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Odd-Even Congruence Labeling of Some Graphs 

 

Rituraj1, Shambhu Kumar Mishra2 

1Research Scholar, Department of Mathematics, Patliputra University, Patna, Bihar, India. rituraj6312@gmail.com 

2Professor, Department of Mathematics, Patliputra University, Patna, Bihar, India 

Shambhumishra5@gmail.com 

 

Article History: 

Received: 15-07-2024 

Revised: 30-08-2024 

Accepted: 11-09-2024 

Abstract:  

Introduction: Graph labeling is one of the fastest growing areas in the field of graph 

theory. A graph labeling is basically an assignment of integers to the nodes or edges or 

both, subject to certain conditions. Different types of graph labeling techniques have 

been developed by several authors.   

Objectives: The applications of labeling of graphs has  diverse fields in the study of 

data base management, secret sharing schemes, physical cosmology, debug circuit 

design, X-ray, crystallography, astronomy, radar, broadcasting network, secret, coding 

theory and much more. The graph labeling techniques  serve as useful  mathematical 

models and  solve various graph related problems. 

Methods: The strength of this research paper is based on Odd-even congruence labeling 

of different types of graphs. Assignment of natural numbers as labels for the edges and 

vertices of a graph. It is based on modular arithmetic property known as congruence 

graph labelling of a graph.For congruence graph labeling it entails the assignment of 

odd integers to vertices and even integers to edges with property of congruence graph 

labeling.  

Results: This labeling method has been identified on friendship graph, shell graph, 

generalized butterfly graph, fan graph, 𝑃2 + 𝑚𝐾1 graph  

Keywords: Labeling, Congruence labeling, Odd-even congruence labeling, friendship 

graph, shell graph, generalized butterfly graph, fan graph, 𝑃2 + 𝑚𝐾1 graph 

1. Introduction 

In this paper all graphs 𝐺 considered as finite, undirected, connected and without loops. Suppose 𝑉(𝐺) 

and 𝐸(𝐺) be the set of vertices and edges of a graph 𝐺 respectively. The cardinality of vertex set is 

denoted by |𝑉(𝐺)| and the edge set is denoted by |𝐸(𝐺)| are called the order and size of the graph  𝐺. 

For standard terminology of Graph Theory we used [1]. For all detailed survey of graph labeling, we 

refer [2]. While studying graph theory, one that has gained a lot of popularity during the last 62 years 

is the concept of labeling of graphs due to its wide range of applications. A labeling of a graph 𝐺 is 

one-to-one mapping that carries the set of graph elements onto a set of numbers, called labels. In 1967, 

Rosa [5] published a pioneering paper on graph labeling problems. Thereafter many types of graph 

labeling methods have been studied by several authors. 

G.Thamizhendhi and K.Kanakambika [6] have introduced Odd-even congruence labeling and they 

proved behaviour of several graphs like bipartite graph, comb graph, star graph, graph acquired by 

connecting two copies of even cycle 𝐶𝑟 by a path 𝑃𝑡 , shadow graph of the path 𝑃𝑡 and the tensor 

product of 𝐾1,𝑡 & 𝑃2 as Odd-even congruence graph. They defined an odd-even congruence graph, if 



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vertex and edge set are assigned by distinct odd and even integers respectively, further 𝑓(𝑢𝑝) ≡

𝑓(𝑢𝑞) (𝑚𝑜𝑑 𝑔(𝑒)) , 𝑢𝑝 and 𝑢𝑞 are adjacent vertices in 𝐺. In this paper we investigate the existence 

of Odd-even congruence labeling for vertex switching graph, friendship graph, shell graph, generalized 

butterfly graph, fan graph, 𝑃2 + 𝑚𝐾1 graph. 

2. Preliminaries  

Definition 2.1.  [6, 7]. A graph obtained by fetching a vertex 𝑥 of 𝐻, eliminating the adjacent edges of 

𝑥 and by adding new edges that are joining 𝑥 to their non-adjacent vertices in 𝐻 is called vertex 

switching 𝐻𝑥 of 𝐻. 

Definition 2.2.  [3]. The friendship graph 𝐹𝑟𝑛 is a collection of 𝑛 triangles with a common vertex. It 

is a planar undirected graph with 2𝑛 + 1 vertices and 3𝑛 edges constructed by joining 𝑛 copies of the 

cycle graph 𝐶3  with a common vertex. 

Definition 2.3.  [10]. A shell 𝑆𝑛 is the graph obtained as a cycle 𝐶𝑛 by taking (𝑛 − 3) concurrent 

chords sharing a common end point called the apex. Shell graph are denoted as 𝐶(𝑛,𝑛−3). A shell 𝑆𝑛 is 

also called fan 𝑓𝑛−1 

Definition 2.4.  [4]. The Generalized butterfly graph denoted by 𝐵𝐹𝑛 obtained by inserting adjoining 

vertices to every wing with assumption that sum of inserting vertices to every wing are same. 

Definition 2.5.  [9]. A fan graph 𝑓𝑛, 𝑛 ≥ 2 obtained by joining all vertices of a path 𝑃𝑛 to a further 

vertex , called the center. 

𝑓𝑛 = 𝐾1 + 𝑃𝑛 ,     |𝑉(𝑓𝑛)| = 𝑛 + 1  and  |𝐸(𝐺)| = 2𝑛 − 1 

3. Main Results 

Theorem 3.1 The graph obtained from switching of any vertex in cycle 𝐶𝑟 is an Odd-even 

congruence graph. 

Proof: 

Suppose 𝑥1, 𝑥2, … … … … … , 𝑥𝑟−1 be the vertices of a cycle 𝐶𝑟. Let 𝐻𝑥1
 is the graph obtained from 

switching of a vertex 𝑥1in 𝐶𝑟. In 𝐻𝑥1
 each vertex 𝑥𝑖 other than 𝑥2 and 𝑥𝑟 join to 𝑥1. 

Here |𝑉(𝐻𝑥1
)| = 𝑟 and |𝐸(𝐻𝑥1

)| = 2𝑟 − 5 

we have 𝑑 = min{2𝑟 ,2(2𝑟 − 5) } 

                    = 2(2𝑟 − 5) 

Let the edge set of 𝐻𝑥1
 be 

𝐸(𝐻𝑥1
 )  =  {𝑥𝑖𝑥𝑖+1 / 2 ≤ 𝑖 ≤ 𝑟 − 1} ∪ {𝑥1𝑥𝑖+1/ 2 ≤ 𝑖 ≤ 𝑟 − 2}   

Where 𝑒𝑖 =  𝑥𝑖𝑥𝑖+1 / 2 ≤ 𝑖 ≤ 𝑟 − 1 and 𝑤𝑖 =  𝑥1𝑥𝑖+1/ 2 ≤ 𝑖 ≤ 𝑟 − 2 

Define a labeling 𝑓 ∶ 𝑉(𝐻𝑥1
)  →  {1,3,7,15,31,63, … . . … . . , 2𝑛+𝑖 − 1} is defined as follows 

 𝑓(𝑥1) = 1 ,      



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 𝑓(𝑥𝑖+1) =  2𝑖+1 − 1 for 1 ≤ 𝑖 ≤ 𝑟 − 1 

The edge labeling  𝑘 ∶ 𝐸(𝐻𝑥1
)  →  {4,6,8,14,16, … . . … . . ,4𝑖 + 2} is defined as  

𝑘(𝑒𝑖) =  2𝑖  for   2 ≤ 𝑖 ≤ 𝑟 − 1                

𝑘(𝑤𝑖) =  2𝑖+1 − 2 for   2 ≤ 𝑖 ≤ 𝑟 − 2     

Clearly  𝑘(𝑒𝑖) divides |(𝑓(𝑥𝑖) − 𝑓(𝑥𝑖+1))|   for  2 ≤ 𝑖 ≤ 𝑟 − 1  and 

𝑘(𝑤𝑖) divides |(𝑓(𝑥1) − 𝑓(𝑥𝑖+1))|   for  2 ≤ 𝑖 ≤ 𝑟 − 2   

Hence the graph obtained from switching of any vertex in cycle 𝐶𝑟 is an Odd-even congruence graph. 

Example: 3.2 

Consider a graph  𝐺 = 𝐻𝑥1 obtained from 𝐶8. 

 

Figure-1 

Fig-1 shows that graph 𝐺 admits odd-even congruence labeling.  

Theorem 3.3 The friendship graph 𝐹𝑟𝑛 is an Odd-even congruence graph. 

Proof: 

Let the vertex of 𝐹𝑟𝑛 be  

𝑉(𝐹𝑟𝑛) =  { 𝑣𝑖  /  𝑖 = 0,1,2,3 … … … … … 2𝑛} with 𝑣0 as the central vertex 

and the edge set be  

𝐸(𝐹𝑟𝑛)  =  {𝑣0𝑣𝑖 , 𝑣𝑖𝑣𝑖+1 , 𝑖 = 1,2 … … … … … 2𝑛}  

Where 𝑒𝑖 = {𝑣0𝑣𝑖  /1 ≤ 𝑖 ≤ 2𝑛} and 𝑤𝑖 =  𝑣𝑖𝑣𝑖+1 for 𝑖 = 1,3,5, … … … 2𝑛 − 1 

With |𝑉| =  2𝑛 + 1  and |𝐸|  = 3𝑛 

For 𝐺 = 𝐹𝑟𝑛 , we have 𝑑 =  𝑚𝑖𝑛 {2(2𝑛 + 1) , 2(3𝑛)}    

                                             = 2(2𝑛 + 1) 

The vertices of 𝐹𝑟𝑛 are labeled as given below. 



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Define the bijection 𝑓 ∶ 𝑉(𝐹𝑟𝑛) → {1,3,7, . . . . . . . . . . . . . . . , 2𝑖 − 1}  as 𝑓(𝑣𝑖−1) = 2𝑖 − 1 for 1 ≤ 𝑖 ≤

2𝑛 + 1  

The edge labeling  𝑘 ∶ 𝐸(𝐹𝑟𝑛)  →  {2,4,6,8, … . . … . . ,4𝑖 + 2} is defined as  

𝑘(𝑒𝑖) =  2𝑖+1 − 2  for 1 ≤ 𝑖 ≤ 2𝑛       

𝑘(𝑤𝑖) =  4𝑖    for 𝑖 = 1,3,5, … … … 2𝑛 − 1 

Clearly  𝑘(𝑒𝑖) divides |(𝑓(𝑣𝑖) − 𝑓(𝑣0))|   for  1 ≤ 𝑖 ≤ 2𝑛  and 

𝑘(𝑤𝑖) divides |(𝑓(𝑣𝑖) − 𝑓(𝑣𝑖+1))|   for  𝑖 = 1,3,5, … … … 2𝑛 − 1 

Hence the graph friendship graph 𝐹𝑟𝑛 is an odd-even congruence graph. 

Example: 3.4 

Consider a graph  𝐺 = 𝐹𝑟𝑛 with 𝑛 = 7  

 

Figure-2 

Fig-2 shows that friendship graph 𝐹𝑟7 admits odd-even congruence labeling.  

Theorem 3.5 The shell graph 𝑆𝑛 is an Odd-even congruence graph.  

Proof: 

Let 𝐺 be a shell graph. 

Define 𝑉(𝐺 ) =  { 𝑢, 𝑣𝑖  /1 ≤ 𝑖 ≤ (𝑛 − 1)}   and 

𝐸(𝐺) =  {𝑒𝑖 =  𝑢𝑣𝑖 ∶ 1 ≤ 𝑖 ≤ 𝑛 − 1; 𝑒𝑖
′ = 𝑣𝑖𝑣𝑖+1: 1 ≤ 𝑖 ≤ 𝑛 − 2} 

are the vertices and edges of a graph 𝐺   

Also  |𝑉(𝐺)| = 𝑛 and |𝐸(𝐺)| = 2𝑛 − 3 

Then 𝑑 =min {2𝑛, 2(2𝑛 − 3)} 

              = 2𝑛 



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Define a bijective function 𝑓 ∶ 𝑉(𝐺)  →  {1,3,7, … . . … . . , 2𝑖+1 − 1} is defined as follows 

𝑓(𝑢) =  1   

𝑓(𝑣𝑖) =  2𝑖+1 − 1  for 1 ≤ 𝑖 ≤ 𝑛 − 1 

The edge labeling  𝑘 ∶ 𝐸(𝐺)  →  {2,4,8 … . . … . . ,4𝑖 + 2} is defined as  

𝑘(𝑒𝑖) =  2𝑖+1 − 2 for 1 ≤ 𝑖 ≤ 𝑛 − 1                  

𝑘(𝑒𝑖
′) =  2𝑖    for 1 ≤ 𝑖 ≤ 𝑛 − 2       

Clearly  𝑘(𝑒𝑖) divides |(𝑓(𝑢) − 𝑓(𝑣𝑖))|   for  1 ≤ 𝑖 ≤ 𝑛 − 1  and 

𝑘(𝑒𝑖
′) divides |(𝑓(𝑣𝑖) − 𝑓(𝑣𝑖+1))|   for  1 ≤ 𝑖 ≤ 𝑛 − 2 

Hence the shell graph 𝑆𝑛  is an odd-even congruence graph. 

Example: 3.6 

Consider a shell graph  𝐺 = 𝑆𝑛 with 𝑛 = 8 

 

Figure-3 

Fig-3 shows that shell graph 𝑆8 admits odd-even congruence labeling.  

Theorem 3.7 A generalized butterfly graph 𝐵𝐹𝑛 admits odd-even congruence graph for 𝑛 ≥ 2. 

Proof: 

Suppose the vertex set of 𝐵𝐹𝑛 be  

Define 𝑉(𝐵𝐹𝑛 ) =  { 𝑣𝑖  /0 ≤ 𝑖 ≤ 2𝑛}   and the edge set of 𝐵𝐹𝑛 be  



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𝐸(𝐵𝐹𝑛) =  { 𝑣𝑖𝑣𝑖+1 /1,2, … … . , 𝑛 − 1, 𝑛 + 1, … … 2𝑛 − 1} ∪ {𝑣0𝑣𝑖  /𝑖 = 1,2, … … . . ,2𝑛} 

Where |𝑉(𝐺)| = 2𝑛 + 1 and |𝐸(𝐺)| = 4𝑛 − 2 

Then 𝑑 =min {2(2𝑛 + 1),2(4𝑛 − 2)} 

              = 2(2𝑛 + 1) 

Define a bijective function 𝑓 ∶ 𝑉(𝐺)  →  {1,3,7, … . . … . . , 2𝑖+1 − 1  } is defined as follows 

𝑓(𝑢) =  1   

𝑓(𝑣𝑖) =  2𝑖+1 − 1  for 1 ≤ 𝑖 ≤ 𝑛 − 1 

The edge labeling  𝑘 ∶ 𝐸(𝐺)  →  {2,4,8 … . . … . . ,4𝑖 + 2} is defined as  

ℎ(𝑣𝑖−1𝑣𝑖) =  2𝑖  for 2 ≤ 𝑖 ≤ 𝑛                  

𝑘(𝑣𝑖−1𝑣𝑖) =  2𝑖  for 𝑛 + 2 ≤ 𝑖 ≤ 2𝑛    

 𝑝(𝑣0𝑣𝑖) =  2𝑖+1 − 2  for 1 ≤ 𝑖 ≤ 2𝑛      

Clearly  ℎ(𝑣𝑖−1𝑣𝑖) divides |(𝑓(𝑣𝑖−1) − 𝑓(𝑣𝑖))|   for  2 ≤ 𝑖 ≤ 𝑛 , 

𝑘(𝑣𝑖−1𝑣𝑖) divides |(𝑓(𝑣𝑖) − 𝑓(𝑣𝑖+1))|   for  𝑛 + 2 ≤ 𝑖 ≤ 2𝑛 and 

𝑝(𝑣0𝑣𝑖) divides |(𝑓(𝑣𝑖) − 𝑓(𝑣0))|   for  1 ≤ 𝑖 ≤ 2𝑛 

Hence the generalized butterfly graph 𝐵𝐹𝑛  is an odd-even congruence graph. 

Example: 3.8 

Consider a generalized butterfly graph 𝐵𝐹𝑛 with 𝑛 = 5 

 

Figure-4 

Fig-4 shows that friendship graph 𝐹𝑟6 admits odd-even congruence labeling.  

Theorem 3.9 The Fan graph 𝐹𝑛 admits odd-even congruence graph. 

Proof: 

Let 𝐺 be a graph of the fan graph 𝐹𝑛 



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Let 𝑉(𝐺) = {𝑣𝑖: 1 ≤ 𝑖 ≤ 𝑛 + 1} be the vertices of 𝐺 and  

𝐸(𝐺) = {𝑣1𝑣𝑖+1: 1 ≤ 𝑖 ≤ 𝑛} ∪ {𝑣𝑖𝑣𝑖+1 ∶ 2 ≤ 𝑖 ≤ 𝑛} be the edges of 𝐺. 

Where  𝑒𝑖 = {𝑣1𝑣𝑖+1: 1 ≤ 𝑖 ≤ 𝑛} and 𝑤𝑖 =  {𝑣𝑖𝑣𝑖+1 /2 ≤ 𝑖 ≤ 𝑛}  

Now |𝑉(𝐺)| = 𝑛 + 1 and |𝐸(𝐺)| = 2𝑛 − 1 

Then 𝑑 =min (2(𝑛 + 1), 2(2𝑛 − 1)) 

              = 2(𝑛 + 1) 

Define the bijection 𝑓 ∶ 𝑉(𝐺)  →  {1,3,7, … . . … . . , 2𝑝 − 1} is defined as follows 

𝑓(𝑣𝑖) =  2𝑖 − 1 for 1 ≤ 𝑖 ≤ 𝑛 + 1 

The edge labeling  𝑘 ∶ 𝐸(𝐺)  →  {2,4,6,8,14, … . . … . . , 2𝑖+1 − 2} is defined as follows 

𝑘(𝑒𝑖) =  2𝑖+1 − 2 for 1 ≤ 𝑖 ≤ 𝑛 

𝑘(𝑤𝑖) =  2𝑖+1 for 2 ≤ 𝑖 ≤ 𝑛 

Clearly  𝑘(𝑒𝑖) divides |(𝑓(𝑣𝑖+1) − 𝑓(𝑣1))|   for  1 ≤ 𝑖 ≤ 𝑛  and 

𝑘(𝑤𝑖) divides |(𝑓(𝑣𝑖+1) − 𝑓(𝑣𝑖))|   for  2 ≤ 𝑖 ≤ 𝑛    

Hence the fan 𝐹𝑛 is an odd-even congruence graph. 

Example: 3.10 

Consider a generalized fan graph 𝐹𝑛 with 𝑛 = 6 

 

Figure-5 

Fig-5 shows that friendship graph 𝐹6 admits odd-even congruence labeling.  

Theorem 3.11 The graph 𝑃2 + 𝑚𝐾1 admits odd-even congruence graph. 

Proof: 

Suppose 𝑃2 is a path having two vertices 𝑢1, 𝑢2. 



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Let 𝑢3, 𝑢4, … … … . . , 𝑢𝑚+2 be the 𝑚 isolated vertices 

Connecting 𝑢1, 𝑢2 with 𝑢𝑖 , 3 ≤ 𝑖 ≤ 𝑚 + 2 

We obtain 𝑃2 + 𝑚𝐾1 

Suppose 𝐺 = 𝑃2 + 𝑚𝐾1 

Let the vertex set of 𝐺 be  

 Let 𝑉(𝐺) = {𝑢𝑖: 1 ≤ 𝑖 ≤ 𝑚 + 2} be the vertices of 𝐺 and  

𝐸(𝐺) = {𝑢1𝑢2} ∪ {𝑢1𝑢𝑖  /3 ≤ 𝑖 ≤ 𝑚 + 2} ∪ {𝑢2𝑢𝑖  /3 ≤ 𝑖 ≤ 𝑚 + 2} be the edges of 𝐺 where 𝑤𝑖 =

𝑢1𝑢𝑖 and 𝑞𝑖 = 𝑢2𝑢𝑖. Also  |𝑉(𝐺)| = 2 + 𝑚 and |𝐸(𝐺)| = 2𝑚 + 1 

Then 𝑑 =min (2(2 + 𝑚 ), 2(2𝑚 + 1)) 

              = 2(2 + 𝑚) 

Define a bijective function ℎ ∶ 𝑉(𝐺)  →  {1,3,7, … . . … . . ,4𝑖 + 3} is defined as follows 

𝑓(𝑢1) =  1 , 𝑓(𝑢2) = 3   

ℎ(𝑢𝑖+2) =  4𝑖 + 3 for 1 ≤ 𝑖 ≤ 𝑚 

The edge labeling  𝑘 ∶ 𝐸(𝐺)  →  {2,4,6,8, … . . … . . ,4𝑖 + 2} is defined as  

𝑓(𝑤) = 2 

𝑘(𝑤𝑖) =  4𝑖 + 2  for 1 ≤ 𝑖 ≤ 𝑚       

𝑘(𝑞𝑖) =  4𝑖   for 1 ≤ 𝑖 ≤ 𝑚       

Clearly 𝑘(𝑤) divides |𝑓(𝑢1) − 𝑓(𝑢2)|   

𝑘(𝑤𝑖) divides |(𝑓(𝑢1) − ℎ(𝑢𝑖))|   for  1 ≤ 𝑖 ≤ 𝑚   and 

𝑘(𝑞𝑖) divides |(𝑓(𝑢2) − ℎ(𝑢𝑖))|   for  1 ≤ 𝑖 ≤ 𝑚    

Hence the graph 𝑃2 + 𝑚𝐾1 is an odd-even congruence graph. 

Example: 3.12 

Consider a graph 𝑃2 + 𝑚𝐾1 with 𝑚 = 6 

 

Figure-6 



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Fig-6 shows that a graph 𝑃2 + 𝑚𝐾1 with 𝑚 = 6 admits odd-even congruence labeling.  

4. Conclusion 

The labelling of graphs is an interesting and vast research area which is very useful and it is extended 

in various topics by several people. In this paper we have studied Odd-even congruence labeling 

behaviour of odd-even congruence labeling for friendship graph, shell graph, generalized butterfly 

graphs, fan graph,  𝑃2 + 𝑚𝐾1 graph etc. To derive similar results for other graph families is an open 

problem. 

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