Antimagic Total Labeling on

Antimagic Total Labeling on 𝒁 _𝒑+𝒒

H. Velwet Getzimah*

and K. Palani

1

Assistant Professor, Department of Mathematics,

Pope’s College (Autonomous), Sawyerpuram-628251, Thoothukudi, Tamilnadu, INDIA.

2

Associate Professor, Department of Mathematics,

A.P.C. Mahalaxmi College for Women, Thoothukudi-628002, Tamilnadu, INDIA.

12

Affiliated to Manonmaniam Sundaranar University, Tirunelveli-627012, Tamilnadu, INDIA.

email: [email protected]., [email protected].

(Received on: October 30, 2018)

ABSTRACT

Let G = (𝑉, 𝐸) be a graph. The Antimagic Total labeling on 𝑍𝑝+𝑞 is a one- one function mapping both the vertices and edges of the graph onto the group of integers modulo 𝑝 + 𝑞 where 𝑝 and 𝑞 are the vertices and edges in the graph. In this paper we introduce Vertex Antimagic Total labeling, Edge Antimagic Total labeling on 𝑍𝑝+𝑞 and discuss these labelings for Cycles, Stars, Complete bipartite graphs, the Subdivision graphs 𝑆(𝐿𝑛) and 𝑆(𝑃_𝑛⊙ 𝐾1). Also we investigate Totally (𝑎, 𝑑) Edge Antimagic graph, Totally Super Vertex, Edge Antimagic graphs and determine the bounds for the vertices as well as the edges under Total labelling.

AMS Subject Classification: 05C78.

Keywords: Antimagic,.Bipartite, Star, Subdivision graphs.

1. INTRODUCTION

Graph labeling is an assignment of labels to the vertices or edges or to both the vertices

and edges of a graph subject to certain conditions. The concept of Antimagic labeling was

introduced by Hartsfield and Ringel in 1990. In these type of labelings either the weight of all

vertices in a graph or the weight of all edges in a graph are distinct positive integers. In

Antimagic Total labeling, labels are assigned to both the vertices and edges subject to certain

conditions. In a Vertex Antimagic Total labeling the weight of a vertex is the sum of the vertex

label and the labels of the edges incident with that vertex. In an Edge Antimagic Total labeling

the weight of an edge is the the sum of the edge label and the labels of the vertices incident with that edge. In this paper we define Vertex, Edge Antimagic Total labeling in relation with the vertices and edges of a graph and investigate the behaviour of some classes of graphs satisfying the above labeling and specify graphs that do not admit these labelings.

2. VERTEX ANTIMAGIC TOTAL LABELING ON 𝒁

Definition 2.1. For a graph 𝐺 with 𝑝 vertices and 𝑞 edges the Vertex Antimagic Total labeling on 𝑍

is a one-one function 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

such that the induced vertex weights defined by 𝑤𝑡(𝑣) = 𝑓(𝑣) + ∑

𝑓(𝑣𝑢) are all distinct positive integers. Here 𝑁(𝑣) is the neighbourhood of the vertex v. A graph G which admits such a labeling is called a Totally Vertex Antimagic graph on 𝑍

.

Definition 2.2. In a Totally Vertex Antimagic graph if the vertex weights form an Arithmetic Progression with the first term as ‘a’ and Common difference as ‘d’ then the labeling is called as an (𝑎, 𝑑) Vertex Antimagic Total labeling on 𝑍

. If such a labeling exists in a graph then the graph is called a Totally (𝑎, 𝑑) Vertex Antimagic graph.

Definition 2.3. A Vertex Antimagic Total labeling on 𝑍

is said to be a Super Vertex Antimagic Total labeling on 𝑍

if the set of labels of the total graph consists of consecutive integers. A graph G which admits such a labeling is called a Totally Super Vertex Antimagic graph.

Theorem 2.4. All Cycles 𝐶

, Paths 𝑃

admit Vertex Antimagic Total labeling on 𝑍

and the Paths 𝑃

are Totally Super Vertex Antimagic graphs for all n.

Proof. Let 𝐺 be a graph which is either a Path 𝑃

or a Cycle 𝐶

with n vertices. Let the vertices be denoted by 𝑣

, 𝑣

, … , 𝑣

and the edges as 𝑒

, 𝑒

, … , 𝑒

where the Path has 𝑛 − 1 edges and the Cycle has 𝑛 edges.

Define 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

as follows:

𝑓(𝑣

) = 𝑖 − 1 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑛 𝑓(𝑒

) = 𝑝 + 𝑞 − 1

𝑓(𝑒

) = 𝑓(𝑒

) − 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 Example:

0 1 2 3 4 5 6 7 8 9

18 17 16 15 14 13 12 11 10

By the above labeling all the vertex weights are distinct. For the Path 𝑃

, except the

weight of the first and the last vertex, the weights of the other vertices form an arithmetic

progression with the first term as 2(𝑝 + 𝑞) − 2 and the common difference as −1. Similarly

for the Cycle 𝐶

, except the weight of the first vertex the weights of the other vertices form an arithmetic progression with the first term as 2(𝑝 + 𝑞) − 2 and the common difference as

−1. In the path the vertex labels as well as the edge labels are all consecutive integers. Hence the Paths 𝑃

are Totally Super Vertex Antimagic graphs for all 𝑛. Therefore all Cycles 𝐶

, Paths 𝑃

admit Vertex Antimagic Total labeling on 𝑍

.

Theorem 2.5. All Complete bipartite graphs 𝐾

admit Vertex Antimagic Total labeling on 𝑍

for 𝑚, 𝑛 ≥ 2 and are Totally Super Vertex Antimagic graphs for all 𝑚, 𝑛.

Proof. Let G be a Complete bipartite graph 𝐾

. A graph is said to be a Complete bipartite graph is if the vertex set V can be partitioned into two subsets V

and V

such that every line joins a point of V

to a point of V

. Let v

,v

, …,v

be the vertices of V

and u

,u

…,u

be the vertices of V

. Let e

,e

…,e

be the edges of 𝐾

. Therefore we have V = V

∪V

. Let

|𝑉

(𝐺)| = m and |𝑉

(𝐺)| = n. Hence |𝑉(𝐺)| = m+n and |𝐸(𝐺)| = mn.

Case (i). Let 𝑚 = 𝑛 = 2.

Define 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

as follows:

𝑓(𝑣

) = 𝑝 + 𝑞 − 𝑖 𝑓𝑜𝑟 𝑖 = 1,2 𝑓(𝑢

) = 𝑚𝑛 + (𝑖 − 1) 𝑓𝑜𝑟 𝑖 = 1,2

Case (ii). Let 𝑚, 𝑛 ≥ 3. Define 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

as follows:

𝑓(𝑣

) = 𝑝 + 𝑞 − 1

𝑓(𝑣

) = 𝑓(𝑣

) − 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑚 𝑓(𝑢

) = 𝑓(𝑣

) − 𝑚

𝑓(𝑢

) = 𝑓(𝑢

) − 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑒

) = 𝑖 − 1 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑚𝑛

By the above labeling all the vertex weights are distinct positive integers. For all complete bipartitite graphs 𝐾

, 𝑚, 𝑛 ≥ 3 the vertex weights of the set V

form an arithmetic progression with the first term as 𝑝 + 𝑞 − 1 +

and common difference as 𝑚𝑛 − 1 for 𝑚 = 𝑛, 𝑚𝑛 − 𝑚 for 𝑚 > 𝑛, 𝑚𝑛 + 𝑚 for 𝑚 < 𝑛. The vertex weights of

the set V

form an arithmetic progression with the first term as 𝑓(𝑢

) + 𝑛 + 2𝑛 + ⋯ + (𝑚 − 1)𝑛 and common difference as 𝑚 − 1. As the vertices and edges of 𝐾

receive their labels as consecutive integers clearly all complete bipartitite graphs are Totally Super Vertex Antimagic graphs for all 𝑚, 𝑛.

Theorem 2.6. All Star graphs 𝐾

admit Vertex Antimagic Total labeling on 𝑍

and they are Totally Super Vertex Antimagic graphs on 𝑍

.

Proof. Let 𝐺 = 𝐾

be a Star graph with 𝑛 edges and 𝑛 + 1 vertices. Let 𝑣

be the root vertex, v

, i = 1 to n be the pendant vertices and e

, i = 1 to n be the the edges.

Define 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

as follows:

𝑓(𝑣

) = 𝑝 + 𝑞 − 1

𝑓(𝑣

) = 𝑛

𝑓(𝑣

) = 𝑓(𝑣

) + 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑒

) = 𝑖 − 1 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 𝑛

By the above labeling all the vertex weights are distinct positive integers. The weight of the vertex 𝑣

= 𝑝 + 𝑞 − 1 +

and the weights of the pendant vertices 𝑣

, 𝑣

, … , 𝑣

are respectively 𝑛, 𝑛 + 2, 𝑛 + 4, … 𝑛 + (2𝑛 − 2). Therefore the weights of the pendant vertices form an arithmetic progression with the first term as n and common difference as 2.

As the vertex labels as well as the edge labels are all consecutive integers the labeling is popularly termed as a Super Vertex Antimagic Total labeling on 𝑍

. Hence all Star graphs 𝐾

are Totally Super Vertex Antimagic graphs on 𝑍

.

Remark 2.7. If a graph G has a vertex 𝑣 such that 𝑣 is incident with every other vertex of the graph with atleast two cycles then G fails to be a Totally Vertex Antimagic graph on 𝑍

. Hence all friendship graphs, wheel graphs do not admit Vertex Antimagic Total labeling on 𝑍

.

Remark 2.8. For all Vertex Antimagic Total labeling on 𝑍

the vertex weights range from 1 ≤ 𝑤𝑡(𝑣) ≤ ∑

𝑝 + 𝑞 − 𝑖 where 𝑤𝑡(𝑣) is the weight of any vertex under total labeling.

3. EDGE ANTIMAGIC TOTAL LABELING ON 𝒁

Definition 3.1 For a graph 𝐺 with 𝑝 vertices and 𝑞 edges the Edge Antimagic Total labeling on 𝑍

is a one-one function 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

such that the induced edge weights defined by 𝑤(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣) + 𝑓(𝑢𝑣) where 𝑢𝑣 ∈ 𝐸(𝐺) are all distinct positive integers. A graph G which admits such a labeling is called a Totally Edge Antimagic graph on 𝑍

.

Definition 3.2. In a Totally Edge Antimagic graph if the edge weights form an Arithmetic Progression with the first term as ‘a’ and Common difference as ‘d’ then the labeling is called as an (𝑎, 𝑑) Edge Antimagic Total labeling on 𝑍

and the graph is called a Totally (𝑎, 𝑑) Edge Antimagic graph.

Definition 3.3. The Edge Antimagic Total labeling on 𝑍

is said to be a Super Edge Antimagic Total labeling on 𝑍

if the set of labels of the total graph consists of consecutive integers and the graph is called a Totally Super Edge Antimagic graph.

Theorem 3.4. All Friendship graphs got by the one point union of t copies of the Cycle 𝐶

admit Edge Antimagic Total labeling on 𝑍

.

Proof. Let 𝐺 = 𝐶

be a friendship graph with the one point union of t copies of the cycle

𝐶

. Let 𝑝, 𝑞 denote the number of vertices and edges in the graph where 𝑝 = 2𝑡 + 1, 𝑞 =

3𝑡. Let 𝑣

denote the central vertex of 𝐺 and 𝑣

, 𝑣

,…, 𝑣

be the other vertices of the graph

in the clockwise direction. Let 𝑒

, 1 ≤ 𝑖 ≤ 2𝑡 be the edges incident with the central vertex 𝑣

in the clockwise direction. Let the other edges of the graph be 𝑒

, 2t+1 ≤ 𝑖 ≤ 3𝑡 such that the edge 𝑒

is incident with the vertices 𝑣

, 𝑣

.

Define 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

as follows:

𝑓(𝑣

) = 0 𝑓(𝑣

) = 2𝑡 + 1

𝑓(𝑣

) = 𝑓(𝑣

) + 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 2𝑡 𝑓(𝑒

) = 𝑖 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ 2𝑡

𝑓(𝑒

) = 4𝑡 + 1 𝑓𝑜𝑟 𝑖 = 2𝑡 + 1

𝑓(𝑒

) = 𝑓(𝑒

) + 1 𝑓𝑜𝑟 2 𝑡 + 2 ≤ 𝑖 ≤ 3𝑡

Here the innermost edges incident with the vertex 𝑣

receive consecutive integers as their labels. By the above labeling all the edge weights are distinct positive integers. Hence all Friendship graphs got by the one point union of t copies of the Cycle 𝐶

admit Edge Antimagic Total labeling on 𝑍

.

Remark 3.5. Under the same labeling scheme mentioned in theorem 2.4 the paths and cycles also satisfy the condition of Edge Antimagic Total labeling on 𝑍

and the paths are also Totally Super Edge Antimagic graphs as both the vertex and edge labels are consecutive integers. The edge weights of the paths form an arithmetic progression with the first term as 𝑝 + 𝑞 and common difference as 1. Hence all paths are Totally (𝑎, 𝑑) Edge Antimagic graphs.

Also the edge weights of the cycles form an arithmetic progression with the first term as 𝑝 + 𝑞 and common difference as 1 except the n

edge.

Remark 3.6. The Complete bipartite graphs 𝐾

satisfy the condition of Edge Antimagic Total labeling on 𝑍

for 𝑚 = 2,3 and 2 ≤ 𝑛 ≤ 4. All Star graphs admit Edge Antimagic Total labeling on 𝑍

under the same labeling scheme mentioned in theorem 2.6 and the edge weights form an arithmetic progression with the first term as 𝑝 + 𝑞 + 𝑛 − 1 and the common difference as 2. Hence the Star graphs 𝐾

are Totally (𝑎, 𝑑) Edge Antimagic graphs and are also Totally Super Edge Antimagic graphs on 𝑍

.

Theorem 3.7. The Subdivision graph 𝑆(𝑃

⊙ 𝐾

) admits Edge Antimagic Total labeling on 𝑍

for all 𝑛 and are Totally Super Edge Antimagic graphs on 𝑍

.

Proof. The graph obtained by subdividing each edge of a graph 𝐺 is called the subdivision graph of 𝐺 and is denoted by 𝑆(𝐺). If 𝑒 = 𝑢𝑣 is an edge of 𝐺 and 𝑤 is a vertex not in 𝐺 then 𝑒 is said to be subdivided when it is replaced by the edges 𝑢𝑤 and 𝑤𝑣. Let the vertex set of the subdivision graph 𝑆(𝑃

⊙ 𝐾

) be 𝑉(𝑆(𝑃

⊙ 𝐾

)) = {𝑢

: 1 ≤ 𝑖 ≤ 2𝑛 − 1} ∪ {𝑤

, 𝑣

: 1 ≤ 𝑖 ≤ 𝑛} and the edge set be 𝐸(𝑆(𝑃

⊙ 𝐾

)) = {𝑢

𝑢

: 1 ≤ 𝑖 ≤ 2𝑛 − 2} ∪ {𝑢

𝑤

: 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑤

𝑣

: 1 ≤ 𝑖 ≤ 𝑛}. Define 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

as follows:

𝑓(𝑢

) = 0

𝑓(𝑢

) = 𝑖 + 𝑗 𝑓𝑜𝑟 3 ≤ 𝑖 ≤ 2𝑛 − 1, 𝑖 𝑖𝑠 𝑜𝑑𝑑 where for each 𝑖, 𝑗 varies from 0,1,2, ….

𝑓(𝑢

) = 𝑝 + 𝑞 − (𝑛 − 1)

𝑓(𝑢

) = 𝑓(𝑢

) + 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 2𝑛 − 2. 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛

𝑓(𝑤

) = 1

𝑓(𝑤

) = 𝑓(𝑤

) + 3 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑣

) = 2

𝑓(𝑣

) = 𝑓(𝑣

) + 3 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑢

𝑢

) = 𝑝 + 𝑛 + 1

𝑓(𝑢

𝑢

) = 𝑓(𝑢

𝑢

) + 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 2𝑛 − 2 𝑓(𝑢

𝑤

) = 𝑝 − 𝑛 + 1

𝑓(𝑢

𝑤

) = 𝑓(𝑢

𝑤

) + 2 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑤

𝑣

) = 𝑝 − 𝑛 + 2

𝑓(𝑤

𝑣

) = 𝑓(𝑤

𝑣

) + 2 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛

By the above labeling all the edge weights of the subdivision graph 𝑆(𝑃

⊙ 𝐾

) are all distinct positive integers. Hence 𝑆(𝑃

⊙ 𝐾

) admits Edge Antimagic Total labeling on 𝑍

for all 𝑛. As the labels are all consecutive integers the subdivision graphs 𝑆(𝑃

⊙ 𝐾

) are Totally Super Edge Antimagic graphs on 𝑍

.

Theorem 3.8. The Subdivision graph 𝑆(𝐿

) where 𝐿

is a ladder on n vertices admits Edge Antimagic Total labeling on 𝑍

and are Totally Super Edge Antimagic graphs on 𝑍

. Proof. Let 𝑆(𝐿

) denote the Subdivision graph of a ladder 𝐿

on n vertices. Let the vertex set be 𝑉(𝑆(𝐿

)) = {𝑢

, 𝑣

, 𝑤

, 𝑎

𝑏

: 1 ≤ 𝑖 ≤ 𝑛, 1 ≤ 𝑗 ≤ 𝑛 − 1 } and 𝐸(𝑆(𝐿

)) = {𝑢

𝑤

, 𝑤

𝑣

: 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑢

𝑎

, 𝑎

𝑢

, 𝑣

𝑏

, 𝑏

𝑣

: 1 ≤ 𝑖 ≤ 𝑛 − 1} be the edge set of 𝑆(𝐿

).

Define 𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → 𝑍

as follows:

𝑓(𝑢

) = 0

𝑓(𝑢

) = 𝑓(𝑢

) + 2 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑣

) = 2𝑛 − 1

𝑓(𝑣

) = 𝑓(𝑣

) + 2 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑎

) = 1

𝑓(𝑎

) = 𝑓(𝑎

) + 2 𝑓𝑜𝑟 2 ≤ 𝑗 ≤ 𝑛 − 1 𝑓(𝑏

) = 2𝑛

𝑓(𝑏

) = 𝑓(𝑏

) + 2 𝑓𝑜𝑟 2 ≤ 𝑗 ≤ 𝑛 − 1 𝑓(𝑤

) = 𝑝 + 𝑞 − 1

𝑓(𝑤

) = 𝑓(𝑤

) − 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑢

𝑤

) = 𝑞 + 2𝑛 − 2

𝑓(𝑢

𝑤

) = 𝑓(𝑢

𝑤

) + 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑤

𝑣

) = 𝑞 + 3𝑛 − 2

𝑓(𝑤

𝑣

) = 𝑓(𝑤

𝑣

) + 1 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 𝑓(𝑢

𝑎

) = 𝑝 − 𝑛

𝑓(𝑢

𝑎

) = 𝑓(𝑢

𝑎

) + 2 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 − 1 𝑓(𝑎

𝑢

) = 𝑝 − 𝑛 + 1

𝑓(𝑎

𝑢

) = 𝑓(𝑎

𝑢

) + 2 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 − 1

𝑓(𝑣

𝑏

) = 𝑞

𝑓(𝑣

𝑏

) = 𝑓(𝑣

𝑏

) + 2 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 − 1 𝑓(𝑏

𝑣

) = 𝑞 + 1

𝑓(𝑏

𝑣

) = 𝑓(𝑏

𝑣

) + 2 𝑓𝑜𝑟 2 ≤ 𝑖 ≤ 𝑛 − 1

By the above labeling all the edge weights of the subdivision graph 𝑆(𝐿

) are all distinct positive integers. The weights of the edges 𝑢

𝑎

, 𝑎

𝑢

form an arithmetic progression with the first term as 𝑝 − 𝑛 + 1 and common difference as 3, the weights of the edges 𝑣

𝑏

, 𝑏

𝑣

form an arithmetic progression with the first term as 𝑝 + 𝑞 − 𝑛 + 1 and common difference as 3, the weights of the edges 𝑢

𝑤

form an arithmetic progression with the first term as 2(𝑝 + 𝑞) − (3𝑛 + 1) and common difference as 2 and , the weights of the edges 𝑤

𝑣

form an arithmetic progression with the first term as 2(𝑝 + 𝑞) − 2 and common difference as 2. As the labels are all consecutive integers the subdivision graphs 𝑆(𝐿

) are Totally Super Edge Antimagic graphs on 𝑍

.

0 1 2 3

4 5 6 7

8

9 10 11

12 13

14 15

16 18

19 20

21

22 23 24 25

26 27

28 29

30 31 32 33 34

35

36

37

38

39

40

41

42

44 43 45 46 47 48

Edge Antimagic Total labeling of 𝑺(𝑳17𝟓) on 𝒁𝒑+𝒒.

Remark 3.9. If a graph G has a vertex 𝑣 such that 𝑑(𝑣) ≥ 6 and 𝑣 is incident with every other vertex of the graph G then the graph fails to be a Totally Edge Antimagic graph on 𝑍

. In such a case it is not possible to get distinct edge weights.

Remark 3.10. For all Edge Antimagic Total labeling on 𝑍

the edge weights range from 3 ≤ 𝑤𝑡(𝑒) ≤ 3(𝑝 + 𝑞 − 2) where 𝑤𝑡(𝑒) is the weight of any edge under total labeling.

4. CONCLUSION

It is interesting to deal with Antimagic labelings related with the vertices and edges of

a graph. The Vertex Antimagic Total labelings on 𝑍

is not satisfied by those graphs in

which the degree of a vertex is exceedingly large or there is a vertex 𝑣 incident with all other vertices of the graph. Also subdivision graphs with more number of vertices and edges do not satisfy this kind of labeling. As every edge in a graph is incident with two vertices, most of the graphs satisfy the Edge Antimagic Total labelings on 𝑍

. Since Antimagic labelings deal with distinct weights related with the vertices and edges of a graph and the weights do not interfere, the study of antimagic labelings can be used in problems related with independent allocations.

REFERENCES

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

4. M. Baca and I. Hollander, On (a,d) – antimagic prisms, Ars Combin., 48, 297-306 (1998).

5. M. Miller, M. Baca, Antimagic valuations of genetralised Petersen graphs, Australian