commit to user
{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
a, a + d, a + 2d, . . . , a + (k − 1)d where a and d are positive integers and k is the
number of subgraphs of G isomorphic to H. Inayah et. al [5] proved that fan Fn
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admits (a, d)-cycle Cn -anti magic covering for some d. In Gallian [2], Susilowati et
al. proved that ladders Pn × P2 admits (a, d)-cycle-anti magic covering for some d.
This research aims to find (a, d)-H-anti magic covering on double cones DCn ,
friendship Dkn , and grid Pn × P3 .
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 Y1 , Y2 , . . . , Yk , such that for every
∑
⊎
Yi = kY ∈ N , and ki=1 Yi = Y . If this is the case for every
i ∈ [1, k], |Yi | = |Yk | ,
i ∈ [1, k] then Yi 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 be positive integers and k ≥ 3 is odd. For every i ∈ [1, k],
define Yi = {ai , bi , ci , di , ei , fi , gi } where
ai =
{
bi =
{
x+
ci =
{
di =
{
for i odd;
y + k − i−1
2
i
k+1
y + 2 − 2 for i even;
y + k−1
− i−1
for i odd;
2
2
for i even;
y + k − 2i
x+
x+
x+
i−1
2
k−1
2
k+1
2
i
2
for i odd;
+
+
i
2
i−1
2
for i even;
for i odd;
for i even;
ei =
{
z+
fi =
{
z+
z+
z+
i−1
2
k−1
2
k−1
2
i
2
for i odd;
+
+
i
for i even;
2
i+1
for i odd;
2
for i even;
gi = r + k − i.
Further, we define the sets
A = {ai |1 ≤ i ≤ k} = [x, x + k − 1];
E = {ei |1 ≤ i ≤ k} = [z, z + k − 1];
B = {bi |1 ≤ i ≤ k} = [x + 1, x + k];
F = {fi |1 ≤ i ≤ k} = [z + 1, z + k];
C = {ci |1 ≤ i ≤ k} = [y + 1, y + k];
G = {gi |1 ≤ i ≤ k} = [r, r + k − 1].
D = {di |1 ≤ i ≤ k} = [y, y + k − 1];
⊎
Since A ⊎ B ⊎ C ⊎ D ⊎ E ⊎ F ⊎ G = Y , we have ki=1 Yi = Y . Since |Yi | = 7 and
∑
+ r + 2x + 2y + 2z is constant, for every i ∈ [1, k], we conclude that
Yi = − 21 + 7k
2
commit to user
Y is k-balanced.
Lemma 2.2. Let x and y be positive integers and k ≥ 5 is odd. Then the multiset
Y = [x + 1, x + k] ⊎ [x + 1, x + k] ⊎ [y, y + k − 1] is k-balanced.
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Proof. Let x and y be positive integers and k ≥ 5 is odd. For every i ∈ [1, k], define
Yi = {ai , bi , ci } where
ai = x + i, 1 ≤ i ≤ k;
bi =
{
x+i+
k+1
,
2
x+1+i−
ci =
1≤i≤
k+1
,
2
k+1
2
{
y − 2i + k,
1≤i≤
y − 2i + 2k,
k+1
2
k−1
;
2
≤ i ≤ k.
k−1
;
2
≤ i ≤ k;
Further, we define the sets
A = {ai |1 ≤ i ≤ k} = [x, x + k − 1];
C = {ci |1 ≤ i ≤ k} = [y + 1, y + k].
B = {bi |1 ≤ i ≤ k} = [x + 1, x + k];
⊎
∑
+2x+y
Since A⊎B ⊎C = Y , we have ki=1 Yi = Y . Since |Yi | = 3 and Yi = 21 + 3k
2
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 X1 , X2 , X3 , . . . , Xk ,
⊎
such that for every i ∈ [1, k], |Xi | = |X|
, ki=1 Xi = X and for i ∈ [1, k − 1],
k
∑
∑
(Xi+1 ) − (Xi ) = δ is satisfied.
Lemma
X=
2.3. Let k ≥ 2 be an integer. If
[1, 8k + 5] ⊎ [2, k] ⊎ [k + 3, 2k + 1] ⊎ [2k + 4, 3k + 2]
⊎[3k + 5, 4k + 3] ⊎ [7k + 5, 8k + 5] ⊎ [7k + 6, 8k + 4]
⊎[7k + 6, 8k + 4] ⊎ {4k + 5, 4k + 8, . . . , 4k + 5 + 3(k − 1)}
⊎{4k + 7, 4k + 10, . . . , 4k + 7 + 3(k − 1)},
if j = 0;
[1, 3k] ⊎ [4, 3k + 3] ⊎ [3k + 4, 6k + 3] ⊎ [6k + 4, 8k + 3]
⊎[6k + 6, 8k + 5] ⊎ {2, 5, 8, . . . , 3k + 2} ⊎ {2, 5, 8, . . . , 3k + 2}, if j = 1.
then X is (2k, 2j + 2)-anti balanced for j = 0, 1.
Proof. For i ∈ [1, k] define Xij = {6m+i, 6m+i+1, 4k−6m−i+5, 4k−6m−i+4, 4k+
2m+3i+2, 8k −i+6, 8k −i+5, 4k +3i+3}, with m = 0, 1 for j = 0 and Xij = {2m+
3i−2, 2m+3i+1, 8k−m−2i+7, 8k−m−2i+5, 3i−1, 3i+2, 6k−3i+5, 3k+2m+3i+1},
with m = 0, 1 for j = 1. It is easy to verify that each i ∈ [1, k], |Xij | = 8, Xi ⊂ X,
⊎
∑
and ki=1 Xi = X. Since (Xij ) = 2m + 32k + 4i + 26 + j(2m − 7k + 4i − 8) for every
∑
∑
i ∈ [1, k] and (Xi+1 ) − (Xi ) = 2j + 2 for every i ∈ [1, k], X is (2k, 2j + 2)-anti
balanced for j = 0, 1.
Lemma 2.4. Let x, t, and k ≥ 2 becommit
positive
integers and t is odd. If X = [x +
to user
1, x + tk] then X is (k, t)-anti balanced.
Proof. For i ∈ [1, k] define Xi = {x + i + jk} with j = 0, 1, . . . , (t − 1). It is easy
⊎
to verify that for every i ∈ [1, k], |Xi | = t, Xi ⊂ X, and ki=1 Xi = X. Since
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∑
(Xi ) =
nt2
2
−
nt
2
+ tx + ti for every i ∈ [1, k] and
i ∈ [1, k], X is (k, t)-anti balanced.
∑
(Xi+1 ) −
∑
(Xi ) = t for every
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 Xi = {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], |Xi | = t, Xi ⊂ X, and ki=1 Xi = X. Since
∑
2
2
2
− j 2k + it − kt2 + kt2 + tx for every i ∈ [1, k] and
(Xi ) = 2j − ij − j2 + ij 2 + jk
2
∑
∑
(Xi+1 )− (Xi ) = 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, t2 )-anti balanced.
Proof. For i ∈ [1, k] define Xi = {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], |Xi | = t,
⊎
∑
2
Xi ⊂ X, and ki=1 Xi = X. Since (Xi ) = 2t − t2 + t2 i + xt for every i ∈ [1, k] and
∑
∑
(Xi+1 ) − (Xi ) = t2 for every i ∈ [1, k], X is (k, t2 )-anti balanced.
Lemma 2.7. Let n ≥ 5 and k be positive integers,with k = 2n and n is odd. If
⊎
X = n1 {1, 2} ⊎ [k + 3, 2k + 2] ⊎ [k + 3, 2k + 2] then X is (k, 1)-anti balanced.
Proof. For i ∈ [1, k] define
{ 21 (3 + i + 6n), 21 (5 + i + 4n), 2},
{ 1 (4 + i + 6n), 1 (6 + i + 4n), 1},
2
2
Xi =
1
{ 2 (3 + i + 4n), 21 (5 + i + 6n), 2},
1
{ 2 (4 + i + 4n), 21 (6 + i + 6n), 1},
for 1 ≡ i (mod 4);
for 2 ≡ i (mod 4);
for 3 ≡ i (mod 4);
for 0 ≡ i (mod 4).
⊎
It is easy to verify that each i ∈ [1, k], |Xi | = 3, Xi ⊂ X, and ki=1 Xi = X,with k =
∑
∑
∑
2n, n ≥ 5 is odd. Since (Xi ) = 6+i+5n for every i ∈ [1, k], (Xi +1)− (Xi ) = 1
for every i ∈ [1, k], X is (k, 1)-anti balanced.
3. Main results
3.1. (a, d) - C3 -anti magic coverings on double cones
Double cones is defined by DCn = Cn +K2 , for n ≥ 3. G ∼
= DCn has |V (DCn )| =
n + 2 and |E(DCn )| = 3n. We derive an upper bound of the difference d for DCn
to be (a, d) - C3 -anti magic covering.
commit to user
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 DCn isomorphic to C3 , say Hi′ , with t = 2n.
Since DCn is (a, d)-C3 -anti magic, the maximum possible Hi′ -weight is (4n + 2 − 6 +
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1)+(4n+2−6+2)+(4n+2−6+3)+(4n+2−6+4)+(4n+2−6+5)+(4n+2) = 24n−3
and the least possible Hi′ -weight is 1 + 2 + 3 + 4 + 5 + 6 = 21.
(2n − 1)d ≤ (24n − 3) − 21
24n − 24
d ≤
2n − 1
Theorem 3.2. Let n ≥ 5 and n be positive integers. Graph DCn is (14 + 7n +
(n+1)
, 1)-C3 -anti
2
magic.
Proof. We define a bijective function ξ : V (DCn ) ∪ E(DCn ) → {1, 2, . . . , 4n + 2}.
Let P be the set of label used to label vertices and edges of subgraph of DCn which
isomorphic to Cn 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
u1 and u2 are labeled using 1 and 2. Let Q be the set of label of the rest. Edges u1 vi
and u2 vi are labeled using integers on interval [2n + 3, 4n + 2] such that every Hi′ weight satisfies Lemma 2.7. Q is (2n, 1)-anti balanced. It is easy to verify that ξ is
a bijective function from V (DCn ) ∪ E(DCn ) to {1, 2, 3, . . . , 4n + 2}. For 1 ≤ i ≤ 2n,
13 + i + 7n + (n+1)
,
for 1 ≡ i (mod 4);
2
13 + i + 7n + (n+1) ,
for 2 ≡ i (mod 4);
2
w(Hi′ )
11 + (1−i)
+ (3−i)
+ 2i + (1−n)
+ 8n, for 3 ≡ i (mod 4);
2
2
2
10 + (2−i) + (4−i) + 2i + (1−n) + 8n, for 0 ≡ i (mod 4).
2
2
2
′
Since 1 ≤ i ≤ 2n, w(Hi+1
) − w(Hi′ ) = 1 and w(H1′ ) = 14 + 7n +
weight, then DCn is (14 + 7n +
(n+1)
, 1)-C3 -anti
2
(n+1)
2
is the least
magic.
3.2. (a, d) - Ck -anti magic coverings on friendship
Friendship is a graph consisting of n cycles with a common vertex. Let H be
Ck with |V (Ck )| = k and |E(Ck )| = k. G ∼
= Dkn has |V (Dkn )| = 1 + n(k − 1) and
|E(Dkn )| = nk. We derive an upper bound of the difference d for Dkn to be (a, d) Ck -anti magic covering.
Theorem 3.3. If G is (a, d)-H-anti magic then d ≤ 2k(2k − 1).
commit
user
Proof. Let t be a number of subgraphs
of Dkntoisomorphic
to Ck , say Hi′ , with t = n.
Since Dkn is (a, d)-Ck -anti magic, the maximum possible Hi′ -weight is (2kn − n + 1 −
k − k + 1) + (2kn − n + 1 − k − k + 2) + (2kn − n + 1 − k − k + 3) + . . . + (2kn −
n + 1 − 1) + (2kn − n + 1) = k(3 − 2k − 2n + 4kn) and the least possible Hi′ -weight
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is 1 + 2 + . . . + 2k = k(2k + 1).
(n − 1)d ≤ k(3 − 2k − 2n + 4kn) − k(2k + 1)
2k(2k − 1)(n − 1)
d ≤
n−1
= 2k(2k − 1)
Theorem 3.4. Let n ≥ 2, j and k be positive integers. Graph Dkn is (a, d)-Ck -anti
magic.
Proof. We define a bijective function ξ : V (Dkn ) ∪ E(Dkn ) → {1, 2, . . . , 1 + n(2k − 1)}.
Then we define 3 cases of labelings.
1. Case d = 2k − 1
The center vertex is labeled with 1. Use the set X = [2, n(2k − 1) + 1] to label the
rest of the vertices and edges such that every Hi′ -weight satisfies Lemma 2.4 with
k = n, x = 1, and t = 2k − 1. X is (n, 2k − 1)-anti balanced. It is easy to verify
that ξ is a bijective function from V (Dkn ) ∪ E(Dkn ) to {1, 2, 3, . . . , 1 + n(2k − 1)}. For
′
1 ≤ i ≤ n, w(Hi′ ) = 1+(2k −1)(1+i+(k −1)n). Since w(Hi+1
)−w(Hi′ ) = 2k −1 and
w(H1′ ) = (2k − 1)(2 + (k − 1)n), then Dkn is (1 + (2k − 1)(2 + (k − 1)n), 2k − 1)-Ck -anti
magic.
2. Case d = j 2 − j + 2k − 1
We label the center vertex with 1. The set X = [2, n(2k − 1) + 1] is used to label
the rest of the vertices and edges such that every Hi′ -weight satisfies Lemma 2.5
with k = n, x = 1, t = 2k − 1, and 1 ≤ j ≤ 2k − 1. X is (n, j 2 − j + 2k − 1)-anti
balanced. It is easy to check that ξ is a bijective function from V (Dkn ) ∪ E(Dkn ) to
{1, 2, 3, . . . , 1 + n(2k − 1)}. For 1 ≤ i ≤ n, w(Hi′ ) = 21 (j − j 2 − 2 + 2i(+j 2 − j − 1) +
′
k(4 + 4i − 6n) + 2n + 4k 2 n + jn − j 2 n) + 1. Since w(Hi+1
) − w(Hi′ ) = j 2 − j + 2k − 1
and w(H1′ ) =
1 2
(j
2
− j + 8k − 4 + (2k − j − 1)(2k + j)n − 2) + 1, then Dkn is
( 21 (j 2 − j + 8k − 4 + (2k − j − 1)(2k + j)n − 2) + 1, j 2 − j + 2k − 1)-Ck -anti magic.
3. Case d = (1 − 2k)2
Put 1 as the label of the center vertex. Use the set X = [2, n(2k − 1) + 1] to label the
rest of the vertices and edges such that every Hi′ -weight satisfies Lemma 2.6 with
k = n, x = 1, and t = 2k − 1. X is (n, (2k − 1)2 )-anti balanced. It is easy to verify
that ξ is a bijective function from V (Dkn ) ∪ E(Dkn ) to {1, 2, 3, . . . , 1 + n(2k − 1)}. For
′
)−w(Hi′ ) = (1−2k)2
Since w(Hi+1
1 ≤ i ≤ n, w(Hi′ ) = (2k −1)(2−i+k(2i−1))+1.
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and w(H1′ ) = 2k 2 + k, then Dkn is (2k 2 + k, (1 − 2k)2 )-Ck -anti magic.
3.3. (a, d)-C4 -anti magic coverings on grid
The cartesian product of simple graphs G and H is the graph G×H whose vertex
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(a, d)-H-Antimagic Coverings on Some Classes of Graphs
set is V (G) × V (H) and whose edge set is the set of all pairs (u1 v1 )(u2 v2 ) such that
either u1 u2 ∈ E(G) and v1 = v2 , or v1 v2 ∈ E(H) and u1 = u2 . A graph grid is
defined by Pn × Pm . Let m = 3. Let H be C4 with |V (C4 )| = 4 and |E(C4 )| = 4.
G∼
= Pn × P3 has |V (Pn × P3 )| = 3n and |E(Pn × P3 )| = 5n − 3. We derive an upper
bound of the difference d for Pn × P3 to be (a, d)-C4 -anti magic covering.
Theorem 3.5. If G is (a, d)-H-anti magic then d ≤
64n−88
.
2n−2
Proof. Let t be a number of subgraphs of Pn × P3 isomorphic to C4 , say Hi′ , with
t = 2(n − 1). Since Pn × P3 is (a, d)-C4 -anti magic, the maximum possible Hi′ -weight
is (8n − 3 − 8 + 1) + (8n − 3 − 8 + 2) + (8n − 3 − 8 + 3) + (8n − 3 − 8 + 4) + (8n −
3 − 8 + 5) + (8n − 3 − 8 + 6) + (8n − 3 − 8 + 7) + (8n − 3) = 64n − 52 and the least
possible Hi′ -weight is 1 + 2 + . . . + 8 = 36.
(2(n − 1) − 1)d ≤ 64n − 52 − 36
64n − 88
d ≤
2n − 3
Theorem 3.6. Let n ≥ 4 be positive integer and n even. Graph Pn ×P3 is (3+ 57n
, 1)2
C4 -anti magic.
Proof. We define a bijective function ξ : V (Pn ×P3 )∪E(Pn ×P3 ) → {1, 2, . . . , 8n−3}.
Let R be the set of labels with R = [1, 8n − 3]. Partition R into 7 sets, A = [1, n],
B = [n+1, 2n], C = [2n+1, 3n], D = [3n+1, 4n], E = [4n+1, 5n], F = [5n+1, 6n−1],
and G = [6n, 8n − 3]. The set A is used to label ui , B for wi , C for ui vi , D for wi vi ,
E for vi , and F for vi vi+1 such that every Hi′ -weight satisfies the Lemma 2.1 with
k = n − 1, z = 4n + 1, and r = 5n + 1. For Hi′ which contains vertices ui , x = 1,
y = 3n + 1 and for Hi′ which contains vertices wi , x = n + 1, y = 2n + 1. Then G is
used to label ui ui+1 and wi wi+1 . R is (2(n − 1), 1)-anti balanced. It is easy to verify
that ξ is a bijective function from V (Pn × P3 ) ∪ E(Pn × P3 ) to {1, 2, 3, . . . , 8n − 3}.
′
Hence, we have w(Hi′ ) = 2+i+ 57n
. Since w(Hi+1
)−w(Hi′ ) = 1 and w(H1′ ) = 3+ 57n
,
2
2
then Pn × P3 is (3 +
57n
, 1)-C4 -anti
2
magic.
Theorem 3.7. Let n ≥ 4 be positive integer and n even. Graph Pn × P3 is (30 +
32(n − 1), 2)-C4 -anti magic.
Proof. We define a bijective function ξ : V (Pn ×P3 )∪E(Pn ×P3 ) → {1, 2, . . . , 8n−3}.
Let Li = Hi′ . Let R be the set of label with R = [1, 8n − 3]. Partition R into 6 sets,
A = [1, n], B = [n+1, 2n], C = [2n+1,commit
3n], Dto=user
[3n+1, 4n], E = [4n+1, 7n−3], and
F = [7n − 2, 8n − 3]. The set A is used to label ui , B for wi , C for vi wi , D for ui vi , E
for ui ui+1 , vi vi+1 , wi wi+1 , and F for vi such that every Li -weight satisfies the Lemma
2.3 with k = n − 1 and j = 0. For Li which contains vertices ui , m = 0 and for Li
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which contain vertices wi , m = 1. R is (2(n−1), 2)-anti balanced. It is easy to verify
that ξ is a bijective function from V (Pn × P3 ) ∪ E(Pn × P3 ) to {1, 2, 3, . . . , 8n − 3}.
1
0
Hence, we have w(Lm
i ) = 2(13 + 2i + 16(n − 1) + m). Since w(Li+1 ) − w(Li ) = 2
and w(L1i ) − w(L0i ) = 2 constitute an arithmatic progression L01 , L11 , L02 , . . . , L1n−1 ,
and w(L01 ) = 30 + 32(n − 1) then Pn × P3 is (30 + 32(n − 1), 2)-C4 -anti magic.
Theorem 3.8. Let n ≥ 4 be positive integer and n even. Graf Pn × P3 is (26 +
25(n − 1), 4)-C4 -anti magic.
Proof. We define a bijective function ξ : V (Pn ×P3 )∪E(Pn ×P3 ) → {1, 2, . . . , 8n−3}.
Let Li = Hi′ . Let R be the set of label with R = [1, 8n − 3]. Partition R into 3 sets,
A = [1, 3n], B = [3n + 1, 6n − 3], and C = [6n − 2, 8n − 3]. The set A is used to
label all the vertices of grid Pn × P3 , the set B is used to label ui ui+1 , vi vi+1 , and
wi wi+1 , C for ui vi and vi wi such that every Li -weight satisfies the Lemma 2.3 with
k = n − 1 and j = 1. For Li which contains vertices ui , m = 0 and for Li which
contains vertices wi , m = 1. R is (2(n − 1), 4)-anti balanced. It is easy to verify
that ξ is a bijective function from V (Pn × P3 ) ∪ E(Pn × P3 ) to {1, 2, 3, . . . , 8n − 3}.
0
1
Hence, we have w(Lm
i ) = 18 + 8i + 25(n − 1) + 4m. Since w(Li+1 ) − w(Li ) = 4 and
w(L1i ) − w(L0i ) = 4 constitute an arithmatic progression L01 , L11 , L02 , . . . , L1n−1 , and
w(L01 ) = 26 + 25(n − 1) then Pn × P3 is (26 + 25(n − 1), 4)-C4 -anti magic.
4. Conclusion
In this section, we conclude that a double cones is (a, d)-C3 -anti magic with d = 1
for n ≥ 5, a friendship is (a, d)-Ck -anti magic with d = {2k − 1, j 2 − j + 2k − 1, (1 −
2k)2 } for n ≥ 2 and k are positive integers, and a grid Pn ×P3 is (a, d)-C4 -anti magic
with d = {1, 2, 4} for n ≥ 4 is positive integers.
References
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Mannhein, 1993.
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[3] Gutiérrez, and Lladó, Magic Coverings, J. Combin. Math. Combin. Comput 55 (2005), 43-46.
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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. Comput 71 (2009), 1662-1680.
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to user
[6] Kotzig, A. and A. Rosa, Magic Valuations
of Finite
Graphs, Canad. Math. Bull 13 (1970),
451-461.
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