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How to cite: Marsidi and I. H. Agustin, “The local antimagic on disjoint union of some family 

graphs”, mantik, vol. 5, no. 2, pp. 69-75, October 2019. 

The Local Antimagic On Disjoint Union of Some Family Graphs 
 

 

Marsidi1,2, Ika Hesti Agustin1,3* 

CGANT Research Group, University of Jember, Indonesia1 

Mathematics Edu. Depart. IKIP PGRI Jember, Jember, Indonesia 2, marsidiarin@gmail.com2 

Mathematics Depart. University of Jember, Jember, Indonesia3, ikahesti.fmipa@unej.ac.id
3 

 
doi: https://doi.org/10.15642/mantik.2019.5.2.69-75 

 

 
Abstrak: Graf 𝐺 pada penelitian ini adalah nontrivial, berhingga, terhubung, sederhana, dan tidak 
terarah. Misal u, 𝑣 adalah dua elemen di himpunan titik dan q adalah kardinalitas himpunan sisi di 
graf 𝐺, fungsi bijektif dari himpunan sisi dipetakan ke bilangan asli q disebut vertex local 
antimagic edge labeling jika untuk dua titik yang bertetangga 𝑣1 dan 𝑣2, bobot 𝑣1 tidak sama 
dengan bobot 𝑣2, dimana bobot 𝑣 (dinotasikan dengan 𝑤(𝑣)) adalah jumlah label sisi yang terkait 
dengan 𝑣. Selanjutnya, setiap vertex local antimagic edge labeling menginduksi pewarnaan titik 
pada graf 𝐺 dimana 𝑤(𝑣) adalah warna pada titik 𝑣. Bilangan kromatik vertex local antimagic 
edge labeling 𝜒𝑙𝑎(𝐺) adalah banyaknya warna minimum yang disebabkan oleh titik local 
antimagic pelabelan sisi pada graf 𝐺. Dalam artikel ini, kita membahas tentang bilangan kromatik 
vertex local antimagic edge labeling pada graf disjoint union dari keluarga graf seperti lintasan, 

lingkaran, bintang, dan friendship, dan menentukan batas bawah dari vertex local antimagic edge 

labeling. Nilai kromatik pada graf disjoint union pada penelitian ini mencapai batas bawah. 

 

Kata kunci: pelabelan local antimagic, nilai kromatik local antimagic, graf disjoint union 

 

 
Abstract: A graph 𝐺 in this paper is nontrivial, finite, connected, simple, and undirected. Graph 𝐺 
consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality 

of edge set in G, a bijective function from the edge set to the first q natural number is called a 

vertex local antimagic edge labelling if for any two adjacent vertices 𝑣1and 𝑣2, the weight of 𝑣1 is 
not equal with the weight of 𝑣2, where the weight of 𝑣 (denoted by 𝑤(𝑣)) is the sum of labels of 
edges that are incident to 𝑣. Furthermore, any vertex local antimagic edge labelling induces a 
proper vertex colouring on where 𝑤(𝑣) is the colour on the vertex 𝑣. The vertex local antimagic 
chromatic number 𝜒𝑙𝑎(𝐺) is the minimum number of colours taken over all colourings induced by 
vertex local antimagic edge labelling of 𝐺. In this paper, we discuss about the vertex local 
antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and 

friendship, and also determine the lower bound of vertex local antimagic chromatic number of 

disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the 

lower bound.   

 

Keywords: local antimagic labelling, local antimagic chromatic number, disjoint union graphs 

 

 

 

  

Jurnal Matematika MANTIK 

Volume 5, Nomor 2, October 2019, pp. 69-75 

 

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

mailto:marsidiarin@gmail.com2
mailto:ikahesti.fmipa@unej.ac.id
http://u.lipi.go.id/1458103791
http://u.lipi.go.id/1457054096


Jurnal Matematika MANTIK 

Vol. 5, No. 2, October 2019, pp. 69-75 

70 

1. Introduction 

Any mathematical object involving points and connection between them 

called a graph. A graph  𝐺(𝑉,𝐸) consists of two sets 𝑉 and 𝐸. The elements of 𝑉 
are called vertices, and the elements of 𝐸 called edges. The all graphs discussed in 
this paper are nontrivial, finite, connected, simple, and undirected [1] – [3]. Graph 

labelling is the assignment of labels to the graph elements such as edges or 

vertices, or both, from the graph. The concept of antimagic labelling of a graph 

was introduced by Hartsfield and Ringel [4]. Every component of the graph, such 

as vertex, edge, and face, has given the distinct label by a natural number. 

Given a graph 𝐺(𝑉,𝐸)and q is the number of edges in G, a bijective function 
from the edges set to the first q natural number is called a vertex local antimagic 

edge labelling if for any two adjacent vertices 𝑣1and 𝑣2, 𝑤(𝑣1) ≠  𝑤(𝑣2), where 
𝑤(𝑣) is the sum of labels of edges that are incident to 𝑣. Furthermore, any vertex 
local antimagic edge labelling induces a proper vertex colouring on where 𝑤(𝑣) is 
the colour on the vertex 𝑣. The minimum number of colours taken over all 
colourings induced by vertex local antimagic edge labelling of 𝐺 denoted by 
𝜒𝑙𝑎(𝐺). 

If a graph 𝐺 has antimagic labelling, it is called antimagic. Since been 
introduced by Hartsfield and Ringel [4], the topic has attracted a lot of attention, 

for details, see Gallian [11]. One of well-known remark and theorem related to 

this is the following. 

 

Remark 1.1. For any graph G, 𝜒𝑙𝑎(𝐺) ≥ 𝜒(𝐺) and the difference 𝜒𝑙𝑎(𝐺) −
𝜒(𝐺)can be arbitrarily large[4]. 

 

Theorem 1.1. For any tree T with l leaves, 𝜒𝑙𝑎(𝑇) ≥ 𝑙 + 1 [5]. 
Arumugam et al. [5] give a lower bound and upper bound of local antimagic 

vertex colouring of 𝐾1 +  𝐻 and also give the exact value of local antimagic 
vertex colouring. Agustin et al. [6] studied the local edge antimagic colouring of 

graphs. The other results about local antimagic of graphs can be seen in [6] –[10]. 

Thus, in this paper, we have found the chromatic number of vertex local 

antimagic chromatic number on disjoint union of some family graphs, namely 

path, cycle, star, and friendship.  Before we present our results, we define a 

definition of disjoint union graph that we discuss in this paper. For any graph 𝐺, 
the graph ⋃ 𝐺𝑖=𝑚  denotes the disjoint union of 𝑚 copies of graph 𝐺. 
 

2. Main Result 

In this section, we present our results by showing the vertex local 

antimagicchromatic number on disjoint union of some family graphs, namely 

path, cycle, star, and friendship in the following theorems. We also present the 

remark and theorem about the lower bound of the disjoint union of graphs.  

 

Remark 2.1. For any disjoint union graph G, we have 𝜒𝑙𝑎(⋃ 𝐺𝑖=𝑚 ) ≥
𝜒𝑙𝑎(𝐺). 

 

Theorem 2.1. For any tree T with kpendant vertices, 𝜒𝑙𝑎(⋃ 𝑇𝑖=𝑚 ) ≥ 𝑘 + 1 



Marsidi and I. H. Agusti 
The local antimagic on disjoint union of some family graphs 

71 

Proof. Let 𝑔 be any local antimagic labelling of disjoint union of 𝑇, then the 
colouring induced by g. The colour of a pendant vertex v is the label on edge that 

is incident with 𝑣. Thus, all the pendant vertices receive distinct colours. 
Furthermore, for any non-pendant vertex incident with an edge 𝑒1 with 𝑓(𝑒1) =
𝑚, the colour assigned to 𝑤 is larger than 𝑚. Since the number of colours in the 
colouring induced by 𝑔 is at least 𝑘 + 1, 𝜒𝑙𝑎(⋃ 𝑇𝑖=𝑚 ) ≥ 𝑘 + 1. 

Theorem 2.2. For the path 𝑃𝑛 with 𝑛 ≥ 3, then we have 𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) =

2𝑚 + 1. 
Proof. The ⋃ 𝑃𝑛𝑗=𝑚  is the disjoint union of path graph. The Vertex set is 

𝑉(⋃ 𝑃𝑛𝑗=𝑚 ) = {𝑣1
1,𝑣2

1,…,𝑣𝑖
𝑗
,…,𝑣𝑛−1

𝑚 ,𝑣𝑛
𝑚; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}} 

and the edge set 𝐸(⋃ 𝑃𝑛𝑗=𝑚 ) = {𝑣1
1𝑣2

1,𝑣2
1𝑣3

1 …,𝑣𝑖
𝑗
𝑣𝑖+1
𝑗
,…,𝑣𝑛−2

𝑚 𝑣𝑛−1
𝑚 ,𝑣𝑛−1

𝑚 𝑣𝑛
𝑚; 𝑖 ∈

{1,2,3,…,𝑛 − 1},𝑗 ∈ {1,2,3,…,𝑚}}. Hence |𝑉(⋃ 𝑃𝑛𝑗=𝑚 )| =

𝑚𝑛,|𝐸(⋃ 𝑃𝑛𝑗=𝑚 )| = 𝑚(𝑛 − 1). Because graph ⋃ 𝑃𝑛𝑗=𝑚  have 2𝑚 leaves, based 

on Theorem 2.1, we have 𝜒𝑙𝑎(⋃ 𝑇𝑖=𝑚 ) ≥ 𝑘 + 1, so the color needed for⋃ 𝑃𝑛𝑗=𝑚  

is2𝑚 + 1. Thus, we have𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≥ 2𝑚 + 1. 

Furthermore, we will prove that the upper bound is 2𝑚 + 1, denoted by 

𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≤ 2𝑚 + 1. We define the labelling on the edges of 𝑃𝑛 by the 

following functions. 

 

𝑓(𝑣𝑖
𝑗
𝑣𝑖+1
𝑗
) =

{
 
 
 
 
 
 

 
 
 
 
 
 

1

2
(𝑖 + 𝑗(𝑛 − 1) − 𝑛 + 1, 𝑖 ∈ {2,4,6,…,𝑛 − 1};                             

𝑗 ∈ {1,2,3,…,𝑚}for 𝑛 odd                   

𝑚(𝑛 − 1) + (
𝑛 − 𝑖

2
) − 𝑗(

𝑛 − 1

2
), 𝑖 ∈ {1,3,5,…,𝑛 − 2};                                              

𝑗 ∈ {1,2,3,…,𝑚}for 𝑛 odd                                    

(
𝑖

2
− 1)𝑚 + 𝑗, 𝑖 ∈ {2,4,6,…,𝑛 − 2};           

𝑗 ∈ {1,2,3,…,𝑚}for 𝑛 even

𝑚𝑛 − 𝑚(
𝑖 + 1

2
) − 𝑗 + 1, 𝑖 ∈ {1,3,5,…,𝑛 − 1};                              

𝑗 ∈ {1,2,3,…,𝑚}for 𝑛 even                   

 

 

Base on the function f, we determine the set of vertex weight on pendant vertices 

as the following. 

𝑊1 = {𝑓(𝑣1
𝑗
; 𝑗 ∈ {1,2,3,…,𝑚}} ∪ {𝑓(𝑣𝑛

𝑗
; 𝑗 ∈ {1,2,3,…,𝑚}} 

 = {𝑚(𝑛 − 1),𝑚(𝑛 − 1) − (
𝑛 − 1

2
),…,(𝑚 + 1)(

𝑛 − 1

2
)} ∪ {

𝑛 − 1

2
,𝑛 − 1,…,

𝑚(𝑛 − 1)

2
} 

The vertex weights on the two degrees vertices are 

𝑊2 = 𝑚(𝑛 − 1) + 1 
𝑊3 = 𝑚(𝑛 − 1) 

Hence from the above, it easy to see that the different vertex weights are2𝑚 + 1, 

so 𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≤ 2𝑚 + 1. Therefore𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≤ 2𝑚 + 1 and 

𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) ≥ 2𝑚 + 1. It concludes that 𝜒𝑙𝑎(⋃ 𝑃𝑛𝑗=𝑚 ) = 2𝑚 + 1.  

 

The illustration of local antimagic labelling of ⋃ 𝑃𝑛𝑗=𝑚  can be seen in Figure 1. 

 



Jurnal Matematika MANTIK 

Vol. 5, No. 2, October 2019, pp. 69-75 

72 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
Figure 1. Illustration of local antimagic labelling of ⋃ 𝑃7𝑚=5  

 

Theorem 2.3. For the cycle 𝐶𝑛 with 𝑛 ≥ 𝑡ℎ𝑟𝑒𝑒 𝑎𝑛𝑑 𝑛 is even integer, then 

we have 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) = 3. 

Proof. The ⋃ 𝐶𝑛𝑗=𝑚  is the disjoint union of cycle graph.The vertex set  

𝑉(⋃ 𝐶𝑛𝑗=𝑚 ) = {𝑣1
1,𝑣2

1,…,𝑣𝑖
𝑗
,…,𝑣𝑛−1

𝑚 ,𝑣𝑛
𝑚; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚} 

and the edge set 𝐸(⋃ 𝑃𝑛𝑗=𝑚 ) = {𝑒1
1,𝑒2

1,𝑒3
1,…,𝑒𝑛

1,…,𝑒𝑖
𝑗
,…,𝑒𝑛−1

𝑚 ,𝑒𝑛
𝑚,𝑤ℎ𝑒𝑟𝑒 𝑒1

1,=

𝑣1
1𝑣2

1,𝑒2
1 = 𝑣2

1𝑣3
1,…,𝑒𝑛

1 = 𝑣𝑛
1𝑣1

1; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}}. Hence 

|𝑉(⋃ 𝑃𝑛𝑗=𝑚 )| = 𝑚𝑛,|𝐸(⋃ 𝑃𝑛𝑗=𝑚 )| = 𝑚𝑛. We will prove that 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) ≥

3. Based on Arumugam theorem [4], the local antimagic chromatic number of 
cycle graph is  𝜒𝑙𝑎(𝐶𝑛) = 3. Furthermore, based on Remark 2.1 we have 

𝜒𝑙𝑎(⋃ 𝐶𝑛𝑖=𝑚 ) ≥ 𝜒𝑙𝑎(𝐶𝑛) so we can see that  𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) ≥ 3. 

Furthermore, we will prove that the upper bound is 3, denoted 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) ≤ 3, 

by defining the labelling as follow. 

 

𝑔(𝑒𝑖
𝑗
) =

{
 
 

 
 
1

2
(𝑖 + 1) +

1

2
(𝑗 − 1), 𝑓𝑜𝑟 𝑖 ∈ {1,3,5,…,𝑛 − 1},𝑗 ∈ {1,2,3,…,𝑚}

𝑛𝑚 + 1 −
𝑖

2
−
𝑛

2
(𝑗 − 1), 𝑓𝑜𝑟 𝑖 ∈ {2,4,6,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}

 

 

Then we have the vertex weights as follow: 

 

𝑊(𝑒𝑖
𝑗
) = {

    𝑛𝑚 + 1,𝑓𝑜𝑟 𝑖 ∈ {1,3,5,…,𝑛 − 1},𝑗 ∈ {1,2,3,…,𝑚}

𝑛𝑚 + 2,𝑓𝑜𝑟 𝑖 = 1,3,5,…,𝑛 − 1,𝑗 ∈ {1,2,3,…,𝑚}

𝑛𝑚 + 2 −
𝑛

2
,𝑓𝑜𝑟 𝑖 = 𝑛,𝑗 ∈ {1,2,3,…,𝑚}

 

 

Hence from the function above, it easy to see that the vertex weight 𝑊(𝑒𝑖
𝑗
) =

{𝑛𝑚 + 1,𝑛𝑚 + 2,𝑛𝑚 + 2 −
𝑛

2
} contains three elements which induce a proper 

vertex colouring of ⋃ 𝐶𝑛𝑗=𝑚 . Thus, it gives 𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) ≤ 3.  It concludes that 

𝜒𝑙𝑎(⋃ 𝐶𝑛𝑗=𝑚 ) = 3.  

 

17 18 

1 2 3 

4 5 6
2 

7 8 9 

10 11 12 

13 14 15 16 

19 20 21 

22 23 24 

25 26 27 

28 29 30 

30 31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

31 

30 30 

30 

30 30 

30 30 

30 30 

30 27 

24 

21 

18 

3 

6

2 

9 

12 

15 



Marsidi and I. H. Agusti 
The local antimagic on disjoint union of some family graphs 

73 

Theorem 2.4. For the star 𝑆𝑛 with 𝑛 ≥ 3, then we have 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) =

𝑛𝑚 + 1, 𝑚 ≠ even integer when 𝑛 is an odd integer. 
Proof. The ⋃ 𝑆𝑛𝑗=𝑚  is the disjoint union of star graph. The vertex set  

𝑉(⋃ 𝑆𝑛𝑗=𝑚 ) = {𝐴,𝑣1
1,𝑣2

1,…,𝑣𝑖
𝑗
,…,𝑣𝑛−1

𝑚 ,𝑣𝑛
𝑚; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}} 

and the edge set 𝐸(⋃ 𝑃𝑛𝑗=𝑚 ) =

{𝐴𝑣1
1,𝐴𝑣2

1,𝐴𝑣3
1 …,,𝐴𝑣𝑖

𝑗
,…,𝐴𝑣1

𝑚,𝐴𝑣2
𝑚,…,𝐴𝑣𝑛−1

𝑚 ,𝐴𝑣𝑛
𝑚; 𝑖 ∈ {1,2,3,…,𝑛},𝑗 ∈

{1,2,3,…,𝑚}}. Hence |𝑉(⋃ 𝑆𝑛𝑗=𝑚 )| = 𝑚𝑛 + 𝑚,   |𝐸(⋃ 𝑆𝑛𝑗=𝑚 )| = 𝑚𝑛. We will 

prove that 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≥ 𝑛𝑚 + 1. Graph ⋃ 𝑆𝑛𝑗=𝑚  have𝑛𝑚 leaves, based on 

Theorem 2.1, we have 𝜒𝑙𝑎(⋃ 𝑇𝑖=𝑚 ) ≥ 𝑘 + 1so the color needed for ⋃ 𝑆𝑛𝑗=𝑚  is 

𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≥ 𝑛𝑚 + 1. 

Furthermore, we will prove that the upper bound is 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≤ 𝑛𝑚 + 1, we 

define the labelling into two cases by the following. 

 

Case 1. For 𝑛 even 

ℎ(𝐴𝑥𝑖
𝑗
) = {

𝑚(𝑖 − 1) + 𝑗;𝑖 ∈ {1,3,5,…,𝑛 − 1},𝑗 ∈ {1,2,3,…,𝑚}

𝑚𝑖 − 𝑗 + 1;𝑖 ∈ {2,4,6,…,𝑛},𝑗 ∈ {1,2,3,…,𝑚}
 

 

Case 2. For 𝑛 odd 

ℎ(𝐴𝑥𝑖
𝑗
) =

{
 
 
 
 
 

 
 
 
 
 

𝑗 + 1

2
;𝑖 = 1,𝑗 ∈ {1,3,5,…,𝑚}

𝑚 + 𝑗 + 1

2
;𝑖 = 1,𝑗 ∈ {2,4,6,…,𝑚 − 1}

𝑗

2
+ 𝑚;𝑖 = 2,𝑗 ∈ {2,4,6,…,𝑚 − 1}

3𝑚 + 𝑗

2
;𝑖 = 2,𝑗 ∈ {1,3,5,…,𝑚}

3𝑚 + 1 − 𝑗;𝑗 ∈ {1,2,3,…,𝑚}

𝑚𝑖 − 𝑗 + 1;𝑖 ∈ {4,6,8,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚}

𝑚(𝑖 − 3) − 𝑗 + 1;𝑖 ∈ {5,7,9,…,𝑛};  𝑗 ∈ {1,2,3,…,𝑚}

 

 

Based on the labelling above, the weight on each pendant vertex is the label 

of the edge which incident with pendant vertex. Therefore, the number of vertex 

weight on the disjoint union of star graph is 𝑛𝑚. The weight of each centre vertex 
is as follow. 

𝑊(𝐴) =
𝑛

2
(𝑛𝑚 + 1) 

Because the number of pendant vertex is 𝑛𝑚, the number of different vertex 

weight on the disjoint union of star graph is 𝑛𝑚 + 1, so 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≤ 𝑛𝑚 +

1.Therefore 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≤ 𝑛𝑚 + 1 and 𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) ≥ 𝑛𝑚 + 1, it 

concludes𝜒𝑙𝑎(⋃ 𝑆𝑛𝑗=𝑚 ) = 𝑛𝑚 + 1  

 

Theorem 2.5. For the friendship ℱ𝑛 with 𝑛 ≥three and n is even integer, then 

we have 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) = 3. 



Jurnal Matematika MANTIK 

Vol. 5, No. 2, October 2019, pp. 69-75 

74 

Proof. The ⋃ ℱ𝑛𝑗=𝑚  is the disjoint union of friendship graph. The vertex set  

𝑉(⋃ ℱ𝑛𝑗=𝑚 ) = {𝐴} ∪ {𝑢1
1,𝑢2

1,…,𝑢𝑖
𝑗
,…,𝑢𝑛−1

𝑚 ,𝑢𝑛
𝑚; 𝑖 ∈ {1,2,3,…,𝑛};  𝑗 ∈

{1,2,3,…,𝑚}} ∪ {𝑣1
1,𝑣2

1,…,𝑣𝑖
𝑗
,…,𝑣𝑛−1

𝑚 ,𝑣𝑛
𝑚; 𝑖 ∈ {1,2,3,…,𝑛};  𝑗 ∈

{1,2,3,…,𝑚}}and the edge set 𝐸(⋃ ℱ𝑛𝑗=𝑚 ) =

{𝐴𝑢1
1,𝐴𝑢2

1,𝐴𝑢3
1 …,,𝐴𝑢𝑖

𝑗
,…,𝐴𝑢1

𝑚,𝐴𝑢2
𝑚,…,𝐴𝑢𝑛−1

𝑚 ,𝐴𝑢𝑛
𝑚; 𝑖 ∈ {1,2,3,…,𝑛};  𝑗 ∈

{1,2,3,…,𝑚}} ∪ {𝐴𝑣1
1,𝐴𝑣2

1,𝐴𝑣3
1 …, ,𝐴𝑣𝑖

𝑗
,…,𝐴𝑣1

𝑚,𝐴𝑣2
𝑚,…,𝐴𝑣𝑛−1

𝑚 ,𝐴𝑣𝑛
𝑚; 𝑖 ∈

{1,2,3,…,𝑛};  𝑗 ∈ {1,2,3,…,𝑚}} ∪

{𝑢1
1𝑣1

1,𝑢2
1𝑣2

1,…,𝑢𝑖
𝑗
𝑣𝑖
𝑗
,…,𝑢𝑛−1

𝑚 𝑣𝑛−1
𝑚 ,𝑢𝑛

𝑚𝑣𝑛
𝑚; }.Hence |𝑉(⋃ ℱ𝑛𝑗=𝑚 )| = 2𝑚𝑛 + 𝑚,   

|𝐸(⋃ ℱ𝑛𝑗=𝑚 )| = 3𝑚𝑛. We will prove that 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≥ 3. Based on 

Arumugam theorem [4], the local antimagic chromatic number of friendship graph 

is  𝜒𝑙𝑎(ℱ𝑛) = 3 . Furthermore, based on Remark 2.1 we have 𝜒𝑙𝑎(⋃ ℱ𝑛𝑖=𝑚 ) ≥

𝜒𝑙𝑎(ℱ𝑛) so we can see that  𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≥ 3. 

Furthermore, we will prove that the upper bound is 3, 

denoted𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≤ 3, we define the labeling by the following. 

𝑓(𝐴𝑢𝑖
𝑗
) = {

𝑚(𝑖 − 1) + 𝑗;𝑖 ∈ {1,3,5,…,𝑛 − 1};  𝑗 ∈ {1,2,3,…,𝑚}

𝑚𝑖 − 𝑗 + 1;𝑖 ∈ {2,4,6,…,𝑛};𝑗 ∈ {1,2,3,…,𝑚}
 

𝑓(𝐴𝑣𝑖
𝑗
) = {

𝑚(𝑖 − 1) + 𝑗 + 2𝑚𝑛 ;𝑖 ∈ {1,3,5,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚}

𝑚𝑖 − 𝑗 + 1 + 2𝑚𝑛;𝑖 ∈ {2,4,6,…,𝑛};  𝑗 ∈ {1,2,3,…,𝑚}
 

𝑓(𝑢𝑖
𝑗
𝑣𝑖
𝑗
) = {

2𝑚𝑛 + 1 − 𝑚(𝑖 − 1) − 𝑗 ; 𝑖 ∈ {1,3,5,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚}

2𝑚𝑛 − 𝑚𝑖 + 𝑗;𝑖 ∈ {2,4,6,…,𝑛};  𝑗 ∈ {1,2,3,…,𝑚}
 

 

Based on the labeling above, we have the vertex weights as follows: 

𝑊1(𝐴) = 3𝑚𝑛
2 + 𝑛 

𝑊2(𝑢𝑖
𝑗
) = 2𝑚𝑛 + 1,𝑓𝑜𝑟 𝑖 ∈ {1,2,3,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚} 

𝑊3(𝑣𝑖
𝑗
) = 4𝑚𝑛 + 1,𝑓𝑜𝑟 𝑖 ∈ {1,2,3,…,𝑛 − 1};𝑗 ∈ {1,2,3,…,𝑚} 

 

Hence from the above, it easy to see that the vertex weight 𝑊 =
{3𝑚𝑛2 + 𝑛, 2𝑚𝑛 + 1,4𝑚𝑛 + 1} contains 3 element which induces a proper 

vertex coloring of ⋃ 𝐶𝑛𝑗=𝑚 , so𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≤ 3.Therefore 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≤ 3 

and 𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) ≥ 3, it concludes𝜒𝑙𝑎(⋃ ℱ𝑛𝑗=𝑚 ) = 3.  ∎ 

 

3. Concluding Remarks 

All results in this paper are the vertex local antimagic chromatic number of 

disjoint union of the path, cycle, star, and friendship graph. The vertex local 

antimagic chromatic number of all graphs in this paper has attended the lower 

bound.  

Open Problem. Find the vertex local antimagic chromatic number on disjoint 

union of cycle and friendship when the number of their vertices is odd, disjoint 

union of a star when the number of copies is even where the number of vertice sis 

odd, and also find the vertex local antimagic of disjoint union of another graph. 

 

  



Marsidi and I. H. Agusti 
The local antimagic on disjoint union of some family graphs 

75 

Acknowledgement 

We gratefully acknowledge the support from CGANT University of Jember and 

IKIP PGRI Jember of the year 2019. 

 

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