Further Results on Bijective Product k-Cordial Labeling

Abstract

A bijective product k-cordial labeling f of a graph G with vertex set V and edge set E is a bijection from V to $\{1,2,\dots ,|V|\}$ such that the induced edge labeling $f^{\times }:E\left(G\right)\to {\mathbb{Z}}_k=\{i|0\le i\le k-1\}$ defined as $f^{\times }\left(uv\right)\equiv f\left(u\right)f\left(v\right)(\mathrm{mod}k)$ for every edge $uv\in E$ satisfies the condition $|e_f^{\times }\left(i\right)-e_f^{\times }\left(j\right)|\le 1$, where $i,j\in {\mathbb{Z}}_k$ and $e_f^{\times }\left(i\right)$ is the number of edges labeled with i under $f^{\times }$. A graph which admits a bijective product k-cordial labeling is called a bijective product k-cordial graph. In this paper, we study bijective product $\pi$-cordiality for paths and cycles, where $\pi$ is an odd prime. We determine bijective product $\pi$-cordiality for paths and cycles for $3\le \pi \le 13$. Also, we establish the bijective product k-cordial labeling of stars. Further, we find the bijective product 4-cordial labeling of bistars and the splitting graphs of stars and bistars.

1. Introduction

All graphs considered here are simple, finite, connected, undirected and of order greater than 1. For the basic notation and terminology of graph theory, we refer the reader to [1]. Graph theory is quickly being integrated into mainstream mathematics due to the fact that other important areas of mathematics, including group theory, number theory, and combinatorics, are related to it. It is also closely connected to applied mathematics, which is used in operations research; genetics; physical, biological, social sciences; engineering; and computer science. The labeling of graphs is perceived to be a primarily theoretical subject in the field of graph theory and discrete mathematics due to the fact that it serves as a model in a wide range of applications in various fields. Labeling is a function that allocates the elements of a graph to real numbers, usually positive integers. In 1967, Rosa [2] published a pioneering paper on graph labeling problems. An enormous body of literature has grown around graph labeling over the past five decades. More than 350 labeling techniques have been studied in nearly 3600 research papers. Gallian [3], in his survey, beautifully classified all the labelings under suitable headings and subheadings. One of the popular labelings called ’cordial labeling’ was introduced by Cahit [4] in 1987. A graph is called cordial if it is possible to label its vertices with 0s and 1s so that when the edges are labeled with the difference of the labels at their endpoints, the number of vertices (edges) labeled with 1s and 0s differs at most by one. After this, many labeling schemes were also introduced with minor variations in the context of cordiality.

In this paper, we consider the complete residue class of the ring of integers modulo k as ${\mathbb{Z}}_k=\{0,1,\dots ,k-1\}$, $k\ge 2$.

A new labeling called ‘product cordial labeling’ was explored by Sundaram et al. [5] in 2004, with slight deviations in the idea of cordiality. A product cordial labeling f of a graph G is $f:V\left(G\right)\to {\mathbb{Z}}_2$ such that if each edge $uv$ is assigned the label $f\left(u\right)f\left(v\right)$, $|v_f\left(0\right)-v_f\left(1\right)|\le 1$ and $|e_f\left(0\right)-e_f\left(1\right)|\le 1$, where $v_f\left(i\right)$ and $e_f\left(i\right)$ denote the number of vertices and edges, respectively, labeled with $i=0,1$. In 2012, Ponraj et al. [6] extended this concept and introduced k-product cordial labeling as follows: Let $f:V\left(G\right)\to {\mathbb{Z}}_k$, where $1\le k\le |V(G)|$. Each edge $uv$ is assigned the label $f\left(u\right)f\left(v\right)\in {\mathbb{Z}}_k$. f is called a k-product cordial labeling if $|v_f\left(i\right)-v_f\left(j\right)|\le 1$ and $|e_f\left(i\right)-e_f\left(j\right)|\le 1$, $i,j\in {\mathbb{Z}}_k$, where $v_f\left(i\right)$ and $e_f\left(i\right)$ denote the number of vertices and edges, respectively, labeled with $i\in {\mathbb{Z}}_k$. Thus, the concept of two-product cordiality is the same as that of product cordiality. Inspired by the concept of ‘k-product cordial labeling’ and the results in [5,6], we continue our study on ‘k-product cordial labeling’ and establish that several families of graphs admit k-product cordial labeling. Also, we show that some families of graphs do not admit k-product cordial labeling. In addition, we introduce two new concepts, namely bijective product k-cordial labeling and bijective product square k-cordial labeling [7]. Suppose G is a graph of order $n\ge 2$. Let $f:V\left(G\right)\to \{1,2,\dots ,n\}$ be a bijection. For each edge $uv$, define $f^{\times }\left(uv\right)\equiv f\left(u\right)f\left(v\right)\left(\mathrm{mod}k\right)$, $k\ge 2$. f is called a bijective product k-cordial labeling if $|e_f^{\times }\left(i\right)-e_f^{\times }\left(j\right)|\le 1$ for all $i,j\in {\mathbb{Z}}_k$, where $e_f^{\times }\left(x\right)$ denotes the number of edges labeled by x under $f^{\times }$. The graph G, which admits a bijective product k-cordial labeling, is called a bijective product k-cordial graph. The following results are obtained in [7].

Theorem 1

([7], Theorem 2.3). Let G be a unicyclic graph of order n and $k\ge 2$. If G is bijective product k-cordial and the girth of G is greater than $n-⌊{\displaystyle \frac{n}{k}}⌋$, then $n\not\equiv 0\left(\mathrm{mod}k\right)$.

Corollary 1

([7], Corollary 2.4). For $k\ge 2$, if $n\equiv 0\left(\mathrm{mod}k\right)$, then $C_n$ is not bijective product k-cordial.

Motivated by the concept and results in [7], we further study bijective product p-cordiality for paths and cycles when p is a prime. When $p=2$, it is an obvious case. Therefore, we only consider that p is an odd prime. Also, we prove that the star graph is bijective product k-cordial. Further, we establish that the bistar and splitting graphs of stars and bistars admit bijective product 4-cordial labelings.

2. Preliminaries

Let G be a graph of order $n\ge 2$ with the vertex set $\{u_i|1\le i\le n\}$. Suppose $f:V\left(G\right)\to S$ is a vertex labeling on G. We use an n-tuple $f=(f\left(u_1\right),f\left(u_2\right),\dots ,f\left(u_n\right))$ to present the labeling f.

Suppose $f:V\left(G\right)\to S$ is a vertex labeling on a graph G, where $S\subseteq \mathbb{Z}$ (or $S={\mathbb{Z}}_p$). For a fixed integer $p\ge 2$, we define an edge labeling $f_p^{+}$ for G. Since p is fixed, we write $f^{+}$ instead of $f_p^{+}$. Here $f^{+}:E\left(G\right)\to {\mathbb{Z}}_p$ is defined by $f^{+}\left(uv\right)\equiv f\left(u\right)+f\left(v\right)\left(\mathrm{mod}p\right)$. For $i\in {\mathbb{Z}}_p$, we use $v_f\left(i\right)$ (or $v\left(i\right)$) and $e_f^{+}\left(i\right)$ (or $e^{+}\left(i\right)$) to denote the number of vertices labeled by i under f and the number of edges labeled by i under $f^{+}$, respectively.

When $G=P_n=u_1{u}_2\dots u_n$, we use $F=(F\left(u_1{u}_2\right),F\left(u_2{u}_3\right),\dots ,F\left(u_{n-1}{u}_n\right))$ to present an edge labeling F for $P_n$. When $G=C_n=u_1{u}_2\dots u_n{u}_1$, similarly we use $F=(F\left(u_1{u}_2\right),F\left(u_2{u}_3\right),\dots ,F\left(u_{n-1}{u}_n\right),F\left(u_n{u}_1\right))$ to present an edge labeling F for $C_n$.

Suppose $A=(a_1,\dots ,a_p)$ and $B=(b_1,\dots ,b_p)$ are two (ordered) sequences of integers and p is a fixed integer. We denote $A\equiv B\left(\mathrm{mod}p\right)$ (or $A\equiv B$) when $a_i\equiv b_i\left(\mathrm{mod}p\right)$ for all i.

Example 1.

Consider $P_6$, $p=6$ and $S={\mathbb{Z}}_6$. The bijective labelings f, $g_1$ and $g_2$ for $P_6$ are as follows:

$f=(4,1,5,2,0,3)$, $g_1=(3,0,4,1,5,2)$ and $g_2=(0,3,1,4,2,5)$; $f^{+}\equiv (5,0,1,2,3)$ and $g_1^{+}=g_2^{+}\equiv (3,4,5,0,1)$.

Lemma 1.

Let $P_{2m}=u_1{u}_2\dots u_{2m}$, $m\ge 1$.

• There is a bijective labeling $g_1:V\left(P_{2m}\right)\to {\mathbb{Z}}_{2m}$ such that $g_1\left(u_1\right)=m$, $g_1\left(u_{2m}\right)=m-1$ and $g_1^{+}\left(E\left(P_{2m}\right)\right)={\mathbb{Z}}_{2m}\setminus \{m-1\}$.
• There is a bijective labeling $g_2:V\left(P_{2m}\right)\to {\mathbb{Z}}_{2m}$ such that $g_2\left(u_1\right)=0$, $g_2\left(u_{2m}\right)=2m-1$ and $g_2^{+}\left(E\left(P_{2m}\right)\right)={\mathbb{Z}}_{2m}\setminus \{m-1\}$.

Proof. 

Define the labelings $g_1$ and $g_2$ by $g_1=(m,0,m+1,1,\dots ,2m-1,m-1)$—that is, the $(2i-1)$-th term of $g_1$ is $m+i-1$ and the $\left(2i\right)$-th term of $g_1$ is $i-1$ for $1\le i\le m$—and $g_2=(0,m,1,m+1,\dots ,m-1,2m-1)$; that is, swap the $(2i-1)$-th and $\left(2i\right)$-th terms of $g_1$ to obtain $g_2$.

We have $g_1^{+}=g_2^{+}=(m,m+1,m+2,\dots ,3m-3,3m-2)\equiv (m,m+1,m+2,\dots ,m-3,m-2)$. Hence $g_1^{+}\left(E\left(P_{2m}\right)\right)=g_2^{+}\left(E\left(P_{2m}\right)\right)={\mathbb{Z}}_{2m}\setminus \{m-1\}$. □

Figure 1 shows the $g_1$ and $g_2$ labelings for $P_6$.

Lemma 2.

Let $k,m\ge 1$. There exists a labeling $g:V\left(P_{2mk}\right)\to {\mathbb{Z}}_{2m}$ such that

• $g\left(u_{2mk}\right)=m-1$;
• g is a k-to-1 labeling; that is, $v\left(i\right)=k$ for $i\in {\mathbb{Z}}_{2m}$;
• $e^{+}\left(i\right)=k$ for $i\in {\mathbb{Z}}_{2m}\setminus \{m-1\}$ and $e^{+}(m-1)=k-1$.

Proof. 

First we separate $P_{2mk}$ into k ‘consecutive’ subpaths of order $2m$ each. Let $R_i=u_1^i\dots u_{2m}^i$, $1\le i\le k$ and $P_{2mk}=R_1{R}_2\dots R_k$ (as combined sequences of vertices).

Let $g_1$ and $g_2$ be the labelings defined in Lemma 1.

(a)

Suppose $k=2h-1$, where $h\ge 1$. We label $R_{2i-1}$ by $g_1$ for $1\le i\le h$ and $R_{2i}$ by $g_2$ for $1\le i\le h-1$. Since $g_1\left(u_{2m}^{2i-1}\right)=m-1$, $g_2\left(u_1^{2i}\right)=0$, $g_2\left(u_{2m}^{2i}\right)=2m-1$, and $g_1\left(u_1^{2i+1}\right)=m$, the induced label for the edge $u_{2m}^{2i-1}{u}_1^{2i}$ or $u_{2m}^{2i}{u}_1^{2i+1}$ is $m-1$, for $1\le i\le h-1$. This is the required labeling.

(b)

Suppose $k=2h$, where $h\ge 1$. Similar to the above case, we label $R_{2i-1}$ by $g_2$ and $R_{2i}$ by $g_1$ for $1\le i\le h$. This is the required labeling.

□

Corollary 2.

Let $k,m\ge 1$. There exists a labeling $g:V\left(P_{2mk+1}\right)\to {\mathbb{Z}}_{2m}$ such that

• $v\left(i\right)=k$ for $i\in {\mathbb{Z}}_{2m}\setminus \left\{0\right\}$ and $v\left(0\right)=k+1$;
• $e^{+}\left(i\right)=k$ for $i\in {\mathbb{Z}}_{2m}$.

Proof. 

We label the first $2mk$ vertices of $P_{2mk+1}$ as defined in Lemma 2 and label the last vertex of $P_{2mk+1}$ by 0. Since $g\left(u_{2mk}\right)=m-1$ and the last vertex of $P_{2mk+1}$ is labeled by 0, the last edge is labeled by $m-1$. Thus $e^{+}(m-1)=k$. This is a required labeling. □

Example 2.

From Lemma 2 we have the following labelings for $P_{18}$ and $P_{12}$, where $2m=6$.

	Vertex labeling	Induced edge labeling
$P_{18}$	(3, 0, 4, 1, 5, 2, 0, 3, 1, 4, 2, 5, 3, 0, 4, 1, 5, 2)	(3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1)
$P_{12}$	(0, 3, 1, 4, 2, 5, 3, 0, 4, 1, 5, 2)	(3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1)

By Corollary 2 we have the following labelings for $P_{19}$ and $P_{13}$.

	Vertex labeling	Induced edge labeling
$P_{19}$	(3, 0, 4, 1, 5, 2, 0, 3, 1, 4, 2, 5, 3, 0, 4, 1, 5, 2, 0)	(3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2)
$P_{13}$	(0, 3, 1, 4, 2, 5, 3, 0, 4, 1, 5, 2, 0)	(3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2)

3. Bijective Product Prime-Cordial Labelings for Paths

The following results were obtained in our previous paper [7].

Theorem 2

([7], Theorem 2.2). A graph G is bijective product 2-cordial if and only if G admits a 2-product cordial labeling g with $v_g\left(1\right)\ge v_g\left(0\right)$.

Theorem 3

([7], Theorem 2.7). A graph G is bijective product 3-cordial if and only if G admits a 3-product cordial labeling g with $v_g\left(1\right)\ge v_g\left(0\right)$ and $v_g\left(2\right)\ge v_g\left(0\right)$.

The next theorem is a generalization of Theorems 2 and 3.

Theorem 4.

A graph G is bijective product p-cordial if and only if G admits a p-product cordial labeling g with

$$v_g\left(1\right)\ge v_g\left(2\right)\ge \dots \ge v_g(p-1)\ge v_g\left(0\right)and{v}_g\left(1\right)-v_g\left(0\right)\le 1.$$

(1)

Proof. 

The necessary part is obvious. So, we only need to prove the supplementary part. Let the order of G be $n=pq+r$, where $0\le r\le p-1$. Let $L_i$ be the set of vertices in G whose label is $i\in {\mathbb{Z}}_p$.

If $r=0$, then $|L_i|=v_g\left(i\right)=q$ for all i. If $0<r\le p-1$, then $|L_i|=v_g\left(i\right)=q+1$ if $1\le i\le r$, $|L_j|=v_g\left(j\right)=q$ if $r+1\le j\le p-1$, and $|L_0|=v_g\left(0\right)=q$.

For each $i\in {\mathbb{Z}}_p$, we label all the vertices in $L_i$ by k, where $k\equiv i\left(\mathrm{mod}p\right)$ in $\{1,2,\dots ,n\}$. Then this labeling is bijective product p-cordial. □

Lemma 3.

Suppose $P_n=v_1\dots v_n$ admits a p-product cordial labeling g with $g\left(v_1\right)=0$ and $e_g\left(0\right)\le e_g\left(i\right)$ for all $i\in {\mathbb{Z}}_p\setminus \left\{0\right\}$. Then g can be extended as a p-product cordial labeling of $C_n$.

Proof. 

Note that $C_n$ is obtained from $P_n$ by adding the edge $v_n{v}_1$. Let $g^{\prime }:V\left(C_n\right)\to {\mathbb{Z}}_p$ be the same labeling as g. Then $g^{\prime }\left(v_n\right)g^{\prime }\left(v_1\right)=0$. Thus, $e_{g^{\prime }}\left(i\right)=e_g\left(i\right)$ for $i\in {\mathbb{Z}}_p\setminus \left\{0\right\}$ and $e_{g^{\prime }}\left(0\right)=e_g\left(0\right)+1$. From our hypothesis, we have $|e_{g^{\prime }}\left(i\right)-e_{g^{\prime }}\left(j\right)|\le 1$ for $i\ne j$ in ${\mathbb{Z}}_p$. Hence $g^{\prime }$ is a p-product cordial labeling of $C_n$. □

Theorem 5.

Suppose π is an odd prime. $P_{\pi k}$ admits a π-product cordial labeling ρ, which satisfies condition (1) in Theorem 4 and $\rho \left(v_1\right)=0$. Hence $P_{\pi k}$ is bijective product π-cordial for any positive integer k.

Proof. 

Suppose $P=P_{\pi k}=v_1\dots v_k{u}_1{u}_2\dots u_{(\pi -1)k}$. Let $Q=v_1\dots v_k$ and $R=u_1{u}_2\dots u_{(\pi -1)k}$.

Let g be the labeling for R defined in Lemma 2 by letting $2m=\pi -1$.

Since $\pi$ is prime, there is a primitive root of $\pi$, say a. Define $\rho :V\left(P\right)\to {\mathbb{Z}}_{\pi }$ by $\rho \left(v_i\right)=0$, $1\le i\le k$ and $\rho \left(u_j\right)=a^{g\left(u_j\right)}$, $1\le j\le (\pi -1)k$. Now ${\rho }^{\times }\left(u_j{u}_{j+1}\right)=\rho \left(u_j\right)\rho \left(u_{j+1}\right)=a^{g\left(u_j\right)+g\left(u_{j+1}\right)}=a^{g^{+}\left(u_j{u}_{j+1}\right)}$, $1\le j\le (\pi -1)k-1$.

Then $v_{\rho }\left(x\right)=k$ for $x\in {\mathbb{Z}}_{\pi }$, $e_{\rho }\left(0\right)=k$ and $e_{\rho }\left(a^y\right)=k$ for $y\in \{0,1,\dots ,\pi -1\}\setminus \{{\displaystyle \frac{\pi -1}{2}}-1\}$, and $e_{\rho }(a^{{\displaystyle \frac{\pi -1}{2}}-1})=k-1$. Thus, $\rho$ is a $\pi$-product cordial labeling for $P_{\pi k}$, which satisfies condition (1) in Theorem 4. Hence, $P_{\pi k}$ is bijective product $\pi$-cordial. □

Corollary 3.

Suppose π is an odd prime and $k\ge 1$. $P_{\pi k+1}$, $P_{\pi k+2}$ and $P_{\pi (k+1)-1}$ admit a π-product cordial labeling ρ, which satisfies condition (1) in Theorem 4 and $\rho \left(v_1\right)=0$. Moreover, $P_{\pi -1}$ also admits a π-product cordial labeling ρ, which satisfies condition (1) in Theorem 4.

Hence $P_{\pi h+1}$, $P_{\pi h+2}$ and $P_{\pi h-1}$ are bijective product π-cordial for $h\ge 1$.

Proof. 

(a)

Suppose $P=P_{\pi k+1}=v_1\dots v_k{u}_1{u}_2\dots u_{(\pi -1)k}{u}_{(\pi -1)k+1}$. Let $Q=v_1\dots v_k$ and $R=u_1{u}_2\dots u_{(\pi -1)k}{u}_{(\pi -1)k+1}$. Let g be the labeling for R defined in Corollary 2 by letting $2m=\pi -1$.

Similar to Theorem 5, we define a labeling $\rho :V\left(P\right)\to {\mathbb{Z}}_{\pi }$. In this case, $\rho \left(u_{(\pi -1)k+1}\right)=a^0=1$ and $\rho \left(u_{(\pi -1)k}\right)=a^{{\displaystyle \frac{\pi -1}{2}}-1}$. Then $v_{\rho }\left(x\right)=k$ for $x\in {\mathbb{Z}}_{\pi }\setminus \left\{1\right\}$ and $v_{\rho }\left(1\right)=k+1$, $e_{\rho }\left(0\right)=k$ and $e_{\rho }\left(a^y\right)=k$ for $y\in \{0,1,\dots ,\pi -1\}$. The previous two sentences mean that $e_{\rho }\left(x\right)=k$ for $x\in {\mathbb{Z}}_{\pi }$. Thus, $\rho$ is a $\pi$-product cordial labeling for $P_{\pi k+1}$, which satisfies condition (1). From Theorem 4, $P_{\pi k+1}$ is bijective product $\pi$-cordial.

(b)

Let $\rho$ be the $\pi$-product cordial labeling of $P_{\pi k+1}$ defined in Case (a). We extend $\rho$ to $P_{\pi k+2}$ by labeling the last vertex of $P_{\pi k+2}$ by 2. We still denote the $\pi$-product cordial labeling of $P_{\pi k+2}$ as $\rho$. Then $v_{\rho }\left(x\right)=k$ for $x\in {\mathbb{Z}}_{\pi }\setminus \{1,2\}$; $v_{\rho }\left(1\right)=v_{\rho }\left(2\right)=k+1$ and $e_{\rho }\left(x\right)=k$ for $x\in {\mathbb{Z}}_{\pi }\setminus \left\{2\right\}$; and $e_{\rho }\left(2\right)=k+1$. Thus, $\rho$ is a $\pi$-product cordial labeling for $P_{\pi k+2}$, which satisfies condition (1). From Theorem 4, $P_{\pi k+2}$ is bijective product $\pi$-cordial.

(c)

Let $\rho$ be the $\pi$-product cordial labeling for $P_{\pi (k+1)}$ defined in Theorem 5, where $k\ge 1$. Now, we delete the first vertex of $P_{\pi (k+1)}$, which is labeled by 0. We still denote the $\pi$-product cordial labeling of $P_{\pi (k+1)-1}$ as $\rho$. Namely, $v_{\rho }\left(0\right)=k$ and $v_{\rho }\left(x\right)=k+1$ for $x\in {\mathbb{Z}}_{\pi }\setminus \left\{0\right\}$, $e_{\rho }\left(0\right)=k$ and $e_{\rho }\left(a^y\right)=k+1$ for $y\in \{0,1,\dots ,\pi -1\}\setminus \{{\displaystyle \frac{\pi -1}{2}}-1\}$, and $e_{\rho }(a^{{\displaystyle \frac{\pi -1}{2}}-1})=k$, where a is a primitive root of $\pi$.

Note that the labeling $\rho$ defined in all the above cases satisfies the condition that $\rho \left(v_1\right)=0$ if $v_1$ is the first vertex of the corresponding path.

It is easy to see that the labeling defined in Case (c) still works when $k=0$. Thus, we obtain a $\pi$-product cordial labeling for $P_{\pi -1}$. But, in this case, all the labels of vertices in $P_{\pi -1}$ are non-zero.

From Theorem 4, $P_{\pi h+1}$, $P_{\pi h+2}$ and $P_{\pi h-1}$ are bijective product $\pi$-cordial for $h\ge 1$. □

By combining Theorem 4, Lemma 3 and Corollary 3, we obtain the following.

Corollary 4.

Suppose π is an odd prime. The cycles $C_{\pi k+1}$, $C_{\pi k+2}$ and $C_{\pi (k+1)-1}$ are bijective product π-cordial for $k\ge 1$.

Remark 1.

The cycle $C_{\pi -1}$ may not be bijective product π-cordial. For example, it is easy to check that $C_4$ is not bijective product 5-cordial.

Example 3.

Suppose that $\pi =7$. Now $3^2=9\equiv 2\left(\mathrm{mod}7\right)$; $3^3=27\equiv 6\left(\mathrm{mod}7\right)$; $3^4=81\equiv 4\left(\mathrm{mod}7\right)$; $3^5=243\equiv 5\left(\mathrm{mod}7\right)$; and $3^6=729\equiv 1\left(\mathrm{mod}7\right)$. So $a=3$ is a primitive root of 7.

From Theorem 5, $P_{14}=QR$, where $Q\cong P_2$ and $R\cong P_{12}$. From Example 2, the labeling for R is $(3^0,3^3,3^1,3^4,3^2,3^5,3^3,3^0,3^4,3^1,3^5,3^2)\equiv (1,6,3,4,2,5,6,1,4,3,5,2)\left(\mathrm{mod}7\right)$. The induced edge labeling is $(6,4,5,1,3,2,6,4,5,1,3)\left(\mathrm{mod}7\right)$.

For the graph $P_{14}$, the vertex labeling

$\rho =(0,0,1,6,3,4,2,5,6,1,4,3,5,2)$ (which is a 7-product cordial labeling) and the induced edge labeling ${\rho }^{\times }=(0,0,6,4,5,1,3,2,6,4,5,1,3)$. From Theorem 4, we obtain a bijective product 7-cordial labeling $(14,7,1,6,3,4,2,5,13,8,11,10,12,9)$ for $P_{14}$.

Similarly the bijective product 7-cordial labelings for $P_{15}$, $P_{16}$ and $P_{13}$ are

$$\begin{array}{cc}& (14,7,1,6,3,4,2,5,13,8,11,10,12,9,15),\hfill \\ \hfill & (14,7,1,6,3,4,2,5,13,8,11,10,12,9,15,16),\hfill \\ \hfill & (7,1,6,3,4,2,5,13,8,11,10,12,9),\hfill \end{array}$$

with induced edge labelings

$$\begin{array}{cc}& (0,0,6,4,5,1,3,2,6,4,5,1,3,2),\hfill \\ \hfill & (0,0,6,4,5,1,3,2,6,4,5,1,3,2,2),\hfill \\ \hfill & (0,6,4,5,1,3,2,6,4,5,1,3),\hfill \end{array}$$

respectively.

So (7, 14, 1, 6, 3, 4, 2, 5, 13, 8, 11, 10, 12, 9, 15), (7, 14, 1, 6, 3, 4, 2, 5, 13, 8, 11, 10, 12, 9, 15, 16) and (7, 1, 6, 3, 4, 2, 5, 13, 8, 11, 10, 12, 9) are also bijective product 7-cordial labelings for $C_{15}$, $C_{16}$ and $C_{13}$, respectively. Their induced edge labelings are (0, 0, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1, 3, 2, 0), (0, 0, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1, 3, 2, 2, 0) and (0, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1, 3, 0).

Next, we restate the result proven in [7] as a corollary combining the results of Corollaries 1, 3 and 4.

Corollary 5.

$P_n$ is bijective product 3-cordial for $n\ge 2$ and $C_m$ is bijective product 3-cordial if and only if $m\not\equiv 0\left(\mathrm{mod}3\right)$, where $m\ge 3$.

4. Bijective Product $\mathbf{\pi }$-Cordial Labeling of Path and Cycle, Where $\mathit{\pi }=\mathbf{5},\mathbf{7},\mathbf{11},\mathbf{13}$

In this section, we study bijective product $\pi$-cordiality for paths and cycles for some prime $\pi \ge 5$.

Theorem 6.

Let $n=\pi k+r$, where $2\le r\le \pi -2$. Let $P_n=x_1\dots x_{\pi k}{y}_1{y}_2\dots y_r$. Suppose there is a bijective product π-cordial labeling f for $P_r=y_1{y}_2\dots y_r$ with $f\left(y_1\right)=1$. Then there is a π-product cordial labeling g for $P_n$ such that $e_g\left(0\right)\le e_g\left(i\right)$ for all $i\in {\mathbb{Z}}_{\pi }\setminus \left\{0\right\}$. Moreover, $P_n$ and $C_n$ are bijective product π-cordial.

Proof. 

Let $\rho$ be the $\pi$-product cordial labeling for $x_1\dots x_{\pi k}{y}_1$ defined in Corollary 3, Case (a). Note that $\rho \left(y_1\right)=1$, $v_{\rho }\left(i\right)=k$ for $i\in {\mathbb{Z}}_{\pi }\setminus \left\{1\right\}$, $v_{\rho }\left(1\right)=k+1$, and $e_{\rho }\left(i\right)=k$ for all $i\in {\mathbb{Z}}_{\pi }$.

Since $r<\pi$ and f is a bijective product $\pi$-cordial labeling for $P_r=y_1{y}_2\dots y_r$, $v_f\left(i\right)=1$ with $1\le i\le r$, $v_f\left(i\right)=0$ for $r+1\le i\le \pi -1$, $v_f\left(0\right)=0$, $0\le e_f^{\times }\left(i\right)\le 1$ for $1\le i\le \pi -1$, and $e_f^{\times }\left(0\right)=0$. Since $\rho \left(y_1\right)=f\left(y_1\right)=1$, we can combine the labelings $\rho$ and f for $P_n$. Let this combined labeling be g. It is easy to check that g is a $\pi$-product cordial labeling for $P_n$ since $v_{\rho }$ and $v_f$ satisfy condition (1). Moreover, $y_1$ is a common vertex in two paths. So the combine labeling g satisfies condition (1). Hence, from Theorem 4, g is a bijective product $\pi$-cordial labeling of $P_n$ and $C_n$. □

Now, we consider $P_n$ or $C_n$. Let $n=\pi k+r$, with $0\le r\le \pi -1$, where $\pi$ is an odd prime. From Theorem 5 and Corollaries 3, 4 and 5, we need to find results for $3\le r\le \pi -2$ and $\pi \ge 5$. From Theorem 6, it suffices to find a bijective product $\pi$-cordial labeling f for $P_r=y_1{y}_2\dots y_r$ with $f\left(y_1\right)=1$ and $3\le r\le \pi -2$. We denote f as $(f\left(y_1\right),\dots ,f\left(y_r\right))$ and $f^{\times }=(f^{\times }\left(y_1{y}_2\right),\dots ,f^{\times }\left(y_{r-1}{y}_r\right))$.

Suppose $\pi =5$. Then there is only one case, namely when $r=3$. Let $f=(1,2,3)$. Then $f^{\times }=(2,1)$. So f is a bijective product 5-cordial labeling for $P_3$. From Theorem 6, for $k\ge 0$, $P_{5k+3}$ and $C_{5k+3}$ are bijective product 5-cordial (since $C_3$ is clearly bijective product 5-cordial).

Lemma 4.

Let π be an odd prime and $s\in \mathbb{N}$. Suppose $(s-1)s<\pi$. Then $P_r=y_1{y}_2\dots y_r$ admits a bijective product π-cordial labeling f with $f\left(y_1\right)=1$, for $2\le r\le s$.

Proof. 

We define $f\left(y_i\right)=i$ for $1\le i\le r$. For $1\le i\le r-1$, $i(i+1)\le (r-1)r\le (s-1)s<\pi$. Thus, $f^{\times }\left(y_i{y}_{i+1}\right)$ are distinct modulo $\pi$. Hence, f is a bijective product $\pi$-cordial labeling. □

In the

Table 1. Labelings f and $f^{\times }$ for $P_r$ and $C_r$ when $\pi \in \{7,11,13\}$ and $3\le r\le \pi -2$.

$\mathit{\pi }$	r	f for ${\mathit{P}}_{\mathit{r}}$ or ${\mathit{C}}_{\mathit{r}}$	${\mathit{f}}^{\times }$ for ${\mathit{P}}_{\mathit{r}}$	${\mathit{f}}^{\times }$ for ${\mathit{C}}_{\mathit{r}}$
7	4	1, 2, 3, 4	2, 6, 5	2, 6, 5, 4
	5	1, 2, 5, 3, 4	2, 3, 1, 5	2, 3, 1, 5, 4
11	4	1, 2, 3, 4	2, 6, 1	2, 6, 1, 4
	5	1, 2, 3, 4, 5	2, 6, 1, 9	2, 6, 1, 9, 5
	6	1, 2, 3, 6, 5, 4	2, 6, 7, 8, 9	2, 6, 7, 8, 9, 4
	7	1, 7, 2, 3, 4, 6, 5	7, 3, 6, 1, 2, 8	7, 3, 6, 1, 2, 8, 5
	8	1, 8, 7, 2, 3, 6, 4, 5	8, 1, 3, 6, 7, 2, 9	8, 1, 3, 6, 7, 2, 9, 5
	9	1, 2, 7, 9, 3, 4, 5, 8, 6	2, 3, 8, 5, 1, 9, 7, 4	2, 3, 8, 5, 1, 9, 7, 4, 6
13	5	1, 2, 3, 4, 5	2, 6, 12, 7	2, 6, 12, 7, 5
	6	1, 2, 4, 5, 6, 3	2, 8, 7, 4, 5	2, 8, 7, 6, 4, 3
	7	1, 2, 4, 3, 6, 5, 7	2, 8, 12, 5, 4, 9	2, 8, 12, 5, 4, 9, 7
	8	1, 2, 8, 4, 3, 6, 5, 7	2, 3, 6, 12, 5, 4, 9	2, 3, 6, 12, 5, 4, 9, 7
	9	1, 4, 2, 8, 3, 6, 9, 5, 7	4, 8, 3, 11, 5, 2, 6, 9	4, 8, 3, 11, 5, 2, 6, 9, 7
	10	1, 7, 10, 9, 4, 6, 5, 3, 2, 8	7, 5, 12, 10, 11, 4, 2, 6, 3	7, 5, 12, 10, 11, 4, 2, 6, 3, 8
	11	1, 5, 7, 6, 9, 3, 10, 2, 4, 8, 11	5, 9, 3, 2, 1, 4, 7, 8, 6, 10	5, 9, 3, 2, 1, 4, 7, 8, 6, 10, 11

, we list the labelings f and $f^{\times }$ for each r, where $3\le r\le \pi -2$ and $(r-1)r\ge \pi \ge 7$. Here, we omit the parentheses.

By combining with Theorem 6 and Lemma 4, we obtain the following theorem.

Theorem 7.

Suppose $\pi \in \{3,5,7,11,13\}$.

(a)

$P_n$ is bijective product π-cordial for $n\ge 2$.

(b)

Suppose $m>\pi$ and $m\neg \equiv 0\left(\mathrm{mod}\pi \right)$. $C_m$ is bijective product π-cordial.

From the above table and Remark 1 we obtain the following theorem.

Theorem 8.

The cycle $C_3$ is bijective product 5-cordial, but $C_4$ is not bijective product 5-cordial. For $\pi \in \{7,11,13\}$, $C_m$ is bijective product π-cordial for $3\le m<\pi$.

Conjecture 1.

$P_n$ is bijective product $\pi$-cordial for $n\ge 2$, where $\pi$ is an odd prime.

Conjecture 2.

$C_m$ is bijective product $\pi$-cordial for $m>\pi$ and $m\not\equiv 0\left(\mathrm{mod}\pi \right)$, where $\pi$ is an odd prime.

5. Bijective Product $\mathbf{k}$-Cordial Labeling of Star and Symmetric Bistar

We denote the complete bipartite graph with partition sizes m and n, with $m,n\ge 1$, by $K_{m,n}$. The graph $K_{1,n}$ is called a star graph. A bistar graph$B_{m,n}$ is the graph obtained by joining the apex vertices of the stars $K_{1,m}$ and $K_{1,n}$ by an edge. The graph $B_{n,n}$ is called a symmetric bistar graph.

In this section, we assume the vertex set and the edge set of $K_{1,n}$ to be $V\left(K_{1,n}\right)=\left\{u\right\}\cup \{u_i|1\le i\le n\}$ and $E\left(K_{1,n}\right)=\{u{u}_i|1\le i\le n\}$, respectively. Also we assume the vertex set and the edge set of $B_{n,n}$ to be $V\left(B_{n,n}\right)=\{u,v\}\cup \{u_i,v_i|1\le i\le n\}$ and $E\left(B_{n,n}\right)=\left\{uv\right\}\cup \{u{u}_i,v{v}_i|1\le i\le n\}$, respectively.

For convenience, we use $[a,b]$ to denote the set of integers from a to b. We also denote $S\equiv T\left(\mathrm{mod}m\right)$ if two multisets $S=\{s_i|1\le i\le n\}$ and $T=\{t_i|1\le i\le n\}$ satisfy $s_i\equiv t_i\left(\mathrm{mod}k\right)$ for each i, $1\le i\le n$.

Suppose G is a graph of size q. Let $q=kQ+r$, where $0\le r<k$. Note that ${\displaystyle [a,a+q-1]=\left(\bigcup _{j=0}^{Q-1}[a+jk,a+jk-k+1]\right)\cup [a+q-r,a+q-1]}$ ($[a+q-r,a+q-1]$ does not appear when $r=0$). Since $[a+jk,a+jk-k+1]\equiv [0,k-1]\left(\mathrm{mod}k\right)$ for $0\le j\le Q-1$, $[a,a+q-1]\equiv \left([0,k-1]\times Q\right)\cup [a+q-r,a+q-1]\left(\mathrm{mod}k\right)$, $[a+q-r,a+q-1]$ is congruent to a subset of ${\mathbb{Z}}_k$.

Theorem 9.

The star graph $K_{1,n}$ is bijective product k-cordial for all $n\ge 1$ and $k\ge 2$.

Proof. 

Define $f:V\left(K_{1,n}\right)\to [1,n+1]$ by $f\left(u\right)=1$, $f\left(u_i\right)=i+1$ for $1\le i\le n$. Clearly the set of induced edge labels is $[2,n+1]$ before taking modulo k. Thus f is a bijective product k-cordial labeling of $K_{1,n}$ for $k\ge 2$. Hence, the star graph $K_{1,n}$ is bijective product k-cordial for all $n\ge 1$ and $k\ge 2$. □

Example 4.

A bijective product 8-cordial labeling of $K_{1,9}$ is shown in Figure 2.

Theorem 10.

The symmetric bistar graph $B_{n,n}$ is bijective product 4-cordial for all $n\ge 1$.

Proof. 

We define a bijective product 4-cordial labeling f for $B_{n,n}$.

For $n=1$, we define $f\left(u\right)=1$, $f\left(v\right)=3$, $f\left(u_1\right)=2$ and $f\left(v_1\right)=4$.

For $n=2$, we define $f\left(u\right)=1$, $f\left(v\right)=3$, $f\left(u_1\right)=2$, $f\left(u_2\right)=5$, $f\left(v_1\right)=4$ and $f\left(v_2\right)=6$.

For $n\ge 3$, we define $f\left(u\right)=1$, $f\left(u_1\right)=2$, $f\left(u_2\right)=3$, $f\left(u_3\right)=4$, $f\left(v\right)=5$, $f\left(v_i\right)=n+2+i$ ($1\le i\le n$) and $f\left(u_j\right)=2+j$ (if $n\ge 4$, $4\le j\le n$).

Then the set $\{f^{\times }\left(u{u}_j\right)|1\le j\le n\}\cup \left\{f^{\times }\left(uv\right)\right\}=[2,n+2]$ before taking modulo 4. Since $f\left(v\right)=5\equiv 1\left(\mathrm{mod}4\right)$, the set $\{f^{\times }\left(v{v}_i\right)|1\le i\le n\}\equiv [n+3,2n+2]\left(\mathrm{mod}4\right)$. Thus the set of all induced edge labels is congruent to $[2,2n+2]$ modulo 4. Hence f is a bijective product 4-cordial labeling.

Hence, the symmetric bistar graph $B_{n,n}$ is bijective product 4-cordial for all $n\ge 1$. □

Example 5.

A bijective product 4-cordial labeling of $B_{6,6}$ is shown in Figure 3.

The splitting graph $S\left(G\right)$ of a graph G is a graph obtained by adding a new vertex $v^{\prime }$ to each vertex v of G such that $v^{\prime }$ is adjacent to every vertex that is adjacent to v in G.

Theorem 11.

The splitting graph of star graph $S\left(K_{1,n}\right)$ is bijective product 4-cordial for all $n\ge 1$.

Proof. 

Let the vertex and edge set of $S\left(K_{1,n}\right)$ be $V\left(S\left(K_{1,n}\right)\right)=\{u,v,u_i,v_i|1\le i\le n\}$ and $E\left(S\left(K_{1,n}\right)\right)=\{v{u}_i,v{v}_i,u{v}_i|1\le i\le n\}$, respectively. Define $f:V\left(S\left(K_{1,n}\right)\right)\to [1,2n+2]$ as follows:

Case 1: 

For $n=1$, define $f\left(v\right)=1$, $f\left(u\right)=4$, $f\left(u_1\right)=2$ and $f\left(v_1\right)=3$.

Case 2: 

For $n=2$, define $f\left(v\right)=1$, $f\left(u\right)=3$, $f\left(u_1\right)=2$, $f\left(u_2\right)=6$, $f\left(v_1\right)=5$ and $f\left(v_2\right)=4$.

Case 3: 

For $n\ge 3$, define $f\left(v\right)=1$, $f\left(u\right)=3$, $f\left(u_1\right)=2$, $f\left(u_2\right)=4$, $f\left(u_3\right)=2n+2$, $f\left(u_i\right)=i+2$ for $4\le i\le n$ if $n\ge 4$, $f\left(v_1\right)=5$, and $f\left(v_j\right)=j+n+1$ for $2\le j\le n$.

Before taking modulo 4, the sets $S_1=\{f^{\times }\left(v{u}_i\right)|1\le i\le n\}=\{2,4,2n+2\}\cup [6,n+2]$ and $S_2=\{f^{\times }\left(v{v}_j\right)|1\le j\le n\}=\left\{5\right\}\cup [n+3,2n+1]$.

Since $f\left(u\right)=3\equiv -1\left(\mathrm{mod}4\right)$,

$$\begin{array}{cc}\hfill S_3& =\{f^{\times }\left(u{v}_j\right)|1\le j\le n\}\equiv [-2n-1,-n-3]\cup \{-5\}\hfill \\ & \equiv [-2n-1+(4n+4),-n-3+(4n+4\left)\right]\cup \left\{3\right\}\hfill \\ & \equiv [2n+3,3n+1]\cup \left\{3\right\}\left(\mathrm{mod}4\right)\hfill \end{array}$$

Now $S_1\cup S_2\cup S_3\equiv [2,3n+1]\left(\mathrm{mod}4\right)$. Hence f is a bijective product 4-cordial labeling for $S\left(K_{1,n}\right)$.

Therefore, the splitting graph of star graph $S\left(K_{1,n}\right)$ is bijective product 4-cordial for all $n\ge 1$. □

Example 6.

A bijective product 4-cordial labeling of $S\left(K_{1,6}\right)$ is shown in Figure 4:

Theorem 12.

The splitting graph of symmetric bistar graph $S\left(B_{n,n}\right)$ is bijective product 4-cordial for all $n\ge 1$.

Proof. 

Let the vertex and edge set of $S\left(B_{n,n}\right)$ be $V\left(S\left(B_{n,n}\right)\right)=\{u,v,u^{\prime },v^{\prime },u_i,v_i,u_i^{\prime },v_i^{\prime }|1\le i\le n\}$ and $E\left(S\left(B_{n,n}\right)\right)=\{uv,u{v}^{\prime },v{u}^{\prime },u{u}_i,u^{\prime }{u}_i,u{u}_i^{\prime },v{v}_i,v^{\prime }{v}_i,v{v}_i^{\prime }|1\le i\le n\},$ respectively. We have the following three cases.

Define $f:V\left(S\left(B_{n,n}\right)\right)\to [1,4n+4]$ as follows:

• For $n=1$, define $f\left(u\right)=1$, $f\left(v\right)=5$, $f\left(u^{\prime }\right)=7$, $f\left(v^{\prime }\right)=6$, $f\left(u_1^{\prime }\right)=8$, $f\left(v_1^{\prime }\right)=2$, $f\left(u_1\right)=3$ and $f\left(v_1\right)=4$. Then the set (multiset) of edge labels is $\{0,0,0,1,1,2,2,3,3\}$.
• For $n=2$, define $f\left(u\right)=1$, $f\left(v\right)=5$, $f\left(u^{\prime }\right)=3$, $f\left(v^{\prime }\right)=7$, $f\left(u_1^{\prime }\right)=2$, $f\left(u_2^{\prime }\right)=4$, $f\left(v_1^{\prime }\right)=6$, $f\left(v_2^{\prime }\right)=8$, $f\left(u_1\right)=9$, $f\left(u_2\right)=10$, $f\left(v_1\right)=11$ and $f\left(v_2\right)=12$. Then the set of edge labels is $\{0,0,0,0,1,1,1,2,2,2,2,3,3,3,3\}$.
• For $n\ge 3$, define $f\left(u\right)=1$, $f\left(v\right)=5$, $f\left(u^{\prime }\right)=3$, $f\left(v^{\prime }\right)=7$, $f\left(u_1^{\prime }\right)=2$, $f\left(u_2^{\prime }\right)=4$, $f\left(u_3^{\prime }\right)=6$, and $f\left(u_i^{\prime }\right)=i+4$ if $n\ge 4$, $4\le i\le n$.
  $f\left(v_i^{\prime }\right)=i+n+4$ for $1\le i\le n$, $f\left(u_i\right)=i+(2n+4)$ for $1\le i\le n$ and
  $f\left(v_i\right)=i+(3n+4)$ for $1\le i\le n$.
  • Firstly, we have $f^{\times }\left(u{u}_1^{\prime }\right)=2$, $f^{\times }\left(u{u}_2^{\prime }\right)=4$, $f^{\times }\left(u{u}_3^{\prime }\right)=6$, $f^{\times }\left(uv\right)=5$, $f^{\times }\left(u{v}^{\prime }\right)=7$ and $f^{\times }\left(v{u}^{\prime }\right)=3$.
    When $n\ge 4$, the set $\{f^{\times }\left(u{u}_i^{\prime }\right)|4\le i\le n\}\equiv [8,n+4]$.
    Thus $\{f^{\times }\left(u{u}_i^{\prime }\right)|1\le i\le n\}\cup \{f^{\times }\left(uv\right),f^{\times }\left(u{v}^{\prime }\right),f\left(v{u}^{\prime }\right)\}\equiv [2,n+4]$ for $n\ge 3$.
  • The set $\{f^{\times }\left(u{u}_i\right)|1\le i\le n\}=[2n+5,3n+4]$.
  • Since $f\left(v\right)=5\equiv 1\left(\mathrm{mod}4\right)$, the set $\{f^{\times }\left(v{v}_i^{\prime }\right)|1\le i\le n\}\equiv [n+5,2n+4]\left(\mathrm{mod}4\right)$ and the set $\{f^{\times }\left(v{v}_i\right)|1\le i\le n\}\equiv [3n+5,4n+4]\left(\mathrm{mod}4\right)$.
  • Since $f\left(u^{\prime }\right)=3\equiv -1\left(\mathrm{mod}4\right)$, $\{f^{\times }\left(u^{\prime }{u}_i\right)|1\le i\le n\}\equiv [-3n-4,-2n-5]\equiv [n,2n-1]\left(\mathrm{mod}4\right)$.
  • Since $f\left(v^{\prime }\right)=7\equiv -1\left(\mathrm{mod}4\right)$, $\{f^{\times }\left(v^{\prime }{v}_i\right)|1\le i\le n\}\equiv [-4n-4,-3n-5]\equiv [0,n-1]\left(\mathrm{mod}4\right)$.
  Now the set of all induced edge labels is congruent to $A=[0,2n-1]\cup [2,4n+4]\equiv \{2,3,4\}\cup [0,2n-1]\cup [5,4n+4]$ modulo 4.
  Suppose $n=2k+1$, $k\ge 1$. Then $A\equiv [0,4k-1]\cup [5,8k+8]\cup \{2,3,4,4k,4k+1\}\equiv \left([0,3]\times (3k+1)\right)\cup \{0,1,2,3,0\}\left(\mathrm{mod}4\right)$, where $[0,3]\times m$ means m copies of $[0,3]$.
  Suppose $n=2k$, $k\ge 1$. Then $A\equiv [0,4k-1]\cup [5,8k+4]\cup \{2,3,4\}\equiv \left([0,3]\times 3k\right)\cup \{0,2,3\}\left(\mathrm{mod}4\right)$.

Hence f is a bijective product 4-cordial labeling for $S\left(B_{n,n}\right)$. Therefore, the splitting graph of symmetric bistar graph $S\left(B_{n,n}\right)$ is bijective product 4-cordial for all $n\ge 1$. □

Example 7.

A bijective product 4-cordial labeling of $S\left(B_{4,4}\right)$ is shown in Figure 5:

6. Conclusions

In this paper, we establish the bijective product $\pi$-cordial labeling for (1) a path of order $\pi k$, $\pi k+1$, $\pi k+2$ and $\pi k-1$ and (2) a cycle of order $\pi k$, $\pi k+1$, $\pi k+2$ and $\pi (k+1)-1$, where $\pi$ is an odd prime and $k\ge 1$. Also, we prove the existence of bijective product $\pi$-cordial labeling for (1) a path and (2) an m-cycle when m is greater than $\pi$ and m is not a multiple of $\pi$, where $\pi =5,7,11$ and 13. Further, we show that the star graph admits bijective product k-cordial labeling. Moreover, we determine the bijective product 4-cordial labeling of a symmetric bistar, a splitting graph of the star graph and a symmetric bistar graph. Researchers may further explore this topic and try to find more families of graphs that admit/do not admit bijective product k-cordial labeling. It is a fact that only limited studies have been carried out on this new topic; therefore there is more scope for further research on ‘Bijective Product k-Cordial Labeling’. Finally, we conclude this paper with the following open problem.

Open Problem:

Find a bijective product k-cordial labeling for a path, cycle and bistar, where $k\ge 4$. And find a bijective product 3-cordial labeling for a bistar.