On the study of Rainbow Antimagic Coloring of Special Graphs


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(4) (2023), Pages 585-596 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: October 19, 2022 Reviewed: February 20, 2023 Accepted: March 20, 2023 
DOI: http://dx.doi.org/10.18860/ca.v7i4.17836  

 On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik1,2,*,  Riniatul Nur Wahidah 1,2, Ermita Rizki Albirri 1,2 , Sharifah Kartini Said 
Husain 3 

1Mathematics Edu. Depart. University of Jember, Indonesia  
2PUI-PT Combinatorics and Graph, CGANT, University of Jember, Indonesia 
3Institute for Mathematical Research, University Putra Malaysia, Malaysia 

 

Email: d.dafik@unej.ac.id  

ABSTRACT  

Let 𝐺 be a connected graph with vertex set 𝑉(𝐺) and edge set 𝐸(𝐺). The bijective function 𝑓:𝑉(𝐺) →
{1,2,…,|𝑉(𝐺)|} is said to be a labeling of graph where 𝑤(𝑥𝑦) = 𝑓(𝑥) + 𝑓(𝑦) is the associated weight for 
edge 𝑥𝑦 ∈ 𝐸(𝐺). If every edge has different weight, the function 𝑓 is called an edge antimagic vertex 
labeling. A path 𝑃 in the vertex-labeled graph 𝐺, with every two edges 𝑥𝑦,𝑥′𝑦′ ∈  𝐸(𝑃) satisfies 𝑤(𝑥𝑦) ≠
𝑤(𝑥′𝑦′) is said to be a rainbow path. The function 𝑓 is called a rainbow antimagic labeling of 𝐺, if for every 
two vertices 𝑥,𝑦 ∈  𝑉(𝐺), there exists a rainbow 𝑥 − 𝑦 path. Graph 𝐺 admits the rainbow antimagic 
coloring, if we assign each edge 𝑥𝑦 with the color of the edge weight  𝑤(𝑥𝑦). The smallest number of colors 
induced from all edge weights of edge antimagic vertex labeling is called a rainbow antimagic connection 
number of 𝐺, denoted by 𝑟𝑎𝑐(𝐺). In this paper, we study rainbow antimagic connection numbers of 
octopus graph 𝑂𝑛, sandat graph 𝑆𝑡𝑛, sun flower graph 𝑆𝑓𝑛, volcano graph 𝑉𝑛 and semi jahangir graph 𝐽𝑛. 
 
Copyright © 2023 by Authors, Published by CAUCHY Group. This is an open access article under the CC 
BY-SA License (https://creativecommons.org/licenses/by-sa/4.0/) 
  
Keywords: antimagic labeling; rainbow coloring; rainbow antimagic connection number; special graphs 

INTRODUCTION 

The definition of graph used in this paper follows from Chartrand and Zhang [9]. 
In the latest days, graph theory has many applications, one of them is graph coloring. 
The application of graph coloring can be found in many area, such as data mining, image 
segmentation, clustering, image capturing, networking. Chartrand, et al. [10] extended 
the graph coloring concept into a rainbow coloring of graph. Let 𝑐:𝐸(𝐺) →
 {1,2, . . . ,𝑘},𝑘 ∈ ℕ be the edge coloring of a connected graph where the two adjacent 
edges may have the same color. If for every two vertices 𝑥,𝑦 ∈ 𝑉(𝐺), there exists a 
rainbow 𝑥 − 𝑦 path, if no two edges of the 𝑥 − 𝑦 path are the same color, then the path is 
called a rainbow path. A coloring of graph 𝐺 is said to be rainbow connection, if for every 
two vertices 𝑥,𝑦 ∈ 𝑉(𝐺)  have a rainbow 𝑥 − 𝑦 path. 

The edge colored 𝐺 which every two different vertices have a rainbow connection 
is called rainbow coloring of graph, see [10]. Some results in regards to the concept of 
rainbow coloring of graphs can  been found by Nabila, et al [21] and Ma, et al. [19]. Some 
other type of rainbow coloring are rainbow vertex coloring and rainbow total coloring. 
Some relevant results of rainbow vertex coloring can be found in Lie. H, et al. [15], 

http://dx.doi.org/10.18860/ca.v7i4.17836
https://creativecommons.org/licenses/by-sa/4.0/


On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 586 

Bustan et al. [8] and Li. X  et al. [17], while some results of total rainbow coloring can be 
found results in Lie. H et al. [16] and Ma. Y et al.[20].  

Furthermore, the other concepts in graph theory is graph labeling, one of the 
concept of graph labeling is an antimagic labeling of graph 𝐺, defined by Hartsfield and 
Ringel [13]. Baca et al. has found some  antimagic labeling results in [4], [5], [6]. 
Moreover, some results on antimagic labeling have been contributed by Dafik 𝑒𝑡 𝑎𝑙. in 
[11]. In addition, the research on antimagic labeling can also be found in several papers  
[2], [22], [25]. 

Arumugam et al. [3], defined a new concept by combining graph coloring and 
graph labeling. The bijective function 𝑓 ∶  𝐸(𝐺) → {1,2, . . . , |𝐸(𝐺)|}, the vertex weight of 
the vertex 𝑥 is 𝑤(𝑥) = ∑ 𝑓(𝑥𝑦) 𝑥𝑦∈𝐸(𝑥) and 𝐸(𝑥) is the set of edges incident to 𝑥 for every 

𝑥 ∈ 𝑉(𝐺). If for every two adjacent vertices 𝑥,𝑦 ∈ 𝑉 (𝐺),𝑤(𝑥) ≠ 𝑤(𝑦), then the bijective 
function 𝑓 is called a local antimagic labeling. So, each local antimagic label is a vertex 
coloring in 𝐺 with vertex 𝑥 colored with 𝑤(𝑥). Based on the definition of Arumugam [3], 
Dafik 𝑒𝑡 𝑎𝑙. [12] defined the combination of the concepts of antimagic labeling and 
rainbow coloring into a new concept called rainbow antimagic coloring. 

In this study, we will study the combination of rainbow coloring and antimagic 
labeling, and it tends to the new notion, namely a rainbow antimagic coloring. The lower 
bound of the rainbow antimagic connection number has been determined in Septory et 
al. stated in the following lemma. 

 
Lemma 1. Let 𝐺 be any connected graph. Let 𝑟𝑐(𝐺) and Δ(𝐺) be the rainbow connection 
number of 𝐺 and the maximum degree of 𝐺,  𝑟𝑎𝑐(𝐺) ≥ 𝑚𝑎𝑥 {𝑟𝑐(𝐺),Δ(𝐺)}.  

 
While Dafik et al. also characterised the existence of rainbow 𝑢 − 𝑣 path of any 

graph of 𝑑𝑖𝑎𝑚(𝐺) ≤ 2 in the following theorem. 
 

Theorem 1. Let 𝐺 be a connected graph of diameter 𝑑𝑖𝑎𝑚(𝐺) ≤ 2. Let 𝑓 be any bijective 
function from 𝑉(𝐺) to the set {1,2,…, |𝑉(𝐺)| }, there exists a rainbow 𝑥 − 𝑦 path. 

 
Some other results in regards on this notion can be read on [1], [7], [12], [14], 

[23] and [24]. In this paper, we will study the rainbow antimagic connection number of 
octopus graph 𝑂𝑛, sandat graph 𝑆𝑡𝑛, sun flower graph 𝑆𝑓𝑛, volcano graph 𝑉𝑛 and semi 
jahangir graph 𝑆𝐽𝑛. 
 
METHOD 

To determine the number of rainbow antimagic coloring of graph, we use the 
following steps: 
1. For any graph 𝐺, identify the set of vertices 𝑉(𝐺) and set of edges 𝐸(𝐺). 
2. Analyze the lower bound of rainbow antimagic connection number (𝑟𝑎𝑐) based on 

Lemma:  𝑟𝑎𝑐(𝐺) ≥ max {𝑟𝑐(𝐺),Δ(𝐺)}. 
3. Label the vertices of the graph 𝐺 with the function: 𝑉(𝐺) → {1,2,3, . . . , |𝑉(𝐺)|}. 
4. Determine the edge weight based on the sum of vertex label which incident with the 

edge. To calculate edge weight we give the function,  𝑤(𝑢𝑣) = 𝑓(𝑢) + 𝑓(𝑣) for 
𝑢,𝑣 𝜖 𝑉(𝐺). 

5. Verify that every two vertex in the graph 𝐺 have rainbow paths. If not, repeat the step 
3. 

6. Determine the upper bound of 𝑟𝑎𝑐(𝐺) from the number of different edge weight. 
7. The exact value of rainbow antimagic connection number can be determined if lower 



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 587 

bound is the same with upper bound of rainbow antimagic connection number. 
 

  



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 588 

RESULTS AND DISCUSSION 
 

In this section, we will show our new results on those graph above stated in a 
theorem. We start to write the theorem, provide the cardinality of the graph, obtain 
lower and upper bound, establish the rainbow antimagic connection number and show 
the existence of rainbow path for any to vertices and finally conclude the proof. 
 
Theorem 2.  For  𝑛 ≥ 3 , 𝑟𝑎𝑐(𝑂𝑛) =  2𝑛 . 
 
𝑃𝑟𝑜𝑜𝑓.  The octopus graph 𝑂𝑛 is a graph with vertex set 𝑉( 𝑂𝑛 ) = {𝑥} ∪ { 𝑦𝑖, 𝑧𝑖,1 ≤  𝑗 ≤
 𝑛}, and edge set 𝐸(𝑂𝑛) = {𝑥𝑦𝑖,𝑥𝑧𝑖,1 ≤  𝑖 ≤  𝑛} ∪ {𝑦𝑖𝑦𝑖+1, 1 ≤ 𝑖 ≤ 𝑛 − 1}. The 
cardinality of vertex set is |𝑉(𝑂𝑛)| = 2𝑛 + 1 and the cardinality of edge set is |𝐸(𝑂𝑛)| =
3𝑛 − 1. Based on definition of octopus graph, the graph 𝑂𝑛 has maximum degree of  
Δ (𝑂𝑛) = 2𝑛. 

To prove the rainbow antimagic connection number of 𝑂𝑛, the first step is to 
determine the lower bound of 𝑟𝑎𝑐(𝑂𝑛). Based on Lemma 1. we have 𝑟𝑎𝑐(𝑂𝑛) ≥ Δ(𝑂𝑛).  
Since, the labels of the vertices with the bijection 𝑓:𝑉(𝑂𝑛) → {1,2,…,|𝑉(𝑂𝑛)|}, we have  
𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). It implies for each edge 𝑢𝑥,𝑣𝑥 ∈
𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). Thus 𝑟𝑎𝑐 (𝑂𝑛) ≥ 2𝑛.    

The second step is to determine the upper bound of 𝑟𝑎𝑐(𝑂𝑛). Define the vertex 
labeling 𝑓 ∶  𝑉(𝑂𝑛) → {1,2, . . . ,2𝑛 + 1 } as follows.  

 
𝑓(𝑥) = 1 

𝑓(𝑦𝑖) = {

3+𝑖

2
   , for 𝑖 is odd                    

3+𝑛+𝑖

2
, for 𝑖 is even,𝑛 is odd  

2+𝑛+𝑖

2
, for 𝑖 is even,𝑛 is even 

 

𝑓(𝑧𝑖) = 𝑛 + 𝑖 + 1 , for 1 ≤ 𝑖 ≤ 𝑛 
 
The edge weight 𝑓 can be expressed as 
 

𝑤(𝑥𝑧𝑖) = 2 + 𝑛 + 𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 

𝑤(𝑥𝑦𝑖) = {

5+𝑖

2
     , for 𝑖 is odd                    

5+𝑛+𝑖

2
, for 𝑖 is even,𝑛 is odd  

4+𝑛+𝑖

2
, for 𝑖 is even,𝑛 is even 

 

𝑤(𝑦𝑖𝑦𝑖+1) = {

7+2𝑖+𝑛

2
, for 1 ≤ 𝑖 ≤ 𝑛,𝑛 is odd  

6+2𝑖+𝑛

2
, for 1 ≤ 𝑖 ≤ 𝑛,𝑛 is even

 

 
The next step is to count the number of different edge weights inducing the 

rainbow antimagic coloring on the graph 𝑂𝑛. The edge weights are included in the sets 
𝑤(𝑥𝑦𝑖) = {3,4,5,…,𝑛 + 2} and 𝑤(𝑥𝑧𝑖) = {𝑛 + 3,𝑛 + 4,𝑛 + 5,…,2𝑛 + 2}. The number of 
distinct colors of 𝑤(𝑥𝑦𝑖) ∪  𝑤(𝑥𝑧𝑖) is 2𝑛. To prove this number, we use the formula of an 
arithmetic sequence formula. The following is an illustration of determining the number  
of distinct colors.  

𝑈𝑠 = 𝑎 + (𝑠 − 1)𝑑

2𝑛 + 2 = 3 + (𝑠 − 1)1
2𝑛 + 2 = 3 + 𝑠 − 1

𝑠 = 2𝑛

 



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 589 

 
It implies that the edge weight 𝑓 ∶  𝑉(𝑂𝑛) → {1,2, . . . ,2𝑛 + 1} induces a rainbow 

antimagic coloring of 2𝑛 colors. Therefore 𝑟𝑎𝑐 (𝑂𝑛 ) ≤ 2𝑛. Combining two bounds, we 
have the exact value of 𝑟𝑎𝑐 (𝑂𝑛) = 2𝑛. The last is to show the existence of the rainbow 
𝑥 − 𝑦 path of 𝑂𝑛. According to the Theorem 2, since 𝑑𝑖𝑎𝑚(𝑂𝑛) = 2, for every two 
vertices of the 𝑥,𝑦 ∈ 𝑉(𝐺) there is a rainbow 𝑥 − 𝑦 path. It completes the proof. 
 

The illustration of a rainbow antimagic coloring of octopus graph 𝑂𝑛  can be seen 
in Figure 1. 

 
 
 
 
 
 
 
 
 
 
 

 
 
 

 
Figure 1. The illustration of rainbow antimagic coloring of octopus graph 𝑂7 

 
Theorem 3.  For  𝑛 ≥ 3, 𝑟𝑎𝑐(𝑆𝑡𝑛) =  3𝑛 . 
 
𝑃𝑟𝑜𝑜𝑓.  The sandat graph 𝑆𝑡𝑛 is a graph with vertex set 𝑉( 𝑆𝑡𝑛 ) = {𝑎} ∪ { 𝑥 𝑖,𝑦𝑖,𝑧𝑖,1 ≤
 𝑖 ≤  𝑛} and edge set 𝐸(𝑆𝑡𝑛) = {𝑎𝑥𝑖,𝑎𝑦𝑖,𝑎𝑧𝑖,𝑥𝑖𝑦𝑖,𝑦𝑖𝑧𝑖 1 ≤  𝑖 ≤  𝑛}. The cardinality of 

vertex set is |𝑉(𝑆𝑡𝑛)| = 3𝑛 + 1 and the cardinality of edge set is |𝐸(𝑆𝑡𝑛)| = 5𝑛. Based on 
definition of sandat graph, the graph 𝑆𝑡𝑛 has maximum degree of  Δ (𝑆𝑡𝑛) = 3𝑛. 

To prove the rainbow antimagic connection number of 𝑆𝑡𝑛 , the first step is to 
determine the lower bound of 𝑟𝑎𝑐(𝑆𝑡𝑛). Based on Lemma 1. we have 𝑟𝑎𝑐(𝑆𝑡𝑛) ≥
Δ(𝑆𝑡𝑛).  Since, the labels of the vertices with the bijection 𝑓:𝑉(𝑆𝑡𝑛) → {1,2,…,|𝑉(𝑆𝑡𝑛)|}, 
we have  𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). It implies for each edge 𝑢𝑥,𝑣𝑥 ∈
𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). Thus 𝑟𝑎𝑐 (𝑆𝑡𝑛) ≥ 3𝑛.    

The second step is to determine the upper bound of 𝑟𝑎𝑐(𝑆𝑡𝑛). Define the vertex 
labeling 𝑓 ∶  𝑉(𝑆𝑡𝑛) → {1,2, . . . ,3𝑛 + 1 } as follows.  

 
𝑓(𝑎) = 2 

𝑓(𝑥𝑖) = 3𝑛 + 3 − 2𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 

𝑓(𝑦𝑖) = {
1       , for 𝑖 = 𝑛    
𝑖 + 1 ,  for ≤ 𝑖 ≤ 𝑛

 

𝑓(𝑧𝑖) = 3𝑛 + 2 − 2𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 
 
The edge weight 𝑓 can be expressed as 
 

𝑤(𝑎𝑥𝑖) = 3𝑛 + 5 − 2𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 590 

𝑤(𝑎𝑦𝑖) = {
3       , for  i = 1       
𝑖 + 3 , for 2 ≤ 𝑖 ≤ 𝑛

 

𝑤(𝑎𝑧𝑖) = 3𝑛 + 4 − 2𝑖 , for 1 ≤ 𝑖 ≤ 𝑛 

𝑤(𝑥𝑖𝑦𝑖) = {
3𝑛 + 2      , for 𝑖 = 1        
3𝑛 + 4 − 𝑖 , for 2 ≤ 𝑖 ≤ 𝑛

 

𝑤(𝑦𝑖𝑧𝑖) = {
3𝑛 + 1       , for 𝑖 = 1        
3𝑛 + 3 − 𝑖 , for 2 ≤ 𝑖 ≤ 𝑛

 

 
The next step is to count the number of different edge weights inducing the 

rainbow antimagic coloring on the graph 𝑆𝑡𝑛. The edge weights are included in the sets 
𝑤(𝑎𝑥𝑖) ∪ 𝑤(𝑎𝑦𝑖) ∪ 𝑤(𝑎𝑧𝑖) ∪ 𝑤(𝑥𝑖𝑦𝑖) ∪ 𝑤(𝑦𝑖𝑧𝑖) = {5,6,7,…,3𝑛 + 3 }. The number of 
distinct colors of 𝑤(𝑎𝑥𝑖) ∪ 𝑤(𝑎𝑦𝑖) ∪ 𝑤(𝑎𝑧𝑖) ∪ 𝑤(𝑥𝑖𝑦𝑖) ∪ 𝑤(𝑦𝑖𝑧𝑖) is 3𝑛. Based on edge 
weights the number of edge wights is determined in the same way in Theorem 2. 

It implies that the edge weight 𝑓 ∶  𝑉(𝑆𝑡𝑛) → {1,2, . . . ,3𝑛 + 1} induces a rainbow 
antimagic coloring of 3𝑛 colors. Therefore 𝑟𝑎𝑐 (𝑆𝑡𝑛 ) ≤ 3𝑛. Combining two bounds, we 
have the exact value of 𝑟𝑎𝑐 (𝑆𝑡𝑛) = 3𝑛. The last is to show the existence of the rainbow 
𝑥 − 𝑦 path of 𝑆𝑡𝑛. According to the Theorem 1, since 𝑑𝑖𝑎𝑚(𝑆𝑡𝑛) = 2, for every two 
vertices of the 𝑥,𝑦 ∈ 𝑉(𝐺) there is a rainbow 𝑥 − 𝑦 path. It completes the proof. 
 

The illustration of a rainbow antimagic coloring of sandat graph 𝑆𝑡𝑛  can be seen 
in Figure 2. 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
Figure 2. The illustration of rainbow antimagic coloring of sandat graph 𝑆𝑡6. 

 

Theorem 4.  For  𝑛 ≥ 4, 𝑟𝑎𝑐(𝑆𝑓𝑛) =  3𝑛 . 
 
𝑃𝑟𝑜𝑜𝑓.  The sunflower graph 𝑆𝑓𝑛 is a graph with vertex set 𝑉( 𝑆𝑓𝑛 ) = {𝑐} ∪
{ 𝑥 𝑖,𝑦𝑖,𝑧𝑖,1 ≤  𝑖 ≤  𝑛} and edge set 𝐸(𝑆𝑓𝑛) = {𝑐𝑥𝑖,𝑐𝑦𝑖,𝑐𝑧𝑖,𝑦𝑖𝑧𝑖,𝑧𝑖𝑧𝑖+1,1 ≤  𝑖 ≤  𝑛}. The 

cardinality of vertex set is |𝑉(𝑆𝑓𝑛)| = 3𝑛 + 1 and the cardinality of edge set is |𝐸(𝑆𝑓𝑛)| =
5𝑛. Based on definition of sunflower graph, the graph 𝑆𝑓𝑛 has maximum degree of  
Δ (𝑆𝑓𝑛) = 3𝑛. 

To prove the rainbow antimagic connection number of 𝑆𝑓𝑛 , the first step is to 
determine the lower bound of 𝑟𝑎𝑐(𝑆𝑓𝑛). Based on Lemma 1. we have 𝑟𝑎𝑐(𝑆𝑓𝑛) ≥ Δ(𝑆𝑓𝑛).  
Since, the labels of the vertices with the bijection 𝑓:𝑉(𝑆𝑓𝑛) → {1,2,…,|𝑉(𝑆𝑓𝑛)|}, we have  



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 591 

𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). It implies for each edge 𝑢𝑥,𝑣𝑥 ∈
𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). Thus 𝑟𝑎𝑐 (𝑆𝑓𝑛) ≥ 3𝑛.    

The second step is to determine the upper bound of 𝑟𝑎𝑐(𝑆𝑓𝑛). Define the vertex 
labeling 𝑓 ∶  𝑉(𝑆𝑓𝑛) → {1,2, . . . ,3𝑛 + 1 } as follows.  

 
𝑓(𝑐) = 1 

𝑓(𝑥𝑖) = 2𝑛 + 𝑖 + 1 , for 1 ≤ 𝑖 ≤ 𝑛 
𝑓(𝑦𝑖) = 2𝑛 − 𝑖 + 2 , for 1 ≤ 𝑖 ≤ 𝑛 
𝑓(𝑧𝑖) = 𝑖 + 1 , for 1 ≤ 𝑖 ≤ 𝑛 

The edge weight 𝑓 can be expressed as 
 

𝑤(𝑐𝑥𝑖) = 2𝑛 + 𝑖 + 2 , for 1 ≤ 𝑖 ≤ 𝑛 
𝑤(𝑐𝑦𝑖) = 2𝑛 − 𝑖 + 3 , for 1 ≤ 𝑖 ≤ 𝑛 
𝑤(𝑐𝑧𝑖) = 𝑖 + 2 , for 1 ≤ 𝑖 ≤ 𝑛 
𝑤(𝑦𝑖𝑧𝑖) = 2𝑛+3 , for 1 ≤ 𝑖 ≤ 𝑛 

𝑤(𝑧𝑖𝑧𝑖+1) = {
2𝑖 + 3 , for 1 ≤ 𝑖 ≤ 𝑛 − 1
 𝑛 + 3  , for 𝑖 = 𝑛                

 

 
The next step is to count the number of different edge weights inducing the 

rainbow antimagic coloring on the graph 𝑆𝑓𝑛. The edge weights are included in the sets 
𝑤(𝑐𝑥𝑖) ∪ 𝑤(𝑐𝑦𝑖) ∪ 𝑤(𝑐𝑧𝑖) = {3,4,5,…,3𝑛 + 2 },𝑤(𝑦𝑖𝑧𝑖) = {2𝑛 + 3} and 𝑤(𝑧𝑖𝑧𝑖+1) =
{𝑛 + 3} ∪ {5,7,9,…2𝑛 + 1}. The number of distinct colors of 𝑤(𝑐𝑥𝑖) ∪ 𝑤(𝑐𝑦𝑖) ∪ 𝑤(𝑐𝑧𝑖) ∪
𝑤(𝑦𝑖𝑧𝑖) ∪ 𝑤(𝑧𝑖𝑧𝑖+1) is 3𝑛. Based on edge weights the number of edge wights is 
determined in the same way in Theorem 2. 

It implies that the edge weight 𝑓 ∶  𝑉(𝑆𝑓𝑛) → {1,2, . . . ,3𝑛 + 1} induces a rainbow 
antimagic coloring of 3𝑛 colors. Therefore 𝑟𝑎𝑐 (𝑆𝑓𝑛 ) ≤ 3𝑛. Combining two bounds, we 
have the exact value of 𝑟𝑎𝑐 (𝑆𝑓𝑛) = 3𝑛. The last is to show the existence of the rainbow 
𝑥 − 𝑦 path of 𝑆𝑓𝑛. According to the Theorem 1, since 𝑑𝑖𝑎𝑚(𝑆𝑓𝑛) = 2, for every two 
vertices of the 𝑥,𝑦 ∈ 𝑉(𝐺) there is a rainbow 𝑥 − 𝑦 path. It completes the proof. 
 

The illustration of a rainbow antimagic coloring of sunflower graph 𝑆𝑓𝑛  can be 
seen in Figure 3. 
 



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 592 

 
 

Figure 3. The illustration of rainbow antimagic coloring of sunflower graph 𝑆𝑓6. 
 

Theorem 5.  For  𝑛 ≥ 3, 𝑟𝑎𝑐(𝑉𝑛) = 𝑛 + 2 . 
 
𝑃𝑟𝑜𝑜𝑓.  The volcano 𝑉𝑛 is a graph with vertex set 𝑉( 𝑉𝑛 ) = {𝑥1,𝑥2,𝑥3} ∪ {𝑦𝑖,1 ≤  𝑖 ≤  𝑛} 
and edge set 𝐸(𝑉𝑛) = {𝑥1𝑥2,𝑥2𝑥3,𝑥3𝑥1} ∪ {𝑥𝑖𝑦𝑖,1 ≤  𝑖 ≤  𝑛}. The cardinality of vertex set 
is |𝑉(𝑉𝑛)| = 𝑛 + 3 and the cardinality of edge set is |𝐸(𝑉𝑛)| = 𝑛 + 3. Based on definition 
of volcano graph, the graph 𝑉𝑛 has maximum degree of  Δ (𝑉𝑛) = 𝑛 + 2. 

To prove the rainbow antimagic connection number of 𝑉𝑛 , the first step is to 
determine the lower bound of 𝑟𝑎𝑐(𝑉𝑛). Based on Lemma 1. we have 𝑟𝑎𝑐(𝑉𝑛) ≥ Δ(𝑉𝑛).  
Since, the labels of the vertices with the bijection 𝑓:𝑉(𝑉𝑛) → {1,2,…, |𝑉(𝑉𝑛)|}, we have  
𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). It implies for each edge 𝑢𝑥,𝑣𝑥 ∈
𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). Thus 𝑟𝑎𝑐 (𝑉𝑛) ≥ 𝑛 + 2.    

The second step is to determine the upper bound of 𝑟𝑎𝑐(𝑉𝑛). Define the vertex 
labeling 𝑓 ∶  𝑉(𝑉𝑛) → {1,2, . . . ,𝑛 + 3 } as follows.  

𝑓(𝑥1) = 1 
𝑓(𝑥2) = 2 
𝑓(𝑥3) = 3 

𝑓(𝑦𝑖) = 𝑖 + 3 
 
The edge weight 𝑓 can be expressed as 

𝑤(𝑥1𝑥2) = 3 
𝑤(𝑥2𝑥3) = 5 
𝑤(𝑥1𝑥3) = 4 
𝑤(𝑥𝑖𝑦𝑖) = 𝑖 + 4 

 
The next step is to count the number of different edge weights inducing the 

rainbow antimagic coloring on the graph 𝑉𝑛. The edge weights are included in the sets 



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 593 

𝑤(𝑥1𝑥2) ∪ 𝑤(𝑥2𝑥3) ∪ 𝑤(𝑥1𝑥3) = {3,4,5 }  and 𝑤(𝑥𝑖𝑦𝑖) = {5,6,7,…,𝑛 + 4}. The number of 
distinct colors of 𝑤(𝑥1𝑥2) ∪ 𝑤(𝑥2𝑥3) ∪ 𝑤(𝑥1𝑥3) ∪  𝑤(𝑥𝑖𝑦𝑖) is 𝑛 + 2. Based on edge 
weights the number of edge wights is determined in the same way in Theorem 2. 

It implies that the edge weight 𝑓 ∶  𝑉(𝑉𝑛) → {1,2, . . . ,𝑛 + 3} induces a rainbow 
antimagic coloring of 𝑛 + 2 colors. Therefore 𝑟𝑎𝑐 (𝑉𝑛 ) ≤ 𝑛 + 2. Combining two bounds, 
we have the exact value of 𝑟𝑎𝑐 (𝑉𝑛) = 𝑛 + 2. The last is to show the existence of the 
rainbow 𝑥 − 𝑦 path of 𝑉𝑛. According to the Theorem 1, since 𝑑𝑖𝑎𝑚(𝑉𝑛) = 2, for every 
two vertices of the 𝑥,𝑦 ∈ 𝑉(𝐺) there is a rainbow 𝑥 − 𝑦 path. It completes the proof. 

The illustration of a rainbow antimagic coloring of volcano graph 𝑉𝑛  can be seen 
in Figure 4. 

 
 
 
 
 
 
 
 
 
 
 

 
 

 
 

Figure 4. The illustration of rainbow antimagic coloring of volcano graph 𝑉7. 
 
Theorem 6.  For  𝑛 ≥ 3, 𝑟𝑎𝑐(𝑆𝐽𝑛) =  𝑛 . 
 
𝑃𝑟𝑜𝑜𝑓.  The semi jahangir graph 𝑆𝐽𝑛 is a graph with vertex set 𝑉( 𝑆𝐽𝑛 ) = {𝑎} ∪ { 𝑥𝑖,1 ≤
 𝑖 ≤  𝑛} ∪ { 𝑦𝑖,1 ≤  𝑖 ≤  𝑛 − 1} and edge set (𝑆𝐽𝑛) = {𝑎𝑥𝑖,1 ≤  𝑖 ≤  𝑛} ∪ {𝑥𝑖𝑦𝑖,𝑦𝑖𝑥𝑖+1,1 ≤
 𝑖 ≤  𝑛 − 1}. The cardinality of vertex set is |𝑉(𝑆𝐽𝑛)| = 2𝑛 and the cardinality of edge set 
is |𝐸(𝑆𝐽𝑛)| = 3𝑛 − 2. Based on definition of semi jahangir graph, the graph 𝑆𝐽𝑛 has 
maximum degree of  Δ (𝑆𝐽𝑛) = 𝑛. 

To prove the rainbow antimagic connection number of 𝑆𝐽𝑛 , the first step is to 
determine the lower bound of 𝑟𝑎𝑐(𝑆𝐽𝑛). Based on Lemma 1. we have 𝑟𝑎𝑐(𝑆𝐽𝑛) ≥ Δ(𝑆𝐽𝑛).  
Since, the labels of the vertices with the bijection 𝑓:𝑉(𝑆𝐽𝑛) → {1,2,…,|𝑉(𝑆𝐽𝑛)|}, we have  
𝑓(𝑢) ≠ 𝑓(𝑣) for every vertex 𝑢,𝑣 ∈ 𝑉 (𝐺). It implies for each edge 𝑢𝑥,𝑣𝑥 ∈
𝐸 (𝐺),𝑤 (𝑢𝑥) ≠ 𝑤 (𝑣𝑥). Thus 𝑟𝑎𝑐 (𝑆𝐽𝑛) ≥ 𝑛.    

The second step is to determine the upper bound of 𝑟𝑎𝑐(𝑆𝐽𝑛). Define the vertex 
labeling 𝑓 ∶  𝑉(𝑆𝐽𝑛) → {1,2, . . . ,2𝑛} as follows.   

 

𝑓(𝑎) = {
𝑛      , for 𝑛 is odd 
 𝑛 + 1 , for 𝑛 is even

 

𝑓(𝑥𝑖) = {
2         , for 𝑖 = 1                
2𝑖 + 2 , for 2 ≤ 𝑖 ≤ 𝑛 − 1
4         , for 𝑖 = 𝑛                

 

𝑓(𝑦𝑖) = {
2𝑛 − 2𝑖 + 1 , for 1 ≤ 𝑖 ≤ ⌈

𝑛

2
⌉ − 1 

 2𝑛 − 2𝑖 − 3 , for ⌈
𝑛

2
⌉ ≤ 𝑖 ≤ 𝑛 − 2

 



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 594 

𝑓(𝑦𝑛−1) = {
𝑖 − 1 , for 𝑛 is odd  
 𝑖         , for 𝑛 is even

 

 
The edge weight 𝑓 can be expressed as  

𝑤(𝑎𝑥𝑖) = {
𝑛 + 2 , for 𝑛 is odd 
 𝑛 + 3 , for 𝑛 is even

 

 

𝑤(𝑎𝑥𝑖) = {
𝑛 + 2𝑖 + 2 , for 𝑛 is odd,2 ≤ 𝑖 ≤ 𝑛 − 1  
 𝑛 + 2𝑖 + 3 , for 𝑛 is even,2 ≤ 𝑖 ≤ 𝑛 − 1

 

𝑤(𝑎𝑥𝑛) = {
𝑛 + 4 , for 𝑛 is odd  
 𝑛 + 5 , for 𝑛 is even

 

𝑤(𝑥𝑖𝑦𝑖) =

{
 

 
2𝑛 + 1 , for 𝑖 = 1                   

2𝑛 + 3 , for 2 ≤ 𝑖 ≤ ⌈
𝑛

2
⌉ − 1

2𝑛 − 1 , for ⌈
𝑛

2
⌉ ≤ 𝑖 ≤ 𝑛 − 2

 

𝑤(𝑎𝑥𝑛−1𝑦𝑛−1) = {
3𝑛 − 2 , for 𝑛 is odd  
 3𝑛 − 1 , for 𝑛 is even

 

𝑤(𝑦𝑖𝑥𝑖+1) =

{
 

 2𝑛 + 5 , for 2 ≤ 𝑖 ≤ ⌈
𝑛

2
⌉ − 1

2𝑛 + 1 , for ⌈
𝑛

2
⌉ ≤ 𝑖 ≤ 𝑛 − 2

2𝑛 − 5 , for 𝑖 = 𝑛 − 1            

 

 
The next step is to count the number of different edge weights inducing the 

rainbow antimagic coloring on the graph 𝑆𝐽𝑛. The edge weights are included in the sets 
𝑤(𝑥𝑖𝑦𝑖) ∪ 𝑤(𝑦𝑖𝑥𝑖+1) = {2𝑛 + 1,2𝑛 + 3,2𝑛 + 5 } and 𝑤(𝑎𝑥𝑖) = {𝑛 + 3,𝑛 + 4,𝑛 +
5,…,3𝑛 + 1}. The number of distinct colors of 𝑤(𝑥𝑖𝑦𝑖) ∪ 𝑤(𝑦𝑖𝑥𝑖+1) ∪ 𝑤(𝑎𝑥𝑖) is  𝑛. Based 
on edge weights the number of edge wights is determined in the same way in Theorem 
2. 

It implies that the edge weight 𝑓 ∶  𝑉(𝑆𝐽𝑛) → {1,2, . . . ,3𝑛 − 2} induces a rainbow 
antimagic coloring of 𝑛 colors. Therefore 𝑟𝑎𝑐 (𝑆𝐽𝑛 ) ≤ 𝑛. Combining two bounds, we 
have the exact value of 𝑟𝑎𝑐 (𝑆𝐽𝑛) = 𝑛. The last is to show the existence of the rainbow 
𝑥 − 𝑦 path of 𝑆𝐽𝑛. Suppose we take any 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛), there are two possibilities for 𝑥,𝑦, 
namely: 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) where 𝑑(𝑥,𝑦) ≤ 2 or 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) where 𝑑(𝑥,𝑦) ≥ 3. Suppose 
𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) where 𝑑(𝑥,𝑦) ≤ 2, based on Theorem 1, we must have the rainbow 𝑥 − 𝑦 
path. For 𝑥,𝑦 ∈ 𝑉(𝑆𝐽𝑛) where 𝑑(𝑥,𝑦) ≥ 3, we have two case: First case for path 𝑥𝑖 − 𝑦𝑗   

we use the path 𝑥𝑖,𝑎,𝑥𝑗,𝑦𝑗 or 𝑥𝑖,𝑎,𝑥𝑗+1,𝑦𝑗. Second case for path 𝑦𝑖 − 𝑦𝑗 we use the path 

𝑦𝑖,𝑥𝑖,𝑎,𝑥𝑗,𝑦𝑗 or 𝑦𝑖,𝑥𝑖,𝑎,𝑥𝑗+1,𝑦𝑗 or 𝑦𝑖,𝑥𝑖+1,𝑎,𝑥𝑗,𝑦𝑗 or 𝑦𝑖,𝑥𝑖+1,𝑎,𝑥𝑗+1,𝑦𝑗. Thus, for 𝑥,𝑦 ∈

 𝑉(𝑆𝐽𝑛) there is a rainbow 𝑥 − 𝑦 path. It completes the proof.The illustration of a 
rainbow antimagic coloring of semi jahangir graph 𝑆𝐽𝑛  can be seen in Figure 5. 



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 595 

 
Figure 5. The illustration of rainbow antimagic coloring of semi jahangir graph 𝑆𝐽6. 

 
CONCLUDING REMARKS  

Based on these results, the authors get the results of the rainbow antimagic 
connection number on several graphs. The authors finds the exact value of the octopus 
graph 𝑂𝑛, sandat graph 𝑆𝑡𝑛, sunflower graph 𝑆𝑓𝑛, volcano graph 𝑉𝑛 and semi jahangir 
graph 𝑆𝐽𝑛. 

Based on the results of this study, this study raises an open problem. Determine 
the exact value of the rainbow antimagic connection number of operation of graphs. 

REFERENCES 

 

[1]  Al Jabbar Z L, Dafik Adawiyah R, Albirri E R, Agustin I H, On rainbow antimagic 
coloring of some special graph, Journal of Physics: Conference Series. 1465 (1) 
(2020).  

[2]  Arumugam S, Miller M, Phanalasy O, Ryan J, Antimagic labeling of generalized 
pyramid graphs, Acta Mathematica Sinica, English Series. 30 (2) (2014), 283–290. 

[3]  Arumugam S, Premalatha K, Baˇca M, Semaniˇcov´a-Feˇnovˇc´ıkov´a A, Local 
antimagic vertex coloring of a graph, Graphs Combin. 33 (2017), 275-285. 

[4]  Baˇca M, Antimagic labelings of antiprisms, Journal of combinatorial mathematics 
and combinatorial computing. 35 (2000), 217-224. 

[5]  Baˇca M, Baskoro E T, Jendrol S, Miler M, Antimagic labelings of hexagonal plane 
maps, Utilitas mathematica. 66 (2004), 231-238. 

[6]  Baˇca M, Phanalasy O, Ryan J, Semanicov´a-Fenoˇc´ıkov´a A, Antimagic labelings of 
joint graphs, Mathematics in Computer Science. 9 (2) (2015), 139–143. 

[7] Budi H S, Dafik, Tirta I M, Agustin I H, Kristiana A I, On rainbow antimagic coloring 
of graphs, Journal of Physics: Conf. Series, 1832 (2021), 012016. 

[8] Bustan A W, Salman A N M, The rainbow vertex connection number of star wheel 
graphs, AIP Conference Proceedings. (2019), 112–116. 

[9] Chartrand G, Lesniak L, Zhang P, Graphs & Digraphs, sixth ed., Taylor & Francis 
Group, New York, 1 (2016). 

[10] Chartrand G, Johns G L, Mckeon K A, Zhang P, Rainbow connection in graphs, Math. 
Bohemica 133 (2008), 85-98. 



On the study of Rainbow Antimagic Coloring of Special Graphs 

Dafik 596 

[11] Dafik, Miler M, Ryan J, Baˇca M, Antimagic labeling of the union of two stars, 
Australasian Journal of combinatorics. 42 (2018), 35-44. 

[12] Dafik, Susanto F, Alfarisi R, Septory B J, Agustin I H, Venkatachalam M, On rainbow 
antimagic coloring of graphs. Advanced Mathematical Models and Aplication. 6 (3) 
(2021) 278-291. 

[13] Hartsfield N, Ringel G, Pearls in Graph Theory, Academic Press, San Diego, 1990. 

[14] Joedo J C, Dafik, Kristiana A I, Agustin I H, Nisviasari R, On the rainbow antimagic 
coloring of vertex almagamation of graphs, Journal of Physics: Conf. Series 2157 
(2022), 012014. 

[15] Lei H, Liu H, Magnant C, Shi Y, Total rainbow connection of digraphs, Discrete 
Applied Mathematics. 236 (2018), 288–305. 

[16] Lei H, Li S, Liu H, Shi Y, Rainbow vertex connection of digraphs, Journal of 
Combinatorial Optimization. 35 (1)(2018), 86–107. 

[17] Li X, Liu S (n.d.), Rainbow vertex-connection number of 2-connected graphs.  
arxiv:1110v1[math.CO]. 

[18] Ma Y, Total Rainbow Connection Number and Complementary Graph, Results in 
Mathematics, 70 (1–2) (2016), 173–182. 

[19] Ma Y, Lu Z, Rainbow connection numbers of Cayley graphs, Journal of Combinatorial 
Optimization,34 (1) (2017), 182–193. 

[20] Ma Y, Nie K, Jin F, Wang C, Total rainbow connection numbers of some special 
graphs, Applied Mathematics and Computation,360 (2019), 213–220. 

[21] Nabila S, Salman A N M, The Rainbow Connection Number of Origami Graphs and 
Pizza Graphs, Procedia Computer Science, 74 (1) (2015), 162–167. 

[22] Ryan J, Phanalasy O, Miller M, Rylands L, On antimagic labeling for generalized web 
and flower graphs, Lecture Notes in Computer Science (including subseries Lecture 
Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 6460 LNCS 
(2011), 303–313. 

[23] Sulistiyono B, Slamin, Dafik, Agustin I H, Alfarisi R, On rainbow antimagic coloring 
of some graphs, Journal of Physics: Conference Series. 1465 (1) (2020). 

[24] Septory B J, Utoyo M I, Dafik, Sulistiyono B, Agustin I H, On rainbow antimagic 
coloring of special graphs. Journal of Physics: Conference Series. 1836 (2021) 
012016. 

[25] Wang T M, Zhang G H, On antimagic labeling of regular graphs with particular 
factors, Journal of Discrete Algorithms, 23 (2013), 76–82.