On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(1) (2019), Pages 40-48 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: November 19, 2019 Reviewed: November 26, 2019       Accepted: November 30, 2019 
DOI: http://dx.doi.org/10.18860/ca.v6i1.8054  

On  the Local Edge Antimagic Coloring of Corona Product of 
Path and Cycle 

Siti Aisyah1, Ridho Alfarisi3, Rafiantika M. Prihandini2, Arika Indah Kristiana2,  
Ratna Dwi C.1  

1Mathematics Department, Kaltara University of North Kalimantan 
 2 Mathematics Education Department, Jember University of Jember 

3 Primary School Department, Jember University of Jember 
Email: aisyah_rasyid84@yahoo.com, alfarisi.fkip@unej.ac.id, arikakristiana@gmail.com, 

r_christyanti@yahoo.com  

ABSTRACT  

Let 𝐺(𝑉,𝐸) be a nontrivial and connected graph of vertex set 𝑉 and edge set  𝐸. A bijection 𝑓:𝑉(𝐺) →
{1,2,3,…,|𝑉(𝐺)|} is called a local edge antimagic labeling if for any two adjacent edges 𝑒1 and 
𝑒2,𝑤(𝑒1) ≠  𝑤(𝑒2), where 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺),𝑤(𝑒) = 𝑓(𝑢) + 𝑓(𝑣). Thus, the local edge antimagic 
labeling induces a proper edge coloring of 𝐺 if each edge 𝑒 assigned the color  𝑤(𝑒). The color of any 
edge 𝑒 =  𝑢𝑣 is assigned by 𝑤(𝑒) which is defined by the sum of both vertices labels 𝑓(𝑢) and  𝑓(𝑣). 
The local edge antimagic chromatic number, denoted by 𝛾𝑙𝑎𝑒(𝐺) is the minimum number of colors 
taken over all colorings induced by local edge antimagic labeling of 𝐺. In our paper, we present the 
local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle 
corona path, path corona path, and cycle corona cycle. 

Keywords: Local antimagic; edge coloring; corona product; path; cycle. 

INTRODUCTION 

The local antimagic vertex coloring of a graph 𝐺 introduced by Arumugam et. al  [1]. 
Furthermore, Agustin, et. al [2] defined local edge antimagic coloring of the graph. A bijection 
𝑓:𝑉(𝐺) → {1,2,3, . . . , |𝑉(𝐺)|}, is called a local edge antimagic labeling if every two adjacent 
edges 𝑒1 and 𝑒2, 𝑤(𝑒1) ≠  𝑤(𝑒2), where 𝑒 = 𝑢𝑣 ∈  𝐸(𝐺), and 𝑤(𝑒) = 𝑓(𝑢) + 𝑓(𝑣). Thus, the 
local edge antimagic labeling induces a proper edge coloring of 𝐺 if any edge 𝑒 is assigned 
the color 𝑤(𝑒). The color of each edge 𝑒 =  𝑢𝑣 are assigned by 𝑤(𝑒) which is defined by the 
sum of label both and vertices 𝑓(𝑢) and 𝑓(𝑣). The local edge antimagic chromatic number, 
denoted by 𝛾𝑙𝑎𝑒(𝐺), is the minimum number of colors taken over all colorings induced by 
local edge antimagic labeling of graph 𝐺. Agustin, et. al [2] establish the local edge antimagic 
chromatic number of path graph and cycle graph. Dettlaff, et. al in [3], a corona graph of 𝐺 
and 𝐻, denoted by 𝐺 ⊙ 𝐻, is obtained by joining each vertex of Hi to the vertex uj of G.  

Ramya in [4], discussed an acyclic coloring of a corona graph and Yero in [5] studied 
coloring, location, and domination of a corona graph. Kristiana, et.al in [6] found the lower 
bound of the r-dynamic chromatic number of a corona product by wheel graphs. Many 
papers presented a corona product topics for example in [7], [2], and [8]. However, the local 
edge antimagic coloring of corona product still has nothing to discuss. In our paper, we 
investigate the local edge antimagic coloring of corona product of path and cycle, namely 

http://dx.doi.org/10.18860/ca.v6i1.8054
mailto:aisyah_rasyid84@yahoo.com
mailto:alfarisi.fkip@unej.ac.id
mailto:arikakristiana@gmail.com
mailto:r_christyanti@yahoo.com


On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle 

 

Siti Aisyah 41 

path corona cycle, cycle corona path, path corona path, and cycle corona cycle. The results 
of local edge antimagic labeling are as follows. 

Conjecture 1.1. Every connected graph other than 𝐾2 is local antimagic. 
Observation 1.1. [6]For any graph 𝐺, 𝛾𝑙𝑎𝑒(𝐺) ≥ 𝛾(𝐺) − 1. 
Observation 1.2. [8]For any graph G, χla(G) ≥ χ(G), where χ(G) is a vertex chromatic 

number of G. 
Observation 1.3. [6] For any graph 𝐺, 𝛾𝑙𝑎𝑒(𝐺) ≥ 𝛾(𝐺), where 𝛾(𝐺) is an edge chromatic 

number of G. 
Theorem 1.1. [6] If Δ(𝐺) is maximum degrees of 𝐺, then we have 𝛾𝑙𝑎𝑒(𝐺) ≥ Δ(𝐺). 

Proposition 1.1. [6] Let G be a connected graph, we have  
a) If 𝐺 ≅ 𝑃𝑛 ,  then 𝛾𝑙𝑎𝑒(𝐺) = 2. 
b) If 𝐺 ≅ 𝐶𝑛 ,  then 𝛾𝑙𝑎𝑒(𝐺) = 3. 
c) If 𝐺 ≅ 𝐿𝑛 ,  then 𝛾𝑙𝑎𝑒(𝐺) = 3. 
d) If 𝐺 ≅ 𝐾𝑛 ,  then 𝛾𝑙𝑎𝑒(𝐺) = 2𝑛 − 3. 
e) If 𝐺 ≅ 𝑊𝑛 ,  then 𝛾𝑙𝑎𝑒(𝐺) = 𝑛 + 2. 
f) If 𝐺 ≅ 𝑆𝑛 ,  then 𝛾𝑙𝑎𝑒(𝐺) = 𝑛. 
g) If 𝐺 ≅ 𝐹𝑛 ,  then 𝛾𝑙𝑎𝑒(𝐺) = 2𝑛 + 1. 
h) If  𝐺 ⊙ 𝑚𝐾1, then 𝛾𝑙𝑎𝑒(𝐺 ⊙ 𝑚𝐾1) = 𝛾𝑙𝑎𝑒(𝐺) + 𝑚. 

RESULTS AND DISCUSSION  

In our paper, we consider the local edge antimagic chromatic number of a corona 
product of path and cycle, including path corona cycle, cycle corona path, path corona path, 
cycle corona cycle. Furthermore, we determine the exact values of local edge antimagic 
chromatic number of corona product in the following theorems. 

Theorem 2.1. The local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝑃𝑚 for 𝑛 odd and 𝑛,𝑚 ≥
3  is  𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) = 2(𝑛 + 1) + 𝑚.  
Proof. The graph 𝑃𝑛 ⊙ 𝑃𝑚 is a connected graph with vertex set 𝑉(𝑃𝑛 ⊙ 𝑃𝑚) = {𝑥𝑖:1 ≤  𝑖 ≤
 𝑛} ∪ {𝑥𝑗

𝑖:1 ≤  𝑗 ≤ 𝑚;  1 ≤  𝑖 ≤ 𝑛 }   and edge set 𝐸(𝑃𝑛 ⊙ 𝑃𝑚) = {𝑥𝑖𝑥𝑖+1:1 ≤  𝑖 ≤  𝑛 − 1} ∪

{𝑥𝑗
𝑖𝑥𝑗+1
𝑖 : 1 ≤  𝑗 ≤ 𝑚 − 1; 1 ≤  𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑥𝑗

𝑖: 1 ≤  𝑗 ≤ 𝑚; 1 ≤  𝑖 ≤ 𝑛}. The cardinality of the 

vertex set is |𝑉(𝑃𝑛 ⊙ 𝑃𝑚)| = 𝑛 + 𝑚𝑛 and the cardinality of the edge set is |𝐸(𝑃𝑛 ⊙ 𝑃𝑚)| =
2𝑚𝑛 − 1. We define a bijection 𝑓:𝑉(𝑃𝑛 ⊙ 𝑃𝑚) → {1,2,3,…, |𝑉(𝑃𝑛 ⊙ 𝑃𝑚)|} for the graph 𝑃𝑛 ⊙
𝑃𝑚 to be local edge antimagic labeling as follows. 

𝑓(𝑥𝑖) = {

𝑖 + 1

2
, if 𝑖 is odd

𝑛 − (
𝑖 − 2

2
), if 𝑖 is even

 



On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle 

 

Siti Aisyah 42 

𝑓(𝑥𝑗
𝑖) =

{
 
 
 

 
 
 𝑛 + 1 + (

𝑖 − 2

2
) + (

𝑗 − 2

2
)𝑛,                            if 𝑖 and 𝑗 are even

2𝑛 + 𝑛⌈
𝑚

2
⌉ + 1 + (

𝑖 − 2

2
) − 𝑛(

𝑗 − 1

2
), if 𝑖 is even and 𝑗 is odd

2𝑛 − (
𝑖 − 1

2
) + (

𝑗 − 2

2
)𝑛,                                  if  𝑖 is odd and 𝑗 is even

𝑚𝑛 + 𝑛 − (
𝑖 − 1

2
) − 𝑛(

𝑗 − 1

2
),                    if 𝑖 and 𝑗 are odd

 

It is clear that 𝑓 is a local antimagic labeling of 𝑃𝑛 ⊙ 𝑃𝑚 and the edge weights are as 
follows: 

𝑤(𝑥𝑖𝑥𝑖+1) = {
𝑛 + 1, if 𝑖 is odd
𝑛 + 2, if 𝑖 is even

 

𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 ) =

{
 

 
𝑚𝑛 + 3𝑛 − (𝑖 − 1), if 𝑖 and j are odd

𝑚𝑛 + 2𝑛 − (𝑖 − 1), if 𝑖 is odd and  𝑗 is even
𝑚𝑛 + 𝑛 + 𝑖,                 if 𝑖 is even and  𝑗 is odd

𝑚𝑛 + 𝑖,                if 𝑖  and 𝑗 are even

 

𝑤(𝑥𝑖𝑥𝑗
𝑖) = {

𝑚𝑛 + 1 + 𝑛(
𝑗 − 3

2
), if 𝑗 is odd

2𝑛 + 1 + 𝑛(
𝑗 − 2

2
), if 𝑗 is even

 

Hence, we get that the upper bound of the local edge antimagic chromatic number of 
𝑃𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) ≤ 2(𝑛 + 1) + 𝑚. Furthermore, we prove that lower bound of the 
local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) ≥ 2(𝑛 + 1) + 𝑚. By 
contradiction, we assume that 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) < 2(𝑛 + 1) + 𝑚. Without lost of generality, we 

assume that 𝑤(𝑥𝑖𝑥𝑖+1) ≠ 𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 ) ≠ 𝑤(𝑥𝑖𝑥𝑗

𝑖). Based on Proposition 1, 𝛾𝑙𝑎𝑒(𝑃𝑛) = 2 and 

𝛾𝑙𝑎𝑒(𝑃𝑚) = 2 then we get |{𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛)}| = 2, |{𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 )}| = 𝑚 and |{𝑤(𝑒);𝑒 ∈

𝐸((𝑃𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| = 2(𝑛 − 1), |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| = 1 such that 
|𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝑃𝑚 )| = |{𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛)}| 

+|{𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 )}| + |{𝑤(𝑒);𝑒 ∈  𝐸((𝑃𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| 

|𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝑃𝑚 )| = 2 + 𝑚 + 2(𝑛 − 1) + 1 
|𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝑃𝑚 )| = 𝑚 + 2𝑛 + 1 

If |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| = 1, then we obtain at least two edges which have same edge 
weight, it is a contradiction. Thus, we receive that the lower bound of the local edge 
antimagic chromatic number of  𝑃𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) ≥ 𝑚 + 2𝑛 + 2 = 2(𝑛 + 1) + 𝑚. It 
concludes that the local antimagic edge chromatic number of 𝑃𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝑃𝑚) =
2(𝑛 + 1) + 𝑚.∎            

Theorem 2.2  The local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝐶𝑚 for 𝑛,𝑚 odd and 
𝑛,𝑚 ≥ 4 is  𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) = 2 + 3𝑛 + 𝑚.  
Proof. The graph 𝑃𝑛 ⊙ 𝐶𝑚 is a connected graph with vertex set 𝑉(𝑃𝑛 ⊙ 𝐶𝑚) = {𝑥𝑖:1 ≤  𝑖 ≤
 𝑛} ∪ {𝑥𝑗

𝑖:1 ≤  𝑗 ≤ 𝑚;  1 ≤  𝑖 ≤ 𝑛 }   and edge set 𝐸(𝑃𝑛 ⊙ 𝐶𝑚) = {𝑥𝑖𝑥𝑖+1:1 ≤  𝑖 ≤  𝑛 − 1} ∪

{𝑥𝑗
𝑖𝑥𝑗+1
𝑖 : 1 ≤  𝑗 ≤ 𝑚 − 1; 1 ≤  𝑖 ≤ 𝑛} ∪ {𝑥𝑚

𝑖 𝑥1
𝑖 ,1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑥𝑗

𝑖: 1 ≤  𝑗 ≤ 𝑚;  1 ≤  𝑖 ≤ 𝑛}. 

The cardinality of the vertex set is |𝑉(𝑃𝑛 ⊙ 𝐶𝑚)| = 𝑛 + 𝑚𝑛 and the cardinality of the edge 
set is  |𝐸(𝑃𝑛 ⊙ 𝐶𝑚)| = 2𝑚𝑛 + 𝑛 − 1. We define a function bijection 𝑓:𝑉(𝑃𝑛 ⊙ 𝐶𝑚) →
{1,2,3,…,|𝑉(𝑃𝑛 ⊙ 𝐶𝑚)|} for the graph 𝑃𝑛 ⊙ 𝐶𝑚 to be local edge antimagic labeling as follows. 
 



On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle 

 

Siti Aisyah 43 

𝑓(𝑥𝑖) = {

𝑖 + 1

2
,           if 𝑖 is odd

𝑛 −
𝑖 − 2

2
, if 𝑖 is even

 

 

𝑓(𝑥𝑗
𝑖) =

{
 
 
 

 
 
 𝑚𝑛 + 1 + (

𝑖 − 2

2
) − (

𝑗 − 2

2
)𝑛,                            if 𝑖  and 𝑗 are even

𝑛 + 1 + (
𝑖 − 2

2
) + (

𝑗 − 1

2
)𝑛,                               if 𝑖 is even and 𝑗 is odd

𝑚𝑛 + 𝑛 − (
𝑖 − 1

2
) − (

𝑗 − 2

2
)𝑛,                              if  𝑖 is odd and 𝑗 is even

2𝑛 − (
𝑖 − 1

2
) + 𝑛(

𝑗 − 1

2
),                                     if 𝑖 and 𝑗 are odd

 

 

𝑓(𝑥𝑚
𝑖 ) = {

𝑛⌈
𝑚

2
⌉ + 𝑛 − (

𝑖 − 1

2
), if 𝑖 is odd

𝑛⌈
𝑚

2
⌉ + 1 + (

𝑖 − 2

2
), if 𝑖 is even

 

 
It is easy to see that 𝑓 is a local edge antimagic labeling of 𝑃𝑛 ⊙ 𝐶𝑚 and the edge 

weights are as follows: 
 

𝑤(𝑥𝑖𝑥𝑖+1) = {
𝑛 + 1, if 𝑖 is odd
 𝑛 + 2, if 𝑖 is even

 

 

𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 ) =

{
 

 
𝑚𝑛 + 4𝑛 − (𝑖 − 1),                              if 𝑖  and 𝑗 are odd

𝑚𝑛 + 3𝑛 − (𝑖 − 1),                             if 𝑖 is odd and 𝑗 is even

𝑚𝑛 + 2(𝑛 + 1) + (𝑖 − 2),                if 𝑖 is even and 𝑗 is odd

𝑚𝑛 + 𝑛 + 2 + (𝑖 − 2),                      if 𝑖 and 𝑗 are even

 

 

𝑤(𝑥𝑚
𝑖 𝑥1

𝑖) = {
𝑛⌈
𝑚

2
⌉ + 3𝑛 − (𝑖 − 1), if 𝑗 is odd

𝑛⌈
𝑚

2
⌉+ 𝑛 + 2 + (𝑖 − 2), if 𝑗 is even

 

 

𝑤(𝑥𝑖𝑥𝑗
𝑖) = {

2𝑛 + 1 + 𝑛(
𝑗 − 1

2
), if 𝑗 is odd

𝑚𝑛 + 𝑛 − 𝑛(
𝑗 − 2

2
), if 𝑗 is even

 

𝑤(𝑥𝑖𝑥𝑚
𝑖 ) = {

𝑛⌈
𝑚

2
⌉+ 𝑛 + 1 − (

𝑖 − 1

2
), if 𝑗 is odd

𝑛⌈
𝑚

2
⌉ + 2 + (

𝑖 − 2

2
), if 𝑗 is even

 

 
Hence, we get that the upper bound of the local edge antimagic chromatic number of 

𝑃𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) ≤ 2 + 3𝑛 + 𝑚. Furthermore, we prove that the lower bound of the 



On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle 

 

Siti Aisyah 44 

local edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) ≥ 2 + 3𝑛 + 𝑚. By 
contradiction, we assume that 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) < 2 + 3𝑛 + 𝑚 . Without  of generality, 
𝑤(𝑥𝑖𝑥𝑖+1) ≠ 𝑤(𝑥𝑗

𝑖𝑥𝑗+1
𝑖 ) ≠ 𝑤(𝑥1

𝑖𝑥𝑚
𝑖 ) ≠ 𝑤(𝑥𝑖𝑥𝑗

𝑖). Based on  Proposition 1 that 𝛾𝑙𝑎𝑒(𝑃𝑛) = 2 

and 𝛾𝑙𝑎𝑒(𝐶𝑚) = 3 then we get |{𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛)}| = 2, |{𝑤(𝑥𝑖𝑥𝑗
𝑖)}| = 𝑚 and |{𝑤(𝑒);𝑒 ∈

𝐸((𝐶𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| = 3(𝑛 − 1), |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 2 such that 
|𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝐶𝑚 )| = |{𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛)}| + |{𝑤(𝑥𝑖𝑥𝑗

𝑖)}| + |{𝑤(𝑒);𝑒 ∈  𝐸((𝐶𝑚)𝑖),1 ≤

𝑖 ≤ 𝑛 − 1}|+|{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| |𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝐶𝑚 )| 
= 2 + 𝑚 + 3(𝑛 − 1) + 2 |𝑤(𝑒);𝑒 ∈ 𝐸(𝑃𝑛 ⊙ 𝐶𝑚 )| = 𝑚 + 3𝑛 + 1 
If |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 2, then we obtain at least two edges which have the same edge 
weight, which is a contradiction. Accordingly,  the lower bound of the local edge antimagic 
chromatic number of  𝑃𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) ≥ 𝑚 + 3𝑛 + 2. It concludes that the local 
edge antimagic chromatic number of 𝑃𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝑃𝑛 ⊙ 𝐶𝑚) = 2 + 3𝑛 + 𝑚.                                  
∎   

                                                                       
Theorem 2.3. The local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝐶𝑚 for 𝑛,𝑚 even and 
 𝑛,𝑚 ≥ 4 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) = 3(𝑛 + 1) + 𝑚.  
Proof. The graph 𝐶𝑛 ⊙ 𝐶𝑚 is a connected graph with vertex set 𝑉(𝐶𝑛 ⊙ 𝐶𝑚) = {𝑥𝑖:1 ≤  𝑖 ≤
 𝑛} ∪ {𝑥𝑗

𝑖:1 ≤  𝑗 ≤ 𝑚;  1 ≤  𝑖 ≤ 𝑛 }   and edge set 𝐸(𝐶𝑛 ⊙ 𝐶𝑚) = {𝑥𝑖𝑥𝑖+1:1 ≤  𝑖 ≤  𝑛 − 1} ∪

{𝑥1𝑥𝑛} ∪ {𝑥𝑗
𝑖𝑥𝑗+1
𝑖 : 1 ≤  𝑗 ≤ 𝑚 − 1; 1 ≤  𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑥𝑗

𝑖: 1 ≤  𝑗 ≤ 𝑚; 1 ≤  𝑖 ≤ 𝑛} ∪ {𝑥𝑚
𝑖 𝑥1

𝑖 ,1 ≤

𝑖 ≤ 𝑛}. The cardinality of the vertex set is |𝑉(𝐶𝑛 ⊙ 𝐶𝑚)| = 𝑛 + 𝑚𝑛 and the cardinality of the 
edge set is |𝐸(𝐶𝑛 ⊙ 𝐶𝑚)| = 2𝑚𝑛 + 𝑛. We define a function bijection 𝑓:𝑉(𝐶𝑛 ⊙ 𝐶𝑚) →
{1,2,3,…,|𝑉𝐶𝑛 ⊙ 𝐶𝑚)|} for the graph 𝐶𝑛 ⊙ 𝐶𝑚 to be local edge antimagic labeling as follows. 
 

𝑓(𝑥𝑖) = {

𝑖 + 1

2
, if 𝑖 is odd

𝑛 −
𝑖 − 2

2
, if 𝑖 is even

 

 

𝑓(𝑥𝑗
𝑖) =

{
 
 
 

 
 
 𝑚𝑛 + 1 + (

𝑖 − 2

2
) − (

𝑗 − 1

2
)𝑛,                            if 𝑖  and 𝑗 are even

𝑛 + 1 + (
𝑖 − 2

2
) + (

𝑗 − 1

2
)𝑛,                               if 𝑖 is even and 𝑗 is odd

𝑚𝑛 + 𝑛 − (
𝑖 − 1

2
) − (

𝑗 − 2

2
)𝑛,                              if  𝑖 is odd and 𝑗 is even

2𝑛 − (
𝑖 − 1

2
) + 𝑛(

𝑗 − 1

2
),                                     if 𝑖 and 𝑗 are odd

 

 

𝑓(𝑥𝑚
𝑖 ) = {

𝑛⌈
𝑚

2
⌉ + 𝑛 − (

𝑖 − 1

2
),           𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

𝑛⌈
𝑚

2
⌉ + 1 + (

𝑖 − 2

2
),             𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 

It is easy to see that 𝑓 is a local edge antimagic labeling of 𝐶𝑛 ⊙ 𝐶𝑚 and the edge 
weights are as follows: 



On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle 

 

Siti Aisyah 45 

𝑤(𝑥𝑖𝑥𝑖+1) = {
𝑛 + 1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑
𝑛 + 2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 

𝑤(𝑥1𝑥𝑛) =
𝑛 + 2

2
 

 

𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 ) =

{
 

 
𝑚𝑛 + 4𝑛 − (𝑖 − 1),                              𝑖𝑓 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑑𝑑

𝑚𝑛 + 3𝑛 − (𝑖 − 1),                             𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛

𝑚𝑛 + 2(𝑛 + 1) + (𝑖 − 2),                𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑜𝑑𝑑

𝑚𝑛 + 𝑛 + 2 + (𝑖 − 2),                      𝑖𝑓 𝑖  𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑒𝑣𝑒𝑛

 

 

𝑤(𝑥𝑚
𝑖 𝑥1

𝑖) = {
𝑛⌈
𝑚

2
⌉ + 3𝑛 − (𝑖 − 1),𝑖𝑓 𝑗 𝑖𝑠 𝑜𝑑𝑑

𝑛⌈
𝑚

2
⌉ + 𝑛 + 2 + (𝑖 − 2), 𝑖𝑓 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 

𝑤(𝑥𝑖𝑥𝑗
𝑖) = {

2𝑛 + 1 + 𝑛(
𝑗 − 1

2
), 𝑖𝑓 𝑗 𝑖𝑠 𝑜𝑑𝑑

𝑚𝑛 + 𝑛 − 𝑛(
𝑗 − 2

2
),𝑖𝑓 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 

𝑤(𝑥𝑖𝑥𝑚
𝑖 ) = {

𝑛⌈
𝑚

2
⌉+ 𝑛 + 1 − (

𝑖 − 1

2
),𝑖𝑓 𝑗 𝑖𝑠 𝑜𝑑𝑑

𝑛⌈
𝑚

2
⌉+ 2 + (

𝑖 − 2

2
),𝑖𝑓 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 
Hence, we get that the upper bound of the local edge antimagic chromatic number of 𝐶𝑛 ⊙
𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) ≤ 3(𝑛 + 1) + 𝑚. Furthermore, we prove that lower bound of the local 
edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) ≥ 3(𝑛 + 1) + 𝑚. By 
contradiction, we assume that 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) < 3(𝑛 + 1) + 𝑚 . Without lost of generality, 

we gives that 𝑤(𝑥𝑖𝑥𝑖+1) ≠ 𝑤(𝑥1𝑥𝑛) ≠  𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 ) ≠ 𝑤(𝑥1

𝑖𝑥𝑚
𝑖 ) ≠ 𝑤(𝑥𝑖𝑥𝑗

𝑖). Based on 

Proposition 1 that 𝛾𝑙𝑎𝑒(𝐶𝑛) = 3 and 𝛾𝑙𝑎𝑒(𝐶𝑚) = 3 then we get |{𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛)}| = 3, 
|{𝑤(𝑥𝑖𝑥𝑗

𝑖)}| = 𝑚 and |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| = 3(𝑛 − 1), |{𝑤(𝑒);𝑒 ∈

𝐸((𝐶𝑚)𝑛)}| = 2 such that 
|𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛 ⊙ 𝐶𝑚 )|

= |{𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛)}| + |{𝑤(𝑥𝑖𝑥𝑗
𝑖)}| + |{𝑤(𝑒);𝑒 ∈  𝐸((𝐶𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}|

+ |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 3 + 𝑚 + 3(𝑛 − 1) + 2 = 𝑚 + 3𝑛 + 2 
 

If  |{𝑤(𝑒);𝑒 ∈ 𝐸((𝐶𝑚)𝑛)}| = 2, then we obtain at least two edges which have same 
edge weight, which is a contradiction. Thus, we receive that the lower bound of the local 
edge antimagic chromatic number of  𝐶𝑛 ⊙ 𝐶𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) ≥ 𝑚 + 3𝑛 + 3. It concludes 
that the local edge antimagic chromatic number of  𝐶𝑛 ⊙ 𝐶𝑚 is  𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝐶𝑚) = 3(𝑛 + 1) +
𝑚.∎      

                                       
  



On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle 

 

Siti Aisyah 46 

Theorem 2.4. The local edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝑃𝑚  for 𝑛 odd and 𝑛,𝑚 ≥
3 is𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) = 3 + 2𝑛 + 𝑚.  
Proof. The graph 𝐶𝑛 ⊙ 𝑃𝑚 is a connected graph with vertex set 𝑉(𝐶𝑛 ⊙ 𝑃𝑚) = {𝑥𝑖:1 ≤  𝑖 ≤
 𝑛} ∪ {𝑥𝑗

𝑖:1 ≤  𝑗 ≤ 𝑚;  1 ≤  𝑖 ≤ 𝑛 }   and edge set 𝐸(𝐶𝑛 ⊙ 𝑃𝑚) = {𝑥𝑖𝑥𝑖+1:1 ≤  𝑖 ≤  𝑛 − 1} ∪

{𝑥1𝑥𝑛} ∪ {𝑥𝑗
𝑖𝑥𝑗+1
𝑖 : 1 ≤  𝑗 ≤ 𝑚 − 1; 1 ≤  𝑖 ≤ 𝑛} ∪ {𝑥𝑖𝑥𝑗

𝑖: 1 ≤  𝑗 ≤ 𝑚; 1 ≤  𝑖 ≤ 𝑛}. The 

cardinality of the vertex set is |𝑉(𝐶𝑛 ⊙ 𝑃𝑚)| = 𝑛 + 𝑚𝑛 and the cardinality of the edge set is 
|𝐸(𝐶𝑛 ⊙ 𝑃𝑚)| = 2𝑚𝑛 + 𝑛 − 1. We define a function bijection 𝑓:𝑉(𝐶𝑛 ⊙ 𝑃𝑚) →
{1,2,3,…,|𝑉(𝐶𝑛 ⊙ 𝑃𝑚)|} for the graph 𝐶𝑛 ⊙ 𝑃𝑚 to be local edge antimagic labeling as follows. 
 

𝑓(𝑥𝑖) = {

𝑖 + 1

2
,           𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

𝑛 −
𝑖 − 2

2
, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 

𝑓(𝑥𝑗
𝑖) =

{
 
 
 

 
 
 𝑛 + 1 + (

𝑖 − 2

2
) + (

𝑗 − 2

2
)𝑛,                            𝑖𝑓 𝑖 𝑎𝑛𝑑  𝑗 𝑎𝑟𝑒 𝑒𝑣𝑒𝑛

2𝑛 + 𝑛⌈
𝑚

2
⌉ + 1 + (

𝑖 − 2

2
) − 𝑛(

𝑗 − 1

2
), 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑜𝑑𝑑

2𝑛 − (
𝑖 − 1

2
) + (

𝑗 − 2

2
)𝑛,                                  𝑖𝑓  𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛

𝑚𝑛 + 𝑛 − (
𝑖 − 1

2
) − 𝑛(

𝑗 − 1

2
),                    𝑖𝑓 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑑𝑑

 

 
 

It is easy to see that 𝑓 is a local edge antimagic labeling of 𝑃𝑛 ⊙ 𝐶𝑚 and the edge 
weights are as follows: 

 

𝑤(𝑥𝑖𝑥𝑖+1) = {
𝑛 + 1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑
𝑛 + 2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 

𝑤(𝑥1𝑥𝑛) =
𝑛 + 2

2
 

 

(𝑥𝑖𝑥𝑖+1) = {
𝑛 + 1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑
𝑛 + 2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 

𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 ) =

{
 

 
𝑚𝑛 + 3𝑛 − (𝑖 − 1),𝑖𝑓 𝑖 𝑎𝑛𝑑  𝑗 𝑎𝑟𝑒 𝑜𝑑𝑑

𝑚𝑛 + 2𝑛 − (𝑖 − 1), 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛
𝑚𝑛 + 𝑛 + 𝑖,                 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑜𝑑𝑑
𝑚𝑛 + 2 + 𝑖,                𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 

𝑤(𝑥𝑖𝑥𝑗
𝑖) = {

𝑚𝑛 + 1 + 𝑛(
𝑗 − 3

2
),𝑖𝑓 𝑗 𝑖𝑠 𝑜𝑑𝑑

2𝑛 + 1 + 𝑛(
𝑗 − 2

2
),𝑖𝑓 𝑗 𝑖𝑠 𝑒𝑣𝑒𝑛

 

 



On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle 

 

Siti Aisyah 47 

Hence, we get that the upper bound of the local edge antimagic chromatic number of 𝐶𝑛 ⊙
𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) ≤ 3 + 2𝑛 + 𝑚. furthermore, we prove that the lower bound of the local 
edge antimagic chromatic number of 𝐶𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) ≥ 3 + 2𝑛 + 𝑚. By 
contradiction, we assume that 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) < 3 + 2𝑛 + 𝑚 . Without lost of generality, we 

gives that 𝑤(𝑥𝑖𝑥𝑖+1) ≠  𝑤(𝑥𝑗
𝑖𝑥𝑗+1
𝑖 ) ≠ 𝑤(𝑥1

𝑖𝑥𝑚
𝑖 ) ≠ 𝑤(𝑥𝑖𝑥𝑗

𝑖). Based on Proposition 1 that 

𝛾𝑙𝑎𝑒(𝐶𝑛) = 3 and 𝛾𝑙𝑎𝑒(𝑃𝑚) = 2 then we get |{𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛)}| = 3, |{𝑤(𝑥𝑖𝑥𝑗
𝑖)}| = 𝑚 and 

|{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| = 2(𝑛 − 1), |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| = 1 such that 
|𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛 ⊙ 𝑃𝑚 )| = |{𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛)}| + |{𝑤(𝑥𝑖𝑥𝑗

𝑖)}| + 

|{𝑤(𝑒);𝑒 ∈  𝐸((𝑃𝑚)𝑖),1 ≤ 𝑖 ≤ 𝑛 − 1}| + |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| 
|𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛 ⊙ 𝑃𝑚 )| = 3 + 𝑚 + 2(𝑛 − 1) + 1 

|𝑤(𝑒);𝑒 ∈ 𝐸(𝐶𝑛 ⊙ 𝑃𝑚 )| = 𝑚 + 2𝑛 + 2 
If |{𝑤(𝑒);𝑒 ∈ 𝐸((𝑃𝑚)𝑛)}| = 1, then we obtain at least two edges which have same edge 

weight, Which is a contradiction. Thus, we receive that the lower bound of the local edge 
antimagic chromatic number of  𝐶𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) ≥ 𝑚 + 2𝑛 + 3. It concludes that 
the local edge antimagic chromatic number of  𝐶𝑛 ⊙ 𝑃𝑚 is 𝛾𝑙𝑎𝑒(𝐶𝑛 ⊙ 𝑃𝑚) = 3 + 2𝑛 + 𝑚.∎   

 

CONCLUSIONS 

In this paper we have given the result on the local edge antimagic chromatic number 
of corona product of path and cycle, namely path corona cycle, cycle corona path, path 
corona path, cycle corona cycle. 
Open Problem 1. What is the upper bound of local edge antimagic coloring of corona 
product of a connected graph? 
Open Problem 2. What is the lower bound of local edge antimagic coloring of corona 
product of a connected graph?  

 

ACKNOWLEDGMENT 

We gratefully acknowledge the support from DIKTI Indonesia trough the “Penelitian Dosen 
Pemula” RISTEKDIKTI 2019 research project. 

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On  the Local Edge Antimagic Coloring of Corona Product of Path and Cycle 

 

Siti Aisyah 48 

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