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Equitable Total Coloring of Line Graph of Certain Graphs 

 

R. Sudhakar1, G. Jayaraman2*, A. Punitha3 and R. Arasu4 

123Department of Mathematics, Vels Institute of Science, Technology and Advanced 

Studies, Pallavaram, Chennai-117, India 

4Department of Mathematics, Vel Tech Multi-Tech Dr. Rangarajan Dr. Sakunthala Engineering College, Chennai 600 

062, India 

*Corresponding author Email: jayaram07maths@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract: An equitable total-coloring of a graph G is a proper total-coloring such that the 

number of vertices and edges in any two color classes differ by at most one. In this paper, 

we determined the equitable total chromatic number for line graph of ladder, slanting 

ladder, triangular snake, alternate triangular snake, quadrilateral snake and alternate 

quadrilateral snake 

Introduction: Graph coloring is a fundamental problem in graph theory with applications 

in scheduling, networking, and resource allocation. A total-coloring of a graph G is an 

assignment of colors to both vertices and edges such that adjacent vertices, adjacent edges, 

and incident vertex-edge pairs receive distinct colors. An equitable total-coloring is a 

special type of total-coloring where the sizes of any two color classes differ by at most one. 

The equitable total chromatic number, denoted as ( )''

e G is the minimum number of 

colors required for such a coloring.  

Objectives: To establish the equitable total chromatic number for the line graph of specific 

families  of structured graphs. To develop systematic coloring techniques for achieving an 

equitable total-     coloring of these graphs. To contribute to the broader study of equitable 

colorings in graph theory and expand the known results in this domain. 

Methods: To determine the equitable total chromatic number for the line graphs of the 

given graph families, we employ the following methodology: Graph Construction: We 

formally define the structure of the ladder, slanting ladder, triangular snake, alternate 

triangular snake, quadrilateral snake, and alternate quadrilateral snake, along with their 

corresponding line graphs. Coloring Strategy: We apply systematic coloring techniques 

ensuring that adjacent vertices, adjacent edges, and incident vertex-edge pairs receive 

different colors while maintaining equitable distribution of color classes. Mathematical 

Analysis: We derive lower bounds for ( )''

e G and establish its exact value using 

combinatorial and structural properties of the graphs. Verification and Proof: We validate 

the obtained chromatic numbers through case-based analysis and, where applicable, 

provide rigorous proofs for correctness. 

Results: The study successfully determines the exact value of the equitable total chromatic 

number for the line graphs of the considered structured graphs. The results provide new 

insights into the equitable total-coloring of line graphs of ladder-based and snake-like 

structures, which are commonly encountered in chemical graph theory and network design 

problems.  

Conclusions: This paper establishes the equitable total chromatic number for the line 

mailto:jayaram07maths@gmail.com


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graphs of several structured graphs, contributing to the ongoing research in equitable 

colorings. The findings demonstrate that the structural properties of the base graphs 

significantly influence their equitable total chromatic numbers. These results can be 

extended to other classes of graphs, and future research may explore algorithmic 

approaches for efficient equitable total-coloring in larger and more complex graph families 

 Keywords: Total coloring, equitable total coloring, line graph, ladder graphs, snake 

graphs. 

1. Introduction 

In this paper, we consider only finite undirected graphs without loops or multiple edges. Let G = (V(G), 

E(G)) be a graph with vertex set V (G) and edge set E(G). The total coloring of a graph G is an 

assignment of colors to both the vertices and edges of G, such that no two adjacent or incident vertices 

and edges of G are received the same colors. They both conjectured that for any graph G the following 

inequality holds: ''( ) 1 ( ) ( ) 2G G G +    + , where ( )G is the maximum degree of G. It is clear 

that ( ) 1G +  is the possible lower bound. In 1994, Fu [4] first introduced the concept of equitable total 

coloring and the equitable total chromatic number of a graph. Gong Kun et.al [3] proved some results 

on the equitable total chromatic number of ,n n m nW K F K   and m nS K . Jayaraman et al.[5, 6] 

determined the equitable total chromatic number for the splitting midde, total graph of paths, cycles 

and splitting graph of star graphs.  Veninstine vivik et.al [8] determining the equitable total chromatic 

number for wheel, gear, helm, and sunlet graphs.  Gong Kun et.al [2] derived several findings regarding 

the equitable total chromatic number for the graphs ,n n n nW K F K   and m nS K .  Wang et.al [9] 

addressed the equitable total coloring for the graphs with a maximum degree of 3, while Zhang Zhong-

fu [10] investigated the equitable total coloring of certain join graphs.   

 

2. Preliminaries 

Definition 2.1. The line graph L(G) is defined such that its vertices correspond to the edges of G, and 

two vertices in L(G) are adjacent if their corresponding edges in G share a common vertex. 

Definition 2.2.  The ladder graph formed by taking two parallel paths of length  and connecting 

corresponding vertices with additional edges. 

Definition 2.3. The slanting ladder is a graph that consists of two copies of  having vertex set 

 and the edge set is formed by adjoining and  for all . 

Definition 2.4. A Triangular snake nT [1] is obtained from th 1 2, ,..., nv v v  by joining iv and 1iv + to a new 

vertex iu for 1,2,3,...., 1i n= − . 

Definition 2.5. An alternate triangular snake nAT [1] is a graph having vertex and edge set  

( ) { :1 } :1
2

m l l

m
V AT v l m u l

  
=      

  
 and 

nL ' 'n

nSL nP

{ :1 } { :1 }i iu i n v i n     iu 1iv + 1 1i n  −



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1( ) { :1 1}m l lE AT v v l m+=   −
2 1 2:1 :1

2 2
l l l l

m m
v u l v u l−

      
         

      
 

Definition 2.6. A Quadrilateral snake Q [7] is obtained from a path by replacing every edge by a 

cycle 4.C   

Definition 2.7. An Alternate quadrilateral snake AQ [7] is a graph constructed from a path P  by 

replacing every alternate edges with a quadrilateral cycle 4C , forming a sequence of interspersed 4C

cycles along the path. 

Conjecture 2.8([4]). For any simple graph ,G  ''( ) ( ) 2.e G G   +  

Conjecture 2.9([9]). For every graph G, G has an equitable total k-coloring for each
''max{ ( ),  ( ) 2}k G G  + . 

3. Results and Discussion 

Theorem 3.1: Let ( )nL L represent the line graph of a ladder graph, then ( )'' ( ) 5.e nL L =  

Proof: Let    '( ( )) , :1 1 :1nV L L u v n u n   =   −   and 

   ' '' ' ''( ( )) , :1 2 , , , :1 1nE L L e f n e e f f n      =   −   − , where 1,e u u   += ' ' ,e u u  =
'' '

1,e u u   +=
 

1,f v v   += ' ' ,f u v  = '' '

1.f v u  +=  

We divide the vertex and edge set of ( )nL L  into distinct partition as described below.  

For1 1n  − . 

' ' ' ' ' '

4 9 5 1 2 7 5 3 1 6 5 4 5 10 5 2 7 5 3

1 '' '' '' ' ' ' '' '' ''

3 8 5 2 5 10 5 3 8 5 2 4 9 5 1

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }                  

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − − −

− − −

=
     





 

' ' ' ' ' '

5 10 5 3 8 5 2 2 7 5 3 6 11 5 1 3 8 5 2

2 '' '' '' ' ' ' '' '' ''

4 9 5 1 1 6 5 4 4 9 5 1 5 10 5

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }                 

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − + −

− − −

=
        





 

' ' ' ' ' '

1 6 5 4 4 9 5 1 3 8 5 2 2 7 5 3 4 9 5 1

3 '' '' '' ' ' ' '' '' ''

5 10 5 2 7 5 3 5 10 5 1 6 5 4

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }                  

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − − − −

− −

=
        





 

' ' ' ' ' '

2 7 5 3 5 10 5 4 9 5 1 3 8 5 2 5 10 5

4 '' '' '' ' ' ' '' '' ''

1 6 5 4 3 8 5 2 1 6 5 4 2 7 5 3

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }   

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }               

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − −

− − − −

=
        





 

' ' ' ' ' '

3 8 5 2 1 6 5 4 5 10 5 4 9 5 1 1 6 5 4

5 '' '' '' ' ' ' '' '' ''

2 7 5 3 4 9 5 1 2 7 5 3 3 8 5 2

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }   

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }              

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − − −

− − − −

=
          





 

Based on the above procedure of coloring, it is evident that the color classes 1 2 3 4, , ,T T T T  and 5T  are 

independent sets of ( )nL L  and it holds inequality | | | | 1a bT T−   for a b . This implies that



Communications on Applied Nonlinear Analysis 

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( )'' ( ) 5e nL L  . Further, since 4 = , we have ( ) ( )'' "( ) ( ) 1 4 1 5e n nL L L L =  +  +  .  Hence 

( )'' ( ) 5.e nL L =  

Example 3.1: The graph 7( )L L and its equitable total coloring is shown in figure 1. 

 
 

Figure 1: 7( )L L and its equitable total coloring 

It is clear that the color classes 1 2 3 4, , ,T T T T  and  5T  are independent sets of 7( )L L . For which the color 

classes are partition into 1 2 4 10T T T= = =  and 3 5 11T T= = (see Figure 1) and it holds the inequality 

| | | | 1,i jT T−   for every pair of ( , )i j . This implies that ( )''

7( ) 5e L L  . Moreover, 4 = . We have

( )''

7( )e L L   1 4 1 5+  +  .  Hence ( )''

7( ) 5e L L = . 

Theorem 3.2: Let ( )nL SL represent the line graph of a slanting ladder, then ( )'' ( ) 5.e nL SL =   

Proof: '( ( )) { , , :1 }nV L SL u v u n   =    

   ' '' ' ''( ( )) , :1 2 , , , :1 1nE L SL e f n e e f f n      =   −   − , where 1,e u u   +=  ' ' ,e u u  =  

' ' ,e u u  = '' '

1,e u u   +=  1,f v v   += ' ' ,f u v  = '' '

1.f v u  +=     

We divide the vertex and edge set of ( )nL SL into distinct partition as described below.  

' ' ' ' ' '

4 9 5 1 2 7 5 3 5 10 5 1 6 5 4 3 8 5 2

1 '' '' '' ' ' ' '' '' ''

5 10 5 5 10 5 4 9 5 1 2 7 5 3

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }                   

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − − −

− −

=
    



  

' ' ' ' ' '

5 10 5 3 8 5 2 1 6 5 4 2 7 5 2 4 9 5 1

2 '' '' '' ' ' ' '' '' ''

1 6 5 4 1 6 5 4 5 10 5 3 8 5 2

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }                  

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − − −

− − −

=
       



  

' ' ' ' ' '

1 6 5 4 4 9 5 1 2 7 5 3 3 8 5 2 5 10 5

3 '' '' '' ' ' ' '' '' ''

2 7 5 3 2 7 5 3 1 6 5 4 4 9 5 1

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }     

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }            

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − − −

− − − −

=
              



     

' ' ' ' ' '

2 7 5 3 5 10 5 3 8 5 2 4 9 5 1 1 6 5 4

4 '' '' '' ' ' ' '' '' ''

3 8 5 2 3 8 5 2 2 7 5 3 5 10 5

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }   

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }               

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − − −

− − −

=
            



  



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' ' ' ' ' '

3 8 5 2 1 6 5 4 4 9 5 1 5 10 5 2 7 5 3

5 '' '' '' ' ' ' '' '' ''

4 9 5 1 4 9 5 1 3 8 5 2 1 6 5 4

{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , ,..., }   

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }              

u u u v v v u u u e e e e e e
T

e e e f f f f f f f f f

    

   

− − − −

− − − −

=
          



   

Based on the coloring technique described above, it is evident that the color classes 1 2 3 4, , ,T T T T  and 5T  

are independent sets of ( )nL SL and it holds inequality | | | | 1a bT T−   for a b .  This implies that

( )'' ( ) 5e nL SL  . Further, since 4 = , we have ( )'' ( )e nL SL = ( )" ( )nL SL   1 4 1 5+  +  .  Hence

( )'' ( ) 5e nL SL = . 

Example 3.2: The graph 7( )L SL and its equitable total coloring is shown in figure 2. 

 

Figure 2: 7( )L SL and its equitable total coloring 

The color classes are ( )  7 1 2 3 4 5( ) , , , ,T L SL T T T T T= .  It is clear that the color classes  1 2 3 4, , ,T T T T  and  

5T  are independent sets of 7( )L SL . For which the color classes are partition into 2 3 5 12T T T= = =  

and 1 4 11T T= =  (see Figure 2) and it holds the inequality | | | | 1,i jT T−   for every pair of ( , )i j . This 

implies that ( )''

7( ) 5e L SL  . Moreover, 4 = . We have ( )''

7( )e L SL   1 4 1 5+  +  .  Hence

( )''

7( ) 5e L SL = . 

Theorem 3.3: Let ( )nL T represent the line graph of a triangular snake graph, then 

( )'' ( ) ( ( )) 1.e n nL T L T =  +  

Proof: Let '( ( )) { , , :1 1}nV L T u v u n   =   −  

   ' ' '' '' '''( ( )) , , :1 1 , , , :1 2nE L SL e f f n e e f f n       =   −   − , where  ' ,e u u  =  

' '

1,e u u   += ''

1,e v v   +=
 

,f u v  = ' ' ,f u v  = '' '

1,f u v   += ''

1f v u   +=
 

We divide the vertex and edge set of ( )nL T  into distinct partition as described below.  

For 1 1n  − . 

' ' ' '' '' '' '

1 1 4 3 2 3 6 3 1 4 3 2 2 5 3 1 1{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { , 0(mod3)}nT v v v e e e e e e e e e f n   − − − −= 
 



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' ' ' '' '' '' '

2 2 5 3 1 2 5 3 1 3 6 3 3 6 3 1 1{ , ,..., } { , ,..., } { , ,..., } { , ,..., } { } { , 1(mod3)}nT v v v e e e e e e e e e f f n   − − −= 

' ' ' '' '' ''

3 6 3 4 7 3 1 2 5 3 1 4 7 3 1

3 '

1 2 1

{ , ,..., } { , ,..., } { , ,..., } { , ,..., }

{ } { } { , 0,2(mod3)}                                                     

v v v e e e e e e e e e
T

u f f n

   + − +


= 


 

' ' ' '

4 1 3 2 1 3 4 1 2{ , ,..., } { , ,..., } { }nT u u u f f f f − −=
 

''' ' ' ' ' '

5 2 4 1 2 3 4 2 1 1{ , ,..., } { } { , ,..., } { , 2(mod3)  6} { , 1(mod3)}n n nT u u u f f f f f n and n f n− − −=  = 

' ' ' ''' ''' ''' '' '''

6 2 4 1 3 4 2 1 1{ , ,..., } { , ,..., } { } { }n nT u u u f f f f f− −=  
'' '' '' ''

7 3 7 2 2 3 2 1 1{ , ,..., } { , ,..., } { } { }n nT u u u f f f e e− −=  
…………………………… 

…………………………… 
''' ' ' ' ' '

1 2 4 1 2 3 4 2 1 1{ , ,..., } { } { , ,..., } { , 2(mod3)  6} { , 1(mod3)}n n nT u u u f f f f f n and n f n− − − −=  = 

' ' ' ''' ''' ''' '' '''

2 4 1 3 4 2 1 1{ , ,..., } { , ,..., } { } { }n nT u u u f f f f f − −=  
'' '' '' ''

1 3 7 2 2 3 2 1 1{ , ,..., } { , ,..., } { } { }n nT u u u f f f e e+ − −=  

Based on the coloring technique described above, it is evident that the color classes 1 2 3, , ,...T T T  and 

1T+  are independent sets of ( )nL T  and it holds inequality | | | | 1a bT T−   for a b . This implies that

( )'' ( ) 1e nL T  + . Further, we have ( )'' ( )e nL T = ( )'' ( ) 1nL T  + . Hence ( )'' ( ) 1e nL T = + . 

Example 3.3: The graph 6( )L T  and its equitable total coloring is shown in figure 3. 

 

Figure 3: 6( )L T  and its equitable total coloring 

Theorem 3.4: Let ( )nL AT  represent the line graph of alternate triangular snake graph, then  

( )'' ( ) 5,  7e nL T n =  . 

Proof: ( ( )) { :1 } { :1 1}nV L AT u n v n  =     −  and  

  ' '( ( )) :1 :1 2 { , :1 1}
2

n

n
E L AT e e n f f n     

  
=     −   −  

  
,  

where  2 1 2 ,e u u  −= '

1,e v v   +=  ,f u v  = '

1,f v u  +=  

We divide the vertex and edge set of ( )nL SL  into distinct partition as described below.  



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For1 1n  − . 

4 9 5 1 1 6 5 4 1 6 5 4

1 ' ' ' ' ' '

3 8 5 2 5 10 5 2 7 5 3

{ , ,..., } { , ,..., } { , ,..., }

{ , ,..., } { , ,..., } { , ,..., }   

u u u v v v e e e
T

e e e f f f f f f

  

  

− − −

− −


= 
  

      

2 7 5 3 4 9 5 1 5 10 5

2 ' ' ' ' ' '

1 6 5 4 3 8 5 2 5 10 5

{ , ,..., } { , ,..., } { , ,..., }  

{ , ,..., } { , ,..., } { , ,..., }   

u u u v v v e e e
T

e e e f f f f f f

  

  

− −

− −


= 
  

5 10 5 2 7 5 3 4 9 5 1

3 ' ' ' ' ' '

4 9 5 1 1 6 5 4 3 8 5 2

{ , ,..., } { , ,..., } { , ,..., }    

{ , ,..., } { , ,..., } { , ,..., }   

u u u v v v e e e
T

e e e f f f f f f

  

  

− −

− − −


= 
  

3 8 5 2 5 10 5 3 8 5 2

4 ' ' ' ' ' '

2 7 5 3 4 9 5 1 1 6 5 4

{ , ,..., } { , ,..., } { , ,..., }    

{ , ,..., } { , ,..., } { , ,..., }   

u u u v v v e e e
T

e e e f f f f f f

  

  

− −

− − −


= 
  

 

1 6 5 4 3 8 5 2 2 7 5 3

5 ' ' ' ' ' '

5 10 5 2 7 5 3 4 9 5 1

{ , ,..., } { , ,..., } { , ,..., }    

{ , ,..., } { , ,..., } { , ,..., }       

u u u v v v e e e
T

e e e f f f f f f

  

  

− − −

− −


= 
  

Based on the coloring technique described above, it is evident that the color classes 1 2 3 4, , ,T T T T  and 5T  

are independent sets of ( )nL AT , and it holds the inequality | | | | 1a bT T−   for a b . This implies that

( )'' ( ) 1e nL AT  + . Further, we have ( )'' ( )e nL AT = ( )" ( ) 1nL AT  +   1 4 1 5+  +  .  Hence

( )'' ( ) 5.e nL AT =  

Example 3.4: The graph 8( )L AT and its equitable total coloring is shown in figure 4. 

 

Figure 4: 7( )L AT  and its equitable total coloring 

Theorem 3.5: Let ( )nL Q  represent the line graph of quadrilateral snake graph, then ( )'' ( ) 7.e nL Q =  

Proof: ( ( )) { :1 1} { :1 2 2} { :1 1}nV L Q u n v n z n    =   −   −   −  and  

   ' '' ''' ' '' '''( ( )) , , , :1 2 , , , :1 1nE L Q e e x x n x x e e n        =   −   − , where 1,e z z   += '

2 2 1,e v v   +=  

''

2 1 2 ,e v v  −= '''

2 ,e z v  = 2 1,x u v   −= '

2 ,x u v  = ''

2 1,x v z   += '''

2 1.x z v   +=  

We divide the vertex and edge set of ( )nL Q  into distinct partition as described below. There are two 

cases  

Case (i): Suppose n is even 
' ' ' '' '' ''

1 1 3 1 4 2 2 4 2 2 4 2{ , ,..., } { , } { , ,..., } { , ,..., }n n n n nT z z z u u e e e e e e− − − − −=
 



Communications on Applied Nonlinear Analysis 

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' ' ' '' '' '' '

2 2 4 2 3 1 1 3 3 1 3 1 2{ , ,..., } { , } { , ,..., } { , ,..., } { }n n n n n nT z z z u u e e e e e e x− − − − − −=
 

' '

3 1 3 2 3 3 1 1 3 3{ , ,..., } { , } { , ..., }n n n nT v v v x x e e e− − − −=
 

4 2 4 2 2 3 2 1 2 4 2{ , ,..., } { , , } { , ..., }n n n n nT v v v x x x e e e− − − − −=
 

''

5 { :1 4} { :1 2}T x n x n  =   −   −
 

' '''

6 { :1 4} { :1 2}T x n x n  =   −   −
 

'''

7 { :1 5} { :1 1}T u n e n  =   −   −
 

Case (ii): Suppose n is odd 

' ' ' '' '' '' '

1 1 3 2 3 1 2 4 3 2 4 1 2{ , ,..., } { , } { , ,..., } { , ,..., } { }n n n n n nT z z z u u e e e e e e x− − − − − −=
 

' ' ' '' '' ''

2 2 4 1 4 2 1 3 2 1 3 2{ , ,..., } { , } { , ,..., } { , ,..., }n n n n nT z z z u u e e e e e e− − − − −=
 

' '

3 1 3 2 3 3 1 1 3 2{ , ,..., } { , } { , ..., }n n n nT v v v x x e e e− − − −=
 

4 2 4 2 2 3 2 1 2 4 3{ , ,..., } { , , } { , ..., }n n n n nT v v v x x x e e e− − − − −=
 

''

5 { :1 4} { :1 2}T x n x n  =   −   −
 

' '''

6 { :1 4} { :1 2}T x n x n  =   −   −
 

'''

7 { :1 5} { :1 1}T u n e n  =   −   −
 

Based on the coloring technique described above, clearly the color classes 1 2, ,...,T T  and 7T  are 

independent sets of ( )nL Q , and it holds the inequality | | | | 1a bT T−   for a b . This implies that

( )'' ( ) 7.e nL Q   Further, we have ( )'' ( )e nL Q = ( )" ( ) 1nL Q  + 6 1 7 +  . Hence ( )'' ( ) 7.e nL Q =  

Example 3.5.1: The graph 6( )L Q and its equitable total coloring is shown in figure 5. 

 

Figure 5: 6( )L Q  and its equitable total coloring 

Example 3.5.2: The graph 5( )L Q and its equitable total coloring is shown in figure 6. 

 

Figure 6: 5( )L Q  and its equitable total coloring 



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Theorem 3.6: Let ( )nL AQ  represent the line graph of alternate quadrilateral snake graph, then 

( )'' ( ) 5.e nL AQ =  

Proof: ( ( )) :1 { :1 } { :1 1}
2

n

n
V L AQ u v n z n    

  
=       −  

  
 and  

    ' '' '( ( )) :1 2 , :1 1 , :1
2

n

n
E L AQ e n e e n x x      

  
=   −   −    

  
,  

where  1,e z z   += ' ,e v z  =  ''

1,e z v   += 2 1,x u v   −= '

2 .x u v  =  

We divide the vertex and edge set of ( )nL AQ  into distinct partition as described below.  

 
2

'' '' '' '

1 1 3 1 2 4 1 2 1{ , ,..., } { , ,..., } :1 3
2

nn n

n
T z z z e e e x u − − −

  
=   −  

    

2

'' '' ''

2 2 4 2 2 1 1 3 2{ , ,..., } :1 1 { , ,..., } { }
2

nn n

n
T z z z x e e e u − − −   

  
=   −  

    

   
2 2

' ' ' '

3 2 4 1 3 2 1 3 2 1
, , ,..., { , ..., } { , ..., }n nn nT x x x v v v e e e x− −   −   

=
 

 
2

'

4 2 4 1 2 4 3 2 4 1{ , ,..., } { , ..., } , ,..., { }nn nT v v v e e e x x x e− −   
=

 
'

5 2
{ :1 1} { : 2 2}nT u e n  =   −   −    

Based on the coloring technique described above, clearly the color classes 1 2 3 4, , ,T T T T  and 5T  are 

independent sets of ( )nL AQ , and it holds the inequality | | | | 1a bT T−   for a b . This implies that

( )'' ( ) 5.e nL AQ   Further, we have ( )'' ( )e nL AQ = ( )" ( ) 1nL AQ  + 4 1 5 +  .  Hence

( )'' ( ) 5.e nL AQ =  

Example 3.6.1: The graph 7( )L AQ and its equitable total coloring is shown in figure 7. 

 

Figure 7: 7( )L AQ  and its equitable total coloring 

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https://internationalpubls.com 

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https://doi.org/10.1016/j.dam.2014.11.014
https://doi.org/10.22199/issn.0717-6279-2019-04-0045
https://doi.org/10.14317/jami.2024.353
https://doi.org/10.1007/s003730200051