RETRACTED: On Cubic Roots Cordial Labeling for Some Graphs

Abstract

In this paper we used the cubic roots of unit group together with the concept of cordiality in graph theory to introduce a new method of labeling, this construed cubic cordial labeling can be applied to all paths, cycles, fans and wheel graphs. Moreover, some other properties are investigated and show that the union of any two cycles and the union any two paths are cubic cordial graphs. Also, we study the cubic cordiality for the union of any cycle with a path.

1. Introduction

“Graph labelling” is the process of assigning values to a graph’s vertices under specific restrictions. The majority of graph labelling issues share the following three components: a collection of integers for assigning vertex labels, a rule that provides an edge label, and some requirement(s) that these labels must meet.

Cahit [1] developed cordial labelling, defining it as the presence of a vertex labelling $f:V\to \{0,1\}$ such that the induced labelling $f^{*}:E\to \{0,1\}$, holds for all edges $u,v\in V$ and with the following inequalities holding: $\left|v\right(0)-v(1\left)\right|\le 1$ and $\left|e\right(0)-e(1\left)\right|\le 1$, where $v\left(i\right)$ (respectively $e\left(i\right)$) is the number of vertices (respectively, edges) labeled with $i\in \{0,1\}$.

A mapping $f:V\left(G\right)\to \{0,1,2\}$ is known as 3-Equitable labelling of $G$, and $f\left(v\right)$ is known as the label of the vertex $v$ of $G$ under $f$, as stated in [2]. The induced edge labelling $f^{*}:E\to \{0,1\}$ for an edge $e=uv$ is provided by $f^{*}\left(e\right)=\left(\left|f\left(u\right)-f\left(v\right)\right|\right)mod 3$. Let $e\left(0\right),e\left(1\right)$ and $e\left(2\right)$ represent the number of edges with labels $0$, 1 and $2$, respectively, under $f^{*}$ and $v\left(0\right),v\left(1\right)$ and $v\left(2\right)$ represent the number of $G$’s vertices with labels $0, 1$ and 2, respectively, under $f$. A graph $G$ is said to have a 3-Equitable labelling, if $\left|v\right(i)-v(j\left)\right|\le 1$ and $\left|e\right(i)-e(j\left)\right|\le 1$, where $i\ne j$ and $i,j\in \{0,1,2\}$.

In [3] ELrokh, Al-Shamiri and Abd El-hay proved that $P_n\odot C_m$ is Logically if and only if $gcd(n;m)\ne 1 or 3(mod 4)$. In [4] Badr and et.al, investigated the radio mean square numbers for paths and cycles. Nada, Elrokh and El-hay [5] provided necessary and sufficient conditions for the existence of total cordial labeling for the corona between paths and second power of fan graph. In the study [6], they examine whether certain types of graphs, such as trees, paths, and cycles, have a local super ant magic total chromatic number. They described the chromatic number of powers of paths in the paper [7]. They also discover specific upper and lower bounds for the chromatic number of powers of cycles. Finding the spanning numbers of the join products of six symmetric graphs on six vertices with paths and cycles on $n$ vertices was the major goal of the research [8]. The exact value of the signed, according to the study [9], is the Cartesian product of the directed path $P_m$ and the directed cycle $C_n$ for any positive integers $m$ and $n$.

The strong products of two paths, the strong product of a path and a cycle, and the strong product of two cycles are all shown to have g-extra connectivity in article [10]. In [11], they obtained an $S-(a,0)-EAMT$ and $S-(a^{\prime },2)-EAMT$ labeling of symmetric classes of networks termed as hexagonal lattice $HT{T}_{m,n}$ and prismatic lattice $T_{m,n}$. For more details about the cordial labeling, the reader can refer to [12,13,14,15]. For detail survey on graph labeling one can refer to Gallian [16].

This paper studies each path $P_n$, $n\ge 2$ admits a cubic cordial labeling for all $n$. Each cycle $C_n$, $n\ge 3$ admits a cubic labeling. Each Fan $F_n$, $n\ge 2$ admits a cubic cordial labeling for all $n$. The Wheel graph $W_n,n\ge 3$ admits a cubic cordial labeling except $n\equiv 2 mod 3$ and $n$ even. The union of $P_n\cup P_m$, $n,m\ge 2$ admits a cubic labeling for all $n,m$. The union of $C_n\cup C_m$, $n,m\ge 3$ admits a cubic labeling for all $n,m$. The union of $P_n\cup C_m$, $n\ge 2,m\ge 3$ admits a cubic cordial labeling for all $n,m$.

In Section 2, we introduce some basic definition which are used later and we show that all paths and cycle are cubic cordial. The remaining part of this article is structured as follows: a cubic cordial labeling of cycles, paths, Fan and Wheel graphs are presented in Section 3. A cubic cordial labeling of the union of two cycles, two paths and both are presented in Section 4. Finally, in Section 5, conclusions are drawn.

2. Terminology and Notation

By $A_{3r}$ we mean the labeling $1\omega {\omega }^2\dots 1\omega {\omega }^2$ (r-times) of the cycle $C_{3r}$ or the path $P_{3r}$. We denote by $B_{3r}$ and $D_{3r}$ the labeling ${\omega }^{2}1\omega \dots {\omega }^{2}1\omega (r-times)$ and $1{\omega }^2\omega \dots 1{\omega }^2\omega (r-times)$ respectively of either $C_{3r}$ or $P_{3r}$. Sometimes, we modify $A_{3r}$, $B_{3r}$ and $D_{3r}$ by inserting symbols at either the left or right end, or both; for example, $A_{3r}{\omega }^{2}1$ means the labeling $1{\omega }^2\omega \dots 1{\omega }^2\omega (r-times){\omega }^{2}1$ for $C_{3r+2}$ or the path $P_{3r+2}$. Similarly, $D_{3r}\omega$ is the labeling $1{\omega }^2\omega \dots 1{\omega }^2\omega (r-time)\omega$ of the path $P_{3r+1}$ or the cycle $C_{3r+1}$. We denoted $v\left(1\right)$, $v\left({\omega }^2\right)$ and $v\left(\omega \right)$ to be the number of vertices labeled $i$, ${\omega }^2$ and $\omega$ respectively. Similarly, we denoted $e\left(1\right)$, $e\left({\omega }^2\right)$ and $e\left(\omega \right)$ to be the number of edges labeled $1$, ${\omega }^2$ and $\omega$ respectively, for the graph $G$.

A vertex labeling of graph $G$ of $f:V\to \{1,\omega ,{\omega }^2\}$ with induced edge labeling $f^{*}:E\left(G\right)\to \{1,\omega ,{\omega }^2\}$ defined by

$$\begin{array}{llll}.& v\left(1\right)& v\left(\omega \right)& v\left({\omega }^2\right)\\ u\left(1\right)& 1& \omega & {\omega }^2\\ u\left(\omega \right)& \omega & {\omega }^2& 1\\ u\left({\omega }^2\right)& {\omega }^2& 1& \omega \end{array}$$

This is called a cubic roots cordial labeling if $\left|v\right(x)-v(y\left)\right|\le 1$ and $\left|e\right(x)-e(y\left)\right|\le 1$, $x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$ where $v\left(x\right)$ (respectively $e\left(x\right)$) is the number of vertices (respectively, edges) labeled with $x\in \{1,\omega ,{\omega }^2\}$. A graph $G$ is cubic permuted cordial if it admits a cubic labeling.

Let us assume that the cube roots of a unit is $z$, i.e., $z^3=1$ which has a three roots are $1, \omega =\frac{-1+\sqrt{3}i}{2}, {\omega }^2=\frac{-1-\sqrt{3}i}{2}$. Its know that ${\omega }^2+\omega +1=0$, $\omega {\omega }^2=1$, $\omega .\omega ={\omega }^2$, ${\omega }^2.{\omega }^2=\omega$ [17].

Definition 1.

A vertex labeling of graph $G$ of $f:V\to \{1,\omega ,{\omega }^2\}$ with induced edge labeling $f^{*}:E\left(G\right)\to \{1,\omega ,{\omega }^2\}$ defined by

$$\begin{array}{llll}.& v\left(1\right)& v\left(\omega \right)& v\left({\omega }^2\right)\\ v\left(1\right)& 1& \omega & {\omega }^2\\ v\left(\omega \right)& \omega & {\omega }^2& 1\\ v\left({\omega }^2\right)& {\omega }^2& 1& \omega \end{array}$$

is called a cubic roots labeling if $\left|v\right(x)-v(y\left)\right|\le 1$ and $\left|e\right(x)-e(y\left)\right|\le 1$, $x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$ where $v\left(x\right)$ (respectively $e\left(x\right)$) is the number of vertices (respectively, edges) labeled with $x\in \{1,\omega ,{\omega }^2\}$. A graph $G$ is cubic roots cordial if it admits cubic roots cordial labeling.

3. A Cubic Roots Cordial Labeling of Some Graphs

In this section we shall prove that a cubic roots cordial labeling of paths, cycles, fan and wheel graphs.

Theorem 1.

Each path  $P_n$, $n\ge 2$ admits a cubic roots cordial labeling for all $n$.

Proof. 

Let $V=\{v_1,v_2,\dots ,v_n\}$ be the vertex set of $P_n$ and $E=\{v_r{v}_{r+1};1\le r\le n-1\}$ its edge set. To define vertex labeling $f:V\to \{1,\omega ,{\omega }^2\}$ the following cases show to be considered:

When $n\equiv 0,1,2(mod 3)$

$$f(v_r)=\left\{\begin{array}{ll}1& ;0 mod 3\\ \omega & ;1 mod 3\\ {\omega }^2& ;2 mod 3\end{array}\right..$$

The induced edge labeling $f^{*}$ is given in Section 2 where our roots is $\{1,{\omega }^2,\omega \}$. Let us study the following three cases:

Case (1): $n\equiv 0(mod 3)$, i.e., $n=3r$. The total number of labelled vertices $1’$s is denoted by $v\left(1\right)$ is equal to the entire number of labelled vertices $\omega$ and denoted by $v\left(\omega \right)$ also equivalent to the total number of vertices labeled ${\omega }^2$ denoted by $v(\omega {)}^2$ and this number is $r$, i.e., $v\left(1\right)=v\left(\omega \right)=v\left({\omega }^2\right)=r$. Obviously $\left|v\right(x)-v(y\left)\right|=0,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$. Similarly in this same one can see that number of edges labeled $1’$s denoted by $e\left(1\right)$ the same as the number of edges labeled $\left(\omega \right)$ denoted by $e\left(\omega \right)$ and this number is $r$, while the number of edges labeled ${\omega }^2$ denoted by $e\left({\omega }^2\right)$ and this number is $r-1$. Consequently, the difference between the edges labeled with $1$ $\left|e\right(x)-e(y\left)\right|\le 1,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$.

Case (2): $n\equiv 1(mod 3)$, i.e., $n=3r+1$. The total number of vertices labeled ${\omega }^2$ is denoted by $v\left({\omega }^2\right)$ is the same as the total number of vertices labeled $\omega$ denoted by $v\left(\omega \right)$ and this number is $r$, while the number of vertices of labeled $i$ denoted by $v\left(1\right)$ and this number is $r-1$. Consequently, $\left|v\right(x)-v(y\left)\right|\le 1,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$. Similarly, in this same one can see that the number of edges labeled $1$ denoted by $e\left(1\right)$ is the same as the number of edges labeled $\omega$ denoted by $e\left(\omega \right)$ and the same as the number of edges labeled ${\omega }^2$ denoted by $e\left({\omega }^2\right)$ and this number is $r$. Therefore, $\left|e\right(x)-e(y\left)\right|=0,x\ne y$ and $x,y\in \{i,\omega ,{\omega }^2\}$.

Case (3): $n\equiv 2(mod 3)$, i.e., $n=3r+2$. The total number of vertices labeled $i$ is denoted by $v\left(i\right)$ is the same as the total number of vertices labeled $\omega$ is denoted by $v\left(\omega \right)$ and this number is $r+1$, while the total number of vertices labeled $g$ is denoted by $v\left({\omega }^2\right)$ and this number is $r$. Therefore, $\left|v\right(x)-v(y\left)\right|\le 1,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$. Similarly, in this same one can see that the total number of edges labeled $1$ is denoted by $e\left(1\right)$ is the same as the total number of edges labeled ${\omega }^2$ denoted by $e\left({\omega }^2\right)$ and this number is $r$, while the number of edges labeled $\omega$ denoted by $e\left(\omega \right)$ and this number is $r+1$. Therefore, $\left|e\right(x)-e(y\left)\right|\le 1,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$.

Thus, we have seen in each case $\left|v\right(x)-v(y\left)\right|\le 1$ and $\left|e\right(x)-e(y\left)\right|\le 1$, $x\ne y$ and $x,y\in \{i,\omega ,{\omega }^2\}$. Hence the path $P_n,n\ge 2$ admits a cubic roots cordial labeling. □

Theorem 2.

Each cycle $C_n$, $n\ge 3$ admits a cubic roots cordial labeling.

Proof. 

Let $V=\{v_1,v_2,\dots ,v_n\}$ be the vertex set of $C_n$ an $E=v_r{v}_{r+1};1\le r\le n-1\}\cup v_1{v}_n$ be the edge set of $C_n$. Define vertex labeling $f:V\left(G\right)\to \{1,{\omega }^2,\omega \}$ as given in Table 1. □

Table 1. Labeling of Vertices of $C_n$.

$\mathbf{When} \mathit{n}\equiv 0, 1 (\mathit{m}\mathit{o}\mathit{d} 3)$	$\mathbf{And} \mathbf{when} \mathit{n}\equiv 2 (\mathit{m}\mathit{o}\mathit{d} 3)$
$f(v_r)=\left\{\begin{array}{ll}1& ;0 mod 3\\ \omega & ;1 mod 3\\ {\omega }^2& ;2 mod 3\end{array}\right..$	$f(v_r)=\left\{\begin{array}{ll}{\omega }^2& ;0 mod 3\\ 1& ;1 mod 3\\ \omega & ;2 mod 3\\ \omega & ;r=n-1\\ {\omega }^2& ;r=n\end{array}\right..$

Now, we defined the labeling of the edge set of $C_n$ using the function as follows in Table 2.

Table 2. Labeling of Edge set of $C_n$.

$h^{*}\left(e\right)$	$.$	$v\left(1\right)$	$v\left(\omega \right)$	$v\left({\omega }^2\right)$
	$v\left(1\right)$	$1$	$\omega$	${\omega }^2$
	$v\left(\omega \right)$	$\omega$	${\omega }^2$	$1$
	$v\left({\omega }^2\right)$	${\omega }^2$	$1$	$\omega$

It is follows that $C_n$, $n\ge 3$ admits cubic roots cordial labeling as required in Table 3.

Table 3. Cycle graph’s vertex and edge conditions.

$\mathit{n}$	$\mathit{v}\left(1\right)$	$\mathit{v}\left(\mathit{\omega }\right)$	$\mathit{v}\left({\mathit{\omega }}^2\right)$	$\begin{array}{l}\left|\mathit{v}\right(\mathit{x})-\mathit{v}(\mathit{y}\left)\right|\mathit{x}\ne \mathit{y}\\ \mathbf{a}\mathbf{n}\mathbf{d} \mathit{x},\mathit{y}\in \{1,\mathit{\omega },{\mathit{\omega }}^2\}\end{array}$
$\begin{array}{l}n\equiv 0 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r\end{array}$,	$r$	$r$	$r$	$0,0,0$
$\begin{array}{l}n\equiv 1 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r+1\end{array}$,	$r+1$	$r$	$r$	$1,1,0$
$\begin{array}{l}n\equiv 2 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r+2\end{array}$,	$r$	$r+1$	$r+1$	$1,1,0$
$n$	$e\left(1\right)$	$e\left(\omega \right)$	$e\left({\omega }^2\right)$	$\begin{array}{l}\left|e\right(x)-e(y\left)\right|x\ne y\\ \mathrm{a}\mathrm{n}\mathrm{d} x,y\in \{i,\omega ,{\omega }^2\}\end{array}$
$\begin{array}{l}n\equiv 0 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r\end{array}$,	$r$	$r$	$r$	$0,0,0$
$\begin{array}{l}n\equiv 1 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r+1\end{array}$,	$r+1$	$r$	$r$	$1,1,0$
$\begin{array}{l}n\equiv 2 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r+2\end{array}$,	$r$	$r+1$	$r+1$	$1,1,0$

Hence the cycle graph $C_n,n\ge 3$ a cubic roots cordial labeling.

Theorem 3.

Each Fan $F_n$, $n\ge 2$ admits cubic roots cordial labeling for all $n$.

Proof. 

Let $V=\{u,v_1,v_2,\dots ,v_n\}$ be the vertex set of $F_n$ and $E=\{v_r{v}_{r+1};1\le r\le n-1\}\cup \{u{v}_r;1\le r\le n\}$ its edges set. Define vertex labeling $h:V\to \{1,\omega ,{\omega }^2\}$ as follows in Table 4. □

Table 4. Labeling of Vertices of $F_n$.

$\mathbf{When} \mathit{n}\equiv 0(\mathit{m}\mathit{o}\mathit{d} 3)$	$\mathbf{When} \mathit{n}\equiv 1(\mathit{m}\mathit{o}\mathit{d} 3)$	$\mathbf{And} \mathbf{when} \mathit{n}\equiv 2(\mathit{m}\mathit{o}\mathit{d} 3),$$\mathit{n} \mathbf{odd}$
$h\left(u\right)=1$	$h\left(u\right)={\omega }^2$	$h\left(u\right)={\omega }^2$
$h(v_r)=\left\{\begin{array}{ll}1& ;0 mod 3\\ \omega & ;1 mod 3\\ {\omega }^2& ;2 mod 3\end{array}\right.$	$h(v_r)=\left\{\begin{array}{ll}1& ;0 mod 3\\ \omega & ;1 mod 3\\ {\omega }^2& ;2 mod 3\end{array}\right.$	$h(v_r)=\left\{\begin{array}{ll}1& ;0 mod 3\\ \omega & ;1 mod 3\\ {\omega }^2& ;2 mod 3\end{array}\right.$

The induced edge labeling $h^{*}$ is given in section (2) where our cubic roots is $\{1,\omega ,{\omega }^2\}$ Let us study the following three cases:

Case (1): $n\equiv 0(mod 3)$. The total number of vertices labeled $1$ is denoted by $v\left(1\right)$ is the same as the total number of vertices labeled ${\omega }^2$ and denoted by $v\left({\omega }^2\right)$ and this number is $r$, also the total number of vertices labeled $\omega$ denoted by $v\left(\omega \right)$ and this number is $r+1$, i.e., $v\left(1\right)=v\left({\omega }^2\right)=r,v\left(\omega \right)=r+1$. Obviously, $\left|v\left(x\right)-v\left(y\right)\right|\le 1,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$. Similarly, in this same one can see that the number of edges labeled $1$ denoted by $e\left(1\right)$ the same as the number of edges labeled $\omega$ denoted by $e\left(\omega \right)$ and also equivalent to the number of edges labeled ${\omega }^2$ denoted by $e\left({\omega }^2\right)$ and this number is $2r$. Consequently, $\left|e\left(x\right)-e\left(y\right)\right|=0,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$.

Case (2): $n\equiv 1(mod 3)$. The total number of vertices labeled ${\omega }^2$ is denoted by $v\left({\omega }^2\right)$ is the same as the total number of vertices labeled $1$ denoted by $v\left(1\right)$ and this number is $r$, while the number of vertices of labeled $\omega$ denoted by $v\left(\omega \right)$ and this number is $r+1$. Consequently, $\left|v\right(x)-v(y\left)\right|\le 1,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$. Similarly, in this same one can see that the number of edges labeled $1$ denoted by $e\left(1\right)$ is the same as the number of edges labeled $\omega$ denoted by $e\left(\omega \right)$ and this number is $2r$, while the number of edges labeled ${\omega }^2$ denoted by $e\left({\omega }^2\right)$ and this number is $2r+1$. Therefore, $\left|e\left(x\right)-e\left(y\right)\right|\le 1,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$.

Case (3): $n\equiv 2(mod 3)$. The total number of vertices labeled $1$ is denoted by $v\left(1\right)$ is the same as the total number of vertices labeled $\omega$ is denoted by $v\left(\omega \right)$ and also equivalent to the number of vertices labeled ${\omega }^2$ is denoted by $v\left({\omega }^2\right)$ and this number is $r+1$. Therefore, $\left|v\left(x\right)-v\left(y\right)\right|=0,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$. Similarly, in this same one can see that the number of edges labeled $1$ denoted by $e\left(1\right)$ the same as the number of edges labeled $\omega$ denoted by $e\left(\omega \right)$ and also equivalent to the number of edges labeled ${\omega }^2$ denoted by $e\left({\omega }^2\right)$ and this number is $2r+1$. Consequently, $\left|e\left(x\right)-e\left(y\right)\right|=0,x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$.

Thus, we have seen in each case $\left|v\right(x)-v(y\left)\right|\le 1$ and $\left|e\right(x)-e(y\left)\right|\le 1$, $x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$. Hence the fan $F_n,n\ge 2$ admits a cubic roots cordial labeling.

Theorem 4.

Each wheel $W_n$, $n\ge 3$ admits a cubic roots cordial labeling except $n\equiv 2 mod 3$ and  $n$  even.

Proof. 

Let $V=\{u,v_1,v_2,\dots ,v_n\}$ be the vertex set of $W_n$ and $E=\{v_r{v}_{r+1};1\le r\le n-1\}\cup v_1{v}_n\cup \{u{v}_r;1\le r\le n\}$ be the edge set of $W_n$. Define vertex labeling $h:V\left(G\right)\to \{1,\omega ,{\omega }^2\}$ as given in Table 5. □

Table 5. Labeling of Vertices of $W_n$.

$\mathbf{When} \mathit{n}\equiv 0 (\mathit{m}\mathit{o}\mathit{d} 3)$	$\mathbf{When} \mathit{n}\equiv 1 (\mathit{m}\mathit{o}\mathit{d} 3)$	$\mathbf{And} \mathbf{when} \mathit{n}\equiv 2 (\mathit{m}\mathit{o}\mathit{d} 3),$$\mathit{n} \mathbf{odd}$
$h\left(u\right)=1$	$h\left(u\right)={\omega }^2$	$h\left(u\right)=1$
$h\left(v_r\right)=\left\{\begin{array}{ll}1& ;0 mod 3\\ \omega & ;1 mod 3\\ {\omega }^2& ;2 mod 3\end{array}\right.$	$h\left(v_r\right)=\left\{\begin{array}{ll}1& ;0 mod 3\\ \omega & ;1 mod 3,r\ne n\\ {\omega }^2& ;2 mod 3\\ 1& ;r=n\end{array}\right.$	$h(v_r)=\left\{\begin{array}{ll}{\omega }^2& ;r=0 mod 3,r\ne n-1\\ 1& ;r=1 mod 3,r\ne n\\ \omega & ;r=2 mod 3\\ \omega & ;r=n-1\\ {\omega }^2& ;r=n\end{array}\right.$

Now, we defined the labeling of the edge set of $W_n$ using the function as follows in Table 6.

Table 6. Labeling of Edge set.

$h^{*}\left(e\right)$	$.$	$v\left(1\right)$	$v\left(\omega \right)$	$v\left({\omega }^2\right)$
	$v\left(1\right)$	$1$	$\omega$	${\omega }^2$
	$v\left(\omega \right)$	$\omega$	${\omega }^2$	$1$
	$v\left({\omega }^2\right)$	${\omega }^2$	$1$	$\omega$

It is follows that $W_n$, $n\ge 3$ admits cubic roots cordial labeling as required in Table 7.

Table 7. Wheel graph’s vertex and edge conditions.

$\mathit{n}$	$\mathit{v}\left(1\right)$	$\mathit{v}\left(\mathit{\omega }\right)$	$\mathit{v}\left({\mathit{\omega }}^2\right)$	$\begin{array}{l}\left|\mathit{v}\right(\mathit{x})-\mathit{v}(\mathit{y}\left)\right|\mathit{x}\ne \mathit{y}\\ \mathbf{a}\mathbf{n}\mathbf{d} \mathit{x},\mathit{y}\in \{1,\mathit{\omega },{\mathit{\omega }}^2\}\end{array}$
$\begin{array}{l}n\equiv 0 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r\end{array}$,	$r+1$	$r$	$r$	$1,1,0$
$\begin{array}{l}n\equiv 1 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r+1\end{array}$,	$r+1$	$r$	$r+1$	$1,0,1$
$\begin{array}{l}n\equiv 2 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r+2\end{array},$ $n$ odd	$r+1$	$r+1$	$r+1$	$0,0,0$
$n$	$e\left(1\right)$	$e\left(\omega \right)$	$e\left({\omega }^2\right)$	$\begin{array}{l}\left|e\right(x)-e(y\left)\right|x\ne y\\ \mathrm{a}\mathrm{n}\mathrm{d} x,y\in \{1,\omega ,{\omega }^2\}\end{array}$
$\begin{array}{l}n\equiv 0 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r\end{array}$,	$2r$	$2r$	$2r$	$0,0,0$
$\begin{array}{l}n\equiv 1 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r+1\end{array}$,	$2r+1$	$2r$	$2r+1$	$1,0,1$
$\begin{array}{l}n\equiv 2 mod 3\\ \mathrm{i}.\mathrm{e}.,n=3r+2\end{array},$ n odd	$2r+2$	$2r+1$	$2r+1$	$1,1,0$

Hence the Wheel graph $W_n,n\ge 3$ admits cubic roots cordial labeling except $n\equiv 2 mod 3$ and $n$ even.

4. Cubic Roots Cordial Labeling for Union Graphs

In this section, we will study the cubic roots cordial labeling of a union of two paths and similar study will be done of two union cycles, we end this section by studding the cubic roots cordial labeling the union cycles with paths.

Theorem 5.

The union of $P_n\cup P_m$, $n,m\ge 2$ admits a cubic roots cordial labeling for all  $n,m$.

Proof. 

Let $V=\{v_1,v_2,\dots ,v_n,{v^{\prime }}_1,{v^{\prime }}_2,\dots ,{v^{\prime }}_m\}$ be the vertex set of $P_n\cup P_m$ and $E=E_1\cup E_2$, where $E_1=\{v_r{v}_{r+1};1\le r\le n-1\}$, $E_2=\{{v^{\prime }}_s{v^{\prime }}_{s+1};1\le r\le m-1\}$, be the edge set of $P_n\cup P_m$. Define vertex labeling $f:V\to \{1,{\omega }^2,\omega \}$ of the $P_n\cup P_m$ as given in Table 8. □

Table 8. Labeling of $P_n$ and $P_m$.

$\begin{array}{l}\mathit{n}=3\mathit{r}+\mathit{j}\\ \mathit{r}\ge 0,\\ \mathit{j}=0,1,2\end{array}$	$\begin{array}{l}\mathbf{L}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{g} \mathbf{o}\mathbf{f}\\ {\mathit{P}}_{\mathit{n}}\\ \end{array}$	$\begin{array}{l}\\ \mathit{v}\left(1\right)\\ \end{array}$	$\begin{array}{l}\\ \mathit{v}\left(\mathit{\omega }\right)\\ \end{array}$	$\begin{array}{l}\\ \mathit{v}\left({\mathit{\omega }}^2\right)\\ \end{array}$	$\begin{array}{l}\\ \mathit{e}\left(1\right)\\ \end{array}$	$\begin{array}{l}\\ \mathit{e}\left(\mathit{\omega }\right)\\ \end{array}$	$\begin{array}{l}\\ \mathit{e}\left({\mathit{\omega }}^2\right)\\ \end{array}$
$\begin{array}{l}j=0\\ \end{array}$	$\begin{array}{l}{A}_0=A_{3r}\\ {A^{\prime }}_0=D_{3r}\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r-1\end{array}$	$\begin{array}{l}r-1\\ r\end{array}$
$j=1$	$A_1=D_{3r}\left(\omega \right)$	$r$	$r+1$	$r$	$r$	$r$	$r$
$j=2$	$A_2=A_{3r}\left({\omega }^{2}1\right)$	$r+1$	$r$	$r+1$	$r$	$r+1$	$r$
$\begin{array}{l}m=3s+j\\ s\ge 0,\\ j=0,1,2\end{array}$	$\begin{array}{l}\mathrm{L}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{o}\mathrm{f}\\ P_m\\ \end{array}$	$\begin{array}{l}\\ v^{\prime }\left(1\right)\\ \end{array}$	$\begin{array}{l}\\ v^{\prime }\left(\omega \right)\\ \end{array}$	$\begin{array}{l}\\ v^{\prime }\left({\omega }^2\right)\\ \end{array}$	$\begin{array}{l}\\ e^{\prime }\left(1\right)\\ \end{array}$	$\begin{array}{l}\\ e^{\prime }\left(\omega \right)\\ \end{array}$	$\begin{array}{l}\\ e^{\prime }\left({\omega }^2\right)\\ \end{array}$
$\begin{array}{l}j=0\\ \end{array}$	$\begin{array}{l}{B}_0=A_{3s}\\ {B^{\prime }}_0=D_{3s}\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s-1\end{array}$	$\begin{array}{l}s-1\\ s\end{array}$
$\begin{array}{l}j=1\\ \end{array}$	$\begin{array}{l}{B}_1=A_{3s}\left(1\right)\\ {B^{\prime }}_1=A_{3s}\left({\omega }^2\right)\end{array}$	$\begin{array}{l}s+1\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s+1\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$
$\begin{array}{l}j=2\\ \end{array}$	$\begin{array}{l}{B}_2=A_{3s}\left({\omega }^{2}1\right)\\ {B^{\prime }}_2=D_{3s}\left(\omega 1\right)\end{array}$	$\begin{array}{l}s+1\\ s+1\end{array}$	$\begin{array}{l}s\\ s+1\end{array}$	$\begin{array}{l}s+1\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s+1\\ s\end{array}$	$\begin{array}{l}s\\ s+1\end{array}$

Now, we defined the labeling of the edge set of $P_n$ using the function as follows in Table 9.

Table 9. Labeling of Edge set of $P_n$.

$f^{*}\left(e\right)$	$.$	$v\left(1\right)$	$v\left(\omega \right)$	$v\left({\omega }^2\right)$
	$v\left(1\right)$	$1$	$\omega$	${\omega }^2$
	$v\left(\omega \right)$	$\omega$	${\omega }^2$	$1$
	$v\left({\omega }^2\right)$	${\omega }^2$	$1$	$\omega$

Hence the deduced labeling of the union is shown in Table 10.

Table 10. Vertex and edge conditions of $P_n\cup P_m$.

$\begin{array}{l}\mathit{n}=3\mathit{r}+\mathit{j}\\ \mathit{r}\ge 0,\\ \mathit{j}=0, 1, 2\end{array}$	$\begin{array}{l}\mathit{m}=3\mathit{s}+\mathit{j},\\ \mathit{s}\ge 0,\\ \mathit{j}=0, 1, 2\end{array}$	$\begin{array}{l}\\ {\mathit{P}}_{\mathit{n}}\\ \end{array}$	$\begin{array}{l}\\ {\mathit{P}}_{\mathit{m}}\\ \end{array}$	$\begin{array}{l}\left|\mathit{v}\right(\mathit{x})-\mathit{v}(\mathit{y}\left)\right|\\ ,\mathit{x}\ne \mathit{y} \mathbf{a}\mathbf{n}\mathbf{d}\\ \mathit{x},\mathit{y}\in \{1, \mathit{\omega }, {\mathit{\omega }}^2\}\end{array}$	$\begin{array}{l}\left|\mathit{e}\right(\mathit{x})-\mathit{e}(\mathit{y}\left)\right|\\ ,\mathit{x}\ne \mathit{y} \mathbf{a}\mathbf{n}\mathbf{d}\\ \mathit{x},\mathit{y}\in \{1, \mathit{\omega }, {\mathit{\omega }}^2\}\end{array}$
$0$	$0$	$A_0$	${B^{\prime }}_0$	$0,0,0$	$1,1,0$
$0$	$1$	${A^{\prime }}_0$	$B_1$	$1,1,0$	$1,0,1$
$0$	$2$	${A^{\prime }}_0$	$B_2$	$1,0,1$	$0,0,0$
$1$	$1$	$A_1$	${B^{\prime }}_1$	$1,1,0$	$0,0,0$
$1$	$2$	$A_1$	$B_2$	$0,0,0$	$1,0,1$
$2$	$2$	$A_2$	${B^{\prime }}_2$	$1,1,0$	$1,1,0$

It is obvious that the difference $\left|v\right(x)-v(y\left)\right|$ and $\left|e\right(x)-e(y\left)\right|$,where $x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$ are always do not exceed one. Therefore $P_n\cup P_m$, $n,m\ge 2$ admits a cubic roots cordial labeling.

Now, we denote our attention to study the union of two cycles.

Theorem 6.

The union of  $C_n\cup C_m$ , $n,m\ge 3$ admits a cubic roots cordial labeling for all $n,m$.

Proof. 

Let $V=\{v_1,v_2,\dots ,v_n,{v^{\prime }}_1,{v^{\prime }}_2,\dots ,{v^{\prime }}_m\}$ be the vertex set of $C_n\cup C_m$ and $E=E_1\cup E_2$, where $E_1=\{v_r{v}_{r+1};1\le r\le n-1\}\cup v_1{v}_n$, $E_2=\{{v^{\prime }}_s{v^{\prime }}_{s+1};1\le r\le m-1\}\cup {v^{\prime }}_1{v^{\prime }}_m$ be the edge set of $C_n\cup C_m$. Define $f:V\to \{1,{\omega }^2,\omega \}$ to be the chosen labeling for the vertex set of each $C_n$ and $C_m$ as seen in Table 11. □

Table 11. Labeling of $C_n$ and $C_m$.

$\begin{array}{l}\mathit{n}=3\mathit{r}+\mathit{j},\\ \mathit{j}=0,1,2\end{array}$	$\begin{array}{l}\mathbf{L}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{g} \mathbf{o}\mathbf{f}\\ {\mathit{C}}_{\mathit{n}}\end{array}$	$\begin{array}{l}\mathit{v}\left(1\right)\\ \end{array}$	$\begin{array}{l}\mathit{v}\left(\mathit{\omega }\right)\\ \end{array}$	$\begin{array}{l}\mathit{v}\left({\mathit{\omega }}^2\right)\\ \end{array}$	$\begin{array}{l}\mathit{e}\left(1\right)\\ \end{array}$	$\begin{array}{l}\mathit{e}\left(\mathit{f}\mathit{\omega }\right)\\ \end{array}$	$\begin{array}{l}\mathit{e}\left({\mathit{\omega }}^2\right)\\ \end{array}$
$\begin{array}{l}j=0\\ \end{array}$	$\begin{array}{l}{A}_0=A_{3r}\\ {A^{\prime }}_0=D_{3r}\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$
$\begin{array}{l}j=1\\ \end{array}$	$\begin{array}{l}{A}_1=D_{3r}\left(\omega \right)\\ {A^{\prime }}_1=A_{3r}\left(1\right)\end{array}$	$\begin{array}{l}r\\ r+1\end{array}$	$\begin{array}{l}r+1\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r+1\end{array}$	$\begin{array}{l}r+1\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$
$\begin{array}{l}j=2\\ \end{array}$	$\begin{array}{l}{A}_2=A_{3r}\left({\omega }^{2}1\right)\\ {A^{\prime }}_2=D_{3s}\left(\omega 1\right)\end{array}$	$\begin{array}{l}r+1\\ r+1\end{array}$	$\begin{array}{l}r\\ r+1\end{array}$	$\begin{array}{l}r+1\\ r\end{array}$	$\begin{array}{l}r+1\\ r+1\end{array}$	$\begin{array}{l}r+1\\ r\end{array}$	$\begin{array}{l}r\\ r+1\end{array}$
$\begin{array}{l}m=3s+j,\\ j=0,1,2\end{array}$	$\begin{array}{l}\mathrm{L}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{o}\mathrm{f}\\ C_n\end{array}$	$\begin{array}{l}{v}^{\prime }\left(1\right)\\ \end{array}$	$\begin{array}{l}{v}^{\prime }\left(\omega \right)\\ \end{array}$	$\begin{array}{l}{v}^{\prime }\left({\omega }^2\right)\\ \end{array}$	$\begin{array}{l}{e}^{\prime }\left(1\right)\\ \end{array}$	$\begin{array}{l}{e}^{\prime }\left(\omega \right)\\ \end{array}$	$\begin{array}{l}{e}^{\prime }\left({\omega }^2\right)\\ \end{array}$
$\begin{array}{l}j=0\\ \end{array}$	$\begin{array}{l}{B}_0=A_{3s}\\ {B^{\prime }}_0=D_{3s}\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$
$\begin{array}{l}j=1\\ \end{array}$	$\begin{array}{l}{B}_1=A_{3s}\left(1\right)\\ {B^{\prime }}_1=A_{3s}\left({\omega }^2\right)\end{array}$	$\begin{array}{l}s+1\\ s\end{array}$	$\begin{array}{l}s\\ s\end{array}$	$\begin{array}{l}s\\ s+1\end{array}$	$\begin{array}{l}s+1\\ s\end{array}$	$\begin{array}{l}s\\ s+1\end{array}$	$\begin{array}{l}s\\ s\end{array}$
$\begin{array}{l}\\ j=2\\ \end{array}$	$\begin{array}{l}{B}_2=A_{3s}\left({\omega }^{2}1\right)\\ {B^{\prime }}_2=D_{3s}\left(\omega 1\right)\\ B_2^{\mathrm{”}}=B_{3s}\left(\omega {\omega }^2\right)\end{array}$	$\begin{array}{l}s+1\\ s+1\\ s\end{array}$	$\begin{array}{l}s\\ s+1\\ s+1\end{array}$	$\begin{array}{l}s+1\\ s\\ s+1\end{array}$	$\begin{array}{l}s+1\\ s+1\\ s\end{array}$	$\begin{array}{l}s+1\\ s\\ s+1\end{array}$	$\begin{array}{l}s\\ s+1\\ s+1\end{array}$

So, one can define the edge labeling $f^{*}$ for $C_n\cup C_m$ as follows in Table 9.

We have the vertex labelling based on the previously mentioned labelling pattern of $C_n\cup C_m$ and also the edges labeling of it as indicated in the following Table 12.

Table 12. Vertex and edge conditions of $C_n\cup C_m$.

$\begin{array}{l}\mathit{n}=3\mathit{r}+\mathit{j}\\ \mathit{r}\ge 1,\\ \mathit{j}=0,1,2\end{array}$	$\begin{array}{l}\mathit{m}=3\mathit{s}+\mathit{j}\\ \mathit{s}\ge 1,\\ \mathit{j}=0,1,2\end{array}$	$\begin{array}{l}\\ {\mathit{C}}_{\mathit{n}}\\ \end{array}$	$\begin{array}{l}\\ {\mathit{C}}_{\mathit{m}}\\ \end{array}$	$\begin{array}{l}\left|\mathit{v}\right(\mathit{x})-\mathit{v}(\mathit{y}\left)\right|\\ \mathit{x}\ne \mathit{y} \mathbf{a}\mathbf{n}\mathbf{d}\\ \mathit{x},\mathit{y}\in \{1,\mathit{\omega },{\mathit{\omega }}^2\}\end{array}$	$\begin{array}{l}\left|\mathit{e}\right(\mathit{x})-\mathit{e}(\mathit{y}\left)\right|\\ \mathit{x}\ne \mathit{y} \mathbf{a}\mathbf{n}\mathbf{d}\\ \mathit{x},\mathit{y}\in \{1,\mathit{\omega },{\mathit{\omega }}^2\}\end{array}$
$0$	$0$	$A_0$	$B_0$	$0,0,0$	$0,0,0$
$0$	$1$	$A_0$	$B_1$	$1,1,0$	$1,1,0$
$0$	$2$	$A_0$	$B_2$	$1,0,1$	$0,1,1$
$1$	$1$	$A_1$	$B_1$	$0,1,1$	$0,1,1$
$1$	$2$	${A^{\prime }}_1$	$B{\mathrm{”}}_2$	$0,0,0$	$0,0,0$
$2$	$2$	$A_2$	${B^{\prime }}_2$	$1,1,0$	$1,1,0$

It is obvious that the difference $\left|v\right(x)-v(y\left)\right|$ and $\left|e\right(x)-e(y\left)\right|$, where $x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$ are always donate exceed one. Therefore $C_n\cup C_m$, $n,m\ge 3$ admits a cubic roots cordial labeling.

Finally, we study the cubic roots cordial labeling for $P_n\cup C_m$.

Theorem 7.

The union of  $P_n\cup C_m$, $n\ge 2,m\ge 3$ admits a cubic roots cordial by for all $n,m$.

Proof. 

Let $V=\{v_1,v_2,\dots ,v_n,{v^{\prime }}_1,{v^{\prime }}_2,\dots ,{v^{\prime }}_m\}$ be the vertex set of union of $P_n\cup C_m$ and $E=E_1\cup E_2$, where $E_1=\{v_r{v}_{r+1};1\le r\le n-1\}$, $E_2=\{v_s^{\prime }{v}_{s+1}^{\prime };1\le r\le m-1\}\cup v_1^{\prime }{v}_m^{\prime }$ be the edge set of the union of $P_n\cup C_m$. Define vertex labeling $f:V\to \{1,{\omega }^2,\omega \}$ to be the chosen labeling for the vertex set of each $P_n$ and $C_n$ as seen in Table 13. □

Table 13. Labeling of $P_n$ and $C_m$.

$\begin{array}{l}\mathit{n}=3\mathit{r}+\mathit{j}\\ \mathit{j}=0,1,2\end{array}$	$\begin{array}{l}\mathbf{L}\mathbf{a}\mathbf{b}\mathbf{e}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{g} \mathbf{o}\mathbf{f}\\ {\mathit{P}}_{\mathit{n}}\end{array}$	$\begin{array}{l}\mathit{v}\left(1\right)\\ \end{array}$	$\begin{array}{l}\mathit{v}\left(\mathit{\omega }\right)\\ \end{array}$	$\begin{array}{l}\mathit{v}\left({\mathit{\omega }}^2\right)\\ \end{array}$	$\begin{array}{l}\mathit{e}\left(1\right)\\ \end{array}$	$\begin{array}{l}\mathit{e}\left(\mathit{\omega }\right)\\ \end{array}$	$\begin{array}{l}\mathit{e}\left({\mathit{\omega }}^2\right)\\ \end{array}$
$\begin{array}{l}j=0\\ \end{array}$	$\begin{array}{l}{A}_0=A_{3r}\\ {A^{\prime }}_0=B_{3r}\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r\\ r-1\end{array}$	$\begin{array}{l}r\\ r\end{array}$	$\begin{array}{l}r-1\\ r\end{array}$
$j=1$	$A_1=A_{3r}\left(1\right)$	$r+1$	$r$	$r$	$r$	$r$	$r$
$\begin{array}{l}j=2\\ \end{array}$	$\begin{array}{l}{A}_2=A_{3r}\left(1\omega \right)\\ {A^{\prime }}_2=A_{3r}({\omega }^2\omega )\end{array}$	$\begin{array}{l}r+1\\ r\end{array}$	$\begin{array}{l}r+1\\ r+1\end{array}$	$\begin{array}{l}r\\ r+1\end{array}$	$\begin{array}{l}r\\ r+1\end{array}$	$\begin{array}{l}r\\ r+1\end{array}$	$\begin{array}{l}r+1\\ r-1\end{array}$
$\begin{array}{l}m=3s+j\\ j=0,1,2\end{array}$,	$\begin{array}{l}\mathrm{L}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{o}\mathrm{f}\\ C_m\end{array}$	$\begin{array}{l}{v}^{\prime }\left(1\right)\\ \end{array}$	$\begin{array}{l}{v}^{\prime }\left(\omega \right)\\ \end{array}$	$\begin{array}{l}{v}^{\prime }\left({\omega }^2\right)\\ \end{array}$	$\begin{array}{l}{e}^{\prime }\left(1\right)\\ \end{array}$	$\begin{array}{l}{e}^{\prime }\left(\omega \right)\\ \end{array}$	$\begin{array}{l}{e}^{\prime }\left({\omega }^2\right)\\ \end{array}$
$j=0$	$B_0=A_{3s}$	$s$	$s$	$s$	$s$	$s$	$s$
$\begin{array}{l}\\ j=1\\ \end{array}$	$\begin{array}{l}{B}_1=A_{3s}\left(1\right)\\ {B^{\prime }}_1=A_{3s}\left({\omega }^2\right)\\ B_1^{*}=D_{3s}({\omega }^2)\end{array}$	$\begin{array}{l}s+1\\ s\\ s\end{array}$	$\begin{array}{l}s\\ s\\ s\end{array}$	$\begin{array}{l}s\\ s+1\\ s+1\end{array}$	$\begin{array}{l}s+1\\ s\\ s+1\end{array}$	$\begin{array}{l}s\\ s+1\\ s-1\end{array}$	$\begin{array}{l}s\\ s\\ s+1\end{array}$
$\begin{array}{l}\\ j=2\\ \end{array}$	$\begin{array}{l}{B}_2=B_{3s}\left({\omega }^{2}1\right)\\ {B^{{}^{\prime }}}_2=B_{3s}(\omega {\omega }^2)\\ B_2^{*}=A_{3s}({\omega }^{2}1)\end{array}$	$\begin{array}{l}s+1\\ s\\ s+1\end{array}$	$\begin{array}{l}s\\ s+1\\ s\end{array}$	$\begin{array}{l}s+1\\ s+1\\ s+1\end{array}$	$\begin{array}{l}s\\ s\\ s+1\end{array}$	$\begin{array}{l}s\\ s+1\\ s+1\end{array}$	$\begin{array}{l}s+2\\ s+1\\ s\end{array}$

The edge labeling follows in Table 9. Hence the deduced labeling of the union is shown in Table 14.

Table 14. Vertex and edge conditions of $P_n\cup C_m$.

$\begin{array}{l}\mathit{n}=3\mathit{r}+\mathit{j}\\ \mathit{r}\ge 0,\\ \mathit{j}=0,1,2\end{array}$	$\begin{array}{l}\mathit{m}=3\mathit{s}+\mathit{j},\\ \mathit{s}\ge 1,\\ \mathit{j}=0,1,2\end{array}$	$\begin{array}{l}\\ {\mathit{P}}_{\mathit{n}}\\ \end{array}$	$\begin{array}{l}\\ {\mathit{C}}_{\mathit{m}}\\ \end{array}$	$\begin{array}{l}\left|\mathit{v}\right(\mathit{x})-\mathit{v}(\mathit{y}\left)\right|\\ ,\mathit{x}\ne \mathit{y} \mathbf{a}\mathbf{n}\mathbf{d}\\ \mathit{x},\mathit{y}\in \{1,\mathit{\omega },{\mathit{\omega }}^2\}\end{array}$	$\begin{array}{l}\left|\mathit{e}\right(\mathit{x})-\mathit{e}(\mathit{y}\left)\right|\\ ,\mathit{x}\ne \mathit{y} \mathbf{a}\mathbf{n}\mathbf{d}\\ \mathit{x},\mathit{y}\in \{1,\mathit{\omega },{\mathit{\omega }}^2\}\end{array}$
$0$	$0$	$A_0$	$B_0$	$0,0,0$	$0,1,1$
$0$	$1$	${A^{\prime }}_0$	$B_1$	$1,1,0$	$0,0,0$
$0$	$2$	$A_0$	$B_2$	$1,0,1$	$0,1,1$
$1$	$0$	$A_1$	$B_0$	$1,1,0$	$0,0,0$
$1$	$1$	$A_1$	${B^{\prime }}_1$	$1,0,1$	$1,0,1$
$1$	$2$	$A_1$	${B^{\prime }}_2$	$0,0,0$	$1,1,0$
$2$	$0$	$A_2$	$B_0$	$0,1,1$	$0,1,1$
$2$	$1$	$A_2$	${B^{\prime }}_1$	$0,0,0$	$1,1,0$
$2$	$2$	$A_2$	$B_2^{*}$	$1,1,0$	$0,0,0$

It is obvious that the difference $\left|v\right(x)-v(y\left)\right|$ and $\left|e\right(x)-e(y\left)\right|$, where $x\ne y$ and $x,y\in \{1,\omega ,{\omega }^2\}$ are always donate exceed one. Therefore $P_n\cup C_m$, $n\ge 2,m\ge 3$ admits a cubic roots cordial labeling.

5. Conclusions

We proved that each path $P_n$, $n\ge 2$ admits a cubic roots cordial labeling for all $n$. Each cycle $C_n$, $n\ge 3$ admits a permuted cordial labeling. Each Fan $F_n$, $n\ge 2$ admits permuted cordial labeling for all $n$. The Wheel graph $W_n,n\ge 3$ admits permuted cordial labeling except $n\equiv 2 mod 3$ and $n$ even. The union of $P_n\cup P_m$, $n,m\ge 2$ admits a cubic roots cordial labeling for all $n,m$. The union of $C_n\cup C_m$, $n,m\ge 3$ admits a cubic roots cordial labeling for all $n,m$. The union of $P_n\cup C_m$, $n\ge 2,m\ge 3$ admits a cubic roots cordial labeling for all $n,m$. In the future, we will apply a cubic roots cordial labeling to other types of graphs.