EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 2828-2842
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Local Total Antimagic Chromatic Number for the
Disjoint Union of Star Graphs

Venkatesan Sandhiya1, Moviri Chettiar Nalliah1,∗

1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology,
Vellore, Tamil Nadu, India

Abstract. Let G be a graph with n vertices and m edges without isolated vertices. A local total
antimagic labeling of a graph G is defined as there is a bijection f from the set V (G) ∪ E(G) →
{1, 2, ..., n + m}, such that, any two adjacent vertices, any two adjacent edges, a vertex and an
edge incident to the vertex does not receive the same weight. The vertex weight is the sum of the
edge labels incident to that vertex and the edge weight is the sum of its end vertex labels. The
local total antimagic chromatic number is the minimum number of colors taken over all induced
by local total antimagic colorings (labelings) of G, which is denoted by χlt(G). In this paper, we
determine the local total antimagic chromatic number for the disjoint union of star graphs.

2020 Mathematics Subject Classifications: 05C78, 05C15

Key Words and Phrases: Local antimagic graphs, chromatic number, total coloring

1. Introduction

Let G be a graph with no isolated vertices, neither multiple edges nor loops. The set
of vertices and edges of G(V,E) are denoted by V (G) and E(G) respectively. For graph
theoretic terminology, we refer to Chartrand and Lesniak [9] and [10]. Graph coloring and
graph labeling are the significant research areas in graph theory. The coloring of a graph
G is known as the assignment of colors to vertices or edges or both is called as vertex
coloring or edge coloring or total coloring. In this study, we consider the total coloring of
a graph.

Graph coloring has various applications in network optimization and scheduling prob-
lems. Total Coloring, being a slight variation and generalization of the usual vertex color-
ing. The main application of total coloring is scheduling trains in a region which has high
rail traffic with several interconnected tracks, and the region between two stations either
consisting of dense forests with wild animals frequently found roaming on tracks at a spe-
cific time, or unmanned level-crossing gates with peak hour traffic congestion problems.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5383

Email addresses: vsandhiya5295@gmail.com (V. Sandhiya), nalliah.moviri@vit.ac.in (M. Nalliah)

https://www.ejpam.com 2828 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2829

In order to schedule the appropriate train timetables, one can use the total coloring of the
rail network graph, where the train stations are vertices of the graph and the connecting
tracks form the edges. The scheduling involving multi-factors can be devised using to-
tal coloring. The advancement in signal processing and control systems using Z and L-
transforms are given in [13].

Definition 1. [39] The total coloring of G is defined as a map f : V (G) ∪ E(G) → K,
where K is a set of colors, satisfying the following three conditions:

(i) f(u) ̸= f(v) for any two adjacent vertices u, v ∈ V (G);

(ii) f(e) ̸= f(e′) for any two adjacent edges e, e′ ∈ E(G); and

(iii) f(v) ̸= f(e) for any vertex v ∈ V (G) and an incident edge e = vx.

The minimum number of colors used in any total coloring of G is called the total
chromatic number χt(G). A trivial lower bound for total chromatic number is maximum
degree of the graph G plus one. Behzad [8] and Vizing [38] posed the conjecture indepen-
dently that, for any graph G, χt(G) ≤ ∆(G) + 2. This lower and upper bounds leads to
every graph G has ∆(G) + 1 ≤ χt(G) ≤ ∆(G) + 2.

A graph labeling is an assignment of integers to the vertices or edges or both, subject
to certain conditions. To learn more about graph labeling we refer Gallian’s [12] survey
of graph labeling. Graph labeling have been motivated by practical problems, labeled
graphs serve useful mathematical models for a broad range of applications such as: coding
theory, including the design of good types codes, synch-set codes, missile guidance codes
and convolutional codes with optimal auto correlation properties.

The idea of an antimagic labeling of a graph were proposed by Hartsfield and Ringel
[11].

Definition 2. [11] A bijection f : E → {1, 2, 3, ..., |E|} is called an antimagic labeling of
G, if w(u) ̸= w(v) for any two distinct vertices u, v ∈ V (G). The weight of the vertex u
is defined as w(u) =

∑
e∈E(u) f(e), where E(u) is the set of edges incident to u. A graph

G is antimagic if G admits an antimagic labeling.

Hartsfield and Ringel [11] proposed the following conjectures.

Conjecture 1. [11] Every connected graph other than K2 is antimagic.

Conjecture 2. [11] Every tree other than K2 is antimagic.

Based on the above two conjectures several authors are studied and obtained several re-
sults. In [14–17], the authors studies the b-chromatic number of some special and standard
graphs. Arumugam and Nalliah [3] found the super (a, d)-edge antimagic total labelings of
friendship graphs in 2012. For further results one can refer [4, 5, 24–27]. In 2017, Nalliah
and Arumugam [28] determined the super (a, 3)-edge antimagic total labeling for union of
two stars. For further study see in [31, 33, 34].

In 2017, Arumugam et al.[6] proposed the local antimagic chromatic number using
antimagic labeling and vertex coloring concepts, which is given as follows.



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2830

Definition 3. [6] A local vertex antimagic labeling is a bijection f : E → {1, 2, ..., |E|}
such that w(u) ̸= w(v),for all uv ∈ E(G), where w(u) =

∑
e∈E(u) f(e), and E(u) is the

set of edges incident to u. If a graph G admits a local antimagic labeling, then G is called
local antimagic. The minimum number of colors taken over all colorings induced by local
antimagic labelings of G is called local antimagic chromatic number, denoted by χla(G).

The local antimagic chromatic number has been studied by several authors for the
various families of graphs. In 2022, Lau et.al [22] obtained the local antimagic chromatic
number of lexicographic product graph. Also in [23], the authors are given the complete
characterization of s-bridge graphs with local antimagic chromatic number 2. Nalliah
et.al [29] found the local vertex coloring for generalized friendship graph. Shankar and
Nalliah [37] studied the local vertex antimagic chromatic number of some wheel related
graph. Also in [36], the authors found the local antimagic chromatic number for the
corona product of wheel and null graphs. In 2023, Lau and Nalliah [19] determined the
local antimagic chromatic number of a corona product of graph. In [2], authors found
the degree based topological indices of corona product graphs. For further study see in
[18, 20, 21, 30].

An edge version of local antimagic labeling were introduced by Agustin et al.[1] in
2017, which is given as follows:

Definition 4. [1] A local edge antimagic labeling is defined as there is a bijection f :
V (G) → {1, 2, 3, ..., |V (G)|}, for any two adjacent edges e1 and e2 with their weights
w(e1) ̸= w(e2). The edge weight of an edge e = xy is defined by w(e = xy) = f(x)+ f(y).
If a graph G admits a local edge antimagic labeling, then G is called local edge antimagic.
The minimum number of colors taken over all colorings induced by local edge antimagic
labelings of G is called local edge antimagic chromatic number, denoted by χ′

lea(G).

Agustin et al.[1] found the local edge antimagic chromatic number for the path graph,
star graph, cycle graph and friendship graph. Rajkumar and Nalliah [32] determined the
local edge antimagic chromatic number of wheel graph, helm graph, closed helm graph
and double star graph.

In [35], Sandhiya and Nalliah introduced the concept of local total antimagic labeling
and its related parameter local total antimagic chromatic number.

Definition 5. [35] Let G be a graph with n vertices, m edges and no isolated vertices. A
local total antimagic coloring of a graph G is a bijection f : V (G)∪E(G) → {1, 2, ..., n+m},
if

(i). w(u) ̸= w(v) for any two adjacent vertices u, v ∈ V (G);

(ii). w(e) ̸= w(e′) for any two adjacent edges e = uv, e′ = u′v′ ∈ E(G); and

(iii). w(v) ̸= w(e) for any vertex v ∈ V (G) and for any edge e = uv ∈ E(G) that is
incident to the same vertex v.

The weight of a vertex u is defined by w(u) =
∑

e∈E(u) f(e), where E(u) is the set of all
incident edges of a vertex u, and an edge weight is w(xy) = f(x) + f(y), xy ∈ E(G).



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2831

Definition 6. [35] The local total antimagic chromatic number is the minimum number
of colors taken over all colorings induced by local total antimagic colorings of G, which is
denoted by χlt(G).

Observation 1. [35] If G admits a local total antimagic coloring, then χlt(G) ≥ χt(G).

Observation 2. [35] The graph K2 does not admit a local total antimagic labeling.

Theorem 1. [35] Let G be a graph with at least three vertices and k pendants. Then
χlt(G) ≥ k + 1.

Theorem 2. [35] For the star graph K1,n, we have χlt(K1,n) = n+ 1.

Theorem 3. [35] For the double star graph Sn,n, n > 5, we have χlt(Sn,n) = 2n+ 2.

Martin Bača [7], estimated the bounds of the local antimagic chromatic number for
the disjoint union of multiple copies of graphs and obtained the local antimagic chromatic
number for the disjoint union of stars, cycles and caterpillar graphs. Rajkumar and
Nalliah [31] found the local edge antimagic chromatic number for the disjoint union of
star graphs, ladder graph and cycle graph. The disjoint union of star graphs is used to
model and analyze communication networks, where each star graph represents a central
hub connected to several nodes. Many authors found the different types of antimagic
chromatic numbers for the union of star graphs. In this paper, we determine the local
total antimagic chromatic number of disjoint union of star graph mK1,n.

2. Main Results

Let a and b be two positive integers with a < b, we denote [a, b] = {a, a+1, ..., b−1, b}
and c− set denotes the set of corresponding values taken by the formula f or (w).

Proposition 1. The graph mK2 is not a local total antimagic.

Proof. Let V (mK2) = {ui, vi, 1 ≤ i ≤ m} and E(mK2) = {uivi, 1 ≤ i ≤ m} be the
vertex and edge sets of mK2. Then |V (mK2)| = 2m and |E(mK2)| = m. Suppose the
graph G = mK2 admits a local total antimagic labeling. Then there exists a local total
antimagic labeling f : V (mK2) ∪E(mK2) → {1, 2, 3, ..., 3m}. Let e = uv ∈ E(mK2) with
label f(e). Then w(u) = w(v) = f(e), which is a contradiction that adjacent vertices u
and v received the same weight. Thus, mK2 has no local total antimagic labeling f .

Theorem 4. Let G = mK1,2 be the disjoint union of m copies of star K1,2. Then
χlt(mK1,2) = 2m+ 1.

Proof. Let V (mK1,2) = {cj , uj1, u
j
2, 1 ≤ j ≤ m} and E(mK1,2) = {cjuj1, cju

j
2, 1 ≤ j ≤

m} be the vertex and edge sets of mK1,2. Then |V (mK1,2)| = 3m and |E(mK1,2)| = 2m.
Define a labeling f : V (mK1,2) ∪ E(mK1,2) → {1, 2, 3, ..., 5m} by

f(uji ) =

{
j, i = 1, 1 ≤ j ≤ m, c-set is {1, 2, 3, ...,m− 1,m}
m+ j, i = 2, 1 ≤ j ≤ m, c-set is {m+ 1,m+ 2, ..., 2m}



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2832

f(cj) = 3m+ 1− j, 1 ≤ j ≤ m, c-set is {3m, 3m− 1, ..., 2m+ 1}

f(cju
j
i ) =

{
5m+ 1− j, i = 1, 1 ≤ j ≤ m, c-set is {5m, 5m− 1, ..., 4m+ 1}
3m+ j, i = 2, 1 ≤ j ≤ m, c-set is {3m+ 1, 3m+ 2, ..., 4m}

Then the vertex and edge weights are

w(uji ) = f(cju
j
i ), i = 1, 2 and 1 ≤ j ≤ m

w(cj) = 8m+ 1, 1 ≤ j ≤ m

w(cju
j
1) = 3m+ 1, 1 ≤ j ≤ m

w(cju
j
2) = 4m+ 1, 1 ≤ j ≤ m

The union of all distinct weights of the set C is given as follows:

C ={w(uj1) ∪ w(uj2) ∪ w(cj) ∪ w(cju
j
1) ∪ w(cju

j
2), 1 ≤ j ≤ m}

={3m+ 1, 3m+ 2, ..., 4m} ∪ {4m+ 1, 4m+ 2, ..., 5m} ∪ {8m+ 1} ∪ {3m+ 1} ∪ {4m+ 1}
={3m+ 1, 3m+ 2, ..., 4m, 4m+ 1, ..., 5m} ∪ {8m+ 1}

and hence |C| = 2m+ 1. So that f provides a local total antimagic labeling with 2m+ 1
colors. Thus χlt(mK1,2) ≤ 2m+1. By Theorem 1 [35], we get χlt(mK1,2) ≥ 2m+1. Thus
χlt(mK1,2) = 2m+ 1.

Example 1. The local total antimagic labeling of 5K1,2 with 11 colors are {16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 41} given in Figure 1.

15

1

6

14

2

7

13

3

8

12

4

9

11

5

10

25

16

24

17

23

18

22

19

21

20

Figure 1: Local total antimagic labeling of 5K1,2 with 11 colors {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 41}

Theorem 5. Let mK1,n, n ≥ 3, m ≥ 2 be the disjoint union of m copies of star K1,n.
Then χlt(mK1,n) = mn+ 1.

Proof. Let V (mK1,n) = {cj , uji , 1 ≤ i ≤ n, 1 ≤ j ≤ m} and E(mK1,n) = {cjuji , 1 ≤
i ≤ n, 1 ≤ j ≤ m} be the vertex and edge sets of mK1,n. Then |V (mK1,n)| = m(n+ 1),
|E(mK1,n)| = mn and |V (mK1,n) ∪ E(mK1,n)| = 2mn+m.
Case 1: When n is even
Define a labeling f : V (mK1,n) ∪ E(mK1,n) → {1, 2, 3, ..., 2mn+m} by

f(uji ) = m(i− 1) + j, 1 ≤ i ≤ n, 1 ≤ j ≤ m, c-set is given in Table 1

f(cj) = m(n+ 1) + 1− j, 1 ≤ j ≤ m



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2833

c-set is {mn+m, mn+m− 1, ..., mn+ 2, mn+ 1}

f(cju
j
i ) =


2m(n+ 1) + 1−mi− j, if i is odd, 1 ≤ i ≤ n− 1, 1 ≤ j ≤ m

c- set is given in Table 2

m(2n+ 1)−mi+ j, if i is even, 1 ≤ i ≤ n, 1 ≤ j ≤ m

c- set is given in Table 3

The above labeling f is given in Tables 1, 2 and 3 for easy reading.

Table 1: Labels for the pendant vertices uj
i of mK1,n, where n is even

i u1i u2i u3i ... um−1
i umi

1 1 2 3 ... m-1 m

2 m+1 m+2 m+3 ... 2m-1 2m

... ... ... ... ... ... ...

... ... ... ... ... ... ...

n-1 m(n-2)+1 m(n-2)+2 m(n-2)+3 ... m(n-1)-1 m(n-1)

n m(n-1)+1 m(n-1)+2 m(n-1)+3 ... mn-1 mn

Table 2: Labels for the pendant edges cju
j
i of mK1,n, where n is even and i is odd

i c1u
1
i c2u

2
i c3u

3
i ... cm−1u

m−1
i cmumi

1 2mn+m 2mn+m-1 2mn+m-2 ... 2mn+2 2mn+1

3 2mn-m 2mn-m-1 2mn-m-2 ... 2mn-2m+2 2mn-2m+1

... ... ... ... ... ... ...

... ... ... ... ... ... ...

n-3 mn+5m mn+5m-1 mn+5m-2 ... mn+4m+2 mn+4m+1

n-1 mn+3m mn+3m-1 mn+3m-2 ... mn+2m+2 mn+2m+1

Table 3: Labels for the pendant edges cju
j
i of mK1,n, where n is even and i is even

i c1u
1
i c2u

2
i c3u

3
i ... cm−1u

m−1
i cmumi

2 2mn-m+1 2mn-m+2 2mn-m+3 ... 2mn-1 2mn

4 2mn-3m+1 2mn-3m+2 2mn-3m+3 ... 2mn-2m-1 2mn-2m

... ... ... ... ... ... ...

... ... ... ... ... ... ...

n-2 mn+3m+1 mn+3m+2 mn+3m+3 ... mn+4m-1 mn+4m

n mn+m+1 mn+m+2 mn+m+3 ... mn+2m-1 mn+2m

Then the vertex and edge weights are

w(uji ) = f(cju
j
i ), 1 ≤ i ≤ n, 1 ≤ j ≤ m

w(cj) =
mn(3n+ 2) + n

2
= 2mn+

n

2
[m(3n− 2) + 1], 1 ≤ j ≤ m



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2834

w(cju
j
i ) = mn+mi+ 1, 1 ≤ i ≤ n, 1 ≤ j ≤ m

c-set is{mn+m+ 1, mn+ 2m+ 1, ..., 2mn−m+ 1, 2mn+ 1}

The union of all distinct weights of a set C is given as follows:

C =w(uji ) ∪ w(cj) ∪ w(cju
j
i )

=[mn+m+ 1, 2mn+m] ∪
{
2mn+

n

2
[m(3n− 2) + 1]

}
∪ {mn+m+ 1,mn+ 2m+ 1, ..., 2mn−m+ 1, 2mn+ 1}

=[mn+m+ 1, 2mn+m] ∪
{
2mn+

n

2
[m(3n− 2) + 1]

}
and hence |C| = mn + 1. So that f admits local total antimagic labeling with mn + 1
colors. Thus χlt(mK1,n) ≤ mn + 1. By Theorem 1 [35], we get χlt(mK1,n) ≥ mn + 1.
Thus χlt(mK1,n) = mn+ 1.
Case 2: When n is odd:
Subcase 2(a): n = 3 is odd.
Define a labeling f : V (mK1,3) ∪ E(mK1,3) → {1, 2, 3, ..., 7m} by

f(uji ) =


3m+ 3− 3j, i = 1, 1 ≤ j ≤ m, c-set is {3m, 3m− 3, ..., 9, 6, 3}
3m+ 1− 3j, i = 2, 1 ≤ j ≤ m, c-set is {3m− 2, 3m− 5, ..., 7, 4, 1}
3m+ 2− 3j, i = 3, 1 ≤ j ≤ m, c-set is {3m− 1, 3m− 4, ..., 8, 5, 2}

f(cj) = 3m− 1 + 2j, 1 ≤ j ≤ m;

c-set is {3m+ 1, 3m+ 3, 3m+ 5, ..., 5m− 3, 5m− 1}

f(cju
j
i ) =



7m+ 1− j, i = 1, 1 ≤ j ≤ m,

c-set is {7m, 7m− 1, 7m− 2, ..., 6m+ 2, 6m+ 1}
6m+ 1− j, i = 2, 1 ≤ j ≤ m,

c-set is {6m, 6m− 1, 6m− 2, ..., 5m+ 2, 5m+ 1}
3m+ 2j, i = 3, 1 ≤ j ≤ m,

c-set is {3m+ 2, 3m+ 4, 3m+ 6, ..., 5m− 2, 5m}

Then the vertex and edge weights are

w(uji ) = f(cju
j
i ), 1 ≤ i ≤ 3, 1 ≤ j ≤ m

w(cj) = 16m+ 2, 1 ≤ j ≤ m

w(cju
j
1) = 6m+ 2− j, 1 ≤ j ≤ m,

c-set is {6m+ 1, 6m, 6m− 1, ..., 5m+ 3, 5m+ 2}
w(cju

j
2) = 6m− j, 1 ≤ j ≤ m,

c-set is {6m− 1, 6m− 2, 6m− 3, ..., 5m+ 1, 5m}
w(cju

j
3) = 6m+ 1− j, 1 ≤ j ≤ m,



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2835

c-set is {6m, 6m− 1, 6m− 2..., 5m+ 2, 5m+ 1}

The union of all distinct weights of a set C is given as follows:

C =w(uji ) ∪ w(cj) ∪ w(cju
j
1) ∪ w(cju

j
2) ∪ w(cju

j
3)

={[5m+ 1, 7m] ∪ {3m+ 2, 3m+ 4, 3m+ 6, ..., 5m− 2, 5m}} ∪ {16m+ 2}
∪ [5m+ 2, 6m+ 1] ∪ [5m, 6m− 1] ∪ [5m+ 1, 6m]

=[5m+ 1, 7m] ∪ {3m+ 2, 3m+ 4, 3m+ 6, ..., 5m− 2, 5m} ∪ {16m+ 2}

and hence |C| = 3m+ 1. So that f provides a local total antimagic labeling with 3m+ 1
colors. Thus χlt(mK1,3) ≤ 3m+1. By Theorem 1 [35], we get χlt(mK1,3) ≥ 3m+1. Thus
χlt(mK1,3) = 3m+ 1.
Subcase 2(b): n ≥ 5 is odd
Define a labeling f : V (mK1,n) ∪ E(mK1,n) → {1, 2, 3, ..., 2mn+m} by

f(uji ) = m(n− j) + i, 1 ≤ i ≤ n, 1 ≤ j ≤ m; c-set is given in Table 4

f(cj) =


mn+ 2j, if m is odd and 1 ≤ j ≤ m

c-set is {mn+ 2, mn+ 4, ..., mn+ 2m}
mn− 1 + 2j, if m is even and 1 ≤ j ≤ m

c-set is {mn+ 1, mn+ 3, ..., mn+ 2m− 1}

f(cju
j
i ) =



2m(n+ 1) + 1−mi− j, if i is odd, 1 ≤ i ≤ n− 2 and 1 ≤ j ≤ m

c-set is given in Table 5

m(2n+ 1)−mi+ j, if i is even, 1 ≤ i ≤ n− 3, and 1 ≤ j ≤ m

c-set is given in Table 6

m(n+ 3) + 1− j, if i = n− 1 is even and 1 ≤ j ≤ m

mn− 1 + 2j, if i = n and m is odd, 1 ≤ j ≤ m

c-set is {mn+ 1, mn+ 3, ..., mn+ 2m− 1}
mn+ 2j, if i = n and m is even, 1 ≤ j ≤ m

c-set is {mn+ 2, mn+ 4, ..., mn+ 2m}

The above labeling f is given in Tables 4, 5 and 6 for easy reading.
Then the vertex and edge weights are

w(uji ) = f(cju
j
i ), 1 ≤ i ≤ n and 1 ≤ j ≤ m

w(cj) =

{
3mn2+2mn−m+n−1

2 = 2mn+m+ m[(3n−2)n−3]+(n−1)
2 , if m is odd, 1 ≤ j ≤ m

3mn2+2mn−m+n+1
2 = 2mn+m+ m[(3n−2)n−3]+(n+1)

2 , if m is even, 1 ≤ j ≤ m

w(cju
j
i ) =


2mn+ i+ 2j − nj, if m is odd, 1 ≤ j ≤ m and 1 ≤ i ≤ n

c-set is given in Table 7

2mn− 1 + i+ 2j − nj, if m is even, 1 ≤ j ≤ m and 1 ≤ i ≤ n

c-set is given in Table 8



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2836

Table 4: Labels for the pendant vertices uj
i of mK1,n, where n is odd and m > 1

i u1i u2i u3i ... um−1
i umi

1 (m-1)n+1 (m-2)n+1 (m-3)n+1 ... n+1 1

2 (m-1)n+2 (m-2)n+2 (m-3)n+2 ... n+2 2

... ... ... ... ... ... ...

... ... ... ... ... ... ...

n-1 mn-1 (m-1)n-1 (m-2)n-1 ... 2n-1 n-1

n mn (m-1)n (m-2)n ... 2n n

Table 5: Labels for the pendant edges cju
j
i of mK1,n, where n is odd and i is odd

i c1u
1
i c2u

2
i c3u

3
i ... cm−1u

m−1
i cmumi

1 2mn+m 2mn+m-1 2mn+m-2 ... 2mn+2 2mn+1

3 2mn-m 2mn-m-1 2mn-m-2 ... 2mn-2m+2 2mn-2m+1

... ... ... ... ... ... ...

... ... ... ... ... ... ...

n-4 mn+6m mn+6m-1 mn+6m-2 ... mn+5m+2 mn+5m+1

n-2 mn+4m mn+4m-1 mn+4m-2 ... mn+3m+2 mn+3m+1

The above weights are given in Tables 7 and 8 for easy reading.

For m is odd, the union of all distinct weights of a set Codd is given as follows:

Codd ={w(uji ) ∪ w(cj) ∪ w(cju
j
i ), 1 ≤ j ≤ m and 1 ≤ i ≤ n}

=[mn+ 2m+ 1, 2mn+m] ∪ {mn+ 1,mn+ 3, ...,mn+ 2m− 1}

∪
{
2mn+m+

m[(3n− 2)n− 3] + (n− 1)

2

}
∪ [mn+ 2m+ 1, 2mn+ 2]

=[mn+ 2m+ 1, 2mn+m] ∪ {mn+ 1,mn+ 3, ...,mn+ 2m− 1}

∪
{
2mn+m+

m[(3n− 2)n− 3] + (n− 1)

2

}
|Codd| =m(n− 1) +m+ 1 = mn+ 1

Table 6: Labels for the pendant edges cju
j
i of mK1,n, where n is odd and i is even

i c1u
1
i c2u

2
i c3u

3
i ... cm−1u

m−1
i cmumi

2 2mn-m+1 2mn-m+2 2mn-m+3 ... 2mn-1 2mn

4 2mn-3m+1 2mn-3m+2 2mn-3m+3 ... 2mn-2m-1 2mn-2m

... ... ... ... ... ... ...

... ... ... ... ... ... ...

n-3 mn+4m+1 mn+4m+2 mn+4m+3 ... mn+5m-1 mn+5m

n-1 mn+3m mn+3m-1 mn+3m-2 ... mn+2m+2 mn+2m+1



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2837

Table 7: Weights for the pendant edges cju
j
i of mK1,n, where n is odd and m is odd

i c1u
1
i c2u

2
i c3u

3
i ... cm−1u

m−1
i cmumi

1 mn+(m-1)n+3 mn+(m-2)n+5 mn+(m-3)n+7 ... mn+2m+(n-1) mn+2m+1

2 mn+(m-1)n+4 mn+(m-2)n+6 mn+(m-3)n+8 ... mn+2m+n mn+2m+2

... ... ... ... ... ... ...

... ... ... ... ... ... ...

n-1 2mn+1 mn+(m-1)n+3 mn+(m-2)n+5 ... mn+2m+2n-3 mn+2m+n-1

n 2mn+2 mn+(m-1)n+4 mn+(m-2)n+6 ... mn+2m+2n-2 mn+2m+n

Table 8: Weights for the pendant edges cju
j
i of mK1,n, where n is odd and m is even

i c1u
1
i c2u

2
i c3u

3
i ... cm−1u

m−1
i cmumi

1 mn+(m-1)n+2 mn+(m-2)n+4 mn+(m-3)n+6 ... mn+2m+(n-2) mn+2m

2 mn+(m-1)n+3 mn+(m-2)n+5 mn+(m-3)n+7 ... mn+2m+(n-1) mn+2m+1

... ... ... ... ... ... ...

... ... ... ... ... ... ...

n-1 2mn mn+(m-1)n+2 mn+(m-2)n+4 ... mn+2m+2n-4 mn+2m+n-2

n 2mn+1 mn+(m-1)n+3 mn+(m-2)n+5 ... mn+2m+2n-3 mn+2m+n-1

For m is even, the union of all distinct weights of a set Ceven is given as follows:

Ceven =[mn+ 2m+ 1, 2mn+m] ∪ {mn+ 2,mn+ 4, ...,mn+ 2m}

∪
{
2mn+m+

m[(3n− 2)n− 3] + (n+ 1)

2

}
∪ [mn+ 2m, 2mn+ 1]

[mn+ 2m+ 1, 2mn+m] ∪ {mn+ 2,mn+ 4, ...,mn+ 2m}

∪
{
2mn+m+

m[(3n− 2)n− 3] + (n+ 1)

2

}
|Ceven| =m(n− 1) +m+ 1 = mn+ 1

So that f provides a local total antimagic labeling for mK1,n with mn + 1 colors, and
hence χlt(mK1,n) ≤ mn + 1. By Theorem 1 [35], we get χlt(mK1,n) ≥ mn + 1. Thus
χlt(mK1,n) = mn+ 1.

Example 2. The local total antimagic labeling of 5K1,3 with 16 colors are {17, 19, 21, 23, 25,
26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 82} given in Figure 2.



V. Sandhiya, M. Nalliah / Eur. J. Pure Appl. Math, 17 (4) (2024), 2828-2842 2838

16

15

13

14

18

12

10

11

20

9

7

8

22

6

4

5

24

3

1

2

35

30

17

34

29

19

33

28

21

32

27

23

31

26

25

Figure 2: Local total antimagic labeling of 5K1,3 with 16 colors {17, 19, 21, 23, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 82}

Example 3. The local total antimagic labeling of 4K1,6 with 25 colors are [29 52]∪{243}
given in Figure 3.

28

1

5

9

13

17

21

27

2

6

10

14

18

22

26

3

7

11

15

19

23

25

4

8

12

16

20

24

52

45

44

37

36

29

51

46

43

38

35

30

50

47

42

39

34

31

49

48

41

40

33

32

Figure 3: Local total antimagic labeling of 4K1,6 with 25 colors [29 52] ∪ {243}

Example 4. The local total antimagic labeling of 5K1,7 with 36 colors are {36, 38, 40, 42, 44}∪
[47 75] ∪ {403} are given in Figure 4.

37

29

30

31

32

33

34

35

39

22

23

24

25

26

27

28

41

15

16

17

18

19

20

21

43

8

9

10

11

12

13

14

45

1

2

3

4

5

6

7

75

66

65

56

55

50

36

74

67

64

57

54

49

38

73

68

63

58

53

48

40

72

69

62

59

52

47

42

71

70

61

60

51

46

44

Figure 4: Local total antimagic labeling of 5K1,7 with 36 colors {36, 38, 40, 42, 44} ∪ [47 75] ∪ {403}

3. Conclusions

In this paper, we investigated the local total antimagic labeling and local total an-
timagic chromatic number for the disjoint union of star graphs with some copies of star



REFERENCES 2839

graphs. The disjoint union of star graphs, equipped with an antimagic labeling, has var-
ious applications in network optimization, coding theory, graph decomposition, schedul-
ing, resource allocation, data storage and transportation. These applications leverage the
structural properties of the union of star graphs, combined with the anti-magic labeling,
to solve the complex problems efficiently. The local total antimagic labeling for the other
families of disconnected graphs are still open. In future we will estimate the real time
applications of local total antimagic labeling of both connected and disconnected graphs.

Acknowledgements

The first author is thankful to Vellore Institute of Technology for providing Teaching
Cum Research Associate Fellowship.

References

[1] Ika Hesti Agustin, Mohammad Hasan, Dafik Dafik, Ridho Alfarisi, and Rafi-
antika Megahnia Prihandini. Local edge antimagic coloring of a graph. Far East
Journal of Mathematical Sciences, 102(9):1925–1941, 2017.

[2] Nasir Ali, Zaeema Kousar, Maimoona Safdar, Fikadu Tesgara Tolasa, and Enoch
Suleiman. Mapping connectivity patterns: Degree-based topological indices of corona
product graphs. Journal of Applied Mathematics, 2023(1), 2023.

[3] S Arumugam and M Nalliah. Super (a, d)-edge antimagic total labelings of friendship
graphs. Australasian Journal of Combinatorics, 53:237–244, 2012.

[4] S Arumugam and M Nalliah. Super (a, d)-edge antimagic total labelings of friendship
and generalized friendship graphs. Electronic Notes in Discrete Mathematics, 48:103–
110, 2015.

[5] S Arumugam and M Nalliah. Super (a, 3)-edge-antimagic total labelings for union of
two stars. Utilitas Mathematica, 101, 2016.

[6] S Arumugam, K Premalatha, Martin Bača, and Andrea Semaničová-Feňovč́ıková.
Local antimagic vertex coloring of a graph. Graphs and combinatorics, 33:275–285,
2017.

[7] Martin Bača, Andrea Semaničová-Feňovč́ıková, and Tao-Ming Wang. Local antimagic
chromatic number for copies of graphs. Mathematics, 9(11):12–30, 2021.

[8] M. Behzad. Graphs and their chromatic number. PhD thesis, Michigan State Uni-
versity, 1965.

[9] Gary Chartrand, Linda Lesniak, and Ping Zhang. Graphs and Digraphs. CRC Press,
Chapman & Hall London, 2015.



REFERENCES 2840

[10] Gary Chartrand and Ping Zhang. A first course in graph theory. Dover publication,
Inc, 2013.

[11] N. Hartsfield G. Ringel. Pearls in graph theory. Academic press, USA, 1994.

[12] Joseph A Gallian. A dynamic survey of graph labeling. Electronic Journal of combi-
natorics, 6(25):4–623, 2022.

[13] A Jeeva, Maria Singaraj Rosary, Nasir Ali, and Fikadu Tesgera Tolasa. Advancements
in signal processing and control systems using z and l-transforms. arXiv preprint
arXiv:2407.11063, 2024.

[14] A Jeeva, R Selvakumar, and M Nalliah. The stable b-chromatic graphs. International
Journal of Pure and Applied Mathematics, 106(8):7–12, 2016.

[15] A Jeeva, R Selvakumar, and M Nalliah. The b-chromatic number of some special fam-
ilies of graphs. IOP Conference Series: Materials Science and Engineering, 263(4),
2017.

[16] A Jeeva, R Selvakumar, and M Nalliah. Families of greater achromatic graphs. Indian
Journal of Mathematics, 59(2):255–261, 2017.

[17] A Jeeva, R Selvakumar, and M Nalliah. The b-chromatic number of some standard
graphs. In Applied Mathematics and Scientific Computing: International Conference
on Advances in Mathematical Sciences, 2:573–580, 2019.

[18] GC Lau, K Premalatha, WC Shiu, and M Nalliah. On local antimagic chromatic num-
bers of circulant graphs join with null graphs or cycles. Proyecciones (Antofagasta),
42(5):1307–1332, 2023.

[19] Gee-Choon Lau and M Nalliah. On local antimagic chromatic number of a corona
product graph. arXiv preprint arXiv:2203.09906, 2022.

[20] Gee-Choon Lau, Wai Chee Shiu, M Nalliah, and K Premalatha. Construction of
local antimagic 3-colorable graphs of fixed even size–matrix approach. arXiv preprint
arXiv:2404.18049, 2024.

[21] Gee-Choon Lau, Wai Chee Shiu, K Premalatha, and M Nalliah. Constructions of
local antimagic 3-colorable graphs of fixed odd size— matrix approach. arXiv preprint
arXiv:2403.16484, 2024.

[22] Gee-Choon Lau, Wai-Chee Shiu, K Premalatha, Ruixue Zhang, and M Nalliah. A
note on local antimagic chromatic number of lexicographic product graphs. arXiv
preprint arXiv:2208.14707, 2022.

[23] Gee-Choon Lau, Wai-Chee Shiu, Ruixue Zhang, K Premalatha, and M Nalliah. Com-
plete characterization of s-bridge graphs with local antimagic chromatic number 2.
arXiv preprint arXiv:2210.04394, 2022.



REFERENCES 2841

[24] M Nalliah. Antimagic Labeling of Graphs and Digraphs. PhD thesis, Kalasalingam
University, TamilNadu, India, 2013.

[25] M Nalliah. Further results on antimagic digraphs. IOSR Journal of Mathematics
(IOSR-JM), 11(5):27–32, 2015.

[26] M Nalliah. Antimagic labeling of digraphs. Journal of the Indonesian Mathematical
Society, 22(1):61–69, 2016.

[27] M Nalliah and S Arumugam. Super (a, d)-edge-antimagic total labelings of gener-
alized friendship graphs. Journal of Combinatorial Mathematics and Combinatorial
Computing, 84:81–90, 2013.

[28] M Nalliah and S Arumugam. Super (a, 3)-edge antimagic total labeling for union of
two stars. Theoretical Computer Science and Discrete Mathematics, 10398:203–211,
2017.

[29] M Nalliah, R Shankar, and Tao-Ming Wang. Local antimagic vertex coloring for
generalized friendship graphs. Journal of Discrete Mathematical Sciences and Cryp-
tography, pages 1–16, 2022.

[30] K Premalatha, Gee-Choon Lau, S Arumugam, and WC Shiu. On local antimagic
chromatic number of various join graphs. Communications in Combinatorics and
Optimization, 8(4):693–714, 2023.

[31] S Rajkumar. A Study on Antimagic Total Labeling and Local Edge Antimagic Coloring
of Graphs. PhD thesis, Vellore Institute of Technology, Vellore, TamilNadu, India,
2024.

[32] S Rajkumar and M Nalliah. On local edge antimagic chromatic number of graphs.
Proyecciones (Antofagasta), 41(6):1397–1412, 2022.

[33] S Rajkumar, M Nalliah, and V Madhu. Note on super (a,1)–p3–antimagic total
labeling of star sn. Ural Mathematical Journal, 7(2 (13)):86–93, 2021.

[34] S Rajkumar, M Nalliah, and G Uma Maheswari. Super (a, d)-g + e-antimagic total
labeling of gu[sn]. Proyecciones (Antofagasta), 41(1):249–262, 2022.

[35] V Sandhiya and M Nalliah. Local total anti-magic chromatic number of graphs.
Heliyon, 9(7), 2023.

[36] R Shankar and M Nalliah. Local antimagic chromatic number for the corona product
of wheel and null graphs. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika.
Komp’yuternye Nauki, 32(3):463–485, 2022.

[37] R Shankar and M Nalliah. Local vertex antimagic chromatic number of some wheel
related graphs. Proyecciones (Antofagasta), 41(1):319–334, 2022.



REFERENCES 2842

[38] V Vizing. Some unsolved problems in graph theory. Russian Mathematical Surveys,
23(6):125–141, 1968.

[39] H P Yap. Total coloring of Graphs. Springer, 2006.