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How to cite: N. A. Sudibyo, A. Iswardani , and Y. P. S. R. Hidayat, “Total Vertex Irregularity Strength of 

Disjoint Union of Ladder Rung Graph and Disjoint Union of Domino Graph”, JMM, vol. 6, no. 1, pp. 47-51, 

May 2020. 

Total Vertex Irregularity Strength of Disjoint Union of Ladder 

Rung Graph and Disjoint Union of Domino Graph  
 

 

Nugroho Arif Sudibyo1, Ardymulya Iswardani2, Yohana Putra Surya 

Rahmad Hidayat3 

1Department of Informatics Engineering, Universitas Duta Bangsa, nugroho_arif@udb.ac.id 
2Department of Information Systems, Universitas Duta Bangsa, ardymulya@udb.ac.id 

3SMK Negeri 2 Kudus, yohan.artup@gmail.com 

 
doi: https://doi.org/10.15642/mantik.2020.6.1.47-51 

 

 
Abstrak: Akan diselidiki pelabelan graf yang disebut total vertex irregularity strength (tvs(G)). 

tvs(G) adalah minimum 𝒌 yang graf tersebut memenuhi pelabelan 𝒌-total titik tidak teratur. Pada 
makalah ini, akan ditentukan pelabelan total tak reguler titik dari graf disjoint union of ladder rung 

dan graf disjoint union of domino graph. 
  

Kata kunci: Graf; tvs; Ladder rung; Domino  

 

Abstract: We investigate a graph labeling called the total vertex irregularity strength (tvs(G)). A 

tvs(G) is minimum 𝒌 for which graph has a vertex irregular total 𝒌-labeling. In this paper, we 
determine the total vertex irregularity strength of disjoint union of ladder rung graph and disjoint 

union of domino graph. 
  

Keywords: Graph; tvs; Ladder rung; Domino 

 

  

  

Jurnal Matematika MANTIK 

Volume 6, No.1, May 2020, pp. 47-51 

 

ISSN: 2527-3159 (print) 2527-3167 (online) 

mailto:yohan.artup@gmail.com
http://u.lipi.go.id/1458103791
http://u.lipi.go.id/1457054096


Jurnal Matematika MANTIK 

Volume 6, Issue. 1, May 2020, pp. 47-51 

48 

1. Introduction  

In 1735, graph theory was first introduced by Leonhard Euler to solve the problem of 

the Konigsberg bridge on the river Pregel, Russia [1]. Graph labeling is an interesting topic 

in graph theory so that various types of labeling are researched and developed [2]. Graph 

labeling is the assignment of labels to the graph elements such as edges or vertices, or both, 

from the graph [3]. 

Graph labeling can be applied to various fields including transportation systems, 

communication systems, geographical navigation, radar, and also security systems. For 

example, the design of the code for radar signals and missiles is equivalent to labeling a 

complete graph, where each point is connected to one side which has a label that is always 

different. This side label describes the distance between points, while the point label is the 

position at the time the signal is sent [4]. 

For a graph 𝐺(𝑉,𝐸), a labeling 𝑓:𝑉(𝐺) ∪ 𝐸(𝐺) → {1,2,…,𝑘} to be a vertex  irregular 
total 𝑘-labeling if for any two different vertices x and y, their weights satisfy 𝑤𝑡𝑓(𝑥) ≠

𝑤𝑡𝑓(𝑦) (see [5], [6]). The total vertex irregularity strength is minimum 𝑘 for which graph 

has a vertex irregular total 𝑘-labeling (see [6], [7]).  
The total vertex irregularity strength problem has been investigated for trees [8], 

𝐶𝑛 ∗2 𝐾𝑛 graph [9], regular graph [10], forest graph [11], disjoint union of sun graph [5], 
wheel related graphs [12], graph obtained of a star [13], trees with maximum degree five 

[14], and comb product of two cycles and two stars [15]. In this paper we answer the open 

problem proposed by Baca, et.al [7]. In particular, we determine the total vertex irregularity 

strength of disjoint union of ladder rung graph and disjoint union of domino graph. 

 

2. Preliminaries 

Ball and Coxeter define the ladder graph 𝑛𝑃2, is 𝑛 copies of the path graph 𝑃2 [4]. The 
ladder rung graph can be depicted as in Figure 1. 

 
Figure 1. The ladder rung graph 

 

Theorem 1. The total vertex irregularity strength of disjoint union of ladder rung graph is 

𝑡𝑣𝑠(𝑡𝐿𝑛) = 𝑛𝑡 + 1, for 𝑛 ≥ 1, 𝑡 ≥ 2. 
 

Proof. The disjoint union of ladder rung graph 𝑡𝐿𝑛 has 2𝑛𝑡 vertices. The smallest 𝑤𝑡(𝑡𝐿𝑛) 
must be 2 and the largest 𝑤𝑡(𝑡𝐿𝑛) is at least 𝑛𝑡 + 1. Because of every vertex has degree 
one, then 𝑡𝑣𝑠(𝑡𝐿𝑛) ≥ 𝑛𝑡 + 1. 

To show that 𝑡𝑣𝑠(𝑡𝐿𝑛) ≤ 𝑛𝑡 + 1. The label of vertices of 𝑡𝐿𝑛 are described in the 
following formulas:   

𝜆(𝑢𝑖) = 𝑖,   for 𝑖 ∈ [1,𝑛𝑡], 
𝜆(𝑣𝑖) = 𝑖+1,   for 𝑖 ∈ [1,𝑛𝑡]. 
Then, the label of edges of 𝑡𝐿𝑛 are  

1
u 2u nu

1v 2v nv



N. A. Sudibyo, A. Iswardani , and Y. P. S. R. Hidayat,  

Total Vertex Irregularity Strength of Disjoint Union of Ladder Rung Graph and Disjoint Union of Domino 

Graph 

49 

𝜆(𝑣𝑖𝑣𝑖) = 𝑖+1,   for 𝑖 ∈ [1,𝑛𝑡]. 
The weights of vertices 𝑢𝑖 and 𝑣𝑖 of 𝑡𝐿𝑛 are: 
𝑤𝑡(𝑢𝑖) = 2𝑖 + 1, for 𝑖 ∈ [1,𝑛𝑡], 
𝑤𝑡(𝑣𝑖) = 2(𝑖 + 1), for 𝑖 ∈ [1,𝑛𝑡]. 

The weights calculated at vertices are distinct. So, 𝑡𝑣𝑠(𝑡𝐿𝑛) = 𝑛𝑡 + 1, for 𝑛 ≥ 1, 𝑡 ≥
2. 

 

3. Results 

The domino graph is (2,3)-grid graph [16]. The disjoint union of domino graph can be 

depicted as in Figure 2. 

 

 
Figure 2. The disjoint union of domino graph 

 

Theorem 2. The total vertex irregularity strength of disjoint union of domino graph is 

𝑡𝑣𝑠(𝑛𝐷) = 𝑛 + 1. 
 

Proof. The disjoint union of ladder rung graph 𝑛𝐷 has 6𝑛 vertices. The smallest 𝑤𝑡(𝑛𝐷) 
must be 3 and the largest 𝑤𝑡(𝑛𝐷)at least 3𝑛 + 5. It easy to see that 𝑡𝑣𝑠(𝑛𝐷) ≥ 𝑛 + 1. 

To show that 𝑡𝑣𝑠(𝑛𝐷) ≤ 𝑛 + 1. The label of vertices of 𝑛𝐷 are:   

𝜆(𝑢𝑖,1) = 𝑖,   for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,2) = 𝑖,   for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,3) = 2,   for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,4) = 2,   for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,5) = 𝑖,   for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,6) = 𝑖 + 1,  for 𝑖 ∈ [1,𝑛]. 

Then, the label of edges of 𝑛𝐷 are  

𝜆(𝑢𝑖,1𝑢𝑖,2) = 𝑖,   for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,1𝑢𝑖,3) = 𝑖,   for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,2𝑢𝑖,3) = 𝑖 + 1,  for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,3𝑢𝑖,4) = 𝑖 + 1,  for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,3𝑢𝑖,5) = 𝑖 + 1,  for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,4𝑢𝑖,6) = 𝑖 + 1,  for 𝑖 ∈ [1,𝑛], 

𝜆(𝑢𝑖,5𝑢𝑖,6) = 𝑖 + 1,  for 𝑖 ∈ [1,𝑛]. 

The weights of vertices 𝑢𝑖 and 𝑣𝑖 of 𝑡𝐿𝑛 (respectively) are: 



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50 

𝑤𝑡(𝑢𝑖,1) = 3𝑛,  for 𝑖 ∈ [1,𝑛], 

𝑤𝑡(𝑢𝑖,2) = 3𝑛 + 1, for 𝑖 ∈ [1,𝑛], 

𝑤𝑡(𝑢𝑖,3) = 3𝑛 + 4, for 𝑖 ∈ [1,𝑛], 

𝑤𝑡(𝑢𝑖,4) = 3𝑛 + 5, for 𝑖 ∈ [1,𝑛], 

𝑤𝑡(𝑢𝑖,5) = 3𝑛 + 2, for 𝑖 ∈ [1,𝑛], 

𝑤𝑡(𝑢𝑖,6) = 3(𝑛 + 1), for 𝑖 ∈ [1,𝑛]. 

The weights calculated at vertices are distinct. So, 𝑡𝑣𝑠(𝑛𝐷) = 𝑛 + 1. 
 

4. Conclusions 

The total vertex irregularity strength of disjoint union of ladder rung graph is 

𝑡𝑣𝑠(𝑡𝐿𝑛) = 𝑛𝑡 + 1, for 𝑛 ≥ 1, 𝑡 ≥ 2 and the total vertex irregularity strength of disjoint 
union of domino graph is 𝑡𝑣𝑠(𝑛𝐷) = 𝑛 + 1. 

 

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Graph 

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