Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

2044 
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Rainbow Dynamic Coloring in Few Brick Product Graphs 

 

Gayathri Annasagaram 1, R. Murali 2, and Kulkarni Sunita Jagannatharao 3 

Dr.Ambedkar Institute of Technology, Bangalore, Affiliated to Visvesvaraya Technological University, Belagavi, India.1  

Dr.Ambedkar Institute of Technology, Bangalore, Affiliated to Visvesvaraya Technological University, Belagavi, India.2 

Dr.Ambedkar Institute of Technology, Bangalore, Affiliated to Visvesvaraya Technological University, Belagavi, India.3 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  Consider a connected graph that is nontrivial, defined as a coloring c : V 

(G) → {1, 2, . . . ., k}, k ∈ N of the vertices of G. A rainbow dynamic coloring of a 

graph is a dynamic coloring, and a minimum number of colors required, such that 

every pair of vertices is connected by at least one path whose within vertices have 

different colors. The minimum k for which k-vertex coloring exists is called the 

rainbow dynamic coloring of G, denoted by rdyc(G). In this paper, we determine the 

rdyc of some graphs of brick products C(2n, m, r) associated with odd cycles for m 

= 1. 

Objectives: To find the rdyc of few brick product graphs. 

Conclusions: In this paper, we obtain the rainbow dynamic coloring of fewbrick 

product graphs C(2n, m, r) for m = 1 and r = 3, 5, 7. 

Keywords: rainbow vertex connection number, dynamiccoloring, brick product, 

rainbow dynamic coloring. 

1. Introduction 

All graphs considered in this paper are simple, finite, and undirected. Let G be non-trivial with a vertex 

coloring c : V (G) → {1, 2, . . . ., k}, k ∈ N. In a proper vertex-colored graph G, a path P is in a rainbow 

path if no two vertices P are of the same color, except possibly the end vertices of P. If a rainbow path 

connects every two vertices of G, then G is a rainbow connected to the vertex. A proper vertex coloring 

of a connected graph G  that results in a connected graph with vertex rainbow is a rainbow vertex 

coloring of G. The minimum number of colors needed for the vertex rainbow coloring of G is the 

vertex rainbow connection number rvc(G). Bruce Montgomery introduced a relatively new concept in 

vertex coloring, called dynamic coloring, in 2001 [2].  A dynamic graph coloring is a proper coloring 

of the set of vertex such that each vertex of degree at least two of its neighbors receives at least two 

different colors. Krivelevich and Yuster introduced a rainbow vertex coloring concept in 2010 [3]. A 

rainbow vertex connection number, rvc(G) of a connected graph, is the minimum number of colors 

required to color its vertices. Every pair of vertices is connected by at least one path, which has 

different colors within the vertices. A rainbow dynamic coloring of a graph is not just a theoretical 

concept but a practical one. It is a dynamic coloring, and a minimum number of colors is required such 

that every pair of vertices is connected by at least one path whose within vertices have different colors. 

The minimum k for which k-vertex coloring exists is called the dynamic rainbow coloring of G, 

denoted by rdyc(G). We define a brick product graph associated with an odd cycle. 

 



Communications on Applied Nonlinear Analysis 

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2. Results 

Definition  

Let m, n  and r be positive integers. A cycle of order 2n  is indicated by the notation 𝐶2𝑛 =

𝑣1, 𝑣2, . . . . 𝑣2𝑛−1 ,𝑣2𝑛 = 𝑣1. The brick product (m, r) of 𝐶2𝑛 represented by C(2n, m, r) is defined in 

the following way in two cases.  

1. We need r to be odd and greater than 1 for m = 1. Next, from𝐶2𝑛, C(2n, m, r) can be determined by 

adding chords 𝑣2𝑘(𝑣2𝑘+𝑟) , is obtained by adding chords𝑣2𝑘(𝑣2𝑘+𝑟) , k = 1, 2, ..., n, where the 

computation is performed modulo 2n.  

2. We need m + r to be even for m > 1. Then C(2n, m, r) is obtained by first taking the disjoint union 

of m copies of 𝐶2𝑛, that is, 𝐶2𝑛(1), 𝐶2𝑛(2), ...., 𝐶2𝑛(m) where for every i = 1, 2, ..., m, 𝐶2𝑛(i) = 𝑣𝑖1, 

𝑣𝑖1,,𝑣𝑖1. . . . . . . 𝑣𝑖2𝑛,. Next, an edge (also known as a brick edge) is drawn to join (𝑣𝑖,, 𝑣𝑘,) to (𝑣𝑖+1,, 𝑣𝑘,)  

for each odd i = 1, 2, ...., m − 1 and each even k = 0, 1, 2, ....., 2n − 2,. Similarly, for each even i = 1, 

2, ....m − 1 and each odd k = 1, 2, ..., 2n − 1, an edge (also known as a brick edge) is drawn to join 

(𝑣𝑖+1,, 𝑣𝑘,). Lastly, an edge (referred to as a hooking edge) is created to join (𝑣1,, 𝑣𝑘,) to (𝑣𝑚,, 𝑣𝑘+𝑟,) 

for each odd k = 1, 2, ..., 2n − 1. A flat edge is an edge in C(2n, m, r) that is either a brick or a hooked 

edge. 

Theorem 1. Consider G = (2n, m, r). Then, for r = 3 and m = 1, and n ≥ 3, 

𝑟𝑑𝑦𝑐(𝐺) =

{
 
 
 
 

 
 
 
 
4           𝑓𝑜𝑟  3 ≤ 𝑛 ≤ 7
5         𝑓𝑜𝑟  8 ≤ 𝑛 ≤ 10

⌊
2𝑛

3
⌋ − 1, 2𝑛 ≅ 1(𝑚𝑜𝑑3)

⌊
2𝑛

3
⌋ − 2, 2𝑛 ≅ 0(𝑚𝑜𝑑3)

⌊
2𝑛

3
⌋ − 1, 2𝑛 ≅ 2(𝑚𝑜𝑑3

 

The vertex set of G is defined as V(G) = {v1, v2, . . . . . . , v2n−1, v2n = v1}  and the edge set of G is 

defined as E(G) = {ei: 1 ≤ i ≤ 2n} ∪ {ei′: 1 ≤ i ≤ n} where 𝑒𝑖  represents the cycle edge (𝑣𝑖−1, 𝑣𝑖) 

and 𝑒𝑖′ represents the brick edge (v2k, 𝑣2𝑘+𝑟), 𝑘 = 0,1,2, . . . . . , 𝑛.  In this case, 2k + r is computed 

modulo 2n. Let’s define coloring to G’s vertices in the following manner. 

Case 1. 3 ≤ n ≤ 7  

Coloring is defined by 𝑣𝑖+4𝑘 = 𝑖, 1 ≤ 𝑖 ≤ 4,0 ≤ 𝑘 ≤ ⌈
𝑛

2
⌉ − 1 

Case 2. 8 ≤ n ≤ 10  

for n=8 Coloring is defined by  𝑣𝑖+5𝑘 = 𝑖, 1 ≤ i ≤ 5, 0 ≤ k ≤ 2, 𝑣2𝑛 = 2  

for n = 9, 10 Coloring is defined by 𝑣𝑖+5𝑘 = 𝑖, 1 ≤ i ≤ 5, 0 ≤ k ≤ 3 

Case 3. 2n ≡ 1 (mod 3)  

Coloring is defined by 𝑣𝑖 + ⌈
𝑛

2
⌉ 𝑘 = 𝑖 𝑓𝑜𝑟 1 ≤ 𝑖 ≤ ⌈

𝑛

2
⌉ , 0 ≤ 𝑘 ≤ 3. 



Communications on Applied Nonlinear Analysis 

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It is clear from this color pattern that rdyc(G) =rdyc(G) = ⌊
2𝑛

3
⌋ − 1. 

Case 4. 2n ≡ 0 (mod 3)  

Coloring is defined by 𝑣
𝑖+(⌈

2𝑛

3
⌉−2)𝑘=𝑖

 for 1 ≤ i ≤ ⌈
2n

3
⌉ − 2, 0 ≤ k ≤ 3.   

It is clear from this color pattern that rdyc(G) = ⌊
2n

3
⌋ − 2 

Case 5. 2n ≡ 2 (mod 3)  

Coloring is defined by 𝑣
𝑖+(⌈

2𝑛

3
⌉−1)𝑘=𝑖

 for 1 ≤ i ≤ ⌈
2n

3
⌉ − 1, 0 ≤ k ≤ 3.   

It is clear from this color pattern that rdyc(G) = ⌊
2n

3
⌋ − 1. 

  

   Figure 1: A graph illustrating the way colors are assigned to brick products in C(14,1,3) 

Theorem 2. Consider G = (2n, m, r). Then, for r = 7 and m = 1, and n ≥ 5,  

𝑟𝑑𝑦𝑐(𝐺) =

{
 
 

 
 
4 𝑓𝑜𝑟 𝑛 =  5, 7 ≤ 𝑛 ≤ 12

⌊
𝑛

2
⌋      𝑓𝑜𝑟  𝑛 = 6, 13

⌊
𝑛

2
⌋ − 1  𝑓𝑜𝑟  𝑛 ≥ 14

 

Let us define the vertex set and edge set as in theorem1. Let’s define coloring to G’s vertices in the 

following manner. 

Case 1.  

for n = 5, 7, 9, 11  

Coloring is defined by 𝑣𝑖+4𝑘 = 𝑖, 1 ≤ i ≤ 4, 0 ≤ k ≤ ⌊
n

2
⌋ − 1; vi+4k = i + 1, 1 ≤ i ≤ 2, k = ⌊

n

2
⌋ 

for n = 8, 10, 12  

Coloring is defined by  𝑣𝑖+4𝑘 = 𝑖, 1 ≤ i ≤ 4, 0 ≤ k ≤
n

2
− 1. 

Case 2. n=6,13  



Communications on Applied Nonlinear Analysis 

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Coloring is defined by 𝑣
𝑖+(

𝑛

2
)𝑘=𝑖

 for 1 ≤ i ≤ ⌊
n

2
⌋ , 0 ≤ k ≤ ⌊

n

3
⌋ + 1.  

It is clear from this color pattern that rdyc(G) = ⌊
n

2
⌋ 

Case 3. n ≥ 14 

Coloring is defined by 𝑣
(⌊
𝑛

2
⌋−1)𝑘

= 𝑖 for 1 ≤ i ≤ ⌊
n

2
⌋ − 1, 0 ≤ k ≤ 4.  

It is clear from this color pattern that rdyc(G) = ⌊
n

2
⌋ − 1. 

 

Figure 2: A graph illustrating the way colors are assigned to brick products in C(16,1,5) 

Theorem 3. Consider G = (2n, m, r). Then, for r = 7 and m = 1, and n ≥ 7, 

 

𝑟𝑑𝑦𝑐(𝐺) =

{
 
 
 
 

 
 
 
 
3                         𝑓𝑜𝑟  𝑛 = 7
4 for n = 7,8,10 ≤ n ≤ 14
5        for n = 15,17,16 
6                    for n = 16,18

⌊
𝑛

3
⌋ + 2,    2𝑛 ≅ 2(𝑚𝑜𝑑3)

⌊
𝑛

3
⌋ + 1,   2𝑛 ≅ 1(𝑚𝑜𝑑3)

𝑛

3
+ 1,      2𝑛 ≅ 0(𝑚𝑜𝑑3

 

Let us define the vertex set and edge set as in theorem1. Let’s define coloring to G’s vertices in the 

following 

manner.  

Case 1. n = 9 

Coloring is defined by 𝑣𝑖+3𝑘 = 𝑖, 1 ≤ 𝑖 ≤ 3, 0 ≤ 𝑘 ≤ 5.  

Case 2. n = 7, 8, 10 ≤ n ≤ 14 

Coloring is defined by 𝑣𝑖+4𝑘 = 𝑖, 1 ≤ 𝑖 ≤ 4, 0 ≤ 𝑘 ≤ ⌈
𝑛

2
⌉ − 1. 



Communications on Applied Nonlinear Analysis 

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Case 3. n = 15, 17 

Coloring is defined by 𝑣𝑖+4𝑘 = 𝑖, 1 ≤ 𝑖 ≤ 4, 0 ≤ 𝑘 ≤ ⌈
𝑛

2
⌉ − 1. 

Case 4. n = 16, 18  

for n = 16 Coloring is defined by 𝑣𝑖+6𝑘 = i, 1 ≤ i ≤ 6, 0 ≤ k ≤ 4 ; 𝑣𝑖+6𝑘 = i + 1, 1 ≤ i ≤ 2, k =⌊
𝑛

3
⌋. 

for n = 18, 𝑣𝑖+6𝑘 = i, 1 ≤ i ≤ 6, 0 ≤ k ≤ 5 

Case 5. 2n ≡ 2 (mod 3)  

Coloring is defined by  𝑣
𝑖+(⌊

𝑛

3
⌋+2)𝑘

=i, 1 ≤ i ≤ ⌊
𝑛

3
+ 2⌋ , 0 ≤ k ≤ 4 for 2n ≤ 50, 0 ≤ k ≤ 5 for 2n > 50 

It is clear from this color pattern that rdyc(G) =⌊
𝑛

3
⌋ + 2. 

Case 6. 2n ≡ 1 (mod 3) 

Coloring is defined by  𝑣
𝑖+(⌊

𝑛

3
⌋+1)𝑘

=i, 1 ≤ i ≤ ⌊
𝑛

3
⌋ + 1 , 0 ≤ k ≤ 5  

It is clear from this color pattern that rdyc(G) =⌊
𝑛

3
⌋ + 1 

Case 7. 2n ≡ 0 (mod 3)  

Coloring is defined by 𝑣
𝑖+(

𝑛

3
+1)𝑘

= 𝑖, for 1 ≤ i ≤
n

3
+ 1, 0 ≤ k ≤ 5. 

It is clear from this color pattern that rdyc(G) =⌊
𝑛

3
⌋ + 1. 

 

Figure 3: A graph illustrating the way colors are assigned to brick products in C(18,1,7) 

Refrences 

[1] Alspach, C.C.Chen & and Kevin McAvancy. (1996). On a class of Hamiltonian 

laceable 3-regular graphs,Discrete Math.,151, 19-38. 

[2] G.Chatrand, G.L.Johns, K.A.McKeon & P.Zang.(2008). Rainbow connection in                           

graphs, Math. Bohem 133, 85-98. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

2049 
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[3] M.Krivelevich, & and R.Yuster.(2010). The rainbow connection of a graph is (at most) 

is reciprocal to its minimum degree, J.Graph Theory, 63, 185-191. 

[4]  Kulkarni Sunita Jagannatharao & and R. Murali. (2017). Rainbow connection in some 

brick product graphs, Int. J. of math. and its Appl, 4E(5), 769-77 

[5] Kulkarni Sunita Jagannatharao & and R. Murali. (2019). Rainbow connection insome 

corona product graphs, Malaya Journal of Matematik, 7(1), 127-131. 

[6] Kulkarni Sunita Jagannatharao and R. Murali, ”Star rainbow coloring in brick product 

graphs,” Journal of physics conferences, Vol.1597, no.1,012056, 2020.