Super Total Labeling (a,d)-Edge Antimagic on the Firecracker Graph


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(3) (2020), Pages 133-139 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: August 18, 2020 Reviewed: October 01, 2020 Accepted: November 10, 2020 
DOI: http://dx.doi.org/10.18860/ca.v6i3.10145  

Super Total Labeling (a,d)-Edge Antimagic on the Firecracker Graph 

Juhari 

Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang 
Email: juhari@uin-malang.ac.id    

ABSTRACT 

An An (a, d)-edge antimagic total labeling on (p, q)-graph G is a one-to-one map f  from V (G) ∪ 
E(G) onto the integers  1, 2, . . ., p + q with the property that the edge-weights, w(uv) = f  (u) + 
f(v) + f(uv) where uv ∈ E(G), form an arithmetic progression starting from a and having common 
difference d. Such labeling is called super if the smallest possible labels appear on the vertices. 
In this paper, we investigate the existence of super (a, d)-edge antimagic total labeling of 
Firecracker Graph. 

Keywords: Super (a, d)-Edge-Antimagic Total Labeling; Firecracker Graph (Fn, k) 

INTRODUCTION 

The first label appears in the middle of the year 1960 it started by a Ringel and Rosa 
hypotheses [1]. In 1967 Rosa called this label as liberation β-valuation from a graph with 
e side, if there is the function that mapping one to one from the set of points 𝑉(𝐺) to set 
of integers 0,1,2, … . , 𝑒, so every side XY in graph G gets a different label |𝑓(𝑥) − 𝑓(𝑦)| for 
every edges in graph G.  

One type of graf liberation is super total labeling (𝑎, 𝑑)-antimagic edge  (SEATL), 
where a smallest side dan d different value. This liberation introduced by Simanjutak, 
Bertault and Miller in the year 2000 [1], [2], [3]. The entire release (𝑎, 𝑑)-edge antimagic 
is total labeling in some kinds of graf G that started by labeling all of the graphs first with 
consecutive original numbers, then proceed with buying all sides of the map such that the 
side weights form an arithmetic sequence with the first term and different d [4], [5]. 

The type of star graph, the super total labeling (𝑎, 𝑑)-edge antimagic (SEATL) not 
yet found one of which is a graph firecracker that hasn't been labelled previously. This 
prompted the writer to examine how super total labeling (𝑎 𝑑)-edge antimagic (SEATL) 
on the firecracker graph (Fn, k). Some of the problems formulated are as follows: (1) The 
upper limit d, so the firecracker graph has super total labeling (a, d)-edge antimagic? And 
(2) How bijective function of super total labeling (𝑎, 𝑑)-edge antimagic the firecracker 
graph? 

In order not to be widespread, this research needs to be done, and this research 
needs to be done on total labeling (𝑎, 𝑑)-edge antimagic the firecracker graph (Fn, k) with 
n ≥ 2; k ≥ 3. In this session, n and k are a provision of the definition firecracker graph.  

 
Super Total Labeling (𝒂, 𝒅)-Edge Antimagic. 
A graph is said to have total labeling (𝑎, 𝑑)-edge antimagic if there is a one-to-one mapping 
of one  𝑉(𝐺) ∪ 𝐸(𝐺) to integers. 1,2,3, … , 𝑝 + 𝑞 so the set of side weight 𝑤(𝑢𝑣) = 𝑓(𝑢) +

http://dx.doi.org/10.18860/ca.v6i3.10145
mailto:juhari@uin-malang.ac.id


 Super Total Labeling (a,d)-Edge Antimagic on the Firecracker Graph 

Juhari 134 

𝑓(𝑣) + 𝑓(𝑢𝑣) on all edge G is 𝑎, 𝑎 + 𝑑, … , 𝑎 + (𝑞 − 1)𝑑 for 𝑎 > 0 and 𝑑 >0 both integers 
[6]. The total labeling (𝑎, 𝑑)-edge antimagic called super total labeling (𝑎, 𝑑)- edge 
antimagic if  𝑓(𝑉) = {1,2,3, . . , 𝑝} and 𝑓(𝐸) = {𝑝 + 1, 𝑝 + 2, 𝑝 + 3, … , 𝑝 + 𝑞}. To search 
upper limit different value d super total slowly  (𝑎, 𝑑)-edge antimagic can certain by 
lemma [1], [7]: 

Lemma 1 If a graph (p, q) is super total labeling (a,d)-edge antimagic so 𝑑 ≥  
2𝑝+𝑞−5

𝑞−1
 

 

Prove. 𝑓(𝑉) = {1,2,3, . . , 𝑝} and 𝑓(𝐸) = {𝑝 + 1, 𝑝 + 2, 𝑝 + 3, … , 𝑝 + 𝑞} 
For example,  graph  (𝑝, 𝑞) is super total labeling (𝑎, 𝑑)-edge antimagic by mapping 
𝑓: 𝑉(𝐺) ∪ 𝐸(𝐺) → {1,2,3, … , 𝑝 + 𝑞}. The minimum value that possible from the smallest 
weight side 𝛼(𝑢) + 𝛼(𝑢𝑣) + 𝛼(𝑣) = 1 + (𝑝 + 1) + 2 = 𝑝 + 4 and can be written: 𝑝 + 4 ≤
𝛼. While on the other side,  t h e  maximum value that possible from the biggest weight 
side gained by the sum of  2 smallest labels  and biggest label or can be written (𝑝 − 1) +
(𝑝 + 𝑞) + 𝑝 = 3𝑝 + 𝑞 − 1. Result: 

𝑎 + (𝑞 − 1) 𝑑 ≤ 3𝑝 + 𝑞 − 1 
                                                                               

                                          𝑑 ≤
3𝑝 + 𝑞 − 1 − (𝑝 + 4)

𝑞 − 1
 

                

                      𝑑 ≤
2𝑝 + 𝑞 − 5

𝑞 − 1
 

The equation above has proved and got value𝑑 ≥  
2𝑝+𝑞−5

𝑞−1
 from many kinds or graph 

family.  
 
Firecracker Graph 
Firecracker graph is a graph that gets by star graph combination exactly one leaf of each 
graph is connected [8], [9], [10], usually symbolized 𝐹𝑛,𝑘 where n is the number of 

merged star graphs, while k is the number of points of each connected star graph.  
 

METHODS  

This research uses axiomatic descriptive method, which is by decreasing the existing 
axioms or theorems [11], [12], then applied in super total labeling (𝑎, 𝑑)-edge antimagic 
on the firecracker graph 𝐹𝑛,𝑘 . In addition, some systematic research techniques are as 

follows: (1) Count the number of points v and side e on the firecracker graph 𝐹𝑛,𝑘 ; (2) 
Determine the upper limit of the different d values in the firecracker graph 𝐹𝑛,𝑘  in 

accordance with the Lemma 1; (3) determine or find EAVL label (edge-antimagic vertex 
labeling) or labeling points (𝑎, 𝑑)- edge antimagic of the firecracker graph 𝐹𝑛,𝑘 ;  (4) 
determine the algorithm of the functional function EAVL 𝑓(𝑥𝑖.𝑙 ) on the firecracker graph 
𝐹𝑛,𝑘  by looking at the labeling pattern on the graph firecracker 𝐹𝑛,𝑘  which has been found 

then grouping the numbers on the label of points that form arithmetic rows ; (5) 
determine the algorithm for the functional function of side weights EAVL (𝑤) on the 
firecracker graph 𝐹𝑛,𝑘  by looking at the firecracker graph labeling pattern;(6) label the 
sides of the firecracker graph 𝐹𝑛,𝑘  with SEATL (Super Edge Antimagic Total Labeling) or 

super total labeling (𝑎, 𝑑)-edge antimagic for each corresponding different value d;(7) 
Determine the bijective function  on the firecracker graph  𝐹𝑛,𝑘 ; and (8) Write a 

conclusion. 
 



 Super Total Labeling (a,d)-Edge Antimagic on the Firecracker Graph 

Juhari 135 

RESULTS AND DISCUSSION  

The first step in determining the labeling of super total (𝑎, 𝑑)-edge antimagic 
is to determine the number of points and the number of edges on the graph under 
study, in this case, firecracker graph. After that, select the value of d in the labeling that 
will be examined using lemma 1. The labeling pattern can be determined by detecting the 
way (pattern recognition) after labeling a specific firecracker graph. Next, to determine 
patterns in general, the objective function is found by using the principle of an arithmetic 
sequence. The following will be presented lemma and theorems that have been found. 
Lemma 2. There is a point labeling (𝑎, 1)- edge antimagic on the firecracker graph 𝐹𝑛,𝑘  if 

n odd,  n ≥ 2, and k ≥ 3. 
Prove. First defined 𝑥𝑖.𝑙   is the point in the graph component of the firecracker 𝐹𝑛,𝑘 ,where 

1 ≤ i ≤ n and 0 ≤ l ≤ k − 1. Based on research results,if 𝛼: 𝑉(𝐹𝑛,𝑘 ) → {1,2, … , 𝑛𝑘} so 𝛼 

labelation can be written as follow: 
𝑖;      if i odd (1 ≤ 𝑖 ≤ 𝑛), 

         and l=0 
(𝑛 + 𝑖);    if i even (2 ≤ 𝑖 ≤ 𝑛 − 1), 

          and l=0 
𝑖;     if i even (12 ≤ 𝑖 ≤ 𝑛 − 1), 

       and l=1 
(𝑛 + 𝑖);    if i odd (1 ≤ 𝑖 ≤ 𝑛), 

       and l=1 

𝑛(𝑙 + 1) −
𝑖−1

2
;   if i odd (1 ≤ 𝑖 ≤ 𝑛), 

         and (2 ≤ 𝑙 ≤ 𝑘 − 1) 

(𝑙𝑛 +
𝑛−𝑖−1

2
) + 1;  if i even (2 ≤ 𝑖 ≤ 𝑛 − 1), 

         and (2 ≤ 𝑙 ≤ 𝑘 − 1) 
  

From the equation above 𝛼(𝑥𝑖,𝑙 ) is an objective function that maps  𝑉(𝐹𝑛,𝑘 ) =

{𝑣1, 𝑣2, 𝑣3, … , 𝑣𝑛𝑘 } to the set of integers {1,2, … , 𝑛𝑘}. If 𝑤𝛼  is defindes as the weight edge of 
the labeling point α, so 𝑤𝛼  is formulated:  

𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑛 + 2𝑖)               ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 

𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = (𝑛 + 2𝑖 + 1)    ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1  

𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑙 + 1)𝑛 +
(𝑖+1)

2
 ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑤𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) =  (𝑙 + 1)𝑛 +
(𝑛+𝑖+1)

2
 ; if i even, 1 ≤ 𝑖 ≤ 𝑛 − 1 and  2 ≤ 𝑙 ≤ 𝑘 − 1 

 
Theorem 1. There is super total labeling (2𝑛𝑘 + 𝑛 + 1,0)-edge antimagic on the 
firecracker graph 𝐹𝑛,𝑘  if n odd, n ≥ 2, and k ≥ 3. 

Prove. First define the edge label 𝑓𝛼 : 𝐸(𝐹𝑛,𝑘 ) = {𝑒1, 𝑒2, . . . , 𝑒𝑛𝑘−1} → {𝑛𝑘 + 1, 𝑛𝑘 +

2, … , 2𝑛𝑘 − 1}, so the edge label 𝑓𝛼  for super total labeling (𝑎, 0)- edge antimagic on the 
graph 𝐹𝑛,𝑘  can be formulated as follows: 

𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = 2(𝑛𝑘 − 𝑖) + 1               ; 1 ≤ 𝑖 ≤ 𝑛 and 𝑙 = 1 

𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = 2(𝑛𝑘 − 𝑖)                    ;  1 ≤ 𝑖 ≤ 𝑛 − 1  and 𝑙 = 1 

𝑓𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 +
(1−𝑖)

2
         ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑓𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 +
(−2𝑛−𝑖+6)

2
 ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

 Next, if there is Wα even that defined as weight of super total labeling 

𝛼(𝑥𝑖,𝑙 ) = 
= 



 Super Total Labeling (a,d)-Edge Antimagic on the Firecracker Graph 

Juhari 136 

𝛼(𝑥𝑖,𝑙 ), 𝛼(𝑥𝑖,𝑙 𝑥𝑖,0), and 𝛼(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ), so Wα can be obtained by adding up the edge weight 

formula EAVL 𝑤𝛼  and the formula of edge label 𝑓𝛼 with the terms of boundaries i and l 
which correspond and can be stated as follows: 

𝑊𝛼1 =  𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) +  𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0)        ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 

𝑊𝛼2 =  𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) +  𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0)        ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 

𝑊𝛼3 =  𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑊𝛼4 =  𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 

By substituting the equation above is obtained: 
𝑊𝛼1 = (𝑛 + 2𝑖) + 2(𝑛𝑘 − 𝑖) + 1 
         = 2𝑛𝑘 + 𝑛 + 1  
𝑊𝛼2 =  2(𝑛𝑘 − 𝑖) + (𝑛 + 2𝑖 + 1)    
         = 2𝑛𝑘 + 𝑛 + 1  

𝑊𝛼3 = (2𝑘 − 𝑙)𝑛 +
(1−𝑖)

2
+ (𝑙 + 1)𝑛 +

(𝑖+1)

2
  

         = 2𝑛𝑘 + 𝑛 + 1  

𝑊𝛼4 =  (2𝑘 − 𝑙)𝑛 +
(−2𝑛−𝑖+6)

2
+  (𝑙 + 1)𝑛 +

(𝑛+𝑖+1)

2
  

         = 2𝑛𝑘 + 𝑛 + 1  

Based on the equation above, the set of total labeling edge weights can be written 
as 𝑊𝛼 = {𝑊𝛼1, 𝑊𝛼2, 𝑊𝛼3, 𝑊𝛼4}. It can also be seen that 𝑊𝛼1 =  𝑊𝛼2 = ⋯ =  𝑊𝛼4 = 2𝑛𝑘 +
𝑛 + 1 or can be written as follows: ⋃  4𝑡=1 𝑊𝛼𝑡 = {2𝑛𝑘 + 𝑛 + 1,2𝑛𝑘 + 𝑛 + 1, … , 2𝑛𝑘 + 𝑛 +
1}. From this it can be concluded that the firecracker graph 𝐹𝑛,𝑘  have Super (2𝑛𝑘 + 𝑛 +

1,0)- edge antimagic if n odd, n ≥ 2, and k ≥ 3. 

 
Theorem 2. There are super total labeling ((𝑘 + 1)𝑛 + 3,2)-edge antimagic on the graph 
firecracker 𝐹𝑛,𝑘  if n odd (𝑛 ≥ 2),  and 𝑘 ≥ 3. 

Prove. First define the edge label 𝑓𝛼 : 𝐸(𝐹𝑛,𝑘 ) = {𝑒1, 𝑒2, . . . , 𝑒𝑛𝑘−1} → {𝑛𝑘 + 1, 𝑛𝑘 +

2, … , 2𝑛𝑘 − 1}, so edge label 𝑓𝛼  for super total labeling (𝑎, 2)- edge antimagic on the 
graph 𝐹𝑛,𝑘  can be formulated as follows: 

𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑛𝑘 + 2𝑖 − 1)  ; 1 ≤ 𝑖 ≤ 𝑛 and 𝑙 = 1 

𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = (𝑛𝑘 + 2𝑖)   ; 1 ≤ 𝑖 ≤ 𝑛 − 1  and 𝑙 = 1 

𝑓𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑘 + 𝑙)𝑛 +
(𝑖−1)

2
  ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑓𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) = (𝑘 + 𝑙)𝑛 +
(𝑛+𝑖−1)

2
  ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

Next, Wα  defined as weight of super total labeling edge 𝛼(𝑥𝑖,𝑙 ), 𝛼(𝑥𝑖,𝑙 𝑥𝑖,0), and 

𝛼(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) 

𝑊𝛼1 =  𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) +  𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0)        ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 

𝑊𝛼2 =  𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) +  𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0)        ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 

𝑊𝛼3 =  𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑊𝛼4 =  𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 

By substituting the equation above is obtained: 
𝑊𝛼1 = (𝑛 + 2𝑖) + (𝑛𝑘 + 2𝑖 − 1) 
         = (𝑘 + 1)𝑛 + 4𝑖 − 1  
𝑊𝛼2 =  (𝑛 + 2𝑖 + 1) + (𝑛𝑘 + 2𝑖)    
         = (𝑘 + 1)𝑛 + 4𝑖 + 1  

𝑊𝛼3 = (𝑙 + 1)𝑛 +
(𝑖+1)

2
+ (𝑘 + 𝑙)𝑛 +

(𝑖−1)

2
  



 Super Total Labeling (a,d)-Edge Antimagic on the Firecracker Graph 

Juhari 137 

         = (𝑘 + 2𝑙 + 1)𝑛 + 𝑖  

𝑊𝛼4 =  (𝑙 + 1)𝑛 +
(𝑛+𝑖+1)

2
+ (𝑘 + 𝑙)𝑛 +

(𝑛+𝑖−1)

2
  

         = (𝑘 + 2𝑙 + 2)𝑛 + 𝑖 
Based on the equation above, the set of total labeling edge weights can be written 

as 𝑊𝛼 = {𝑊𝛼1, 𝑊𝛼2, 𝑊𝛼3, 𝑊𝛼4}. It can also be seen that 𝑊𝛼1 =  𝑊𝛼2 = ⋯ =  𝑊𝛼4 = 2𝑛𝑘 +
𝑛 + 1 or can be written as follows: ⋃  4𝑡=1 𝑊𝛼𝑡 = {(𝑘 + 1)𝑛 + 3, (𝑘 + 1)𝑛 + 5, … , (3𝑘 +
1)𝑛 − 1}. From this it can be concluded that the firecracker graph 𝐹𝑛,𝑘 have Super 
((𝑘 + 1)𝑛 + 3,2)- edge antimagic if n odd, n ≥ 2, and k ≥ 3. 

 
Theorem 3. There is super total labeling (2𝑛𝑘 + 1,1)-edge antimagic on the graph 
firecracker 𝐹𝑛,𝑘  if n even (𝑛 ≥ 2),  and 𝑘 ≥ 3. 

Prove. To determine the total super labeling (𝑎, 1)–edge antimagic, defined first 

𝑓𝑎 : 𝐸(𝐹𝑛,𝑘 ) = {𝑒1, 𝑒2, … , 𝑒𝑛𝑘−1} → {𝑛𝑘 + 1, 𝑛𝑘 + 2, … ,2𝑛𝑘 − 1}which is the labeling edge 

label and can be formulated: 
𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑛𝑘 − 4𝑖 + 2)               ; 1 ≤ 𝑖 ≤ 𝑛 and 𝑙 = 1 

𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = (2𝑛𝑘 − 4𝑖)                    ;  1 ≤ 𝑖 ≤ 𝑛 − 1  and 𝑙 = 1 

𝑓𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 + 3𝑛 − 𝑖  ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑓𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0) = (2𝑘 − 𝑙)𝑛 + 2𝑛 − 1  ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

If 𝑊𝛼 defined as the total labeling edge weight 𝛼(𝑥𝑖,𝑙 ), 𝛼(𝑥𝑖,𝑙 𝑥𝑖,0), and 𝛼(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ). 

𝑊𝛼1 =  𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) +  𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0)        ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 

𝑊𝛼2 =  𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) +  𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0)        ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 

𝑊𝛼3 =  𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑊𝛼4 =  𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 

By substituting the equation above is obtained: 
𝑊𝛼1 = (𝑛 + 2𝑖) + (2𝑛𝑘 − 4𝑖 + 2) 
         = ( 2𝑛𝑘 + 𝑛 − 2𝑖 + 2 )  
𝑊𝛼2 =  (2𝑛𝑘 − 4𝑖) + (𝑛 + 2𝑖 + 1)    
         = ( 2𝑛𝑘 + 𝑛 − 2𝑖 + 1 )  

𝑊𝛼3 = (2𝑘 − 𝑙)𝑛 + 3𝑛 − 𝑖 + (𝑙 + 1)𝑛 +
(𝑖+1)

2
  

         = ( 2𝑛𝑘 + 4𝑛 + 
(𝑖+1)

2
 ) 

𝑊𝛼4 =  (2𝑘 − 𝑙)𝑛 + 2𝑛 − 𝑖 +  (𝑙 + 1)𝑛 +
(𝑛+𝑖+1)

2
  

         = ( 2𝑛𝑘 +
7𝑛+1−𝑖

2
 )   

Based on the equation above, the set of total labeling edge weights can be witten with 
𝑊𝛼 = {𝑊𝛼1, 𝑊𝛼2, 𝑊𝛼3, 𝑊𝛼4}. It can also be seen that the smalest edge lies in 𝑊𝛼2 and the 
biggest edge weights lies in 𝑊𝛼1, can be stated that 𝑊𝛼  forming arithmetic lines with initial 
term 2𝑛𝑘 + 1 and different 1 (one), or can be written ⋃  4𝑡=1 𝑊𝛼𝑡 = {2𝑛𝑘 + 1,2𝑛𝑘 +
2,2𝑛𝑘 + 3, … , ((3𝑛𝑘 − 1) + 𝑖)}. So, can be conclude that the firecracker graph 𝐹𝑛,𝑘 have 
Super (2𝑛𝑘 + 1,1)-EAT; n even (𝑛 ≥ 2),  and 𝑘 ≥ 3. 
  



 Super Total Labeling (a,d)-Edge Antimagic on the Firecracker Graph 

Juhari 138 

Theorem 4. There is super total labeling {𝑛𝑘 + 𝑛 + 4,3} –edge antimagic on the 
combination of firecracker graph 𝑚𝐹𝑛𝑘  if 𝑚 ≥ 2, n even (𝑛 ≥ 2),  and 𝑛 ≥ 3. 
Prove. To determine the total super labeling (𝑎, 1)-edge antimagic, defined first 

𝑓𝑎 : 𝐸(𝐹𝑛,𝑘 ) = {𝑒1, 𝑒2, … , 𝑒𝑛𝑘−1} → {𝑛𝑘 + 1, 𝑛𝑘 + 2, … ,2𝑛𝑘 − 1} which is the edge label and 

can be formulated: 
𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0)    = (𝑛𝑘 + 4𝑖 − 2)                  ; 1 ≤ 𝑖 ≤ 𝑛 and 𝑙 = 1 

𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) = (𝑛𝑘 + 4𝑖)                         ;  1 ≤ 𝑖 ≤ 𝑛 − 1  and 𝑙 = 1 

𝑓𝛼3(𝑥𝑖,𝑙 𝑥𝑖,0)    = (2𝑘 − 𝑙)𝑛 + 𝑖 − 1    ; if i odd, 1 ≤ 𝑖 ≤ 𝑛,and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑓𝛼4(𝑥𝑖,𝑙 𝑥𝑖,0)    = (2𝑘 − 𝑙)𝑛 + 𝑛 + 𝑖 − 1  ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 

 If 𝑊𝛼 defined as the total edge labeling weights 𝛼(𝑥𝑖,𝑙 ), 𝛼(𝑥𝑖,𝑙 𝑥𝑖,0), and 𝛼(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ). 

𝑊𝛼1 =  𝑤𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0) +  𝑓𝛼1(𝑥𝑖,𝑙 𝑥𝑖,0)        ; 1 ≤ 𝑖 ≤ 𝑛, and 𝑙 = 1 

𝑊𝛼2 =  𝑤𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0) +  𝑓𝛼2(𝑥𝑖,𝑙 𝑥𝑖,0)        ; 1 ≤ 𝑖 ≤ 𝑛 − 1, and 𝑙 = 1 

𝑊𝛼3 =  𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i odd, 1 ≤ 𝑖 ≤ 𝑛, and 2 ≤ 𝑙 ≤ 𝑘 − 1 

𝑊𝛼4 =  𝑤𝛼3(𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) + 𝑓𝛼3 (𝑥𝑖,𝑙 𝑥𝑖+1,𝑙 ) ; if i even, 2 ≤ 𝑖 ≤ 𝑛 − 1,and 2 ≤ 𝑙 ≤ 𝑘 − 1 

By substituting the equation above is obtained: 
𝑊𝛼1 = (𝑛 + 2𝑖) + (𝑛𝑘 + 4𝑖 + 2) 
         = ( 𝑛𝑘 + 𝑛 + 6𝑖 + 2 )  
𝑊𝛼2 =  (𝑛𝑘 + 2𝑖 + 1) + (𝑛𝑘 + 4𝑖)    
         = ( 2𝑛𝑘 + 𝑛 + 6𝑖 + 1 )  

𝑊𝛼3 = (𝑙 + 1)𝑛 +
(𝑖+1)

2
+ (2𝑘 − 𝑙)𝑛 + 𝑖 − 1  

         = ( 2𝑛𝑘 + 𝑛 + 
(1−𝑖)

2
 ) 

𝑊𝛼4 =  (2𝑘 − 𝑙)𝑛 + 𝑛 − 𝑖 + 1 +  (𝑙 + 1)𝑛 +
(𝑛+𝑖+1)

2
  

         = ( 2𝑛𝑘 +
5𝑛+2𝑖−1

2
 )   

Based on the equation above, the set of total labeling edge weights can be written with 
𝑊𝛼 = {𝑊𝛼1, 𝑊𝛼2, 𝑊𝛼3, 𝑊𝛼4}. It can also be seen that the smalest edge lies in 𝑊𝛼1 and the 
biggest edge weights lies in 𝑊𝛼2, can be stated that 𝑊𝛼  forming arithmetic lines with initial 
term 𝑛𝑘 + 𝑛 + 4 and different 3 (one), or can be written ⋃  4𝑡=1 𝑊𝛼𝑡 = {𝑛𝑘 + 𝑛 + 4, 𝑛𝑘 +
𝑛 + 7, 𝑛𝑘 + 𝑛 + 10, … , ((3𝑛𝑘 + 1)𝑛 + 𝑖)}. So, can be conclude that the firecracker graph 
𝐹𝑛,𝑘 have Super (2𝑛𝑘 + 1,1)-EAT; n even (𝑛 ≥ 2),  and 𝑘 ≥ 3. 

CONCLUSIONS 

Based on the result, can be concluded that firecracker graph 𝐹𝑛,𝑘  have super total labeling 
(𝑎, 𝑑)-edge antimagic, with 𝑑 ∈ {0,1,2,3} and bijective function in some of Lemma and 
Theorem show about super complete labeling (𝑎, 𝑑)-edge antimagic on the firecracker 
graph. The bijective function for each liberation with d different value have shown in 
equation (1),(2),(3) until (10) above.  
Open problem: super total labeling (𝑎, 𝑑)-edge antimagic on the firecracker graph 𝐹𝑛,𝑘  

for d=0 and d=2 where n even and 𝑘 ≥ 3; super total labeling d=1 and d=3 where n odd 
and 𝑘 ≥ 3. 
 

  



 Super Total Labeling (a,d)-Edge Antimagic on the Firecracker Graph 

Juhari 139 

REFERENCES 

[1] Dafik, “Structural Properties and Labeling of Graphs Statement of Authorship,” 
no. November, pp. 1–139, 2007. 

[2] M. Baca, L. Brankovic, M. Lascsakova, O. Phanalasy, and A. Semanicova–
Fenovcıkova, “On d-Antimagic Labelings of Plane Graphs,” Electron. J. Graph 
Theory Appl., vol. 1, no. 1, pp. 28–39, 2013. 

[3] M. A. Muttaqien and A. Suyitno, “Pelabelan Total Sisi Ajaib Pada Graf Double Star 
dan Graf Sun,” Unnes J. Math., vol. 2, no. 2, 2013. 

[4] R. H. Utomo, H. Tjahjana, B. Irawanto, and L. Ratnasari, “Pelabelan Total Super 
Trimagic Sisi Pada Beberapa Graf,” J. Fundam. Math. Appl., vol. 1, no. 1, p. 52, Jun. 
2018. 

[5] T. Utomo and N. Riskiana Dewi, “Dimensi Metrik Graf Amal (nKm),” Limits J. Math. 
Its Appl., vol. 15, no. 1, p. 71, Mar. 2018. 

[6]   V. Swaminathan and P. Jeyanthi, "Super edge-magic strength of firecrackers, 
banana trees and unicyclic graphs," Discrete Math., vol. 306, no. 14, pp. 1624–
1636, 2006. 

[7] S. Chattopadhyay and P. Panigrahi, “Some structural properties of power graphs 
and k -power graphs of finite semigroups,” J. Discret. Math. Sci. Cryptogr., vol. 20, 
no. 5, pp. 1101–1119, Jul. 2017. 

[8] Y.-N. Chen, W.-C.; Lu, H.-I; and Yeh, “Operations of Interlaced Trees and Graceful 
Trees,” Southeast Asian Bull. Math, vol. 21, pp. 337–348, 1997. 

[9]     J. A. Gallian, "A dynamic survey of graph labeling," Electron. J. Comb., vol. 1, no. 
DynamicSurveys, pp. 1–256, 2018. 

[10]  K. A. Sugeng and D. R. Silaban, "On b-edge consecutive edge labeling of some 
regular tree," Indones. J. Comb., vol. 4, no. 1, p. 76, 2020. 

[11] I. Halikin, “Pelabelan Lokal Titik Graf Hasil Diagram Lattice Subgrup Zn,” Al-
Khwarizmi J. Pendidik. Mat. dan Ilmu Pengetah. Alam, vol. 6, no. 1, pp. 47–56, Mar. 
2018. 

[12] A. S. Meliana Deta Anggraeni, Mulyono, “Pelabelan L(3,2,1) dan Pembentukan 
Graf Middle Pada Beberapa Graf Khusus,” vol. 2, no. 1, pp. 76–83, 2013.