EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6755
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On (H1, H2)-Magic Generalized Total Composition

Tita Khalis Maryati1,∗, Fawwaz Fakhrurrozi Hadiputra2, Martin Bača3,
Andrea Semaničová-Feňovč́ıková3,4

1 Department of Mathematics Education, UIN Syarif Hidayatullah Jakarta, Indonesia
2 School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010,
Australia
3 Department of Applied Mathematics and Informatics, Technical University, Košice,
Slovakia
4 Division of Mathematics, Saveetha School of Engineering, SIMATS, Chennai, India

Abstract. Let H1 and H2 be two non-isomorphic graphs. A graph G is said to admit an (H1, H2)-
covering if every edge of G is contained in either a subgraph of G isomorphic to H1 or to H2. We
say that a graph G admitting an (H1, H2)-covering is (H1, H2)-magic if there exists a total labeling
f : V (G) ∪E(G) → [1, |V (G)|+ |E(G)|] such that there exist magic constants c1 and c2 such that
the weight of every subgraph H∗

i of G isomorphic to Hi equals to ci, i = 1, 2. The weight of a
subgraph H is defined as

w(H) =
∑

v∈V (H)

f(v) +
∑

e∈E(H)

f(e).

Moreover, a graph G is called (H1, H2)-supermagic if the vertices are labeled with the numbers
from 1 up to |V (G)|. In this paper, we present some constructions of (H1, H2)-magic graphs.

2020 Mathematics Subject Classifications: 05C78

Key Words and Phrases: Magic labeling, magic covering, generalized total composition, amal-
gamation of graphs

1. Introduction

Let G = (V,E) be a finite, simple, and undirected graph. For two integers a < b, let
[a, b] = {k ∈ Z | a ≤ k ≤ b}.

In 2005, Gutiérrez and Lladó [1] introduced a concept of H-(super)magic graphs. A
graph G is said to admit an H-covering if every edge e ∈ E(G) is contained in some

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6755

Email addresses: tita.khalis@uinjkt.ac.id (T. K. Maryati),
fhadiputra@student.unimelb.edu.au (F. F. Hadiputra), martin.baca@tuke.sk (M. Bača),
andrea.fenovcikova@tuke.sk (A. Semaničová-Feňovč́ıková)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 2 of 13

subgraph of G isomorphic to H. Suppose that G = (V,E) admits an H-covering. A
bijective function f : V (G) ∪ E(G) → [1, |V (G)| + |E(G)|] is called an H-magic labeling
if there exists a positive integer c ∈ N, called a magic constant, such that the weight
w(H∗) =

∑
v∈V (H∗) f(v) +

∑
e∈E(H∗) f(e) = c for every subgraph H∗ of G isomorphic

to H. In addition, if {f(v) | v ∈ V (G)} = [1, |V (G)|] then f is called an H-supermagic
labeling. A graph which admits an H-(super)magic labeling is called an H-(super)magic
graph. To date, there are some results of H-supermagicness in planar graphs [2], grid
graphs [3], polygonal snake graphs [4], edge coronation of graphs [5], and disjoint union of
prisms [6].

It is known that for any graph H other than a K2, we can always find several graphs
which are not H-magic. One possible way to do this is by considering graphs which did
not admit H-covering. Therefore, if we want to have similar magicness property for graphs
which did not admit H-magic, we can consider a relaxation of H-magicness.

Recently, Ashari and Salman [7] introduced a notion of a generalization ofH-(super)ma-
gic labeling, namely an (H1, H2)-(super)magic labeling. Let H1 and H2 be two non-
isomorphic graphs. A graphG is said to admit an (H1, H2)-covering if every edge e ∈ E(G)
is contained in either a subgraph of G isomorphic to H1 or a subgraph of G isomorphic to
H2. Let G admit an (H1, H2)-covering. A bijection f : V (G)∪E(G) → [1, |V (G)|+|E(G)|]
is called an (H1, H2)-magic labeling if there exist two positive integers c1 and c2, called
magic constants, such that for every subgraph H∗

1 of G isomorphic to H1 holds

w(H1) =
∑

v∈V (H1)

f(v) +
∑

e∈E(H1)

f(e) = c1

and for every subgraph H∗
2 of G isomorphic to H2 holds

w(H2) =
∑

v∈V (H2)

f(v) +
∑

e∈E(H2)

f(e) = c2.

Moreover, an (H1, H2)-magic labeling f is called (H1, H2)-supermagic if the vertices are
labeled with the smallest possible numbers, i.e., {f(v) | v ∈ V (G)} = [1, |V (G)|]. A graph
G called is called (H1, H2)-(super)magic if G admits an (H1, H2)-(super)magic labeling.
Some other variations of H-(super)magic valuations of graphs can be seen in [8–11]. For
more insights about graph labeling, please see [12]. Furthermore, there are several recent
applications of graph theory which can be seen in [13, 14].

In this paper, we present several new constructions of (H1, H2)-magic graphs and also
results for Ph-(super)magicness of copies of paths.

2. The (k, θ)-balanced multisets

Maryati et al. [15] presented a characterization of mG being G-supermagic.

Theorem 1. [15] Let m be a positive integer and let G be a graph such that all its
components have at least 2 vertices. Then mG is G-magic if and only if |V (G)|+ |E(G)|
is even or m is odd.



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 3 of 13

The proof of this theorem is based on a technique called (k, θ)-balanced multisets, see
[15, 16]. In this paper, we will also use this method; therefore, we begin by introducing
several definitions and basic properties. Multiset is a generalization of a set, where multiple
instances of each element in a set is allowed. The notion ⊎ combines multisets with
counting repeated occurrences of elements from each multisets, i.e., {a}⊎{a, b} = {a, a, b}.

Let k = hθ for some positive integers h and θ. Let Y be a multiset containing positive
integers.

The multiset Y is said to be (k, θ)-balanced if there exist k submultisets of Y , namely
Yi for i ∈ [1, k], and there exist θ distinct integers aj for j ∈ [1, θ], such that

(i) ⊎k
i=1Yi = Y ,

(ii) |Yi| = |Y |
k for every i ∈ [1, k],

(iii)
∑

b∈Yth+r

b = at+1 for t ∈ [0, θ − 1] and r ∈ [1, h].

For i ∈ [1, k], Yi is called a balanced submultiset of Y . Particularly, a (k, 1)-balanced
multiset is exactly a k-balanced multiset. This method can also be applied to identify
(H1, H2)-magic graphs. Additionally, some established results concerning (k, θ)-balanced
multisets are presented below.

Lemma 1. [15] Let x, y, z and k be non-negative integers and

Y = [x+ 1, x+ k] ⊎ [y + 1, y + k] ⊎ [z + 1, z + k]

be a multiset. Then

(1) for even k ≥ 2, Y is (k, 2)-balanced, with

∑
b∈Yi

b =

{
x+ y + z + 3k

2 + 2, for i ∈ [1, k2 ],

x+ y + z + 3k
2 + 1, for i ∈ [k2 + 1, k],

(2) for odd k ≥ 3, Y is k-balanced, with
∑
b∈Yi

b = x+ y + z + 3
2(k + 1).

3. The Ph-supermagic graphs

Gutiérrez and Lladó [1] characterized the path-supermagicness of the path Pn on n
vertices.

Theorem 2. [1] The path Pn is Ph-supermagic for any integer h ∈ [2, n].

Moreover, Maryati et al. [17] showed that the odd copies of paths with at least 3
vertices is also Ph-supermagic with certain restrictions on h. By mG we denote the union
of disjoint m copies of a graph G.



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 4 of 13

Theorem 3. [17] Let m be odd and n ≥ 3. Then mPn is kPh-supermagic for k ∈ [1,m]
and h ∈

[
⌈n2 ⌉+ 1, n

]
.

They also proposed that mPn is kPh-supermagic also for h ∈
[
2, ⌈n2 ⌉

]
.

In the following theorem, we present a complete characterization when odd copies of
paths is path-supermagic.

Theorem 4. Let n ≥ 5 be a positive integer, m ≥ 3 be a positive odd integer and h ∈ [3, n].
Then, the disjoint union of paths mPn is kPh-supermagic for k ∈ [1,m⌊nh⌋ − 1].

Proof. Let mPn be a graph with the vertex set and the edge set

V (mPn) = {vi,j | i ∈ [1,m], j ∈ [1, n]},
E(mPn) = {vi,jvi,j+1 | i ∈ [1,m], j ∈ [1, n− 1]}.

Let P
(i,l)
h , i ∈ [1,m], l ∈ [1, n− h+ 1] be the subgraph of mPn such that

V (P
(i,l)
h ) = {vi,j | j ∈ [l, l + h− 1]},

E(P
(i,l)
h ) = {vi,jvi,j+1 | j ∈ [l, l + h− 2]}.

According to Theorem 2, there exists a Ph-supermagic labeling g of Pn which induces
the magic constant c. For m odd, define a total labeling f of mPn in the following way

f(vi,j) =


m · (g(vi,j)− 1) + i+ m+1

2 , for j ≡ 1 (mod h), i ∈ [1, m−1
2 ],

m · (g(vi,j)− 1) + i− m−1
2 , for j ≡ 1 (mod h), i ∈ [m+1

2 ,m],

m · (g(vi,j)− 1) + i, for j ̸≡ 1 (mod h), i ∈ [1,m],

f(vi,jvi,j+1) =


m · g(vi,jvi,j+1)− 2i+ 1, for j ≡ 0 (mod h− 1), i ∈ [1, m−1

2 ],

m · (g(vi,jvi,j+1) + 1)− 2i+ 1, for j ≡ 0 (mod h− 1), i ∈ [m+1
2 ,m],

m · g(vi,jvi,j+1)− i+ 1, for j ̸≡ 0 (mod h− 1), i ∈ [1,m].

It is easy to see that f is a bijection and the vertices are labeled with numbers 1, 2, . . . ,mn.

Moreover, for the weight of a subgraph P
(i,l)
h , i ∈ [1,m], l ∈ [1, n−h+1] under the labeling

f we have

w(P
(i,l)
h ) = mc− h(m− 1) + m−1

2 .

Therefore, the weight of every subgraph of mPn which is isomorphic to Ph is constant.
This immediately implies that mPn is kPh-supermagic for k ∈ [1,m⌊nh⌋ − 1]. □

Remark 1. It is also possible to show that mPn is kPh-supermagic for k = m⌊nh⌋ whenever
h does not divide n using the same labeling as in Theorem 2.

Figure 1 illustrates the P3-supermagic labeling of 5P7 obtained from the P3-supermagic
labeling of P7 by the construction described in the proof of Theorem 2.



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 5 of 13

Figure 1: A P3-supermagic labeling of P7 and the corresponding P3-supermagic labeling of 5P7.

4. The (H1, H2)-magic graphs

In this section, we study several graph operations and related results on the magicness
of the obtained graphs. The first operation is a generalized total composition graph. Let
F and G be two graphs. Let Hi, i ∈ [1, n], be a graph which contains 2F as a subgraph.
Let H = {H1, H2, . . . ,Hn}. A generalized total composition G[F ;H] is a graph obtained
from the graph G by replacing each vertex v ∈ V (G) with the graph F , and every edge
e ∈ E(G) with any graph from H. Note that there are many non-isomorphic graphs
G[F ;H]. For convenience, if H = {H1, H2}, then G[F ;H] = G[F ;H1, H2]. For any graph
Γ, let tΓ = |V (Γ)|+ |E(Γ)|.

In the next theorem we give a sufficient condition when a generalized total composition
G[F ;H1, H2] is (H1, H2)-magic.

Theorem 5. Let G and F be connected nontrivial graphs. Let H1 and H2 be non-
isomorphic connected graphs which have size at least 2|E(F )| + 2 and contain 2F as a
subgraph. Let si be the number of subgraphs of G[F ;H1, H2] isomorphic to Hi, i = 1, 2.
Let s1 + s2 = |E(G)|. Let |V (F )| + |E(F )| be even or |V (G)| be odd. If both tH1(s1 − 1)
and tH2(s2 − 1) are even then the graph G[F ;H1, H2] is (H1, H2)-magic.

Proof. Let G be a connected graph of order n. The idea of the proof is to label the
vertices and edges of G[F ;H1, H2] such that its every subgraph isomorphic to F (which
is obtained by replacing a vertex of G) has constant sum of labels, and then ensure that
every pair of vertices or edges in the same ’component’ have the constant sums.

Suppose that either tF is even or n is odd. Then nF is F -magic due to Theorem 1.
Let g : V (nF ) ∪ E(nF ) → [1, ntF ] be a F -magic labeling of nF with the magic constant
c, i.e., the weights of subgraphs F corresponding to the vertices of G are wg(F ) = c.

Let si denote the number of subgraphs of G[F ;H1, H2] isomorphic to Hi, i = 1, 2, and
let s1 + s2 = |E(G)|. Now suppose that tH1(s1 − 1) is even. Consider the following two
cases.

Case 1. When tH1 is even. Let X1 = [ntF +1, s1(tH1 − 2tF )+ntF ]. Create a partition
of X1 into 2-sets, X1

j for j ∈ [1, 12(s1(tH1 − 2tF ))] such that∑
a∈X1

j

a = s1(tH1 − 2tF ) + 2ntF + 1



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 6 of 13

and put s1(tH1 − 2tF ) + 2ntF + 1 = X1.
Case 2. When both s1 and tH1 are odd. Let Y 1 = [ntF + 1, s1(tH1 − 2tF − 3) + ntF ].

Similarly, create a partition of Y 1 into 2-sets, Y 1
k for k ∈ [1, 12(s1(tH1 −2tF −3))] such that∑

a∈Y 1
k

a = s1(tH1 − 2tF − 3) + 2ntF + 1,

and denote s1(tH1 − 2tF − 3) + 2ntF + 1 = Y 1. Next, let

Z1 = [s1(tH1 − 2tF − 3) + ntF + 1, s1(tH1 − 2tF ) + ntF ]

and consider x = s1(tH1 − 2tF − 3) + ntF , y = x + s1, z = x + 2s1. By Lemma 1, Z1 is
s1-balanced. Let Zl be a balanced multisets of Z1 for l ∈ [1, s1]. Hence∑

a∈Z1
l

a = 3(s1(tH1 − 2tF − 2) + ntF + 1
2(s1 + 1)),

let 3(s1(tH1 − 2tF − 2) + ntF + 1
2(s1 + 1)) = Z1.

Furthermore, let

X2 = [s1(tH1 − 2tF ) + ntF + 1, s1(tH1 − 2tF ) + s2(tH2 − 2tF ) + ntF ],

Y 2 = [s1(tH1 − 2tF ) + ntF + 1, s1(tH1 − 2tF ) + s2(tH2 − 2tF − 3) + ntF ],

Z2 = [s1(tH1 − 2tF ) + s2(tH2 − 2tF − 3) + ntF + 1, s1(tH1 − 2tF ) + s2(tH2 − 2tF ) + ntF ].

By a similar approach, when tH2(s2−1) is even, we obtain balanced multisets X2
j , Y

2
k and

Z2
l such that ∑

a∈Xj

a = 2s1(tH1 − 2tF ) + s2(tH2 − 2tF ) + 2ntF + 1 = X2,

∑
a∈Yk

a = 2s1(tH1 − 2tF ) + s2(tH2 − 2tF − 3) + 2ntF + 1 = Y 2,

∑
a∈Zl

a = 2s1(tH1 − 2tF ) + s2(2tH2 − 4tF − 3) + 2ntF + 1 = Z2.

Now, define a total labeling f of G[F ;H1, H2] as follows.

• For v ∈ V (nF ) put f(v) = g(v).

• Assign the elements of X1
j , Y

1
k or Z1

l to label unlabeled vertices and unlabeled edges
of a subgraph isomorphic to H1.

• Likewise, use the elements of X2
j , Y

2
k , Z

2
l to label unlabeled vertices and unlabeled

edges of subgraph isomorphic to H2.



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 7 of 13

It is a routine to check that f is a bijection. To prove that G[F ;H1, H2] is (H1, H2)-
magic, consider a subgraph H∗

1 of G[F ;H1, H2] isomorphic to H1. We get that

wf (H
∗
1 ) =

{
2c+ 1

2X
1(tH1 − 2tF ), if s1 is even,

2c+ 1
2Y

1(tH1 − 2tF − 3) + Z1, if s1 and tH1 are odd.

Moreover, for a subgraph H∗
2 of G[F ;H1, H2] isomorphic to H2, we have

wf (H
∗
2 ) =

{
2c+ 1

2X
2(tH2 − 2tF ), if s2 is even,

2c+ 1
2Y

2(tH2 − 2tF − 3) + Z2, if s2 and tH2 are odd.

Thus it may be concluded that G[F ;H1, H2] is (H1, H2)-magic. □

An illustration of a construction described in the proof of Theorem 5 is given in Figure
2.

Figure 2: The P7[K2;H1, H2] is (H1, H2)-magic, where H1 is a cycle on 6 vertices with a subdivided chord
and H2 is a cycle on 5 vertices with a subdivided chord.

In the next part we present another method of generating (H1, H2)-magic graphs. Let
F1 and F2 be finite graphs containing a graph A as a subgraph. We call A as a connector.
An A-amalgamation of graphs F1 and F2, denoted by Amal(F1, F2;A), is a graph obtained
by taking F1 and F2 and identifying their connectors A. Let H1 be a connected graph
which contains Amal(F1, F2;A) as a proper subgraph and let H2 be a connected graph
which contains F1 ∪ F2 as a proper subgraph. The graph Pn[H1;H2] is constructed as
an alternating sequence of ⌈n/2⌉ copies of the graph H1 and ⌊n/2⌋ copies of the graph
H2. For i = 1, 2, . . . , ⌈n/2⌉ − 1, the ith copy of H1 is connected to the ith copy of H2 by
identifying the connector F2, and the ith copy of H2 is connected to the (i + 1)th copy
of H1 by identifying the connector F1. To aid visualization, a diagram of P3[H1;H2] is
provided in Figure 3.

Let H be a subgraph of G. For the sake of clarity we use the notation tG−H = tG− tH .

Theorem 6. Let F1 and F2 be connected graphs containing a graph A as a proper subgraph
and let F ∗ = Amal(F1, F2;A) contain exactly one subgraph isomorphic to Fi, i = 1, 2.

Let H1 be a connected graph containing F ∗ as a proper subgraph and let H2 be a con-
nected graph containing F1∪F2 as a proper subgraph. Let each of tF1−(F1∩F2)+tF2−(F1∩F2),
tH2−(F1∪F2), and tF1∩F2+tH1−F ∗ be an even number. If there are exactly n+1−i subgraphs
in Pn[H1;H2] isomorphic to Hi, i = 1, 2, then the graph Pn[H1;H2] is (H1, H2)-magic for
n ≥ 2.



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 8 of 13

Figure 3: Visualization of P3[H1;H2].

Proof. The idea of the proof is similar to the proof of Theorem 5. Let
V (Pn) = {v1, v2, . . . , vn} in a natural way. For every Γ ∈ {F1, F2, F1 ∩ F2, H1 − F ∗},
where F ∗ = Amal(F1, F2;A), let Γ(i) be a subgraph isomorphic to Γ which is originated
from vi ∈ V (Pn).

First, let Z = [n(tH1) + 1, n(tH1 + tH2−(F1∪F2))]. Create a partition of Z into 2-sets,

Zk for k ∈ [1, 12n(tH2−(F1∪F2))] such that∑
a∈Zk

a = n(2tH1 + tH2−(F1∪F2)) + 1,

and put n(2tH1 + tH2−(F1∪F2))+1 = Z. Now, we consider several cases based on the parity
of tF1 and tF2 . Since tF1−(F1∩F2) + tF2−(F1∩F2) is even, then both of them have the same
parity.

Case 1.1. When tF1−(F1∩F2) and tF2−(F1∩F2) are even. Let X1 = [1, n(tF1−(F1∩F2))].

Create a partition X1 into 2-sets, X1
i for i ∈ [1, 12n(tF1−(F1∩F2))] such that∑

a∈X1
i

a = n(tF1−(F1∩F2)) + 1,

and denote n(tF1−(F1∩F2)) + 1 = X1. Likewise, let

Y 1 = [n(tF1−(F1∩F2)) + 1, n(tF1−(F1∩F2) + tF2−(F1∩F2))]

and create a partition of Y 1 into 2-sets, Y 1
i for i ∈ [1, 12n(tF2−(F1∩F2))] such that∑

a∈Y 1
i

a = n(2(tF1−(F1∩F2)) + tF2−(F1∩F2)) + 1,

and put n(2(tF1−(F1∩F2)) + tF2−(F1∩F2)) + 1 = Y 1.
Case 1.2. When tF1−(F1∩F2) and tF2−(F1∩F2) are odd. Let X2 = [1, n(tF1−(F1∩F2) − 1)].

Create a partition X2 into 2-sets, X2
i for i ∈ [1, 12n(tF1−(F1∩F2) − 1)] such that∑

a∈X2
i

a = n(tF1−(F1∩F2) − 1) + 1,

and let n(tF1−(F1∩F2) − 1) + 1 = X2. Again, let

Y 2 = [n(tF1−(F1∩F2)) + 1, n(tF1−(F1∩F2) + tF2−(F1∩F2) − 1)]



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 9 of 13

and create a partition of Y 2 into 2-sets, Y 2
i for i ∈ [1, 12n(tF2−(F1∩F2) − 1)] such that∑

a∈Y 2
i

a = n(2(tF1−(F1∩F2)) + tF2−(F1∩F2) − 1) + 1,

and denote n(2(tF1−(F1∩F2)) + tF2−(F1∩F2) − 1) + 1 = Y 2.
Next, we consider the parity of tF1∩F2 and tH1−F ∗ in a similar manner.
Case 2.1. When tF1∩F2 and tH1−F ∗ are even. Let

U1 = [n(tF1−(F1∩F2) + tF2−(F1∩F2)) + 1, n(tF ∗)].

Create a partition of U1 into 2-sets, U1
j for j ∈ [1, 12n(tF1∩F2)] such that∑

a∈U1
j

a = n(tF1−(F1∩F2) + tF2−(F1∩F2) + tF ∗) + 1 = U1.

Similarly, let W 1 = [n(tF ∗) + 1, n(tH1)] and create a partition of W 1 into 2-sets, W 1
j for

j ∈ [1, 12n(tH1−F ∗)] such that∑
a∈W 1

j

a = n(tH1 + tF ∗) + 1 = W 1.

Case 2.2. When tF1∩F2 and tH1−F ∗ are odd. Let

U2 = [n(tF1−(F1∩F2) + tF2−(F1∩F2)) + 1, n(tF ∗ − 1)].

Create a partition of U2 into 2-sets, U2
j for j ∈ [1, 12n(tF1∩F2 − 1)] such that∑

a∈U2
j

a = n(tF1−(F1∩F2) + tF2−(F1∩F2) + tF ∗ − 1) + 1 = U2.

Similarly, let W 2 = [n(tF ∗) + 1, n(tH1 − 1)] and create a partition of W 2 into 2-sets, W 2
j

for j ∈ [1, 12n(tH1−F ∗ − 1)] such that∑
a∈W 2

j

a = n(tH1 + tF ∗ − 1) + 1 = W 2.

Now construct a total labeling f of Pn[H1;H2] as follows.

• According to the parity of tF1∩F2 use either the elements of U1
j or the elements of

U2
j to label vertices and edges of (F1 ∩ F2)

(j). Moreover, if tF1∩F2 is odd, label the

unlabeled vertex or unlabeled edge in (F1 ∩ F2)
(j) with n(tF ∗ − 1) + j.

• According to the parity of tH1−F ∗ use either the elements of W 1
j or W 2

j to label

vertices and edges of (H1 − F ∗)(j). Moreover, if tH1−F ∗ is odd, label the unlabeled
vertex or unlabeled edge in (H1 − F ∗)(j) with n(tH1)− j + 1.



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 10 of 13

• According to the parity of tF1−(F1∩F2) assign the elements of X1
i or X2

i to unlabeled

vertices and unlabeled edges of F
(i)
1 . In addition, if tF1−(F1∩F2) is odd, label the

unlabeled vertex or unlabeled edge in F
(i)
1 with n(tF1−(F1∩F2) − 1) + i.

• According to the parity of tF2−(F1∩F2) assign the elements of Y 1
i or Y 2

i to unla-

beled vertices and unlabeled edges of F
(i)
2 . Similarly, if tF2−(F1∩F2) is odd, label the

unlabeled vertex or unlabeled edge in F
(i)
2 with n(tF1−(F1∩F2) + tF2−(F1∩F2))− i+ 1.

• Lastly, use the elements of Z to label unlabeled vertices and unlabeled edges of a
subgraph isomorphic to H2.

It can be shown that f is a bijection. To show that Pn[H1;H2] is (H1, H2)-magic,

consider a subgraph H
(i)
1 of Pn[H1;H2] isomorphic to H1. If tF1−(F1∩F2) and tF1∩F2 are

even we get that

w(H
(i)
1 ) = 1

2X
1tF1−(F1∩F2) +

1
2Y

1tF2−(F1∩F2) +
1
2U

1tF1∩F2 +
1
2W

1tH1−F ∗ .

If tF1−(F1∩F2) is odd but tF1∩F2 is even, then

w(H
(i)
1 ) = 1

2X
2(tF1−(F1∩F2) − 1) + 1

2Y
2(tF2−(F1∩F2) − 1) + 1

2U
1tF1∩F2 +

1
2W

1tH1−F ∗

+ n(2tF1−(F1∩F2) + tF2−(F1∩F2) − 1) + 1.

If tF1−(F1∩F2) is even but tF1∩F2 is odd, we have

w(H
(i)
1 ) = 1

2X
1tF1−(F1∩F2) +

1
2Y

1tF2−(F1∩F2) +
1
2U

2(tF1∩F2 − 1) + 1
2W

2(tH1−F ∗ − 1)

+ n(2tF ∗ + tH1−F ∗ − 1) + 1.

If tF1−(F1∩F2) and tF1∩F2 are odd, it follows that

w(H
(i)
1 ) = 1

2X
2(tF1−(F1∩F2) − 1) + 1

2Y
2(tF2−(F1∩F2) − 1) + 1

2U
2(tF1∩F2 − 1)

+ 1
2W

2(tH1−F ∗ − 1) + n(2tF1−(F1∩F2) + tF2−(F1∩F2) − 1)

+ n(2tF ∗ + tH1−F ∗ − 1) + 2.

Furthermore, consider H
(i)
2

∼= H2. If tF1−(F1∩F2) and tF1∩F2 are even, then

w(H
(i)
2 ) = 1

2ZtH2−(F1∪F2) +
1
2X

1tF1−(F1∩F2) +
1
2Y

1tF2−(F1∩F2) + U1tF1∩F2 .

If tF1−(F1∩F2) is odd but tF1∩F2 is even, we have

w(H
(i)
2 ) = 1

2ZtH2−(F1∪F2) +
1
2X

2(tF1−(F1∩F2) − 1) + 1
2Y

2(tF2−(F1∩F2) − 1) + U1tF1∩F2

+ n(2tF1−(F1∩F2) + tF2−(F1∩F2) − 1) + 1.



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 11 of 13

If tF1−(F1∩F2) is even but tF1∩F2 is odd, it follows that

w(H
(i)
2 ) = 1

2ZtH2−(F1∪F2) +
1
2X

1tF1−(F1∩F2) +
1
2Y

1tF2−(F1∩F2) + U2(tF1∩F2 − 1)

+ n(2tF ∗ + tH1−F ∗ − 1) + 1.

If tF1−(F1∩F2) and tF1∩F2 are odd, it holds that

w(H
(i)
2 ) = 1

2ZtH2−(F1∪F2) +
1
2X

2(tF1−(F1∩F2) − 1) + 1
2Y

2(tF2−(F1∩F2) − 1)

+ U2(tF1∩F2 − 1) + n(2tF1−(F1∩F2) + tF2−(F1∩F2) − 1)

+ n(2tF ∗ + tH1−F ∗ − 1) + 2.

Therefore, Pn[H1;H2] is (H1, H2)-magic. □

For instance, we present an example of P3[H1;H2] which is (H1, H2)-magic, see Figure
4.

Figure 4: The graphs (a) F1, (b) F2, (c) H1, (d) H2, (e) (H1, H2)-magic labeling of P3[H1;H2]. In addition,
vertices and edges of F1 ∩ F2 are outlined in red.

5. Conclusion

In this paper, we have presented a path-magic family of disjoint union of paths and
two general constructions of (H1, H2)-magic graphs. Our findings partially address the
gaps in the existing knowledge on this emerging topic.

A potential application of (H1, H2)-magic graphs is in the foundational design of struc-
tured networks, where they help enforce a hidden uniformity. By ensuring that specific,
critical patterns within the larger system all share an identical cumulative property, these



T. K. Maryati et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6755 12 of 13

graphs enable a built-in balance and symmetry. This inherent harmony simplifies system-
wide management, promotes fault tolerance by making key components interchangeable,
and provides a robust mathematical framework.

The next research direction in this topic is to systematically investigate the (H1, H2)-
magic properties of graphs formed by a graph operation, such as the Cartesian product
or strong product. While some specific cases are not that hard to be determined, a
general theory is lacking. Establishing necessary and sufficient conditions when a graph
is (H1, H2)-magic would represent a significant advancement.

Acknowledgements

This work was supported by LP2M UIN Syarif Hidayatullah Jakarta Research Fellow-
ship Program 2023 and the Slovak Research and Development Agency under the contract
No. APVV-23-0191 and by VEGA 1/0243/23.

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