EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5887
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Edge k-Product Cordial Labeling of Graphs

N. M. NourEldeen1,6, J. Jenisha2, K. Jeya Daisy3, P. Jeyanthi4,∗,
M. E. Abdel-Aal5

1 Department of Mathematics, College of Science, Taibah University, Madinah,
Kingdom of Saudi Arabia
2 Research Scholar (Reg.No.: 23213042092003), Holy Cross College (Autonomous),
Nagercoil - 629004, Tamilnadu, India, affiliated to Manonmaniam Sundaranar University,
Tirunelveli - 627012, Tamilnadu, India
3 PG and Research Department of Mathematics, Holy Cross College (Autonomous),
Nagercoil - 629004, Tamilnadu, India
4 Research Centre, Department of Mathematics, Govindammal Aditanar College for Women,
Tiruchendur - 628215, Tamilnadu, India.
5 Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt
6 Department of Mathematics, Women’s College of Arts, Sciences and Education,
Ain Shams University, Egypt

Abstract. In this paper, we introduce a new labeling namely ‘edge k-product cordial labeling’ as
follows: For a graph G = (V (G), E(G)) having no isolated vertex, an edge labeling f : E(G) →
{0, 1, ..., k − 1}, where k > 1 is an integer, is said to be an edge k-product cordial labeling if it
induces a vertex labeling f⋆ : V (G) → {0, 1, ..., k − 1} defined by f⋆(v) =

∏
uv∈E(G) f(uv)(mod k)

satisfies |ef (i)− ef (j)| ≤ 1 and |vf⋆(i)− vf⋆(j)| ≤ 1 for i, j ∈ {0, 1, ..., k − 1}, where ef (i) and
vf⋆(i) denote the number of edges and vertices respectively having a label i (i = 0, 1, ..., k − 1).
Further, we study the edge k-product cordial behavior of star, bistar, shadow and splitting graph
of star, path union of star, bistar and cycle graphs.

2020 Mathematics Subject Classifications: 05C78

Key Words and Phrases: Product cordial labeling, k-product cordial labeling, edge k-product
cordial labeling, shadow graph, splitting graph, path union of graph

1. Introduction

In mathematics, the field of graph theory revolves around the examination of graphs,
which are the fundamental objects within discrete mathematics. Over the past six decades,

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5887

Email addresses: neldeen@taibahu.edu.sa (N. M. NourEldeen), jenishaelston@gmail.com (J. Jenisha),
jeyadaisy@holycrossngl.edu.in (K. Jeya Daisy), jeyajeyanthi@rediffmail.com (P. Jeyanthi),
mohamed.abdelghani@fsc.bu.edu.eg (M. E. Abdel-Aal)

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



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 2 of 21

one particular aspect of graph theory called graph labeling, has gained significant popu-
larity due to its diverse applications. Labeling involves assigning real numbers, typically
positive integers, to the elements of a graph. In 1967, Rosa [10] published an influential
paper that laid the groundwork for various graph labeling problems. Since then, numer-
ous authors have delved into researching various graph labeling techniques and a detailed
survey is available in [4]. Among these labeling techniques, ‘cordial labeling’ by Cahit [3]
stands out as a less stringent version compared to graceful and harmonious labeling. Of
these, graceful labeling is more popular since it has various practical applications. Cordial
labeling has potential applications in areas such as network design, error-correcting codes,
and cryptography. In particular, cordial labeling could be useful in designing efficient
communication protocols where balancing two types of nodes (positive and negative) is
essential. In the subsequent years, several variants of cordial labeling, such as, ‘product
cordial labeling’, ‘k-product cordial labeling’, ‘edge product cordial labeling’ and more
are introduced. In ‘edge product cordial labeling’ [11], the roles of vertices and edges in
product cordial labeling [6] are swapped. Building on this notion, several results have
been established. See [2, 5, 9, 12-19]. Researchers have also explored the applications of
specific graph labeling techniques, for instance, use of ‘mean cordial labeling’ in digraph
representations of blood circulation in the human body [1] and the 3-total edge product
cordial labeling (another variant of cordial labeling) in carbon nanotube network [7].

Motivated by the concept of ‘edge product cordial labeling’, and the several results
established on this concept, we take a step further and introduce a new labeling namely
‘Edge k-product cordial labeling’ as follows: For a graph G = (V (G), E(G)) having no
isolated vertex, an edge labeling f : E(G) → {0, 1, ..., k − 1}, where k > 1 is an integer, is
said to be an edge k-product cordial labeling if it induces a vertex labeling f⋆ : V (G) →
{0, 1, ..., k − 1} defined by f⋆(v) =

∏
uv∈E(G) f(uv)(mod k) satisfies |ef (i)− ef (j)| ≤ 1 and

|vf⋆(i)− vf⋆(j)| ≤ 1 for i, j ∈ {0, 1, ..., k − 1}, where ef (i) and vf⋆(i) denote the number
of edges and vertices respectively having a label i (i = 0, 1, ..., k−1). A graph that admits
an edge k-product cordial labeling is called edge k-product cordial graph. In this study,
we explore the edge k-product cordial behavior of some standard graphs. We present our
study as follows: Followed by the introduction, the edge k-product cordial behavior of star,
bistar, complete graph and complete bipartite graph are investigated in the second section.
In the third section, we focus on the edge k-product cordial behavior of the shadow and
splitting graph of star. In the fourth section, we investigate the edge k-product cordial
behavior of the path union of graphs. The definitions of the following graph structures
are also useful for the present study.
Definition 1 [4]. Let G be a graph and G′ be a copy of G. Let v′ be the vertex in G′

corresponding to the vertex v of G. The shadow graph of a graph G, denoted as D2(G) is
a graph obtained by the following operation: Join each vertex v in G to the neighbors of
the vertex v′ in G′ which corresponds to v.
Definition 2 [4]. The splitting graph of a graph G, denoted as S′(G) is the graph ob-
tained from G by taking a new vertex u′ for each u ∈ V (G) and joining u′ to all vertices
of G adjacent to u.
Definition 3 [8]. Let G1, G2, ...., Gn, n ≥ 2, be n copies of a graph G. Let vi ∈



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 3 of 21

V (Gi), i = 1, 2, ..., n be the vertex corresponding to the vertex v ∈ V (G) in the ith copy
of Gi. We denoted by P (n.Gv) the graph obtained by adding the edge vivi+1 to Gi and
Gi+1 , 1 ≤ i ≤ n− 1, and we call P (n.Gv) the path union of n copies of the graph G.

2. Edge k-Product Cordial Labeling of Star, Bistar, Complete Graph
and Complete Bipartite Graph

In this section, first we establish that the star graph K1,n and the bistar graph Bn,n

admit an edge k-product cordial labeling for n ≥ k. In the next two theorems, we give the
necessary condition for the complete graph Kn and the complete bipartite graph Km,n to
admit an edge k-product cordial labeling.

Theorem 1. For n ≥ k, the star K1,n admits an edge k-product cordial labeling.

Proof. Let the vertex set and edge set of K1,n be V (K1,n) = {u, ui; 1 ≤ i ≤ n} and
E(K1,n) = {uui; 1 ≤ i ≤ n} respectively.
Let n ≡ r (mod k) ; 0 ≤ r ≤ k − 1.
Define f : E(K1,n) → {0, 1, 2, ..., k − 1} for n ≥ k as follows:
f(uui) = 0 ; 1 ≤ i ≤ ⌊nk ⌋,

f(uu⌊n
k
⌋+i) =

{
q ; i ≡ q (mod (k − 1)), 1 ≤ q ≤ k − 2
k − 1 ; i ≡ 0 (mod (k − 1))

; 1 ≤ i ≤ n− ⌊nk ⌋.

From this labeling we get,

ef (i) =

{
⌊nk ⌋ ; i = 0 ; r < i ≤ k − 1
⌊nk ⌋+ 1 ; 1 ≤ i ≤ r,

vf∗(i) =

{
⌊nk ⌋ ; r < i ≤ k − 1
⌊nk ⌋+ 1 ; 0 ≤ i ≤ r.

Clearly, |ef (i) − ef (j)| ≤ 1 and |vf∗(i) − vf∗(j)| ≤ 1 for i, j ∈ {0, 1, 2, ..., k − 1}. Hence,
K1,n is an edge k-product cordial graph for n ≥ k.

Example 1. An edge 4-product cordial labeling of K1,9 is given in Figure 1.

0 0
1

1

1

2

2 3

3

0

0
0

1

1

1
2

2

33

Figure 1: Edge 4-product cordial labeling of K1,9



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 4 of 21

Theorem 2. For n ≥ k, the bistar Bn,n admits an edge k-product cordial labeling.

Proof. Let the vertex set and edge set of Bn,n be V (Bn,n) = {u, v, ui, vi; 1 ≤ i ≤ n}
and E(Bn,n) = {uv, uui, vvi; 1 ≤ i ≤ n} respectively.
Let n ≡ r (mod k) ; 0 ≤ r ≤ k − 1.
Define f : E(Bn,n) → {0, 1, 2, ...., k − 1} for n ≥ k as follows:
f(uv) = 0,
f(uui) = 0 ; 1 ≤ i ≤ ⌊nk ⌋,

f(uu⌊n
k
⌋+i) =

{
q ; i ≡ q (mod (k − 1)), 1 ≤ q ≤ k − 2
k − 1 ; i ≡ 0 (mod (k − 1))

; 1 ≤ i ≤ n− ⌊nk ⌋,

f(vvi) = 0 ; 1 ≤ i ≤ ⌊nk ⌋ − 1,

f(vv⌊n
k
⌋) =


0 ; n ≡ 1, 2 (mod 3) , k = 3 ; n ̸≡ 0, 1 (mod k) , k > 3
1 ; n ≡ 0 (mod k) , k ≥ 3
k − 2 ; k = 2 ; n ≡ 1 (mod k) , k > 3,

f(vv⌊n
k
⌋+i) =

{
k − q ; i ≡ q (mod (k − 1)), 1 ≤ q ≤ k − 2
1 ; i ≡ 0 (mod (k − 1))

; 1 ≤ i ≤ n− ⌊nk ⌋.

From this labeling we obtain,

ef (i) =

{
2n
k ; i ̸= 1
2n
k + 1 ; i = 1

;n ≡ 0 (mod k),

ef (i) =

{
2⌊nk ⌋ ; i = 0 , k > 3 ; 2 ≤ i ≤ k − 3
2⌊nk ⌋+ 1 ; i = 1, k − 1, k − 2 ; i = 0 , k = 3

;n ≡ 1 (mod k) , k ≥ 3,

ef (i) =

{
2⌊nk ⌋+ 1 ; i = 0
2⌊nk ⌋+ 2 ; 1 ≤ i ≤ k − 1

;n ≡ k − 1 (mod k),

ef (i) =


2⌊nk ⌋+ 1 ; i = 0, 1 ; 2 ≤ i ≤ r < k − i

2⌊nk ⌋+ 2 ; i ≤ r , k − i ≤ r
;n ̸≡ 0, 1, k − 1 (mod k) , k ≥ 3,; k − i ≤ r < i

vf∗(i) =

{
ef (i) + 1 ; i = 0
ef (i) ; 1 ≤ i ≤ k − 1.

Clearly, |ef (i)− ef (j)| ≤ 1 and |vf⋆(i)− vf⋆(j)| ≤ 1 for i, j ∈ {0, 1, ..., k − 1}. Therefore,
Bn,n is an edge k-product cordial graph if n ≥ k.

Example 2. An edge 4-product cordial labeling of B9,9 is given in Figure 2.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 5 of 21

0

00

0
0 0

0
0

0

1

1

1

1 1

1

1
1

1

1

2
2

2

2

2

2

2
2

2

2

33

3

3

3

3

3

3
3

3

Figure 2: Edge 4-product cordial labeling of B9,9

Theorem 3. A complete graph Kn does not admit edge k-product cordial labeling if 3 ≤
k ≤ n(n−1)

2 .

Proof. Let 3 ≤ k ≤ n(n−1)
2 , then ⌊n(n−1)

2k ⌋ ≥ 1.

Let f be an edge k-product cordial labeling of Kn, then ef (i) = ⌊n(n−1)
2k ⌋ or ⌊n(n−1)

2k ⌋+ 1.
We have the following two cases.
Case (i): For n ≤ k, we have vf∗(i) = 0 or 1. If ef (0) = ⌊n(n−1)

2k ⌋, then vf∗(0) ≥
⌊n(n−1)

2k ⌋ + 1 > 1. Therefore, |vf∗(0) − vf∗(j)| > 1 for some j = 1, 2, ..., k − 1, which is a
contradiction.
Case (ii): For n > k, we have vf∗(i) = ⌊nk ⌋ or ⌊nk ⌋ + 1. If ef (0) = ⌊n(n−1)

2k ⌋, then

vf∗(0) ≥ ⌊n(n−1)
2k ⌋+1 > ⌊nk ⌋+1. Therefore, |vf∗(0)−vf∗(j)| > 1 for some j = 1, 2, ..., k−1,

which is a contradiction.
Hence, Kn is not an edge k-product cordial graph if 3 ≤ k ≤ n(n−1)

2 .

Theorem 4. A complete bipartite graph Km,n with m ≡ r1 (mod k) and n ≡ r2 (mod k)
does not admit edge k-product cordial labeling if r1 + r2 < k ≤ r1r2.

Proof. Let m ≡ r1 (mod k) and n ≡ r2 (mod k). Then |V (Km,n)| = k(⌊mk ⌋ + ⌊nk ⌋) +
r1 + r2 and |E(Km,n)| = k(k⌊mk ⌋⌊

n
k ⌋+ r2⌊mk ⌋+ r1⌊nk ⌋) + r1r2.

Let f be an edge k-product cordial labeling of Km,n. Then ef (i) = k⌊mk ⌋⌊
n
k ⌋ + r2⌊mk ⌋ +

r1⌊nk ⌋+ ⌊ r1r2k ⌋ or k⌊mk ⌋⌊
n
k ⌋+ r2⌊mk ⌋+ r1⌊nk ⌋+ ⌊ r1r2k ⌋+1 and vf∗(i) = ⌊mk ⌋+ ⌊nk ⌋+ ⌊ r1+r2

k ⌋
or ⌊mk ⌋+ ⌊nk ⌋+ ⌊ r1+r2

k ⌋+ 1.
Since r1 + r2 < k ≤ r1r2, we have ⌊ r1+r2

k ⌋ < ⌊ r1r2k ⌋. Now, ef (0) = k⌊mk ⌋⌊
n
k ⌋ + r2⌊mk ⌋ +

r1⌊nk ⌋ + ⌊ r1r2k ⌋ implies vf∗(0) ≥ k⌊mk ⌋⌊
n
k ⌋ + r2⌊mk ⌋ + r1⌊nk ⌋ + ⌊ r1r2k ⌋ + 1 > ⌊mk ⌋ + ⌊nk ⌋ +

⌊ r1+r2
k ⌋+1, which is a contradiction. Hence, Km,n is not an edge k-product cordial graph

if r1 + r2 < k ≤ r1r2.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 6 of 21

3. Edge k-Product Cordial Behavior of Shadow and Splitting Graph of
Star

In order to prove the edge k-product cordial behavior of shadow and splitting graph
of star, we prove the following general result.

Theorem 5. A graph G with k ≤ |V | ≤ |E| does not admit an edge k-product cordial
labeling if |V | ≡ 0(mod k).

Proof. Let G be a graph with k ≤ |V | ≤ |E| and |V | = tk (t ≥ 1). Then |E| = tk + j,

where 0 ≤ j ≤ ⌊ tk(tk−3)
2 ⌋. Let f be an edge k-product cordial labeling of G. Then, ef (i)

is either t+ ⌊ j
k⌋ or t+ ⌊ j

k⌋+ 1 and vf∗(i) = t (i = 0, 1, ..., k − 1). If ef (0) = t+ ⌊ j
k⌋, then

vf∗(0) ≥ t+ ⌊ j
k⌋+ 1 > t, which is not possible. Hence, ef (0) = t+ ⌊ j

k⌋+ 1, which implies

vf∗(0) ≥ t + ⌊ j
k⌋ + 2 > t. Then we get, |vf⋆(0)− vf⋆(j)| > 1 for some j = 1, 2, ..., k − 1,

that is a contradiction. Therefore, G is not an edge k-product cordial graph.

3.1. Shadow Graph of Star

In this subsection, we establish that the shadow graph of a star graph D2(K1,n) does
not admit the edge k-product cordial labeling for k ≤ n. In addition, we investigate the
edge k-product cordial behavior D2(K1,n) for k = 3, 4, 5.

Theorem 6. The graph D2(K1,n) does not admit an edge k-product cordial labeling for
k ≤ n.

Proof. Let k ≤ n. We consider the following two cases.
Case (i): For n = tk + k − 1, we have |V | = 2tk + 2k and |E| = 4tk + 4(k − 1). Clearly,
|V | ≡ 0 (mod k) and k < |V | < |E|. By Theorem 5, D2(K1,n) is not an edge k-product
cordial graph.

Case (ii): For n = tk + r ; t ≥ 1, 0 ≤ r ≤ k − 2, we have |V | = 2tk + 2(r + 1) and
|E| = 4tk + 4r. If f is an edge k-product cordial labeling of D2(K1,n), then

ef (i) =

{
4t ; r = 0
4t+ ⌊4rk ⌋ or 4t+ ⌊4rk ⌋+ 1 ; 1 ≤ r ≤ k − 2,

vf∗(i) =


2t+ 1 ; r = 0 , k = 2
2t or 2t+ 1 ; r = 0 , k ≥ 3
2t+ ⌊2r+2

k ⌋ or 2t+ ⌊2r+2
k ⌋+ 1 ; 1 ≤ r ≤ k − 2.

For the case where r = 0, ef (0) = 4t implies vf∗(0) ≥ 4t+1 > 2t+1 for t ≥ 1. Therefore,
|vf∗(0)−vf∗(j)| > 1 for some j = 1, 2, ..., k−1, which is a contradiction. In the other cases,
if ef (0) = 4t + ⌊4rk ⌋, then vf∗(0) ≥ 4t + ⌊4rk ⌋ + 1. Since ⌊4rk ⌋ ≥ ⌊2r+2

k ⌋ for 1 ≤ r ≤ k − 2,
we get vf∗(0) > 2t + ⌊2r+2

k ⌋ + 1 for t ≥ 1. Therefore, |vf⋆(0)− vf⋆(j)| > 1 for some
j = 1, 2, ..., k − 1, which is a contradiction. Hence, D2(K1,n) is not an edge k-product
cordial graph for k ≤ n.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 7 of 21

Theorem 7. The graph D2(K1,n) admits an edge 3-product cordial labeling if and only if
n = 1.

Proof. Let the vertex set and edge set of D2(K1,n) be V (D2(K1,n)) = {u, v, ui, vi; 1 ≤
i ≤ n} and E(D2(K1,n)) = {uui, vvi, uvi, vui; 1 ≤ i ≤ n} respectively.
Define the edge labeling f : E(D2(K1,1)) → {0, 1, 2} as follows:
f(uu1) = 0, f(vv1) = f(uv1) = 1, f(vu1) = 2.
From this labeling we get, ef (0) = ef (1) − 1 = ef (2) = 1 and vf∗(0) − 1 = vf∗(1) =
vf∗(2) = 1. Hence, D2(K1,1) is an edge 3-product cordial graph.

For n = 2, |V | = 6 and |E| = 8. By Theorem 5, D2(K1,2) is not an edge 3-product
cordial graph. Also, by Theorem 6, D2(K1,n); n ≥ 3 is not an edge 3-product cordial
graph.

Theorem 8. The graph D2(K1,n) does not admit an edge 4-product cordial labeling.

Proof. Let the vertex set and edge set of D2(K1,n) be V (D2(K1,n)) = {u, v, ui, vi; 1 ≤
i ≤ n} and E(D2(K1,n)) = {uui, vvi, uvi, vui; 1 ≤ i ≤ n} respectively.
For n = 1, |V | = 4 and |E| = 4. For n = 3, |V | = 8 and |E| = 12. By Theorem 5,
D2(K1,1) and D2(K1,3) are not edge 4-product cordial graphs.
Let f be an edge 4-product cordial labeling of D2(K1,2). Then, ef (i) = 2 and vf∗(i) = 1
or 2. If ef (0) = 2, then vf∗(0) ≥ 3. Therefore, |vf∗(0) − vf∗(j)| > 1 for some j = 1, 2, 3,
which is a contradiction. Hence, D2(K1,2) is not an edge 4-product cordial graph. Clearly,
by Theorem 6, D2(K1,n) is not an edge 4-product cordial graph if n ≥ 4.

Theorem 9. The graph D2(K1,n) admits an edge 5-product cordial labeling if and only if
n = 2.

Proof. Let the vertex set and edge set of D2(K1,n) be V (D2(K1,n)) = {u, v, ui, vi; 1 ≤
i ≤ n} and E(D2(K1,n)) = {uui, vvi, uvi, vui; 1 ≤ i ≤ n} respectively.
Define the edge labeling f : E(D2(K1,2)) → {0, 1, 2, 3, 4} as f(uu1) = 4, f(uu2) =
1, f(uv1) = 3, f(uv2) = 0, f(vu1) = 2, f(vu2) = 1, f(vv1) = 4, f(vv2) = 3.
From this labeling we get, ef (0) + 1 = ef (1) = ef (2) + 1 = ef (3) = ef (4) = 2 and
vf∗(0)− 1 = vf∗(1) = vf∗(2) = v∗(3) = vf∗(4) = 1.
Hence, D2(K1,2) is an edge 5-product cordial graph.

For n = 1, |V | = |E| = 4. If f is an edge 5-product cordial labeling of D2(K1,1), then ef (i)
and vf∗(i) are either 0 or 1 for i = 0, 1, 2, 3, 4. Clearly, ef (0) = 0 otherwise vf∗(0) = 2.
So, ef (i) = vf∗(i) = 1 (i = 1, 2, 3, 4). In order to get the vertex label 1, there must be
two adjacent edges, say uu1 and uv1 with labels 2 and 3 respectively. To get the vertex
label 2, we must have f(vu1) = 1 and f(vv1) = 4, which results in vf∗(2) = 2, which is a
contradiction. Hence, D2(K1,1) is not an edge 5-product cordial graph.
For n = 3, |V | = 8 and |E| = 12. Let f be an edge 3-product cordial labeling of D2(K1,3).
Then ef (i) = 2 or 3 (i = 0, 1, 2, 3, 4) and vf∗(i) = 1 or 2 (i = 0, 1, 2, 3, 4). Now, ef (0) = 2



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 8 of 21

implies vf∗(0) ≥ 3 > 2. Therefore |vf⋆(0)− vf⋆(j)| > 1 for some j = 1, 2, 3, 4, which is a
contradiction. Hence, D2(K1,3) is not an edge 5-product cordial graph.
For n = 4, |V | = 10 and |E| = 16. By Theorem 5, D2(K1,4) is not an edge 5-product
cordial graph. Also, by Theorem 6, D2(K1,n); n ≥ 5 is not an edge 5-product cordial
graph.

3.2. Splitting Graph of Star

In this subsection, we show that the splitting graph of a star graph S′(K1,n) does not
admit the edge k-product cordial labeling for k ≤ n. Also, we study the edge k-product
cordial behavior of S′(K1,n) for k = 3, 4, 5.

Theorem 10. The graph S′(K1,n) dos not admit an edge k-product cordial labeling for
k ≤ n.

Proof. Let k ≤ n. We consider the following two cases.
Case (i): For n = tk + k − 1, we have |V | = 2tk + 2k and |E| = 3tk + 3(k − 1). Clearly,
|V | ≡ 0 (mod k) and k < |V | < |E|. By Theorem 5, S′(Kn) is not an edge k-product
cordial graph.
Case (ii): For n = tk + r ; t ≥ 1 and 0 ≤ r ≤ k − 2, we have |V | = 2tk + 2(r + 1) and
|E| = 3tk + 3r. If f is an edge k-product cordial labeling of S′(K1,n), then

ef (i) =


3t ; r = 0
3t+ 1 ; r = 1 , k = 3
3t or 3t+ 1 ; r = 1 , k ≥ 4
3t+ ⌊3rk ⌋ or 3t+ ⌊3rk ⌋+ 1 ; 2 ≤ r ≤ k − 2,

; i ∈ {0, 1, ..., k − 1},

vf∗(i) =


2t+ 1 ; r = 0 , k = 2
2t+ 1 or 2t+ 2 ; r = 1 , k = 3
2t+ 1 ; r = 1 , k = 4
2t or 2t+ 1 ; r = 0 , k ≥ 3 ; r = 1 , k ≥ 5
2t+ ⌊2r+2

k ⌋ or 2t+ ⌊2r+2
k ⌋+ 1 ; 2 ≤ r ≤ k − 2.

; i ∈ {0, 1, .., k − 1}.

For the case where 0 ≤ r ≤ 1, ef (0) = 3t implies vf∗(0) ≥ 3t+ 1 > 2t+ 1 for t ≥ 1. Also,
ef (0) = 3t+ 1 implies vf∗(0) ≥ 3t+ 2 > 2t+ 2 for t ≥ 1. Therefore, |vf∗(0)− vf∗(j)| > 1
for some j = 1, 2, ..., k−1, which is a contradiction. In the other cases, if ef (0) = 3t+⌊3rk ⌋,
then vf∗(0) ≥ 3t + ⌊3rk ⌋ + 1. Since ⌊3rk ⌋ ≥ ⌊2r+2

k ⌋ for 2 ≤ r ≤ k − 2, we have vf∗(0) >
2t+ ⌊2r+2

k ⌋+1 for t ≥ 1. Therefore, |vf∗(0)− vf∗(j)| > 1 for some j = 1, 2, .., k− 1, which
is a contradiction. Hence, S′(K1,n) is not an edge k-product cordial graph if k ≤ n.

Theorem 11. The graph S′(K1,n) admits an edge 3-product cordial labeling if and only if
n = 1.

Proof. Let Let the vertex set and edge set of S′(K1,n) be V (S′(K1,n)) = {u, ui, v, vi; 1 ≤
i ≤ n} and E(S′(K1,n)) = {uui, vui, uvi; 1 ≤ i ≤ n} respectively.
Define an edge labeling f : E(S′(K1,1)) → {0, 1, 2} as follows:



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 9 of 21

f(uu1) = 0, f(vu1) = 1, f(uv1) = 2.
From this labeling we get, ef (0) = ef (1) = ef (2) = 1 and vf∗(0)−1 = vf∗(1) = vf∗(2) = 1.
Hence, S′(K1,1) is an edge 3-product cordial graph.

For n = 2, |V | = |E| = 6. By Theorem 5, S′(K1,2) is not an edge 3-product cordial
graph. Also, by Theorem 10, S′(K1,n); n ≥ 3 is not an edge 3-product cordial graph.

Theorem 12. The graph S′(K1,n) admits an edge 4-product cordial labeling if and only if
n = 2.

Proof. Let the vertex set and edge set of S′(K1,n) be V (S′(K1,n)) = {u, ui, v, vi; 1 ≤
i ≤ n} and E(S′(K1,n)) = {uui, vui, uvi; 1 ≤ i ≤ n} respectively.
Define an edge labeling f : E(S′(K1,2)) → {0, 1, 2, 3} as follows:
f(uu1) = 0, f(uu2) = 1, f(vu1) = 3, f(vu2) = 1, f(uv1) = 2, f(uv2) = 2.
From this labeling we get, ef (0) = ef (1) − 1 = ef (2) − 1 = ef (3) = 1 and vf∗(0) − 1 =
vf∗(1) = vf∗(2)− 1 = vf∗(3) = 1. Hence, S′(K1,2) is an edge 4-product cordial graph.

For n = 1, |V | = 4 and |E| = 3. If f is an edge 4-product cordial labeling of S′(K1,1), then
ef (i) is either 0 or 1 for i = 0, 1, 2, 3 and vf∗(i) = 1 for all i = 0, 1, 2, 3. Clearly, ef (0) = 0
otherwise vf∗(0) = 2. Thus, ef (i) = 1 for all i = 1, 2, 3. But ef (2) = 1 implies vf∗(0) = 0
and vf∗(2) = 2. Therefore, |vf∗(0)− vf∗(2)| > 1 which is a contradiction. Hence, S′(K1,1)
is not an edge 4-product cordial graph.
For n = 3, |V | = 8 and |E| = 9. By Theorem 5, S′(K1,3) is not an edge 4-product cordial
graph. Also, by Theorem 10, S′(K1,n) is not an edge 4-product cordial graph if n ≥ 4.

Theorem 13. The graph S′(K1,n) admits an edge 5-product cordial labeling if and only if
n ≤ 3.

Proof. Let the vertex set and edge set of S′(K1,n) be V (S′(K1,n)) = {u, ui, v, vi; 1 ≤
i ≤ n} and E(S′(K1,n)) = {uui, vui, uvi; 1 ≤ i ≤ n} respectively. Define an edge labeling
f : S′(K1,n) → {0, 1, ..., k − 1} for n ≤ 3 as follows:

f(uui) =


0 ; i = 1 , n = 2, 3
1 ; i = 2 , n = 2, 3
2 ; i = n , n = 1, 3,

f(vui) =


1 ; i = 1 , n = 3 ; i = n = 2
3 ; i = n = 3
4 ; i = 1 , n = 1, 2 ; i = 2 , n = 3,

f(uvi) =


1 ; i = n = 1
2 ; i = 1 , n = 2, 3
3 ; i = 2 , n = 2, 3
4 ; i = n = 3.

Clearly, |ef (i)−ef (j)| ≤ 1 and |vf∗(i)−vf∗(j)| ≤ 1 for i, j ∈ {0, 1, 2, 3, 4}. Hence, S′(K1,n)
is an edge 5-product cordial graph if n ≤ 3.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 10 of 21

For n = 4, |V | = 10 and |E| = 12. By Theorem 5, S′(K1,4) is not an edge 5-product
cordial graph. Also, by Theorem 10, S′(K1,n) is not an edge 5-product cordial graph if
n ≥ 5.

4. Edge k-Product Cordial Labeling of Path Union of Graphs

In this section, we explore the edge k-product cordial behavior of the path union of
star, bistar and cycle graphs. In the following general result, we show that the path union
of an edge k-product cordial graph with multiple of k edges also admits an edge k-product
cordial labeling.

Theorem 14. Let G be an edge k-product cordial graph with multiple of k edges. Then
P (n.Gv), where v is a vertex of G such that at least one of its incident edges is labeled
with 0 admits an edge k-product cordial labeling.

Proof. Let the vertex and edge set of P (n.Gv) be V (P (n.Gv)) =
⋃

1≤i≤n V (Gi) and
E(P (n.Gv)) =

⋃
1≤i≤nE(Gi)∪{ei : ei = vivi+1 , vi ∈ V (Gi) , 1 ≤ i ≤ n−1} respectively.

Let g be an edge k-product cordial labeling of G. Since G has kt edges, eg(i) = t for all
i = 0, 1, ..., k − 1 and |vg∗(i)− vg∗(j)| ≤ 1 for i, j ∈ {0, 1, ..., k − 1}.
Define an edge labeling f : E(P (n.Gv)) → {0, 1, ..., k−1} for n ≡ r (mod k) ; 0 ≤ r ≤ k−1
as follows:
f(e) = g(e) ; e ∈ E(Gi) , 1 ≤ i ≤ n,

f(ei) =



0 ; 1 ≤ i ≤ ⌊n−1
k ⌋

1 ; ⌊n−1
k ⌋+ 1 ≤ i ≤ 2⌊n−1

k ⌋
2 ; 2⌊n−1

k ⌋+ 1 ≤ i ≤ 3⌊n−1
k ⌋

:
:
k − 1 ; (k − 1)⌊n−1

k ⌋+ 1 ≤ i ≤ k⌊n−1
k ⌋,

f(ek⌊n−1
k

⌋+i) = j ; i ≡ j (mod k) , 1 ≤ i ≤ n− 1− k⌊n−1
k ⌋.

From this labeling we obtain,

ef (i) =

{
neg(i) + ⌊n−1

k ⌋ ; i ≥ r
neg(i) + ⌊n−1

k ⌋+ 1 ; i < r,
vf∗(i) = vg∗(i).
Clearly, |ef (i) − ef (j)| ≤ 1 and |vf∗(i) − vf∗(j)| ≤ 1 for i, j ∈ {0, 1, 2.., k − 1}. Hence,
P (n.Gv) is an edge k-product cordial graph.

4.1. Path Union of Star

In this subsection, we prove that the path union of a star graph P (n.Kv
1,m), where v

is a root vertex of K1,m admits an edge k-product cordial labeling for n ≡ 0, 1 (mod k).
Also, we show that the graph P (n.Kv

1,m) admits an edge k-product cordial labeling for
n ≡ k − 1 (mod k) if m ≡ 0, k − 1 (mod k).



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 11 of 21

Theorem 15. The path union of star graph P (n.Kv
1,m), where v is a root vertex of K1,m

admits an edge k-product cordial labeling if n ≡ 0, 1 (mod k).

Proof. Let the vertex and edge set of P (n.Kv
1,m) be V (P (n.Kv

1,m)) = {vi, vji : 1 ≤
i ≤ n, 1 ≤ j ≤ m} and E(P (n.Kv

1,m)) = {vivi+1, viv
j
i , vnv

j
n : 1 ≤ i ≤ n − 1, 1 ≤ j ≤ m}

respectively.
If m ≡ 0 (mod k), by Theorems 1 and 14, P (n.Kv

1,m) is an edge k-product cordial graph.
Define f : E(P (n.Kv

1,m)) → {0, 1, 2, ..., k−1} for n ≡ 0, 1 (mod k) andm ≡ r (mod k) ; 1 ≤
r ≤ k − 1 as follows:
f(vivi+1) = 0 ; 1 ≤ i ≤ n− 1,
We consider the following two cases.

Case(i): If n ≡ 0 (mod k), then

f(viv
j
i ) =



0 ; 1 ≤ i ≤ n
k , 1 ≤ j ≤ k⌊mk ⌋ − (k − 1) + r

1 ; 1 ≤ i ≤ n
k , j = k⌊mk ⌋ − (k − 1) + r + 1 ; n

k + 1 ≤ i ≤ 2n
k

2 ; 1 ≤ i ≤ n
k , j = k⌊mk ⌋ − (k − 1) + r + 2 ; 2n

k + 1 ≤ i ≤ 3n
k

:
:

k − 1 ; 1 ≤ i ≤ n
k , j = k⌊mk ⌋+ r ; (k−1)n

k + 1 ≤ i ≤ n.
From this labeling we get,

ef (i) =

{
n⌊mk ⌋+

n
k (1 + r)− 1 ; i = 0

n⌊mk ⌋+
n
k (1 + r) ; 1 ≤ i ≤ k − 1,

vf∗(i) = n⌊mk ⌋+
n
k (1 + r) ; 0 ≤ i ≤ k − 1.

Case (ii): If n ≡ 1 (mod k), then
f(viv

j
i ) ; 1 ≤ i ≤ n− 1 , 1 ≤ j ≤ m as in Case (i),

f(vnv
j
n) =



0 ; 1 ≤ j ≤ ⌊mk ⌋
1 ; ⌊mk ⌋+ 1 ≤ j ≤ 2⌊mk ⌋
2 ; 2⌊mk ⌋+ 1 ≤ j ≤ 3⌊mk ⌋
:
:
k − 1 ; (k − 1)⌊mk ⌋+ 1 ≤ j ≤ k⌊mk ⌋,

f(vnv
k⌊m

k
⌋+j

n ) = j ; 1 ≤ j ≤ r.
From this labeling we have,

ef (i) =

{
⌊nk ⌋+ n⌊mk ⌋+ r⌊nk ⌋ ; i = 0 ; i > r
⌊nk ⌋+ n⌊mk ⌋+ r⌊nk ⌋+ 1 ; 1 ≤ i ≤ r,

vf∗(i) =

{
⌊nk ⌋+ n⌊mk ⌋+ r⌊nk ⌋ ; i > r
⌊nk ⌋+ n⌊mk ⌋+ r⌊nk ⌋+ 1 ; 0 ≤ i ≤ r.

Clearly, |ef (i) − ef (j)| ≤ 1 and |vf∗(i) − vf∗(j)| ≤ 1 for i, j ∈ {0, 1, ..., k − 1}. Hence,
P (n.Kv

1,m) is an edge k-product cordial graph if n ≡ 0, 1 (mod k).

Example 3. An edge 4-product cordial labeling of P (4.Kv
1,5) is shown in Figure 3.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 12 of 21

0 0 0

0
0

0 0

0 0 0 0

1

1

1 1 1 1 1

1 1 1 1 12

2 2 2 2 2 2

2 2 2 2 23

3

3 3 3 3 3

3 3 3 3 3

Figure 3: Edge 4-product cordial labeling of P (4.Kv
1,5)

Theorem 16. The path union star graph P (n.Kv
1,m), where v is a root vertex of K1,m

admits an edge k-product cordial labeling if n ≡ k− 1 (mod k) and m ≡ 0, k− 1 (mod k).

Proof. Let the vertex and edge set of P (n.Kv
1,m) be V (P (n.Kv

1,m)) = {vi, vji : 1 ≤
i ≤ n, 1 ≤ j ≤ m} and E(P (n.Kv

1,m)) = {vivi+1, viv
j
i , vnv

j
n : 1 ≤ i ≤ n − 1, 1 ≤ j ≤ m}

respectively.
If m ≡ 0 (mod k), then by Theorems 1 and 14, P (n.Kv

1,m) is an edge k-product cordial
graph.
Define f : E(P (n.Kv

1,m)) → {0, 1, 2, ..., k−1} for n ≡ k−1 (mod k) and m ≡ k−1 (mod k)
as follows:
f(viv

j
i ) for 1 ≤ i ≤ n− k + 1, 1 ≤ j ≤ m as in Case (i) of Theorem 15,

f(vivi+1) = 0 ; 1 ≤ i ≤ n− 1,

f(viv
j
i ) =



0 ; n− k + 2 ≤ i ≤ n , 1 ≤ j ≤ ⌊mk ⌋
1 ; n− k + 2 ≤ i ≤ n , ⌊mk ⌋+ 1 ≤ j ≤ 2⌊mk ⌋

2 ; n− k + 2 ≤ i ≤ n , 2⌊mk ⌋+ 1 ≤ j ≤ 3⌊mk ⌋

:
:
k − 1 ; n− k + 2 ≤ i ≤ n , (k − 1)⌊mk ⌋+ 1 ≤ j ≤ k⌊mk ⌋

; i = n− k + 2 , m− k + 2 ≤ j ≤ m

; i = n− k + 3 , m− k + 2 ≤ j ≤ m

; i = n , m− k + 2 ≤ j ≤ m.

From this labeling we have,

ef (i) =

{
k⌊nk ⌋⌊

m
k ⌋+ k⌊nk ⌋+ (k − 1)(1 + ⌊mk ⌋)− 1 ; i = 0

k⌊nk ⌋⌊
m
k ⌋+ k⌊nk ⌋+ (k − 1)(1 + ⌊mk ⌋) ; 1 ≤ i ≤ k − 1,

vf∗(i) = k⌊nk ⌋⌊
m
k ⌋+ k⌊nk ⌋+ (k − 1)(1 + ⌊mk ⌋) ; 0 ≤ i ≤ k − 1.

Clearly, |ef (i) − ef (j)| ≤ 1 and |vf∗(i) − vf∗(j)| ≤ 1 for i, j ∈ {0, 1, ..., k − 1}. Hence,
P (n.Kv

1,m) is an edge k-product cordial graph if n ≡ k−1 (mod k) andm ≡ 0, k−1 (mod k).

Example 4. An edge 5-product cordial labeling of P (4.Kv
1,5) is shown in Figure 4.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 13 of 21

0 0 0

0

0

1

1 1

1

1

1

1

1

2
2 2 2 2

2 2 2 2 2

3
3 3 3 3

3 3 3 3 3

4
4 4 4 4

4 4 4 4 4

Figure 4: Edge 5-product cordial labeling of P (4.Kv
1,5)

4.2. Path Union of Bistar

In this subsection, we prove that the path union of a bistar graph P (n.Bv
m,m), where

v is a root vertex of Bm,m admits an edge k-product cordial labeling for n ≡ 0, 1 (mod k).
Also, we show that the graph P (n.Bv

m,m) admits an edge k-product cordial labeling for
n ≡ k − 1 (mod k) if m ≡ 0, k − 1 (mod k).

Theorem 17. The path union of bistar graph P (n.Bv
m,m), where v is a root vertex of

Bm,m admits an edge k-product cordial labeling if n ≡ 0, 1 (mod k).

Proof. Let the vertex and edge set of P (n.Bv
m,m) be V (P (n.Bv

m,m)) = {vi, ui, vji , u
j
i :

1 ≤ i ≤ n, 1 ≤ j ≤ m} and E(P (n.Bv
m,m)) = {vivi+1, viui, viv

j
i , uiu

j
i , vnun, vnv

j
n, unu

j
n :

1 ≤ i ≤ n− 1, 1 ≤ j ≤ m} respectively.
Define f : E(P (n.Bv

m,m)) → {0, 1, 2, ..., k−1} for n ≡ 0, 1 (mod k) andm ≡ r (mod k) ; 0 ≤
r ≤ k − 1 as follows:
f(vivi+1) = 0 ; 1 ≤ i ≤ n− 1,
f(viui) = 0 ; 1 ≤ i ≤ n,
We have the following two cases.
Case (i): If n ≡ 0 (mod k), then
f(viv

j
i ) = 0 ; 1 ≤ i ≤ n , 1 ≤ j ≤ ⌊mk ⌋ − 1,

f(viv
⌊m

k
⌋

i ) =



0 ; 1 ≤ i ≤ (r+1)n
k

1 ; (r+1)n
k + 1 ≤ i ≤ (r+2)n

k

2 ; (r+2)n
k + 1 ≤ i ≤ (r+3)n

k
:
:

k − r − 1 ; (k−1)n
k ≤ i ≤ n

; m ̸≡ k − 1 (mod k),

f(viv
⌊m

k
⌋+j

i ) =


q ; j ≡ q (mod k − 1) , 1 ≤ q ≤ k − 2

k − 1 ; j ≡ 0 (mod k − 1)
; 1 ≤ i ≤ n , 1 ≤ j ≤ (k − 1)⌊mk ⌋,

f(viv
k⌊m

k
⌋+j

i ) =


k − q ; j ≡ q (mod k − 1) , 1 ≤ q ≤ k − 2

1 ; j ≡ 0 (mod k − 1)

; 1 ≤ i ≤ n
k , 1 ≤ j ≤ r,

f(vn
k
+iv

k⌊m
k
⌋+j

n
k
+i ) =


q ; i ≡ q (mod k − 1) , 1 ≤ q ≤ k − 2

k − 1 ; i ≡ 0 (mod k − 1)
; 1 ≤ i ≤ (k−1)n

k , 1 ≤ j ≤ r,



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 14 of 21

f(uiu
j
i ) = f(viv

j
i ) ; 1 ≤ i ≤ n , 1 ≤ j ≤ m.

From this labeling we have,

ef (i) =

{
2n
k + 2n⌊mk ⌋+ 2rn

k − 1 ; i = 0
2n
k + 2n⌊mk ⌋+ 2rn

k ; 1 ≤ i ≤ k − 1,

vf∗(i) = 2n
k + 2n⌊mk ⌋+ 2rn

k ; 0 ≤ i ≤ k − 1.
Case (ii): If n ≡ 1 (mod k), then
f(viv

j
i ) ; 1 ≤ i ≤ n− 1 , 1 ≤ j ≤ m as in Case (i),

f(uiu
j
i ) ; 1 ≤ i ≤ n− 1 , 1 ≤ j ≤ m as in Case (i),

f(vnv
j
n) = 0 ; 1 ≤ j ≤ ⌊mk ⌋,

f(unu
j
n) = 0 ; 1 ≤ j ≤ ⌊mk ⌋ − 1,

f(vnv
⌊m

k
⌋+j

n ) =

{
q ; j ≡ q (mod k − 1) , 1 ≤ q ≤ k − 2
k − 1 ; j ≡ 0 (mod k − 1)

; 1 ≤ j ≤ m− ⌊mk ⌋,

f(unu
⌊m

k
⌋

n ) =


0 ; m ≡ 1, 2 (mod 3) , k = 3 ; m ̸≡ 0, 1 (mod k) , k > 3
1 ; m ≡ 0 (mod k) , k ≥ 3
k − 2 ; k = 2 ; m ≡ 1 (mod k) , k > 3,

f(unu
⌊m

k
⌋+j

n ) =

{
k − q ; j ≡ q (mod k − 1) , 1 ≤ q ≤ k − 2
1 ; j ≡ 0 (mod k − 1)

; 1 ≤ j ≤ m− ⌊mk ⌋.

From this labeling we have,

ef (i) =

{
2⌊nk ⌋+

2mn
k ; i = 0 ; 2 ≤ i ≤ k − 1

2⌊nk ⌋+
2mn
k + 1 ; i = 1

; m ≡ 0 (mod k),

ef (i) =


2⌊nk ⌋+ 2n⌊mk ⌋+ 2⌊nk ⌋ ; i = 0 , k > 3 ; 2 ≤ i ≤ k − 3

2⌊nk ⌋+ 2n⌊mk ⌋+ 2⌊nk ⌋+ 1 ; i = 1, k − 1, k − 2 ; i = 0 , k = 3

; m ≡ 1 (mod k) , k ≥ 3,

ef (i) =

{
2⌊nk ⌋+ 2n⌊mk ⌋+ 2(k − 1)⌊nk ⌋+ 1 ; i = 0
2⌊nk ⌋+ 2n⌊mk ⌋+ 2(k − 1)⌊nk ⌋+ 2 ; 1 ≤ i ≤ k − 1

; m ≡ k − 1 (mod k),

ef (i) =


2⌊nk ⌋+ 2n⌊mk ⌋+ 2r⌊nk ⌋+ 1 ; i = 0, 1 ; 2 ≤ i ≤ r < k − i ; k − i ≤ r < i

2⌊nk ⌋+ 2n⌊mk ⌋+ 2r⌊nk ⌋+ 2 ; i ≤ r , k − i ≤ r

; r ̸= 0, 1, k − 1, k ≥ 3,

vf∗(i) =

{
ef (i) + 1 ; i = 0
ef (i) ; 1 ≤ i ≤ k − 1.

Clearly, |ef (i) − ef (j)| ≤ 1 and |vf∗(i) − vf∗(j)| ≤ 1 for i, j ∈ {0, 1, ..., k − 1}. Hence,
P (n.Bv

m,m) is an edge k-product cordial graph if n ≡ 0, 1 (mod k).

Example 5. An edge 4-product cordial labeling of P (4.Bv
5,5) is shown in Figure 5.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 15 of 21

0 0 0
0 0 0 0

0 0 0 0
0 0 0 0

0

0

0

0

0

0

0

0

1

1
1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1
1

1 1

1

2

2

2

2
2

2
2

2

2

2

2

2

2

2

2

2

2

2 2

2

2

2

2

2

3

3

3

3

3

3

3

3

3

3

3

3

3

3
3

3 3

3 3

3

3

3

3

3

Figure 5: Edge 4-product cordial labeling of P (4.Bv
5,5)

Theorem 18. The path union of bistar graph P (n.Bv
m,m), where v is a root vertex of Bm,m

admits an edge k-product cordial labeling if n ≡ k− 1 (mod k) and m ≡ 0, k− 1 (mod k).

Proof. Let the vertex and edge set of P (n.Bv
m,m) be V (P (n.Bv

m,m)) = {vi, ui, vji , u
j
i :

1 ≤ i ≤ n, 1 ≤ j ≤ m} and E(P (n.Bv
m,m)) = {vivi+1, viui, viv

j
i , uiu

j
i , vnun, vnv

j
n, unu

j
n :

1 ≤ i ≤ n− 1, 1 ≤ j ≤ m} respectively.
Define f : E(P (n.Bv

m,m)) → {0, 1, 2, ..., k − 1} for n ≡ k − 1 (mod k) and m ≡ 0, k −
1 (mod k) as follows:
f(viv

j
i ) ; 1 ≤ i ≤ n− k + 1 , 1 ≤ j ≤ m as in Case (i) of Theorem 17,

f(uiu
j
i ) ; 1 ≤ i ≤ n− k + 1 , 1 ≤ j ≤ m as in Case (i) of Theorem 17,

f(vivi+1) = 0 ; 1 ≤ i ≤ n− 1,
f(viui) = 0 ; 1 ≤ i ≤ n,
We have the following two cases.
Case (i): If m ≡ 0 (mod k), then
f(viv

j
i ) = f(uiu

j
i ) = 0 ; n− k + 2 ≤ i ≤ n , 1 ≤ j ≤ m

k − 1,

f(vn−k+1+iv
m
k
n−k+1+i) = i ; 1 ≤ i ≤ k − 2,

f(vnv
m
k
n ) = f(unu

m
k
n ) = 0,

f(un−k+1+iu
m
k
n−k+1+i) = k − i ; 1 ≤ i ≤ k − 2,

f(viv
j
i ) = f(uiu

j
i ) =


1 ; m

k + 1 ≤ j ≤ 2m
k

2 ; 2m
k + 1 ≤ j ≤ 3m

k
:
:

k − 1 ; (k−1)m
k + 1 ≤ j ≤ m

; n− k + 2 ≤ i ≤ n.

From this labeling we have,

ef (i) =

{
2⌊nk ⌋+

2mn
k + 1 ; i = 0, 1, k − 1

2⌊nk ⌋+
2mn
k + 2 ; 2 ≤ i ≤ k − 2,



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 16 of 21

vf∗(i) =

{
2⌊nk ⌋+

2mn
k + 1 ; i = 1, k − 1

2⌊nk ⌋+
2mn
k + 2 ; i = 0 ; 2 ≤ i ≤ k − 2.

Case (ii): If m ≡ k − 1 (mod k), then
f(viv

j
i ) ; n− k + 2 ≤ i ≤ n , 1 ≤ j ≤ m− k + 1 as in Case (i),

f(uiu
j
i ) ; n− k + 2 ≤ i ≤ n , 1 ≤ j ≤ m− k + 1 as in Case (i),

f(vivi+1) = 0 ; 1 ≤ i ≤ n− 1,
f(viui) = 0 ; 1 ≤ i ≤ n,
f(viv

j
i ) = 0 ; n− k + 2 ≤ i ≤ n , m− k + 2 ≤ j ≤ m− k + ⌊mk ⌋+1,

f(vn−k+1+iv
j
n−k+1+i) = 0 ; 2 ≤ i ≤ k − 2 , m− k + ⌊mk ⌋+ 2 ≤ j ≤ m− k + 2⌊mk ⌋+ 1,

f(vn−k+1+iv
j
n−k+1+i) = i ; 1 ≤ i ≤ k − 1 , m− k + 2⌊mk ⌋+ 2 ≤ j ≤ m,

f(vn−k+1+iv
j
n−k+1+i) = i ; i = 1, k − 1 , m− k + 2 ≤ j ≤ m− k + ⌊mk ⌋+ 1,

f(un−k+1+iu
j
n−k+1+i) = i ; 1 ≤ i, j ≤ k − 1.

From this labeling we have,

ef (i) =

{
2n⌊mk ⌋+ 2n− 1 ; i = 0
2n⌊mk ⌋+ 2n ; 1 ≤ i ≤ k − 1,

vf∗(i) = 2n⌊mk ⌋+ 2n ; 0 ≤ i ≤ k − 1.
Clearly, |ef (i) − ef (j)| ≤ 1 and |vf∗(i) − vf∗(j)| ≤ 1 for i, j ∈ {0, 1, ..., k − 1}. Hence,
P (n.Bv

m,m) is an edge k-product cordial graph if n ≡ k − 1 (mod k) and m ≡ 0, k −
1 (mod k).

Example 6. An edge 3-product cordial labeling of P (5.Bv
3,3) is given in Figure 6.

0 0 0 00 0 0 0 0
0 0 0 0 0

0 0 0 0 0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2 2

2

2

2

2

2

Figure 6: Edge 3-product cordial labeling of P (5.Bv
3,3)

4.3. Path Union of Cycle

In this subsection, we establish the necessary conditions for the path union of a cycle
graph P (n.Cv

m) to admit an edge k-product cordial labeling. Also, we investigate the edge
3-product and 4-product cordial behavior of P (n.Cv

m).
Let v be a vertex of a cycle Cm ; m ≥ 3. According to the symmetry, all P (n.Cv

m) are



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 17 of 21

isomorphic. Hence, we use the notation P (n.Cm).
In order to establish the necessary condition for the path union of a cycle graph

P (n.Cm) to admit an edge k-product cordial labeling, we prove the following general
result.

Theorem 19. Any graph G with ⌊ |V |
k ⌋ < ⌊ |E|

k ⌋ does not an admit edge k-product cordial
labeling.

Proof. Let f be an edge k-product cordial labeling of a graph G with ⌊ |V |
k ⌋ < ⌊ |E|

k ⌋.
Then ef (i) is either ⌊ |E|

k ⌋ or ⌊ |E|
k ⌋+1 and vf∗(i) is either ⌊ |V |

k ⌋ or ⌊ |V |
k ⌋+1 (i = 0, 1, ..., k−1).

If ef (0) = ⌊ |E|
k ⌋, then vf∗(0) ≥ ⌊ |E|

k ⌋+ 1 > ⌊ |V |
k ⌋+ 1, which is a contradiction. Therefore,

ef (0) = ⌊ |E|
k ⌋ + 1, which results vf∗(0) ≥ ⌊ |E|

k ⌋ + 2 > ⌊ |V |
k ⌋ + 1, a contradiction again.

Hence, G is not an edge k-product cordial graph.

Theorem 20. The path union of cycle graph P (n.Cm) does not admit an edge k-product
cordial labeling if n ≥ k.

Proof. For the path union of cycle graph P (n.Cm), we have |V | = nm and |E| =
nm + n − 1. Let f be an edge k-product cordial labeling of P (n.Cm) ; n ≥ k. We have
the following two cases.
Case(i): For n = k, we have ef (i) = m or m+1 (i = 0, 1, 2, ..., k−1) and vf∗(i) = m (i =
0, 1, 2, ..., k − 1). If ef (0) = m, then vf∗(0) ≥ m+ 1, which is a contradiction.
Case (ii): For n ≥ k + 1, we have ⌊nm+n−1

k ⌋ > ⌊nmk ⌋. By Theorem 19, P (n.Cm) is not
an edge k-product cordial graph.
Hence, P (n.Cm) is not an edge k-product cordial graph if n ≥ k.

Theorem 21. The path union of cycle graph P (n.Cm) does not admit an edge k-product
cordial labeling if m is a multiple of k.

Proof. Let m = kt ; t ≥ 1. For the path union of cycle graph P (n.Ckt), we have |V | =
ktn and |E| = ktn+n−1. Let f be an edge k-product cordial labeling of P (n.Ckt) ; n < k.
since n−1 < k, we have ef (i) = tn or tn+1 and vf∗(i) = tn (i = 0, 1, 2, ..., k−1). If ef (0) =
tn, then vf∗(0) ≥ tn+1. Therefore, |vf∗(0)−vf∗(i)| ≥ 2 for some i ∈ {1, 2, ..., k−1}, which
is a contradiction. Thus, f is not an edge k-product cordial labeling of P (n.Ckt) ; n < k.
By Theorem 20, P (n.Cm) ; n ≥ k is not an edge k-product cordial graph. Hence, P (n.Ckt)
is not an edge k-product cordial graph.

Theorem 22. The path union of cycle graph P (n.Cm) admits an edge 3-product cordial
labeling if and only if n = 2 and m ≡ 2 (mod 3).

Proof. Let the vertex and edge set of P (n.Cm) be V (P (n.Cm)) = {vji ; 1 ≤ i ≤ n, 1 ≤
j ≤ m} and E(P (n.Cm)) = {v1i v1i+1, v

j
i v

j+1
i , vjnv

j+1
n , vmi v1i , v

m
n v1n ; 1 ≤ i ≤ n − 1, 1 ≤ j ≤

m− 1} respectively.
Define an edge labeling f : E(P (2.C3t+2)) → {0, 1, 2} for t ≥ 1 as follows:



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 18 of 21

f(v11v
1
2) = 0,

f(v3t+2
i v1i ) = 1 ; 1 ≤ i ≤ 2,

f(vj1v
j+1
1 ) = 0 ; 1 ≤ j ≤ 2t,

f(v2t+j
1 v2t+j+1

1 ) =

{
1 ; j ≡ 0, 3 (mod 4)
2 ; j ≡ 1, 2 (mod 4)

; 1 ≤ j ≤ t+ 1,

f(vj2v
j+1
2 ) =

{
1 ; j ≡ 0, 1 (mod 4)
2 ; j ≡ 2, 3 (mod 4)

; 1 ≤ j ≤ 3t+ 1.

From this labeling we have,

ef (i) =

{
2t+ 1 ; i = 0
2t+ 2 ; i = 1, 2,

vf∗(i) =

{
2t+ 1 ; i = 1, 2
2t+ 2 ; i = 0.

Conversely, let f be an edge 3-product cordial labeling of P (2.C3t+1) ; t ≥ 1. Then,
ef (i) = 2t + 1 (i = 0, 1, 2) and vf∗(i) = 2t or 2t + 1 (i = 0, 1, 2). Clearly, ef (0) = 2t + 1
implies vf∗(0) ≥ 2t + 2 > 2t + 1, which is a contradiction. Hence, P (2.C3t+1) ; t ≥ 1 is
not an edge 3-product cordial graph.
By Theorem 21, P (2.C3t) ; t ≥ 1 is not an edge 3-product cordial graph. Also, by Theorem
20, P (n.Cm) ; n ≥ 3 is not an edge 3-product cordial graph.

Example 7. An edge 3-product cordial labeling of P (n.C8) is shown in Figure 7.

0
0

0

0

0

00

0

0
0

0
1

1
1

1

1

1 1

1

1

1

1

2

2

2

2
2

2 2
2

2

2

2

Figure 7: Edge 3-product cordial labeling of P (n.C8)

Theorem 23. The path union of cycle graph P (n.Cm) admits an edge 4-product cordial
labeling if and only if n = 2 and m = 3, 5.

Proof. Let the vertex and edge set of P (n.Cm) be V (P (n.Cm)) = {vji ; 1 ≤ i ≤ n, 1 ≤
j ≤ m} and E(P (n.Cm)) = {v1i v1i+1, v

j
i v

j+1
i , vjnv

j+1
n , vmi v1i , v

m
n v1n ; 1 ≤ i ≤ n − 1, 1 ≤ j ≤

m− 1} respectively. Define f : E(P (2.Cm) → {0, 1, 2, 3} for m = 3, 5 as follows:
f(v11v

1
2) = 2,

f(vji v
j+1
i ) =


0 ; i = 1 , j = 1
1 ; i = 1 , j = 2
3 ; i = 2 , 1 ≤ j ≤ 2

; m = 3,

f(vji v
j+1
i ) =


0 ; i = 1 , 1 ≤ j ≤ 2
1 ; i = 1 , j = 4 ; i = 2 , 2 ≤ j ≤ 3
2 ; i = 1 , j = 3
3 ; i = 2 , j = 1, 4

; m = 5,



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 19 of 21

f(vmi v1i ) =


1 ; i = 2 , m = 3
2 ; i = 1 , m = 3, 5
3 ; i = 2 , m = 5.

From this labeling we have,

ef (i) =

{
2⌊m4 ⌋ ; i = 0
2⌊m4 ⌋+ 1 ; 1 ≤ i ≤ 3,

vf∗(i) =

{
2⌊m4 ⌋ ; i = 1, 3
2⌊m4 ⌋+ 1 ; i = 0, 2.

Hence, P (2.C3) and P (2.C5) are edge 4-product cordial graphs.

To prove the converse part, we consider the following two cases.
Case(i): If n = 2 and m = 4t + r, where 1 ≤ r ≤ 3, t ≥ 2 for r = 1 and t ≥ 1 for
2 ≤ r ≤ 3, then |V | = 8t+ 2r and |E| = 8t+ 2r + 1.
Let f be an edge 4-product cordial labeling of the graph P (2.Cm). Then we have,

ef (i) =

{
2t or 2t+ 1 ; r = 1
2t+ 1 or 2t+ 2 ; r = 2, 3,

vf∗(i) =


2t or 2t+ 1 ; r = 1
2t+ 1 ; r = 2
2t+ 1 or 2t+ 2 ; r = 3.

For r = 1, we must have the following conditions.
(i) ef (0) = 2t,
(ii) two adjacent edges cannot be labeled with 2,
(iii) 0 must be assigned consecutively
otherwise vf∗(0) ≥ 2t+2. Hence, ef (i) = 2t+1 (i = 1, 2, 3) and 2t+1 non adjacent edges
must be labeled with 2. This implies vf∗(2) ≥ 2t+ 2. By similar argument, for r = 3, we
get ef (0) = 2t+1, ef (i) = 2t+2 (i = 1, 2, 3) and vf∗(2) ≥ 2t+3. For r = 2, ef (0) = 2t+1
implies vf∗(0) ≥ 2t+ 2. Therefore in all the cases, we obtain |vf∗(0)− vf∗(2)| ≥ 2, which
is a contradiction.
Case(ii): If n = 3 and m = 4t + r, where 1 ≤ r ≤ 3, t ≥ 1 for 1 ≤ r ≤ 2 and t ≥ 0 for
r = 3, then |V | = 12t+ 3r and |E| = 12t+ 3r + 2.
Let g be an edge 4-product cordial labeling of the graph P (3.Cm). Then we have,

eg(i) =


3t+ 1 or 3t+ 2 ; r = 1
3t+ 2 ; r = 2
3t+ 2 or 3t+ 3 ; r = 3,

vg∗(i) =


3t or 3t+ 1 ; r = 1
3t+ 1 or 3t+ 2 ; r = 2
3t+ 2 or 3t+ 3 ; r = 3.

For 1 ≤ r ≤ 2, eg(0) = 3t+ r implies vg∗(0) ≥ 3t+ r + 1. For r = 3, as in case(i) we get,
eg(0) = 3t + 2, eg(i) = 3t + 3 (i = 1, 2, 3) and vg∗(i) = 3t + 2 (i = 1, 2, 3), which result
vg∗(2) ≥ 3t + 3. Therefore, in all the cases, we obtain |vg∗(0) − vg∗(2)| ≥ 2, which is a
contradiction.
By Theorem 21, P (n.C4t) ; t ≥ 1 is not an edge 4-product cordial graph. Also, by Theorem
20, P (n.Cm) ; n ≥ 4 is not an edge 4-product cordial graph.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 20 of 21

References

[1] R. Manoharan A. Anto Cathrin Aanisha. Mean Cordial Labeling in Graph Repre-
sentations of Human Anatomy and Circular Systems. Communications on Applied
Nonlinear Analysis, 31(7):453–458, 2024.

[2] C. M. Barasara. Edge and Total Edge Product Cordial Labeling of Some New Graphs.
International Journal of Engineering, Science and Mathematics, 7(2):263–273, 2018.

[3] I. Cahit. Cordial graphs: A Weaker Version of Graceful and Harmonious Graphs.
Ars Combin., 23:201–207, 1987.

[4] J. A. Gallian. A Dynamic Survey of Graph Labeling. The Electronic Journal of
Combinatorics, # DS6, 2023.

[5] E. Roshdy M. Aboshady, R. Elbarkouky and M. Abdel-Azim Seoud. Further Results
on Edge Product Cordial Labeling. Proceedings of the Pakistan Academy of Sciences,
57(4):23–32, 2020.

[6] S. Somasundaram M. Sundaram, R. Ponraj. Product Cordial Labeling of Graphs.
Bulletin of Pure and Applied Sciences, 23E(1):155–163, 2004.

[7] Andrea Semaničová-Feňovč́ıková Martin Bača, Muhammad Irfan. On 3-Total Edge
Product Cordial Labeling of a Carbon Nanotube Network. AKCE International Jour-
nal of Graphs and Combinatorics, 16:310–318, 2019.

[8] Andrea Semaničová-Feňovč́ıková P. Jeyanthi, K. Jeya Daisy. Zk-Magic Labeling of
Path Union of Graphs. CUBO A Mathematical Journal, 21:15–35, 2019.

[9] R. Rajeswari R. Thamizharasi. Edge Product Cordial Labeling and Total Magic
Cordial Labeling of Regular Digraphs. International Journal on Information Sciences
and Computing, 9(2):1–4, 2015.

[10] A. Rosa. On Certain Valuations of the Vertices of a Graph, Theory of Graphs.
Internat. Sympos., pages 349–355, 1966.

[11] C. M. Barasara S. K. Vaidya. Edge Product Cordial Labeling of Graphs. Journal of
Mathematical and Computational Science, 2(5):1436–1450, 2012.

[12] C. M. Barasara S. K. Vaidya. Edge Product Cordial Labeling in the Context of Some
Graph Operations. International Journal of Mathematics and Scientific Computing,
3(1):4–7, 2013.

[13] C. M. Barasara S. K. Vaidya. Some Edge Product Cordial Graphs. International
Journal of Mathematics and Soft Computing, 3(3):49–53, 2013.

[14] C. M. Barasara S. K. Vaidya. Some New Families of Edge Product Cordial Graphs.
Advanced Modeling and Optimization, 15(1):103–111, 2013.

[15] C. M. Barasara S. K. Vaidya. On Edge Product Cordial Labeling of Some Product
Related Graphs. International Journal of Mathematics and its Applications, 2(2):15–
22, 2014.

[16] C. M. Barasara S. K. Vaidya. Product and Edge Product Cordial Labeling of Degree
Splitting Graph of Some Graphs. Advances and Applications in Discrete Mathematics,
15(1):61–74, 2015.

[17] N. B. Patel U. M. Prajapati. Edge Product Cordial Labeling of Some Cycle Related
Graphs. Open Journal of Discrete Mathematics, 6:268–278, 2016.



N. M. NourEldeen et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5887 21 of 21

[18] N. B. Patel U. M. Prajapati. Edge Product Cordial Labeling of Some Graphs. Journal
of Applied Mathematics and Computational Mechanics, 18(1):69–76, 2019.

[19] N. B. Patel U. M. Prajapati. Edge Product Cordial Labeling of Switching Operations
on Some Graphs. TWMS J. App. and Eng. Math., 12(1):191–199, 2022.

• [1–19] are journal articles.