Edge irregular reflexive labeling for the $ r $-th power of the path

Mohamed Basher; Mohamed Basher

doi:10.3934/math.2021604

AIMS Mathematics

2021, Volume 6, Issue 10: 10405-10430. doi: 10.3934/math.2021604

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Research article

Edge irregular reflexive labeling for the $r$ -th power of the path

Mohamed Basher ^,

Department of Mathematics and Computer Science, Faculty of Science, Suez University, Suez, Egypt

Received: 05 May 2021 Accepted: 13 July 2021 Published: 19 July 2021
MSC : 05C12, 05C78, 05C90

Let $G(V, E)$ be a graph, where $V(G)$ is the vertex set and $E(G)$ is the edge set. Let $k$ be a natural number, a total k-labeling $\varphi:V(G)\bigcup E(G)\rightarrow \{0, 1, 2, 3, ..., k\}$ is called an edge irregular reflexive $k$ -labeling if the vertices of $G$ are labeled with the set of even numbers from $\{0, 1, 2, 3, ..., k\}$ and the edges of $G$ are labeled with numbers from $\{1, 2, 3, ..., k\}$ in such a way for every two different edges $xy$ and $x^{'}y^{'}$ their weights $\varphi(x)+\varphi(xy)+\varphi(y)$ and $\varphi(x^{'})+\varphi(x^{'}y^{'})+\varphi(y^{'})$ are distinct. The reflexive edge strength of $G$ , $res(G)$ , is defined as the minimum $k$ for which $G$ has an edge irregular reflexive $k$ -labeling. In this paper, we determine the exact value of the reflexive edge strength for the $r$ -th power of the path $P_{n}$ , where $r\geq2$ , $n\geq r+4$ .

Keywords:

Citation: Mohamed Basher. Edge irregular reflexive labeling for the $r$ -th power of the path[J]. AIMS Mathematics, 2021, 6(10): 10405-10430. doi: 10.3934/math.2021604

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Abstract

1. Introduction

Throughout this paper we consider $G$ a connected, simple, and undirected graph, where $V$ and $E$ are denote to sets of vertices and edges of $G$ with cardinalities $|V|$ and $|E|$ , respectively. Chartrand et al. presented in ^[12] the edge $k$ -labeling of graph $G$ , $\varphi:E(G)\rightarrow \{1, 2, 3, ..., k\}$ such that the aggregate of the labels of edges incident with a vertex is different for all the vertices of $G$ . Such labelings were referred to as irregular assignments and irregular strength, $s(G)$ , of a graph $G$ is known as the smallest $k$ for which $G$ has an irregular assignment using labels at most $k$ . In ^[16] Lahel gives a comprehensive over view of the strength of irregularity. For some studies on irregularity strength see papers by Amar and Tongi ^[5], Dimitiz et al. ^[13], Gyárfás ^[14], and Nierhoff ^[17]. In ^[7] Bača et al. motivated the concept of irregular strength and started to investigate the total edge irregularity strength of a graph. An edge irregular total $k$ -labeling of the graph $G$ is a labeling $\varphi:V(G)\bigcup E(G)\rightarrow \{1, 2, 3, ..., k\}$ such that for every two distinct edges $xy$ , $x^{'}y^{'}$ of $G$ the edge weights $wt_{\varphi}(xy) = \varphi(x)+\varphi(xy)+\varphi(y)\neq$ $wt_{\varphi}(x^{'}y^{'}) = \varphi(x^{'})+\varphi(x^{'}y^{'})+\varphi(y^{'})$ . The total edge irregularity strength, $tes(G)$ is defined as the smallest $k$ for which $G$ has an edge irregular total $k$ -labeling. Some interesting studies on the total edge irregularity strength can be seen in ^{[1,2,3,4,8,9]}. Further, Ryan, Munasinghe, and Tanna in ^[18] introduced the concept of the edge irregular reflexive. For a graph $G(V, E)$ they define an edge labeling $\varphi_{e}:E(G)\rightarrow \{1, 2, 3, ..., k_{e}\}$ and a vertex labeling $\varphi_{v}:V(G)\rightarrow \{0, 2, 4, ..., 2k_{v}\}$ , then defined the labeling $\varphi$ by:

$\begin{align*} \varphi(x) = \left\{ \begin{array}{l} \varphi_{v}(x) , \ \ \ \ \ \ \ \ {{\rm if}\; x \in V(G) }\\ \varphi_{e}(x), \ \ \ \ \ \ \ \ {{\rm if}\; x \in E(G) } \end{array} \right. \end{align*}$

is a total $k$ -labeling, where $k = max\{k_{e}, 2k_{v}\}$ . Moreover, if for every two different edges $xy$ and $x^{'}y^{'}$ of $G$ one has $wt_{\varphi}(xy)\neq wt_{\varphi}(x^{'}y^{'})$ , where $wt_{\varphi}(xy) = \varphi_{e}(xy)+\varphi_{v}(x)+ \varphi_{v}(y)$ , then the total $k$ -labeling $\varphi$ is called an edge irregular reflexive labeling of graph $G$ . The reflexive edge strength, $res(G)$ , is defined as the smallest $k$ for which $G$ has an edge irregular reflexive $k$ -labeling. For more research on reflexive edge strength see ^{[6,10,11,15,20,21,22]}. In this paper, we estimate the exact value of the reflexive edge strength for the $r$ -th power of the path $P_{n}$ , where $r\geq2$ , $n\geq r+4$ .

Definition 1.1. (^[19]) The $r$ -th power of a graph $G$ , denoted by $G^{r}$ , is a graph with the same vertex set of $G$ such that adding edges between the vertices which are at distance at most $r$ , see Figure 1.

Figure 1. The

$3$ -th power of

$P_{8}$ .

DownLoad: Full-Size Img PowerPoint

When we prove the result, we will often use the following lemma, which has been proved in ^[18].

Lemma 1.1. (^[18]). For every graph $G$ ,

$\begin{align*} res(G)\geq\left\{ \begin{array}{l} \lceil\frac{|E(G)|}{3}\rceil , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathit{{{ if}\; |E(G)|\not\equiv {\rm{2,3}} (mod \;6),}}\\ \lceil\frac{|E(G)|}{3}\rceil +1, \ \ \ \ \ \ \ \ \ \ \mathit{{ if |E(G)|\equiv {\rm{2,3}} (mod \;6)}}. \end{array} \right. \end{align*}$

Furthermore, Bača et al. ^[10] proposed the following conjecture:

Conjecture 1.1. (^[10]) Consider the graph $G$ , which has a maximum degree $\triangle = \triangle(G)$ . Hence:

$\begin{equation*} res(G) = max\{\big\lfloor\frac{\triangle+2}{2}\big\rfloor,\big\lceil\frac{|E(G)|}{3}\big\rceil+r\} \end{equation*}$

where $r = 1$ for $|E(G)|\equiv2, 3$ (mod 6), and zero otherwise.

2. The $r$ -th power of the path

The $r$ -th power of a path $P_{n}$ denoted by $P_{n}^{r}$ , $n\geq3, r\geq2$ . Let us denote to the vertex set and edge set of $P_{n}^{r}$ by $V(P_{n}^{r}) = \{x_{i}, 1\leq i \leq n\}$ and $E(P_{n}^{r}) = \overset{r}{\underset{j = 1}\bigcup} \{x_{i}x_{i+j}, 1\leq i \leq n-j\}$ . In the next theorem, we determine the reflexive edge strength of various powers of a path $P_{n}$ .

Theorem 1. For the $r$ -th power of a path $P_{n}$ , $r\geq2$ , $n\geq r+4$ .

$\begin{align*} res(P_{n}^{r}) = \left\{ \begin{array}{l} \lceil\frac{r(2n-r-1)}{6}\rceil , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathit{{{ if}\; \frac{r(2n-r-1)}{2}\not\equiv {\rm{2,3}} (mod \;6),}}\\ \lceil\frac{r(2n-r-1)}{6}\rceil +1, \ \ \ \ \ \ \ \ \ \ \mathit{{{ if}\; \frac{r(2n-r-1)}{2}\equiv {\rm{2,3}} (mod \;6)}}. \end{array} \right. \end{align*}$

Proof. Note that the $r$ -th power of $P_{n}$ has $\frac{r(2n-r-1)}{2}$ edges. The lower bound for $res$ of the $r$ -th power of $P_{n}$ is as follow from the Lemma 1. $res(P_{n}^{r})\geq k = \lceil\frac{r(2n-r-1)}{6}\rceil +1$ if $\frac{r(2n-r-1)}{2}\equiv 2, 3$ (mod 6) and $res(P_{n}^{r})\geq k = \lceil\frac{r(2n-r-1)}{6}\rceil$ if $\frac{r(2n-r-1)}{2}\not\equiv 2, 3$ (mod 6). Moreover, we prove that:

$\begin{align*} res(P_{n}^{r})\leq\left\{ \begin{array}{l} \lceil\frac{r(2n-r-1)}{6}\rceil , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{\rm if}\; \frac{r(2n-r-1)}{2}\not\equiv 2,3 {\rm (mod \;6)}, }\\ \lceil\frac{r(2n-r-1)}{6}\rceil +1, \ \ \ \ \ \ \ \ \ \ { {\rm if} \frac{r(2n-r-1)}{2}\equiv 2,3 {\rm (mod \;6)} }. \end{array} \right. \end{align*}$

Let $k^{*} = max\{\lfloor\frac{2k+r(r+1)-4}{2r}\rfloor, 3\}$ and for $n\geq r+4$ , we recognise two cases.

Case 1. When $\lceil\frac{n-k^{*}}{2}\rceil\geq r$ .

Construct the total $k$ -labeling $\varphi$ of $P_{n}^{r}$ in the following way:

The corresponding labeling for $P^{3}_{8}$ is illustrated in Figure 2. Otherwise we have the following labeling:

$\begin{align*} \varphi(x_{i}) = \left\{ \begin{array}{l} 0 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 1\leq i \leq k^{*} }\\ 2\lfloor\frac{rk^{*}}{4}\rfloor, \ \ \ \ \ \ \ { k^{*}+1\leq i \leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil }\\ k, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+1\leq i \leq n } \end{array} \right. \end{align*}$

Figure 2. A reflexive irregular

$6$ -labeling of

$P^{3}_{8}$ .

DownLoad: Full-Size Img PowerPoint

Furthermore, the labels of edges are defined as the following :

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} i , \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-1 }\\ \frac{r(2k^{*}-r-1)}{2}-2\lfloor\frac{rk^{*}}{4}\rfloor+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { i = k^{*} }\\ (r-1)k^{*}-4\lfloor\frac{rk^{*}}{4}\rfloor+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+1\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil -1 }\\ \frac{r(2k^{*}-r-1)}{2}+r\lceil\frac{n-k^{*}}{2}\rceil-\\-k-2\lfloor\frac{rk^{*}}{4}\rfloor+1,\ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ i = k^{*} +\lceil\frac{n-k^{*}}{2}\rceil }\\ \frac{r(2n-r-1)}{2}-\frac{\lfloor\frac{n-k^{*}}{2}\rfloor(\lfloor\frac{n-k^{*}}{2}\rfloor-3)}{2}-\\-n-2k+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+1\leq i \leq n-1 } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+2}) = \left\{ \begin{array}{l} k^{*}+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-2 }\\ (r-1)k^{*}-\frac{r(r+1)}{2}-\\-2\lfloor\frac{rk^{*}}{4}\rfloor+i+3, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-1\leq i\leq k^{*} }\\ (r-1)k^{*}-4\lfloor\frac{rk^{*}}{4}\rfloor+\\+\lceil\frac{n-k^{*}}{2}\rceil+i-1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil -2 }\\ (r-1)(k^{*}+\lceil\frac{n-k^{*}}{2}\rceil)-\frac{r(r+1)}{2}-\\-2\lfloor\frac{rk^{*}}{4}\rfloor-k+i+3,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-1\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil }\\ \frac{r(2n-r-1)}{2}-\frac{\lfloor\frac{n-k^{*}}{2}\rfloor(\lfloor\frac{n-k^{*}}{2}\rfloor-5)}{2}-\\-n-2k+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+1\leq i \leq n-2 } \end{array} \right. \end{align*}$

For $3\leq j\leq r$ , there exist three subcases:

Subcase 1.1. If $\lfloor\frac{n-k^{*}}{2}\rfloor > r$ , hence the edges label as following:

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-j+1}x_{k^{*}})+i, \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-j }\\ \varphi(x_{k^{*}}x_{k^{*}+j-1})-k^{*}+j+i, \ \ \ \ \ \ \ \ \ \ { k^{*}-j+1\leq i\leq k^{*} }\\ \varphi(x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-j+1}x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil})-\\-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil -j }\\ \varphi(x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil}x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+j-1})-\\-\lceil\frac{n-k^{*}}{2}\rceil-k^{*}+j+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-j+1\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil }\\ \varphi(x_{n-j+1}x_{n})-\lceil\frac{n-k^{*}}{2}\rceil-k^{*}+i, \ \ \ \ \ \ { k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+1\leq i \leq n-j } \end{array} \right. \end{align*}$

Subcase 1.2. If $\lfloor\frac{n-k^{*}}{2}\rfloor = r$ , here for $3\leq j\leq r-1$ the edge labels are common the pervious subcase, and for $j = r$ we define edge labels as follows:

$\begin{align*} \varphi(x_{i}x_{i+r}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-r+1}x_{k^{*}})+i, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-r }\\ \varphi(x_{k^{*}}x_{k^{*}+r-1})-k^{*}+r+i, \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}-r+1\leq i\leq k^{*} }\\ \varphi(x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil}x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+r-1})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+1\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil }. \end{array} \right. \end{align*}$

Subcase 1.3. If $\lfloor\frac{n-k^{*}}{2}\rfloor = r-1$ , for $3\leq j\leq r-2$ the edge labels are common the Subcase 1.1 now, we construct edge labels only for $j = r-1$ and $j = r$ as follows:

$\begin{align*} \varphi(x_{i}x_{i+r-1}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-r+2}x_{k^{*}})+i, \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-r+1 }\\ \varphi(x_{k^{*}}x_{k^{*}+r-2})-k^{*}+r+i-1, \ \ \ \ \ \ \ { k^{*}-r+2\leq i\leq k^{*} }\\ \varphi(x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-r+2}x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil})-\\-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-r+1 }\\ \varphi(x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil}x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+r-2})-\\-k^{*}-\lceil\frac{n-k^{*}}{2}\rceil+r+i-1,\ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-r+2\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-1 . } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+r}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-r+1}x_{k^{*}})+i, \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-r }\\ \varphi(x_{k^{*}}x_{k^{*}+r-1})-k^{*}+r+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-r+1\leq i\leq k^{*} }\\ \varphi(x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil}x_{k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+r-1})-k^{*}+i,\ \ \ \ \ \ \ \ { k^{*}+1\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-1 . } \end{array} \right. \end{align*}$

An explanation of above labeling is depicted in . Evidently the vertices of $P^{r}_{n}$ labeled with even numbers. Hence we will compute the weights of edges under the labeling $\varphi$ :

Figure 3. A reflexive irregular

$10$ -labeling of

$P^{3}_{12}$ .

DownLoad: Full-Size Img PowerPoint

The edge set of $P_{n}^{r}$ can be divided into five mutually separated subsets, $A_{s}, 1\leq s\leq 5$ as follows:

For $1\leq j \leq r$ ,

● $A_{1} = \{x_{i}x_{i+j}:1\leq i\leq k^{*}-j \}$ : The set of all edges which have endpoints tagged with $0$ ,

● $A_{2} = \{x_{i}x_{i+j}:k^{*}-j\leq i\leq k^{*}\}$ : The set of all edges which have endpoints tagged with $0$ and $2\lfloor\frac{rk^{*}}{4}\rfloor$ ,

● $A_{3} = \{x_{i}x_{i+j}:k^{*}\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-j \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{rk^{*}}{4}\rfloor$ ,

● $A_{4} = \{x_{i}x_{i+j}:k^{*}+\lceil\frac{n-k^{*}}{2}\rceil-r\leq i\leq k^{*}+\lceil\frac{n-k^{*}}{2}\rceil\}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{rk^{*}}{4}\rfloor$ and $k$ ,

● $A_{5} = \{x_{i}x_{i+j}:k^{*}+\lceil\frac{n-k^{*}}{2}\rceil+1\leq i\leq n-j\}$ : The set of all edges which have endpoints tagged with $k$ .

Therefore, the edge weights of $P_{n}^{r}$ under the labeling $\varphi$ are the following:

1. The edge weights of the set $A_{1}$ , get the consecutive numbers from the set $\{1, 2, ..., \frac{r(2k^{*}-r-1)}{2}\}$ ,

2. The edge weights of the set $A_{2}$ , receive the consecutive numbers from the set $\{\frac{r(2k^{*}-r-1)}{2}+1, ..., rk^{*}\}$ ,

3. The edge weights of the set $A_{3}$ , get the consecutive numbers from the set $\{rk^{*}+1, ..., \frac{r(2k^{*}-r-1)}{2}+r\lceil\frac{n-k^{*}}{2}\rceil\}$ ,

4. The edge weights of the set $A_{4}$ , get the consecutive numbers from the set $\{\frac{r(2k^{*}-r-1)}{2}+r\lceil\frac{n-k^{*}}{2}\rceil+1, ..., \frac{r(2n-r-1)}{2}-\frac{\lfloor\frac{n-k^{*}}{2}\rfloor(\lfloor\frac{n-k^{*}}{2}\rfloor-1)}{2}\}$ ,

5. Finally, the edge weights of the set $A_{5}$ , receive the consecutive numbers from the set $\{\frac{r(2n-r-1)}{2}-\frac{\lfloor\frac{n-k^{*}}{2}\rfloor(\lfloor\frac{n-k^{*}}{2}\rfloor-1)}{2}+1, ..., \frac{r(2n-r-1)}{2}\}$ .

An explanation of above corresponding weights is depicted in . It is easy to check that the weights of the edges are different numbers from the set $\{1, 2, 3, ..., \frac{r(2n-r-1)}{2}\}$ .

Figure 4. The edge weights of

$P^{3}_{12}$ .

DownLoad: Full-Size Img PowerPoint

Case 2. When $\lceil\frac{n-k^{*}}{2}\rceil < r$ .

In this case we have three subcases.

Subcase 2.1. If $k^{*} < \frac{n}{2}$ . We can define the total $k$ -labeling $\varphi$ of $P_{n}^{r}$ as follows:

The corresponding labelings for $P^{3}_{7}$ , $P^{5}_{9}$ and $P^{4}_{10}$ are illustrated in Figure 7, 8 and 9 respectively. Otherwise we have following labeling:

$\begin{align*} \varphi(x_{i}) = \left\{ \begin{array}{l} 0 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 1\leq i \leq k^{*} }\\ 2\lceil\frac{k}{4}\rceil, \ \ \ \ \ \ \ \ \ { k^{*}+1\leq i \leq n-k^{*}+1 }\\ k, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { n-k^{*}+2\leq i \leq n } \end{array} \right. \end{align*}$

Figure 5. A reflexive irregular

$22$ -labeling of

$P^{7}_{13}$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. The edge weights of

$P^{7}_{13}$ .

DownLoad: Full-Size Img PowerPoint

Figure 7. A reflexive irregular

$6$ -labeling of

$P^{3}_{7}$ .

DownLoad: Full-Size Img PowerPoint

Figure 8. A reflexive irregular

$10$ -labeling of

$P^{5}_{9}$ .

DownLoad: Full-Size Img PowerPoint

Figure 9. A reflexive irregular

$10$ -labeling of

$P^{4}_{10}$ .

DownLoad: Full-Size Img PowerPoint

Moreover, to define edge labels for $P_{n}^{r}$ we have two subcases:

Subcase 2.1.1 If $k^{*}\geq r$ , then we construct the edge labels as follows:

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-1 }\\ \frac{r(2k^{*}-r-1)}{2}-2\lceil\frac{k}{4}\rceil+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*} }\\ (r-1)k^{*}-4\lceil\frac{k}{4}\rceil+i,\ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i\leq n-k^{*} }\\ rk^{*}+\frac{(n-2k^{*}+1)(n-2k^{*})}{2}-\\-2\lceil\frac{k}{4}\rceil-k+1,\ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = n-k^{*} +1 }\\ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-1)(k^{*}-4)}{2}-\\-n-2k+i, \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}+2\leq i \leq n-1 } \end{array} \right. \end{align*}$

For $1 < j < n-2k^{*}+1$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-j+1}x_{k^{*}})+i, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-j }\\ \varphi(x_{k^{*}}x_{k^{*}+j-1})-k^{*}+i+j, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-j+1\leq i\leq k^{*} }\\ \varphi(x_{n-k^{*}-j+1}x_{n-k^{*}})-k^{*}+i,\ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i\leq n-k^{*}-j }\\ \varphi(x_{n-k^{*}}x_{n-k^{*}+j-1}+k^{*}-n-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ \ { n-k^{*}-j+1\leq i\leq n-k^{*} }\\ \varphi(x_{n-j+1}x_{n})-k^{*}-n+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ n-k^{*}+1\leq i \leq n-j .} \end{array} \right. \end{align*}$

$\begin{align} \varphi(x_{i}x_{i+n-2k^{*}+1}) = \left\{ \begin{array}{l} \varphi(x_{3k^{*}-n}x_{k^{*}})+i, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq 3k^{*}-n-1 }\\ \varphi(x_{k^{*}}x_{n-k^{*}})+n-3k^{*}+i+1, \ \ \ \ \ \ \ \ \ { 3k^{*}-n\leq i\leq k^{*} }\\ \varphi(x_{n-k^{*}}x_{2n-3k^{*}})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+1\leq i\leq n-k^{*}+1 }\\ \varphi(x_{2k^{*}}x_{n})-n+k^{*}+i-1,\ \ \ \ \ { \ \ \ \ \ \ \ \ \ n-k^{*}+2\leq i \leq 2k^{*}-1 }. \end{array} \right. \end{align}$

(2.1)

$\begin{align*} \varphi(x_{i}x_{i+n-2k^{*}+2}) = \left\{ \begin{array}{l} \varphi(x_{3k^{*}-n-1}x_{k^{*}})+i, \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq 3k^{*}-n-2 }\\ \varphi(x_{k^{*}}x_{n-k^{*}-1})+n-3k^{*}+i+2, \ \ \ \ \ \ \ \ \ \ { 3k^{*}-n-1\leq i\leq k^{*}-1 }\\ \frac{r(2k^{*}-r-1)}{2}+\frac{(n-2k^{*}+1)(2r-n+2k^{*})}{2}-\\-k+1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*} }\\ \varphi(x_{n-k^{*}}x_{2n+3k^{*}})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i\leq n-k^{*}+1 }\\ \varphi(x_{2k^{*}-1}x_{n})-n+k^{*}+i,\ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}+2\leq i \leq 2k^{*}-2 }\\ \end{array} \right. \end{align*}$

Now, for $n-2k^{*}+3 \leq i \leq r$ we have three subcases:

Subcase 2.1.1.1 If $k^{*}\geq r+2$ , hence the labels of edges are defined as follows:

$\begin{align} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-j+1}x_{k^{*}})+i, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-j }\\ \varphi(x_{n-k^{*}-j+2}x_{n-k^{*}+1})-k^{*}+j+i, \ \ \ \ \ \ \ \ \ { \ \ k^{*}-j+1\leq i\leq n-k^{*}-j+1 }\\ \varphi(x_{k^{*}}x_{k^{*}+j-1})-n+k^{*}+j+i-1,\ \ \ \ \ \ \ { n-k^{*}-j+2\leq i\leq k^{*} }\\ \varphi(x_{n-k^{*}+1}x_{n-k^{*}+j})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+1\leq i \leq n-k^{*}+1 }\\ \varphi(x_{n-j+1}x_{n})-n+k^{*}+i-1,\ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ n-k^{*}+2\leq i \leq n-j }. \end{array} \right. \end{align}$

(2.2)

Subcase 2.1.1.2 If $k^{*} = r+1$ , thus the edges $n-2k^{*}+3 \leq j\leq r-1$ , $x_{i}x_{i+j}$ , $1\leq i \leq n-j$ are labeled by Eq (2.2) and for $j = r$ the edges are labeled as follows:

$\begin{align*} \varphi(x_{i}x_{i+r}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-r+1}x_{k^{*}})+i, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-r }\\ \varphi(x_{n-k^{*}-r+2}x_{n-k^{*}+1})-k^{*}+r+i, \ \ \ \ \ \ \ \ \ \ \ { k^{*}-r+1\leq i\leq n-k^{*}-r+1 }\\ \varphi(x_{k^{*}}x_{k^{*}+r-1})-n+k^{*}+r+i-1,\ \ \ \ \ \ \ \ { n-k^{*}-r+2\leq i\leq k^{*} }\\ \varphi(x_{n-k^{*}+1}x_{n-k^{*}+r})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i \leq n-k^{*}+1 }. \end{array} \right. \end{align*}$

Subcase 2.1.1.3 If $k^{*} = r$ , then the edges $x_{i}x_{i+j}$ , $n-2k^{*}+3 \leq j\leq r-2$ , $1\leq i \leq n-j$ are labeled by Eq (2.2) and for $j = r-1$ and $j = r$ the edges are labeled as follows:

$\begin{align*} \varphi(x_{i}x_{i+r-1}) = \left\{ \begin{array}{l} \varphi(x_{2}x_{k^{*}})+1, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = 1 }\\ \varphi(x_{n-2k^{*}+3}x_{n-k^{*}+1})+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 2\leq i\leq n-2k^{*}+2 }\\ \varphi(x_{k^{*}}x_{2k^{*}-2})-n+2k^{*}+i-2,\ \ \ \ \ \ \ \ \ \ \ \ { n-2k^{*}+3\leq i\leq k^{*} }\\ \varphi(x_{n-k^{*}+1}x_{n-1})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i \leq n-k^{*}+1 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+r}) = \left\{ \begin{array}{l} \varphi(x_{n-2k^{*}+2}x_{n-k^{*}+1})+i, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq n-2k^{*}+1 }\\ \varphi(x_{k^{*}}x_{2k^{*}-1})-n+2k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { n-2k^{*}+2\leq i\leq k^{*} }\\ \varphi(x_{n-k^{*}+1}x_{n})-k^{*}+i,\ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i \leq n-k^{*} }. \end{array} \right. \end{align*}$

Subcase 2.1.2 If $k^{*} = r-1$ , hence we can defined the edge labels as follows:

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-1 }\\ \frac{k^{*}(k^{*}-1)}{2}-2\lceil\frac{k}{4}\rceil+1, \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*} }\\ {k^{*}}^{2}-4\lceil\frac{k}{4}\rceil+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+1\leq i\leq n-k^{*} }\\ \frac{n(n+1)+2k^{*}(3k^{*}-2n)+2}{2}-2\lceil\frac{k}{4}\rceil-k,\ \ \ \ \ \ \ \ \ \ \ \ \ \ { i = n-k^{*} +1 }\\ k^{*}(n-k^{*}+1)-2k+i-3, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { n-k^{*}+2\leq i \leq n-1 . } \end{array} \right. \end{align*}$

For $j = n-2k^{*}+1$ , we used the Eq (2.1) to label the edges and for $j = n-2k^{*}+2$ the edge labels given by:

$\begin{align*} \varphi(x_{i}x_{i+n-2k^{*}+2}) = \left\{ \begin{array}{l} \varphi(x_{3k^{*}-n-1}x_{k^{*}})+i, \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq 3k^{*}-n-2 }\\ \varphi(x_{k^{*}}x_{n-k^{*}-1})+n-3k^{*}+i+2, \ \ \ \ \ \ \ { \ \ 3k^{*}-n-1\leq i\leq k^{*}-1 }\\ \frac{k^{*}(6n-7k^{*}-1)-n(n-1)+2}{2}-k,\ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*} }\\ \varphi(x_{n-k^{*}}x_{2n+3k^{*}})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+1\leq i\leq n-k^{*}+1 }\\ \varphi(x_{2k^{*}-1}x_{n})-n+k^{*}+i-1,\ \ \ \ \ \ \ \ \ \ \ { \ \ \ n-k^{*}+2\leq i \leq 2k^{*}-2 .} \end{array} \right. \end{align*}$

Further, the edges for $n-2k^{*}+3 \leq j\leq r-3$ are labeled by Eq (2.2) and for $j = r-2$ , $j = r-1$ , and $j = r$ the edges are labeled as follows:

$\begin{align*} \varphi(x_{i}x_{i+r-2}) = \left\{ \begin{array}{l} \varphi(x_{2}x_{k^{*}})+1, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = 1 }\\ \varphi(x_{n-2k^{*}+3}x_{n-k^{*}+1})+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 2\leq i\leq n-2k^{*}+2 }\\ \varphi(x_{k^{*}}x_{2k^{*}-2})-n+2k^{*}+i-2,\ \ \ \ \ \ \ \ \ \ \ \ \ { n-2k^{*}+3\leq i\leq k^{*} }\\ \varphi(x_{n-k^{*}+1}x_{n-1})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+1\leq i \leq n-k^{*}+1 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+r-1}) = \left\{ \begin{array}{l} \varphi(x_{n-k^{*}-r+3}x_{n-k^{*}+1})+i, \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ 1\leq i \leq n-2k^{*}+1 }\\ \varphi(x_{k^{*}}x_{2k^{*}-1})-n+2k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ { n-2k^{*}+2\leq i\leq k^{*} }\\ \varphi(x_{n-k^{*}+1}x_{n})-k^{*}+i,\ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ k^{*}+1\leq i \leq n-k^{*} }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+r}) = \left\{ \begin{array}{l} \varphi(x_{n-2k^{*}+1}x_{n-k^{*}+1})+i, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq n-2k^{*} }\\ \varphi(x_{k^{*}}x_{2k^{*}})+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-2k^{*}+1\leq i\leq k^{*} }. \end{array} \right. \end{align*}$

An explanation of above labeling is depicted in Figure 5.

Also in this case the vertices are labeled with even number. Now we will estimate the weights of edges under the labeling $\varphi$ :

We split the edge set of $P_{n}^{r}$ into six mutually separated subsets, $A_{s}$ , $1\leq s \leq6$ as follows: In Subcase 2.1.1.1, and Subcase 2.1.1.2 we have:

● $A_{1} = \{x_{i}x_{i+j}:1\leq j\leq r, 1\leq i\leq k^{*}-j \}$ : The set of all edges which have endpoints tagged with $0$ ,

● $A_{2} = \{x_{i}x_{i+j}:1\leq j\leq n-2k^{*}+1, k^{*}-j+1\leq i\leq k^{*} \}\bigcup\{x_{i}x_{i+j}:n-2k^{*}+2\leq j\leq r, k^{*}-j+1\leq i\leq n-k^{*}-j+1 \}$ : The set of all edges which have endpoints tagged with $0$ and $2\lceil\frac{k}{4}\rceil$ ,

● $A_{3} = \{x_{i}x_{i+j}:1\leq j\leq n-2k^{*}, k^{*}+1\leq i\leq n-2k^{*}-j+1 \}$ : The set of all edges which have endpoints tagged with $2\lceil\frac{k}{4}\rceil$ ,

● $A_{4} = \{x_{i}x_{i+j}:1\leq j\leq n-2k^{*}+1, n-k^{*}-j+2\leq i\leq n-k^{*}+1 \}\bigcup\{x_{i}x_{i+j}:n-2k^{*}+2\leq j\leq r, k^{*}+1\leq i\leq n-k^{*}+1 \}$ : The set of all edges which have endpoints tagged with $2\lceil\frac{k}{4}\rceil$ and $k$ ,

● $A_{5} = \{x_{i}x_{i+j}:1\leq j\leq k^{*}-2, n-k^{*}+2\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $k$ ,

● $A_{6} = \{x_{i}, x_{i+j}, n-2k^{*}+2\leq j\leq r, n-k^{*}-j+2\leq i\leq k^{*} \}$ : The set of all edges which have endpoints tagged with $0$ and $k$ .

In Subcase 2.1.1.3, $A_{1} = \{x_{i}x_{i+j}:1\leq j\leq r-1, 1\leq i\leq k^{*}-j \}$ and other subsets $A_{s}$ , $2\leq s \leq6$ as in the Subcase 2.1.1.1, and in Subcase 2.1.1.2, $A_{1} = \{x_{i}x_{i+j}:1\leq j\leq k^{*}-1, 1\leq i\leq k^{*}-j \}$ , $A_{2} = \{x_{i}x_{i+j}:1\leq j\leq n-2k^{*}+1, k^{*}-j+1\leq i\leq k^{*} \}\bigcup\{x_{i}x_{i+j}:n-2k^{*}+2\leq j\leq k^{*}, k^{*}-j+1\leq i\leq n-k^{*}-j+1 \}\bigcup\{x_{1}x_{n-k^{*}+1} \}$ and other subsets $A_{s}$ , $3\leq s \leq6$ as in the Subcase 2.1.1.1. Therefore, we obtain the edge weights for the Subcases 2.1.1.1, 2.1.1.2, and 2.1.1.3 as follows:

1. The edges of the set $A_{1}$ , obtain weights from the set of sequential integers $\{1, 2, ..., \frac{r(2k^{*}-r-1)}{2}\}$ ,

2. The edges of the set $A_{2}$ , obtain weights from the set of sequential integers $\{\frac{r(2k^{*}-r-1)}{2}+1, ..., \frac{r(2k^{*}-r-1)}{2}+\frac{(n-2k^{*}+1)(2r-n+2k^{*})}{2} \}$ ,

3. The edges of the set $A_{3}$ , obtain weights from the set of sequential integers $\{rk^{*}+1, ..., rk^{*}+\frac{(n-2k^{*}+1)(n-2k^{*})}{2} \}$ ,

4. The edges of the set $A_{4}$ , obtain weights from the set of sequential integers $\{rk^{*}+\frac{(n-2k^{*}+1)(n-2k^{*})}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(k^{*}-1)(k^{*}-4)}{2}+k^{*}+1) \}$ ,

5. The edges of the set $A_{5}$ , get weights from the set of sequential integers $\{\frac{r(2n-r-1)}{2}-\frac{(k^{*}-1)(k^{*}-4)}{2}+k^{*}+2, ..., \frac{r(2n-r-1)}{2}\}$ ,

6. Finally, the edges of the set $A_{6}$ , get weights from the set of sequential integers $\{\frac{r(2k^{*}-r-1)}{2}+\frac{(n-2k^{*}+1)(2r-n+2k^{*})}{2}+1, ..., rk^{*}\}$ .

An explanation of above corresponding weights is depicted in Figure 6.

In the Subcase 2.1.2, the edge weights are obtained as follows :

1. The edges of the set $A_{1}$ , get weights from the set of sequential integers $\{1, 2, ..., \frac{k^{*}(k^{*}-1)}{2}\}$ ,

2. The edges of the set $A_{2}$ , get weights from the set of sequential integers $\{\frac{k^{*}(k^{*}-1)}{2}+1, ..., \frac{k^{*}(6n-7k^{*}-1)-n(n-1)+2}{2}-1)\}$ ,

3. The edges of the set $A_{3}$ , admit weights from the set of sequential integers $\{{k^{*}}^{2}+k^{*}+1, ..., \frac{n(n+1)+2k^{*}(3k^{*}-2n)+2}{2}-1\}$ ,

4. The edges of the set $A_{4}$ , admit weights from the set of sequential integers $\{\frac{n(n+1)+2k^{*}(3k^{*}-2n)+2}{2}, ..., k^{*}(n-k^{*})+n-2 \}$ ,

5. The edges of the set $A_{5}$ , receive weights from the set of sequential integers $\{k^{*}(n-k^{*})+n-1, ..., \frac{r(2n-r-1)}{2}\}$ ,

6. Finally, The edges of the set $A_{6}$ , receive weights from the set of sequential integers $\{\frac{k^{*}(6n-7k^{*}-1)-n(n-1)+2}{2}, ..., {k^{*}}^{2}+k^{*}\}$ .

It is not hard to see that the weights of edges are distinct numbers from the set $\{1, 2, 3, ..., \frac{(r(2n-r-1)}{2}\}$ .

Subcase 2.2. If $k^{*} = \frac{n}{2}$ . Define the total $k$ -labeling $\varphi$ of $P_{n}^{r}$ as follows:

The corresponding labeling for $P^{4}_{8}$ is illustrated in Figure 10. Otherwise we have following labeling:

$\begin{align*} \varphi(x_{i}) = \left\{ \begin{array}{l} 0 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 1\leq i \leq k^{*}-1 }\\ 2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor, \ \ \ { \ \ \ \ \ \ \ \ \ k^{*}\leq i \leq k^{*}+1 }\\ k, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq n } \end{array} \right. \end{align*}$

Figure 10. A reflexive irregular

$8$ -labeling of

$P^{4}_{8}$ .

DownLoad: Full-Size Img PowerPoint

Now we define the edge labels as follows:

For $1\leq j \leq 3$ we have two subcases.

Subcase 2.2.1 If $\frac{(k^{*}-1)(k^{*}+2)}{2}\geq k$ ,

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-2 }\\ \frac{(k^{*}-1)(k^{*}-2)}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+1, \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}-1 }\\ \frac{k^{*}(4r-k^{*}-1)-r(r+1)}{2}-4\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+1,\ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ i = k^{*} }\\ \frac{k^{*}(4r-k^{*}-1)-r(r+1)}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor-k+2,\ \ \ \ \ \ \ \ \ \ \ \ { i = k^{*}+1 }\\ \frac{r(2n-r-1)}{2}-\frac{{k^{*}}^{2}-k^{*}+4}{2}-2k+i, \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq n-1 } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+2}) = \left\{ \begin{array}{l} k^{*}+i-2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-3 }\\ \frac{{k^{*}}^{2}-5k^{*}+2}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+4+i, \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}-2\leq i \leq k^{*}-1 }\\ \frac{k^{*}(4r-k^{*}-3)-r(r+1)}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+i+3,\ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ k^{*} \leq i \leq k^{*}+1 }\\ (1+r)n-\frac{r(r+1)}{2}-\frac{{k^{*}}^{2}+k^{*}+8}{2}-2k+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i \leq n-2 } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+3}) = \left\{ \begin{array}{l} 2k^{*}+i-5 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-4 }\\ \frac{{k^{*}}^{2}-5k^{*}+2}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+7+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-3\leq i \leq k^{*}-2 }\\ \frac{{k^{*}}^{2}+k^{*}-2}{2}-k+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { i = k^{*}-1 }\\ \frac{k^{*}(4r-k^{*}-3)-r(r+1)}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor-\\-k+i+5,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*} \leq i \leq k^{*}+1 }\\ (2+r)n-\frac{r(r+1)}{2}-\frac{{k^{*}}^{2}+3k^{*}+14}{2}-2k+i, \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i \leq n-3 } \end{array} \right. \end{align*}$

Subcase 2.2.2 If $\frac{(k^{*}-1)(k^{*}+2)}{2} < k$ ,

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-2 }\\ \frac{(k^{*}-1)(k^{*}-2)}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+1, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}-1 }\\ (r-k^{*})(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}-\\-4\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+k+1,\ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*} }\\ (r-k^{*})(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}-\\-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+2,\ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}+1 }\\ (r-k^{*}+2)(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}-\\-k^{*}-k+i, \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq n-1 } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+2}) = \left\{ \begin{array}{l} k^{*}+i-2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-3 }\\ \frac{(k^{*}-1)(k^{*}-2)}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor-\\-k^{*}+i+4, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}-2\leq i\leq k^{*}-1 }\\ (r-k^{*})(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}-\\-4\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor+k-k^{*}+i+3,\ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ k^{*}\leq i\leq k^{*}+1 }\\ (r-k^{*}+2)(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}+\\+n-2k^{*}-k+i-2, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq n-2 } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+3}) = \left\{ \begin{array}{l} 2k^{*}+i-5 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-4 }\\ \frac{(k^{*}-1)(k^{*}-2)}{2}-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor-\\-k^{*}+i+7, \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}-3\leq i\leq k^{*}-2 }\\ 1,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}-1 }\\ (r-k^{*})(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}-\\-2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor-k^{*}+i+5, \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}\leq i\leq k^{*}+1 }\\ (r-k^{*}+2)(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}+\\+2n-3k^{*}-k+i-5, \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq n-3 } \end{array} \right. \end{align*}$

For $4\leq j\leq k^{*}-2$ (if $k^{*}\geq 6$ ),

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-j}x_{k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq k^{*}-j-1 }\\ \varphi(x_{k^{*}-j-2}x_{k^{*}+1})-k^{*}+j+i+1 , \ \ \ \ \ \ \ \ \ \ \ { k^{*}-j\leq i \leq k^{*}-j+1 }\\ \varphi(x_{k^{*}-1}x_{k^{*}+j-2})-k^{*}+j+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-j+1\leq i \leq k^{*}-1 }\\ \varphi(x_{k^{*}+1}x_{k^{*}+j})-k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}\leq i \leq k^{*}+1 }\\ \varphi(x_{n-j-1}x_{n})-k^{*}+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i \leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}-1}) = \left\{ \begin{array}{l} \varphi(x_{3}x_{k^{*}+1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq 2 }\\ \varphi(x_{k^{*}-1}x_{2k^{*}-3})+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 3\leq i \leq k^{*}-1 }\\ \varphi(x_{k^{*}+1}x_{2k^{*}-1})-k^{*}+i+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}\leq i \leq k^{*}+1 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}}) = \left\{ \begin{array}{l} \varphi(x_{2}x_{k^{*}+1})+1 , \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = 1 }\\ \varphi(x_{k^{*}-1}x_{2k^{*}-2})+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 2\leq i \leq k^{*}-1 }\\ \varphi(x_{k^{*}+1}x_{2k^{*}})-k^{*}+i+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}\leq i \leq k^{*}+1 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}+1}) = \varphi(x_{k^{*}-1}x_{2k^{*}-1})+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}\leq i \leq n-k^{*}-1 }. \end{align*}$

For $k^{*}+2\leq j\leq 2r-k^{*}+1$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \varphi(x_{n-j+1}x_{n})+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 1\leq i \leq n-j }. \end{align*}$

Thus the vertices are labeled with even numbers. Now we will calculate the weights of edges under the labeling $\varphi$ :

Like in the previous case the edge set of $P_{n}^{r}$ can be partied into six mutually separated subsets $A_{s}, 1\leq i \leq 6$ as follows:

● $A_{1} = \{x_{i}x_{i+j}:1\leq j\leq k^{*}-2, 1\leq i\leq k^{*}-j-1 \}$ : The set of all edges which have endpoints tagged with $0$ ,

● $A_{2} = \{x_{i}x_{i+j}:1\leq j\leq 2, k^{*}-j\leq i\leq k^{*}-1 \}\bigcup\{x_{i}x_{i+j}:3\leq j\leq k^{*}-1, k^{*}-j\leq i\leq k^{*}-j+1 \}\bigcup \{x_{1}x_{k^{*}+1} \}$ : The set of all edges which have endpoints tagged with $0$ and $2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor$ ,

● $A_{3} = \{x_{k^{*}}x_{k^{*}+1} \}$ : The set of only one edge which has endpoints $2\lfloor\frac{(k^{*}-1)(k^{*}-2)}{4}\rfloor$ ,

● $A_{4} = \{x_{k^{*}}x_{k^{*}+j}, x_{k^{*}+1}x_{k^{*}+j+1}:3\leq j\leq n-1, \}\bigcup \{x_{k^{*}+1}x_{k^{*}+2}, x_{k^{*}}x_{n} \}$ : The set of all edges which have endpoints tagged with $2\lceil\frac{k}{4}\rceil$ and $k$ ,

● $A_{5} = \{x_{i}x_{i+j}:1\leq j\leq k^{*}-2, k^{*}+2\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $k$ ,

● $A_{6} = \{x_{i}x_{i+j}, 3\leq j\leq r, k^{*}-j+2\leq i\leq k^{*}-1 \}$ : The set of all edges which have endpoints tagged with $0$ and $k$ .

Accordingly we obtain the edge weights as follows:

In the Subcase 2.2.1,

1. The edge weights of the set $A_{1}$ , admit the successive numbers from the set $\{1, 2, ..., \frac{(k^{*}-1)(k^{*}-2)}{2} \}$ ,

2. The edge weights of the set $A_{2}$ , receive the successive numbers from the set $\{ \frac{(k^{*}-1)(k^{*}-2)}{2} +1, ..., \frac{k^{*}(k^{*}+1)-2}{2} \}$ ,

3. The edge weight of the set $A_{3}$ , admits the number $\{ \frac{k^{*}(4r-k^{*}-1)-r(r+1)}{2}+1 \}$ ;

4. The edge weights of the set $A_{4}$ , admit the successive numbers from the set $\{\frac{k^{*}(4r-k^{*}-1)-r(r+1)}{2}+2, ..., \frac{r(2n-r-1)}{2}-\frac{{k^{*}}^{2}-3k^{*}}{2} \}$ ,

5. The edge weights of the set $A_{5}$ , receive the successive numbers from the set $\frac{r(2n-r-1)}{2}-\frac{{k^{*}}^{2}-3k^{*}}{2}+1, ..., \frac{r(2n-r-1)}{2} \}$ ,

6. Finally, the edge weights of the set $A_{6}$ , receive the successive numbers from the set $\{ \frac{{k^{*}}^{2}+k^{*}-2}{2}+1, ..., \frac{k^{*}(4r-k^{*}-1)-r(r+1)}{2} \}$ .

In the Subcase 2.2.2,

1. The edge weights of the set $A_{1}$ , admit the successive numbers from the set $\{1, 2, ..., \frac{(k^{*}-1)(k^{*}-2)}{2} \}$ ,

2. The edge weights of the set $A_{2}$ , receive the successive numbers from the set $\{ \frac{(k^{*}-1)(k^{*}-2)}{2}+1, ..., \frac{(k^{*}-1)(k^{*}+2)}{2} \}$ ,

3. The edge weight of the set $A_{3}$ , admits the number $\{ (r-k^{*})(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}+k+1 \}$ ,

4. The edge weights of the set $A_{4}$ , admit the successive numbers from the set $\{(r-k^{*})(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}+k, ..., (r-k^{*}+2)(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}+k+1 \}$ ,

5. The edge weights of the set $A_{5}$ , receive the successive numbers from the set $\{(r-k^{*}+2)(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}+k+2, ..., \frac{(2r-k^{*}+2)(k^{*}-1)}{2}+\frac{(2k^{*}-r-1)(r-2)}{2}+k+1 \}$ ,

6. The edge weights of the set $A_{6}$ , receive the successive numbers from the set $\{ k+1, ..., (r-k^{*})(k^{*}-1)+\frac{(2k^{*}-r-1)(r-2)}{2}+k \}$ .

Hence the edge weights in the Subcases 2.2.1 and 2.2.2 are distinct numbers from the sets $\{1, 2, 3, ..., \frac{r(2n-r-1)}{2}\}$ and $\{1, 2, 3, ..., \frac{(2r-k^{*}+2)(k^{*}-1)}{2}+\frac{(2k^{*}-r-1)(r-2)}{2}+k+1 \}$ respectively.

Subcase 2.3 If $k^{*} > \frac{n}{2}$ , then we have the following subcases:

Subcase 2.3.1 If $n$ is odd, we construct the total $k$ -labeling $\varphi$ of $P_{n}^{r}$ as follows:

$\begin{align*} \varphi(x_{i}) = \left\{ \begin{array}{l} 0 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq n-k^{*}-1 }\\ 2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor, \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}\leq i \leq \frac{n+1}{2} }\\ 2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor, \ \ \ \ \ \ \ { \frac{n+3}{2} \leq i \leq k^{*}+1 }\\ k, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq n }. \end{array} \right. \end{align*}$

Now, to define the edge labeling we have to consider the following two subcases:

Subcase 2.3.1.1 If $2k^{*}-n = 1$ ,

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq k^{*}-3 }\\ \frac{(k^{*}-2)(k^{*}-3)}{2}-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor+1, \ \ \ \ \ \ \ \ \ \ { i = k^{*}-2 }\\ \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{k^{*}-1}x_{k^{*}}) = \left\{ \begin{array}{l} \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-\\-4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-2, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ {\rm if}\; 4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor\geq k }\\ \frac{(k^{*}-2)(k^{*}+3)}{2}-4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor+1, \ \ \ \ \ \ \ \ \ \ \ \ \ { {\rm if} \; 4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor < k }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+7)}{2}-\\-4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-1, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*} }\\ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-\\-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-k+1, \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}+1 }\\ \frac{r(2n-r-1)}{2}-\frac{{k^{*}}^2-3k^{*}+8}{2}-2k+i, \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i\leq n-1 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+2}) = \left\{ \begin{array}{l} k^{*}-3+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq k^{*}-4 }\\ \frac{{k^{*}}^2-7k^{*}+16}{2}-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-3\leq i\leq k^{*}-2 }\\ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+7)}{2}-\\-4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor, \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}-1 }\\ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-\\-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-k+1, \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*} }\\ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-\\-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-k+2, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ i = k^{*}+1 }\\ \frac{(2n-r)(r+1)}{2}-\frac{{k^{*}}^2-k^{*}-6}{2}-2k+i, \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ k^{*}+2\leq i\leq n-2 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+3}) = \left\{ \begin{array}{l} k^{*}+i-7, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq k^{*}-5 }\\ \frac{{k^{*}}^2-7k^{*}+22}{2}-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor+i, \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ k^{*}-4\leq i\leq k^{*}-3 }\\ \frac{(k^{*}-2)(k^{*}-3)}{2}-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor+1, \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}-2 }\\ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+5)}{2}-\\-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-k+i+3, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ k^{*}-1\leq i\leq k^{*} }\\ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-\\-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-k+3, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ i = k^{*}+1 }\\ \frac{(2n-r)(r+1)}{2}-\frac{{k^{*}}^2+k^{*}+18}{2}+n-2k+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i\leq n-3 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+4}) = \left\{ \begin{array}{l} k^{*}+i-12, \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq k^{*}-6 }\\ \frac{{k^{*}}^2-7k^{*}+28}{2}-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-5\leq i\leq k^{*}-4 }\\ \frac{(k^{*}-2)(k^{*}-3)}{2}-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor+2, \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}-3 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{k^{*}-2}x_{k^{*}+2}) = \left\{ \begin{array}{l} \frac{(k^{*}-2)(k^{*}+3)}{2}-k+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { {\rm if} \; 4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor\geq k }\\ \frac{(k^{*}-2)(k^{*}+3)}{2}-k+2, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { {\rm if} \; 4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor\ < k }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+4}) = \left\{ \begin{array}{l} \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+5)}{2}-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-\\-k+i+5, \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}-1\leq i\leq k^{*} }\\ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-\\-k+4, \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}+1 }\\ \frac{(2n-r)(r+1)}{2}-\frac{{k^{*}}^2+3k^{*}+26}{2}+2n-2k+i, \ \ \ { \ \ \ \ \ \ \ k^{*}+2\leq i\leq n-4 }. \end{array} \right. \end{align*}$

For $5\leq j\leq k^{*}-3$ (if $k^{*}\geq 8$ ),

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-j-1}x_{k^{*}-2})+i , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq k^{*}-j-2 }\\ \varphi(x_{k^{*}-j+1}x_{k^{*}})-k^{*}+j+i+2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-j-1\leq i \leq k^{*}-j }\\ \varphi(x_{k^{*}-j+2}x_{k^{*}+1})+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}-j+1 }\\ \varphi(x_{k^{*}-2}x_{k^{*}+j-3})-k^{*}+j+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-j+2\leq i \leq k^{*}-2 }\\ \varphi(x_{k^{*}}x_{k^{*}+j-1})-k^{*}+i+2, \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ k^{*}-1\leq i \leq k^{*} }\\ \varphi(x_{k^{*}+1}x_{k^{*}+j})+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ i = k^{*}+1 }\\ \varphi(x_{n-j+1}x_{n})-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ k^{*}+2\leq i\leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}-2}) = \left\{ \begin{array}{l} \varphi(x_{3}x_{k^{*}})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ 1\leq i\leq 2 }\\ \varphi(x_{4}x_{k^{*}+1})+1, \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ i = 3 }\\ \varphi(x_{k^{*}-2}x_{2k^{*}-5})-3 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 4\leq i\leq k^{*}-2 }\\ \varphi(x_{k^{*}}x_{2k^{*}-3})-k^{*}+i+2, \ \ \ \ \ \ { k^{*}-1\leq i \leq k^{*} }\\ \varphi(x_{k^{*}+1}x_{2k^{*}-2})+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { i = k^{*}+1 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}-1}) = \left\{ \begin{array}{l} \varphi(x_{2}x_{k^{*}})+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ i = 1 }\\ \varphi(x_{3}x_{k^{*}+1})+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ i = 2 }\\ \varphi(x_{k^{*}-2}x_{2k^{*}-4})+i-2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 3\leq i\leq k^{*}-2 }\\ \varphi(x_{k^{*}}x_{2k^{*}-2})-k^{*}+i+2, \ \ \ \ \ \ \ \ \ \ { k^{*}-1\leq i \leq k^{*} }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}}) = \left\{ \begin{array}{l} \varphi(x_{2}x_{k^{*}+1})+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ i = 1 }\\ \varphi(x_{k^{*}-2}x_{2k^{*}-3})+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 2\leq i \leq k^{*}-2 }\\ \varphi(x_{k^{*}}x_{2k^{*}-1})+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { i = k^{*}-1 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}+1}) = \varphi(x_{k^{*}-2}x_{2k^{*}-2})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ 1\leq i \leq n-k^{*}-1 }. \end{align*}$

For $k^{*}+2\leq j\leq r$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \varphi(x_{n-j+1}x_{n})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ 1\leq i \leq n-j }. \end{align*}$

An explanation of above labeling is depicted in . In this case, we split edge set of the $P_{n}^{r}$ into nine mutually separated subsets as follows:

Figure 11. A reflexive irregular

$24$ -labeling of

$P^{9}_{13}$ .

DownLoad: Full-Size Img PowerPoint

● $A_{1} = \{x_{i}x_{i+j}:1\leq j \leq k^{*}-3, 1\leq i\leq k^{*}-j-2 \}$ : The set of all edges which have endpoints tagged with $0$ ,

● $A_{2} = \{x_{i}x_{i+j}:2\leq j\leq k^{*}-2, k^{*}-j\leq i\leq k^{*}-j+1 \}\bigcup\{x_{k^{*}-2}x_{k^{*}-1}, x_{1}x_{k^{*}} \}$ : The set of all edges which have endpoints tagged with $0$ and $2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor$ ,

● $A_{3} = \{x_{k^{*}-1}x_{k^{*}}\}$ : The set of only one edge which has endpoints labeled with $2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor$ ,

● $A_{4} = \{x_{k^{*}-1}x_{k^{*}+1}, x_{k^{*}-1}x_{k^{*}+1}\}$ : The set of two edges which has endpoints labeled with $2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor$ and $2k^{*}+2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-4$ ,

● $A_{5} = \{x_{i}x_{i+j}:4\leq j \leq k^{*}+1, k^{*}-j+2\leq i\leq k^{*}-2 \}\bigcup \{ x_{i}x_{i+j}:k^{*}+1\leq j \leq r, 1\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $0$ and $k$ ,

● $A_{6} = \{x_{k^{*}-1}x_{k^{*}+j-1}, x_{k^{*}}x_{k^{*}+j}:3\leq j \leq k^{*}-1 \}\bigcup \{x_{2}x_{k^{*}+2}, x_{k^{*}-1}x_{n} \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor$ and $k$ ,

● $A_{7} = \{x_{k^{*}+1}x_{k^{*}+j}:1\leq j \leq k^{*}-1 \}$ : The set of all edges which have endpoints tagged with $2k^{*}+2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-4$ and $k$ ,

● $A_{8} = \{x_{k^{*}-j+1}x_{k^{*}+1}: 3\leq j \leq k^{*} \}$ : The set of all edges which have endpoints tagged with $0$ and $2k^{*}+2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor-4$ ,

● $A_{9} = \{x_{i}x_{i+j}:1\leq j\leq k^{*}-3, k^{*}+2\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $k$ .

Observe that under the total $k$ -labeling $\varphi$ the edge (edges):

1. from the set $A_{1}$ , receive the weights from the successive numbers $\{1, 2, ..., \frac{(k^{*}-2)(k^{*}-3)}{2}\}$ ,

2. from the set $A_{2}$ , receive the weights from the successive numbers $\{\frac{(k^{*}-2)(k^{*}-3)}{2}+1, ..., \frac{(k^{*}-1)(k^{*}+2)}{2}\}$ ,

3. from the set $A_{3}$ , receives the weight $\{ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-2 \}$ if $4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor\geq k$ or receives the weight $\{ \frac{(k^{*}-2)(k^{*}+3)}{2}+1 \}$ if $4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor < k$ ,

4. from the set $A_{4}$ , admit the two weights $\{ \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-1, \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2} \}$ ,

5. from the set $A_{5}$ , receive the weights from the successive numbers $\{\frac{(k^{*}-1)(k^{*}+2)}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}-3\}$ , if $4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor\geq k$ or receive the weights from the set $\{\frac{(k^{*}-2)(k^{*}+3)}{2}+2, ..., \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}\}$ , if $4\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor < k$ ,

6. from the set $A_{6}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+3)}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+1)}{2}\}$ ,

7. from the set $A_{7}$ , receive the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}-1)}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}-3)}{2}\}$ ,

8. from the set $A_{8}$ , admit the weights from the successive numbers $\{\frac{(k^{*}-2)(k^{*}+1)}{2}+1, ..., \frac{(k^{*}-2)(k^{*}+3)}{2}\}$ ,

9. from the set $A_{9}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}-3)}{2}, ..., \frac{r(2n-r-1)}{2}\}$ .

An explanation of above corresponding weights is depicted in Figure 12.

Figure 12. The edge weights of

$P^{9}_{13}$ .

DownLoad: Full-Size Img PowerPoint

Subcase 2.3.1.2 If $2k^{*}-n\neq1$ , hence we define the edge labeling as follows:

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq n-k^{*}-2 }\\ \frac{(n-k^{*}-1)(n-k^{*}-2)}{2}-2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+1, \ \ \ \ \ \ \ \ \ \ \ \ { i = n-k^{*}-1 }. \end{array} \right. \end{align*}$

For $n-k^{*}\leq i\leq\frac{n-1}{2}$ ,

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} \frac{r(2n-r-1)}{2}-\frac{(k^{*}-2)(k^{*}+1)}{2}-\\-4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-n+i, \ \ \ \ \ { \ \ \ \ \ \ \ \ {\rm if} \; 4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor \geq k }\\ \frac{(n-k^{*}-1)(3k^{*}-n)}{2}-\\-4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{ {\rm if}}\; 4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor < k } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} \frac{r(2n-r-1)}{2}-\frac{(4k^{*}(n-1)-n(n-4)-3)}{8}-\\-4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor+1, \ \ \ \ \ \ \ \ \ \ \ \ { i = \frac{n+1}{2} }\\ \frac{r(2n-r-1)}{2}-\frac{(n^{2}-4n+3)}{8}-4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-\\-4\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor-\frac{n+3}{2}+i+1,\ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{n+3}{2}\leq i\leq k^{*} }\\ \frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(k^{*}-1)}{2}-2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-\\-2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor-k+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}+1 }\\ \frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(n-k^{*}-2)}{2}-2k-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i\leq n-1 }. \end{array} \right. \end{align*}$

For $1 < j \leq \frac{2k^{*}-n-1}{2}$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{n-k^{*}-j}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ 1\leq i\leq n-k^{*}-j-1 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-n+k^{*}+\\+j+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}-j\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n+1}{2}-j+1}x_{\frac{n+1}{2}})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n+1}{2}-j }\\ \varphi(x_{\frac{n+1}{2}-j+1}x_{\frac{n+1}{2}})-\frac{n+1}{2}+j+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \frac{n+1}{2}-j+1\leq i\leq \frac{n+1}{2} }\\ \varphi(x_{k^{*}-j+2}x_{k^{*}+1})-\frac{n+3}{2}+i+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \frac{n+3}{2}\leq i \leq k^{*}-j+1 }\\ \varphi(x_{k^{*}+1}x_{k^{*}+j})-k^{*}+j+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}-j+2\leq i \leq k^{*}+1 }\\ \varphi(x_{n-j+1}x_{n})-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i \leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+\frac{2k^{*}-n+1}{2}}) = \left\{ \begin{array}{l} \varphi(x_{\frac{n-3}{2}}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ 1\leq i\leq \frac{3n-4k^{*}-3}{2} }\\ \varphi(x_{n-k^{*}-1}x_{\frac{n+1}{2}})-\frac{3n-4k^{*}-3}{2}+i , \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \frac{3n-4k^{*}-1}{2}\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{2n-2k^{*}+2}{2}}x_{\frac{n+1}{2}})+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ i = n-k^{*} }\\ \varphi(x_{n-k^{*}+1})-n+k^{*}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ n-k^{*}+1\leq i\leq \frac{n+1}{2} }\\ \varphi(x_{k^{*}+1}x_{\frac{4k^{*}-n+1}{2}})-\frac{n+1}{2}+i , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \frac{n+3}{2}\leq i \leq k^{*}+1 }\\ \varphi(x_{\frac{3n-2k^{*}+1}{2}}x_{n})-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ k^{*}+2\leq i \leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+\frac{2k^{*}-n+3}{2}}) = \left\{ \begin{array}{l} \varphi(x_{\frac{3n-4k^{*}-3}{2}}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq \frac{3n-4k^{*}-5}{2} }\\ \varphi(x_{n-k^{*}-1}x_{\frac{n-1}{2}})-\frac{3n-4k^{*}-5}{2}+i , \ \ \ \ \ \ \ \ \ \ { \ \ \ \frac{3n-4k^{*}-3}{2}\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n+1}{2}}x_{k^{*}+1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n-1}{2} }\\ \frac{4k^{*}(2k^{*}+5)-12k^{*}n+3n(n-4)+17}{8}+\\+\frac{r(2n-r-1)}{2}-2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-k, \ \ \ \ \ \ \ \ { i = \frac{n+1}{2} }\\ \varphi(x_{k^{*}+1}x_{\frac{4k^{*}-n+3}{2}})-\frac{n+1}{2}+i , \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{n+3}{2}\leq i\leq k^{*}+1 }\\ \varphi(x_{\frac{3n-2k^{*}-1}{2}}x_{n})-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq \frac{3n-2k^{*}-3}{2} }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+\frac{2k^{*}-n+5}{2}}) = \left\{ \begin{array}{l} \varphi(x_{\frac{3n-4k^{*}-5}{2}}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ 1\leq i\leq \frac{3n-4k^{*}-7}{2} }\\ \varphi(x_{n-k^{*}-1}x_{\frac{n+1}{2}})-\frac{3n-4k^{*}-5}{2}+i+1 , \ \ \ \ \ \ \ \ \ \ { \frac{3n-4k^{*}-5}{2}\leq i\leq n-k^{*}-2 }\\ \frac{(n-k^{*}-1)(k^{*}+1)}{2}-2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-\\-2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor+1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { i = n-k^{*}-1 }\\ \varphi(x_{\frac{n-1}{2}}x_{k^{*}+1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n-3}{2} }\\ \varphi(x_{\frac{n+1}{2}}x_{k^{*}+2})-\frac{n-3}{2}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \frac{n-1}{2}\leq i\leq \frac{n+1}{2} }\\ \varphi(x_{k^{*}+1}x_{\frac{4k^{*}-n+5}{2}})-\frac{n+1}{2}+i , \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \frac{n+3}{2}\leq i\leq k^{*}+1 }\\ \varphi(x_{\frac{3n-2k^{*}-3}{2}}x_{n})-k^{*}+i-1, \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq \frac{3n-2k^{*}-5}{2} }. \end{array} \right. \end{align*}$

For $\frac{2k^{*}-n+7}{2}\leq j \leq 2k^{*}-n+2$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{n-k^{*}-j}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ 1\leq i\leq n-k^{*}-j-1 }\\ \varphi(x_{\frac{n+1}{2}-j+1}x_{\frac{n+1}{2}})-n+k^{*}+j+i+1 , \ \ \ \ \ \ \ \ \ \ \ { n-k^{*}-j\leq i\leq \frac{n+1}{2}-j }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-\frac{n+1}{2}+j+i , \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \frac{n+3}{2}-j \leq i\leq n-k^{*}-1 }\\ \varphi(x_{k^{*}-j+2}x_{k^{*}+1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq k^{*}-j+1 }\\ \varphi(x_{\frac{n+1}{2}}x_{\frac{n+1}{2}+j-1})-k^{*}+j+i-1 , \ \ \ \ { \ \ \ \ \ \ \ \ \ \ k^{*}-j+2\leq i\leq \frac{n+1}{2} }\\ \varphi(x_{k^{*}+1}x_{k^{*}+j})-\frac{n+1}{2}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{n+3}{2}\leq i\leq k^{*}+1 }\\ \varphi(x_{n-j+1}x_{n})-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+2k^{*}-n+3}) = \left\{ \begin{array}{l} \varphi(x_{2n-3k^{*}-3}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ { \ \ \ \ \ \ \ 1\leq i\leq 2n-3k^{*}-4 }\\ \varphi(x_{\frac{3n-4k^{*}-3}{2}-j+1}x_{\frac{n+1}{2}})+\\+3k^{*}-2n+i+4 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ 2n-3k^{*}-3\leq i\leq \frac{3n-4k^{*}-5}{2} }\\ \varphi(x_{n-k^{*}-1}x_{k^{*}+1})+\\+i-\frac{3n-4k^{*}-5}{2} , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \frac{3n-4k^{*}-3}{2}\leq i\leq n-k^{*}-2 } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{n-k^{*}-1}x_{k^{*}+2}) = \left\{ \begin{array}{l} \frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}-k+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { {\rm if} \; \ \ \ 4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor \geq k }\\ \frac{(12nk^{*}-8{k^{*}}^{2}-12k^{*}+8n-3n^{2}-5)}{8}-\\-k+1, \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm if}\; \ \ \ 4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor < k } \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+2k^{*}-n+3}) = \left\{ \begin{array}{l} \varphi(x_{\frac{n+1}{2}}x_{\frac{4k^{*}-n+5}{2}})-\\-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n+1}{2} }\\ \varphi(x_{k^{*}+1}x_{3k^{*}-n+3})-\\-\frac{n+1}{2}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \frac{n+3}{2}\leq i\leq k^{*}+1 }\\ \varphi(x_{2n-2k^{*}-2}x_{n})-\\-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i \leq 2n-2k^{*}-3 }. \end{array} \right. \end{align*}$

For $2k^{*}-n+4 \leq j \leq n-k^{*}-2$ (if $2n-3k^{*}-5 > 0$ ),

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{n-k^{*}-j}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ 1\leq i\leq n-k^{*}-j-1 }\\ \varphi(x_{\frac{n+1}{2}-j+1}x_{\frac{n+1}{2}})-n+\\+k^{*}+j+i+1 , \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq \frac{n+1}{2}-j }\\ \varphi(x_{k^{*}-j+2}x_{k^{*}+1})-\frac{n+1}{2}+\\+j+i , \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{n+3}{2}-j\leq i\leq k^{*}-j+1 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-\\-k^{*}+j+i-1, \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}-j+2\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n+1}{2}}x_{\frac{n+1}{2}+j-1})+k^{*}-\\-n+i+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n+1}{2} }\\ \varphi(x_{\frac{k^{*}+1}{2}}x_{k^{*}+j})-\frac{n+1}{2}+i, \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \frac{n+3}{2}\leq i\leq k^{*}+1 }\\ \varphi(x_{n-j+1}x_{n})-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i\leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+n-k^{*}-1}) = \left\{ \begin{array}{l} \varphi(x_{\frac{2k^{*}-n+5}{2}}x_{\frac{n+1}{2}})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 1\leq i\leq \frac{3k^{*}-n+3}{2} }\\ \varphi(x_{2k^{*}-n+3}x_{k^{*}+1})-\\-\frac{2k^{*}-n+5}{2}+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \frac{2k^{*}-n+5}{2}\leq i\leq \frac{4k^{*}-2n+4}{2} }\\ \varphi(x_{n-k^{*}-1}x_{2n-2k^{*}-3})-\\-2k^{*}+n+i-2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 2k^{*}-n+3 \leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n+1}{2}}x_{\frac{3n-2k^{*}-3}{2}})-n+\\+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ n-k^{*} \leq i\leq \frac{n+1}{2} }\\ \varphi(x_{k^{*}+1}x_{n-1})-\frac{n+1}{2}+\\+k^{*}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \frac{n+3}{2} \leq i\leq n-j }. \end{array} \right. \end{align*}$

For $n-k^{*}\leq j \leq \frac{n-3}{2}$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{\frac{n+3}{2}-j}x_{\frac{n+1}{2}})+i , \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq \frac{n+1}{2}-j }\\ \varphi(x_{k^{*}-j+2}x_{k^{*}+1})-\frac{n+1}{2}+j+i , \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \frac{n+3}{2}-j\leq i\leq k^{*}-j+1 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-k^{*}+j+i-1 , \ \ \ \ \ \ \ { \ \ \ \ \ k^{*}-j+2\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n+1}{2}}x_{\frac{n-1}{2}+j})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n+1}{2} }\\ \varphi(x_{k^{*}+1}x_{k^{*}-j})-\frac{n-1}{2}+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \frac{n+3}{2}\leq i \leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+\frac{n-1}{2}}) = \left\{ \begin{array}{l} \varphi(x_{2}x_{\frac{n+1}{2}})+i , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = 1 }\\ \varphi(x_{\frac{2k^{*}-n+5}{2}}x_{k^{*}+1})+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ 2\leq i\leq \frac{2k^{*}-n+3}{2} }\\ \varphi(x_{n-k^{*}-1}x_{\frac{3n-2k^{*}-5}{2}})-\frac{2k^{*}-n+3}{2}+i , \ \ \ \ \ \ \ { \ \ \ \frac{2k^{*}-n+5}{2}\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n+1}{2}}x_{n-1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n+1}{2} }. \end{array} \right. \end{align*}$

For $\frac{n+1}{2} \leq j \leq k^{*}$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-j+2}x_{k^{*}+1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \leq i \leq k^{*}-j+1 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-k^{*}+j+i-1 , \ \ \ \ \ \ \ \ \ { k^{*}-j+2\leq i\leq n-k^{*}-1 }\\ \varphi(x_{n-j+1}x_{n})-n+k^{*}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq n-j }. \end{array} \right. \end{align*}$

For $k^{*}+1 \leq j \leq r$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})+i , \ \ \ \ \ \ \ { \ \ \ 1\leq i\leq n-j }. \end{align*}$

Note that in this subcase the edge set of $P_{n}^{r}$ can be partied into ten mutually separated subsets as follows:

● $A_{1} = \{x_{i}x_{i+j}:1\leq j \leq n-k^{*}-2, 1\leq i\leq n-k^{*}-j-1 \}$ : The set of all edges which have endpoints tagged with $0$ ,

● $A_{2} = \{x_{i}x_{i+j}:1\leq j\leq \frac{2k^{*}-n+3}{2}, n-k^{*}-j\leq i\leq n-k^{*}-1\}\bigcup\{x_{i}x_{i+j}:\frac{2k^{*}-n+5}{2}\leq j\leq n-k^{*}-2, n-k^{*}-j\leq i\leq \frac{n+1}{2}-j\}\bigcup\{x_{i}x_{i+j}:n-k^{*}-1\leq j\leq \frac{n-1}{2}, 1\leq i\leq \frac{n+1}{2}-j \}$ : The set of all edges which have endpoints tagged with $0$ and $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor$ ,

● $A_{3} = \{x_{i}x_{i+j}:1\leq j \leq \frac{2k^{*}-n+1}{2}, n-k^{*}\leq i\leq\frac{n+1}{2}-j \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(k^{*}-2)}{4}\rfloor$ ,

● $A_{4} = \{x_{i}x_{i+j}:1\leq j \leq \frac{2k^{*}-n+1}{2}, \frac{n+3}{2}-j\leq i\leq\frac{n+1}{2} \}\bigcup\{x_{i}x_{i+\frac{2k^{*}-n+3}{2}}:n-k^{*}\leq i \leq \frac{n-1}{2} \}\bigcup\{x_{i}x_{i+j}:\frac{2k^{*}-n+5}{2}\leq j \leq 2k^{*}-n+1, n-k\leq i\leq k^{*}-j+1 \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor$ and $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor$ ,

● $A_{5} = \{x_{i}x_{i+j}:2k^{*}-n+3\leq j \leq k^{*}+1, k^{*}-j+2\leq i\leq n-k^{*}-1 \}\bigcup\{x_{i}x_{i+j}:k^{*}+2\leq j \leq r, 1\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $0$ and $k$ ,

● $A_{6} = \{x_{i}x_{i+j}:\frac{2k^{*}-n+3}{2}\leq j \leq 2k^{*}-n+2, k^{*}-j+2\leq i\leq \frac{n+1}{2} \}\bigcup\{x_{i}x_{i+j}:2k^{*}-n+3 \leq j \leq \frac{n-1}{2}, n-k^{*}\leq i\leq \frac{n+1}{2} \}\bigcup\{x_{i}x_{i+j}: \frac{n+1}{2}\leq j \leq k^{*}, n-k^{*}\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor$ and $k$ ,

● $A_{7} = \{x_{i}x_{i+j}:1\leq j \leq \frac{2k^{*}-n+1}{2}, k^{*}+2\leq i \leq k^{*}+j+1 \}\bigcup\{x_{i}x_{i+j}:\frac{2k^{*}-n+3}{2}\leq j \leq n-k^{*}-1, \frac{n+3}{2}\leq i \leq k^{*}+1 \}\bigcup\{x_{i}x_{i+j}:n-k^{*}\leq j \leq \frac{n-3}{2}, \frac{n+3}{2}\leq i \leq n-j \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor$ and $k$ ,

● $A_{8} = \{x_{i}x_{i+j}:\frac{2k^{*}-n+5}{2}\leq j\leq 2k^{*}-n+2, \frac{n+3}{2}-j\leq i \leq n-k^{*}-1 \} \bigcup\{x_{i}x_{i+j}:2k^{*}-n+3\leq j\leq \frac{n+1}{2}, \frac{n+3}{2}-j\leq i \leq k^{*}-j+1\}\bigcup\{x_{i}x_{i+j}:\frac{n+3}{2}\leq j\leq k^{*}, 1\leq i \leq k^{*}-j+1 \}$ : The set of all edges which have endpoints tagged with $0$ and $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor$ ,

● $A_{9} = \{x_{i}x_{i+j}:1\leq j \leq \frac{(2k^{*}-n-1)}{2}, \frac{n+3}{2}\leq i\leq k^{*}+j-1\}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor$ ,

● $A_{10} = \{x_{i}x_{i+j}:1\leq j\leq n-k^{*}-2, k^{*}+2\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $k$ .

It is obvious that under the total $k$ -labeling $\varphi$ the edge (edges):

1. from the set $A_{1}$ , receive the weights from the successive numbers $\{1, 2, ..., \frac{(n-k^{*}-1)(n-k^{*}-2)}{2}\}$ ,

2. from the set $A_{2}$ , receive the weights from the successive numbers $\{\frac{(n-k^{*}-1)(n-k^{*}-2)}{2}+1, ..., \frac{(n-k^{*}-1)(k^{*}+1)}{2}\}$ ,

3. from the set $A_{3}$ , obtain the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{k^{*}(k^{*}+1)}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(4k^{*}(n-1)-n(n-4)-3)}{8} \}$ if $4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor\geq k$ or obtain the weights from the set $\{\frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}+1, ..., \frac{(12nk^{*}-8{k^{*}}^{2}-12k^{*}+8n-3n^{2}-5)}{8}\}$ if $4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor < k$ ,

4. from the set $A_{4}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(4k^{*}(n-1)-n(n-4)-3)}{8}+1, ..., \frac{r(2n-r-1)}{2}-\frac{12k^{*}n-4k^{*}(2k^{*}+5)-3n(n-4)-9}{8}\}$ ,

5. from the set $A_{5}$ , admit the weights from the successive numbers $\{\frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(4k^{*}(n-1)-n(n-4)-3)}{8} \}$ , if $4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor\geq k$ or admit the weights from the set $\{ \frac{(12nk^{*}-8{k^{*}}^{2}-12k^{*}+8n-3n^{2}-5)}{8}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(4k^{*}(n-1)-n(n-4)-3)}{8}\}$ , if $4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor < k$ ,

6. from the set $A_{6}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{12k^{*}n-4k^{*}(2k^{*}+5)-3n(n-4)-9}{8}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(n^{2}+4n+3)}{8}\}$ ,

7. from the set $A_{7}$ , obtain the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(k^{*}-1)}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(n-k^{*}-2)}{2}\}$ ,

8. from the set $A_{8}$ , obtain the weights from the successive numbers $\{\frac{(n-k^{*}-1)(k^{*}+1)}{2}+1, ..., \frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}\}$ ,

9. from the set $A_{9}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(n^{2}+4n+3)}{8}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(k^{*}-1)}{2}\}$ ,

10. from the set $A_{10}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(n-k^{*}-2)}{2}+1, ..., \frac{r(2n-r-1)}{2}\}$ .

Subcase 2.3.2 If $n$ is even, therefore we construct the total $k$ -labeling $\varphi$ of $P_{n}^{r}$ as follows:

$\begin{align*} \varphi(x_{i}) = \left\{ \begin{array}{l} 0 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq n-k^{*}-1 }\\ 2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}\leq i \leq \frac{n}{2} }\\ 2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+2)}{4}\rfloor, \ \ \ \ \ { \frac{n+2}{2} \leq i \leq k^{*}+1 }\\ k, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}+2\leq i \leq n }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+1}) = \left\{ \begin{array}{l} i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq n-k^{*}-2 }\\ \frac{(n-k^{*}-1)(n-k^{*}-2)}{2}-2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+1, \ \ \ \ \ \ \ \ \ \ \ \ { i = n-k^{*}-1 }\\ \frac{(n-k^{*}-1)(3k^{*}-n)}{2}-4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ { n-k^{*}\leq i\leq\frac{n}{2}-1 }\\ \frac{r(2n-r-1)}{2}-\frac{n(4k^{*}-n+2)}{8}-4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-\\-2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+2)}{4}\rfloor+1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = \frac{n}{2} }\\ \frac{r(2n-r-1)}{2}-\frac{(2k^{*}-n+2)(3n-2k^{*}-4)}{8}-\frac{k^{*}(n-k^{*}-1)}{2}-\\-4\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-4\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+3)}{4}\rfloor-\\-\frac{n}{2}+i,\ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{n}{2}+1\leq i\leq k^{*} }\\ \frac{r(2n-r-1)}{2}-\frac{k^{*}(n-k^{*}-1)}{2}-2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-\\-2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+2)}{4}\rfloor-k+1, \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = k^{*}+1 }\\ \frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(n-k^{*}-2)}{2}-2k-k^{*}+i-1, \ \ \ \ \ \ \ \ { \ k^{*}+2\leq i\leq n-1 }. \end{array} \right. \end{align*}$

For $2\leq j\leq k^{*}-\frac{n}{2}$ (if $k^{*}-\frac{n}{2}\neq1$ ),

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{n-k^{*}-j}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq n-k^{*}-j-1 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-n+k^{*}+\\+j+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}-j\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}-j+1}x_{\frac{n}{2}})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ n-k^{*} \leq i\leq \frac{n}{2}-j }\\ \varphi(x_{\frac{n}{2}}x_{\frac{n}{2}+j-1})-\frac{n}{2}+j+i , \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{n}{2}-j+1\leq i\leq \frac{n}{2} }\\ \varphi(x_{k^{*}-j+2}x_{k^{*}+1})-\frac{n}{2}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \frac{n}{2}+1\leq i\leq k^{*}-j+1 }\\ \varphi(x_{k^{*}+1}x_{k^{*}+j})-k^{*}+j+i-1 , \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ k^{*}-j+2\leq i\leq k^{*}+1 }\\ \varphi(x_{n-j+1}x_{n})-k^{*}+i-1, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i \leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}-\frac{n}{2}+1}) = \left\{ \begin{array}{l} \varphi(x_{\frac{3n}{2}-2k^{*}-1}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ 1\leq i\leq \frac{3n}{2}-2k^{*}-2 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}})-\\-\frac{3n}{2}+2k^{*}+i+2 , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{3n}{2}-2k^{*}-1\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}}x_{k^{*}})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ n-k^{*} \leq i\leq \frac{n}{2} }\\ \varphi(x_{k^{*}+1}x_{2k^{*}-\frac{n}{2}+1})-\frac{n}{2}+i , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \frac{n}{2}+1\leq i\leq k^{*}+1 }\\ \varphi(x_{\frac{3n}{2}-k^{*}}x_{n})-k^{*}+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i\leq \frac{3n}{2}-k^{*}-1 }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+k^{*}-\frac{n}{2}+2}) = \left\{ \begin{array}{l} \varphi(x_{\frac{3n}{2}-2k^{*}-2}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ 1\leq i\leq \frac{3n}{2}-2k^{*}-3 }\\ \varphi(x_{\frac{n}{2}-n-k^{*}-2}x_{\frac{n}{2}})-\frac{3n}{2}+\\+2k^{*}+i+3 , \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{3n}{2}-2k^{*}-2\leq i\leq n-k^{*}-2 }\\ \frac{k^{*}(n-k^{*}-1)}{2}-2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-\\-2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+2)}{4}\rfloor+1,\ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ i = n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}}x_{k^{*}+1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ { n-k^{*}\leq i\leq \frac{n}{2}-1 }\\ \frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}-\\-2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor-k+1,\ \ \ \ \ \ \ \ \ \ \ \ \ { i = \frac{n}{2} }\\ \varphi(x_{k^{*}+1}x_{2k^{*}-\frac{n}{2}+2})-\frac{n}{2}+i , \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \frac{n}{2}+1\leq i\leq k^{*}+1 }\\ \varphi(x_{\frac{3n}{2}-k^{*}-1}x_{n})-k^{*}+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i\leq \frac{3n}{2}-k^{*}-2 }. \end{array} \right. \end{align*}$

For $k^{*}-\frac{n}{2}+3\leq j \leq 2k^{*}-n+2$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{n-k^{*}-j}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq n-k^{*}-j-1 }\\ \varphi(x_{\frac{n}{2}-j+1}x_{\frac{n}{2}})-n+k^{*}+j+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ { n-k^{*}-j\leq i\leq \frac{n}{2}-j }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-\frac{n}{2}+j+i , \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \frac{n}{2}-j+1\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}}x_{\frac{n}{2}+j-1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n}{2} }\\ \varphi(x_{k^{*}+1}x_{k^{*}+j})-\frac{n}{2}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \frac{n}{2}+1\leq i\leq k^{*}+1 }\\ \varphi(x_{n-j+1}x_{n})-k^{*}+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ k^{*}+2\leq i\leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+2k^{*}-n+3}) = \left\{ \begin{array}{l} \varphi(x_{2n-3k^{*}-3}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ 1\leq i\leq 2n-3k^{*}-4 }\\ \varphi(x_{\frac{3n}{2}-2k^{*}-2}x_{\frac{n}{2}})-2n+3k^{*}+\\+i+4 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2n-3k^{*}-3\leq i\leq \frac{3n}{2}-2k^{*}-3 }\\ \varphi(x_{n-k^{*}-1}x_{k^{*}+1})-\frac{3n}{2}+2k^{*}+\\+i+3 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{3n}{2}-2k^{*}-2\leq i\leq n-k^{*}-2 }\\ \frac{4k^{*}(n-k^{*}-1)+(2k^{*}-n+2)(3n-2k^{*}-4)}{8}+\\-k+1, \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i = n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}}x_{2k^{*}-\frac{n}{2}+2})-n+k^{*}+i+1 , \ \ \ \ \ \ \ { n-k^{*}\leq i\leq \frac{n}{2} }\\ \varphi(x_{k^{*}+1}x_{3k^{*}-n+3})-\frac{n}{2}+i , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \frac{n}{2}+1\leq i\leq k^{*}+1 }\\ \varphi(x_{2n-2k^{*}-2}x_{n})-k^{*}+i-1 , \ \ \ \ \ \ \ \ { \ \ \ \ \ k^{*}+2\leq i\leq 2n-2k^{*}-3 }. \end{array} \right. \end{align*}$

For $2k^{*}-n+4\leq j\leq n-k^{*}-2$ (if $2n-3k^{*}-5 > 0$ ),

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{n-k^{*}-j}x_{n-k^{*}-1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 1\leq i\leq n-k^{*}-j-1 }\\ \varphi(x_{\frac{n}{2}-j+1}x_{\frac{n}{2}})-n+k^{*}+j+\\+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}-j\leq i\leq \frac{n}{2}-j }\\ \varphi(x_{k^{*}-j+2}x_{k^{*}+1})-\frac{n}{2}+j+i , \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \frac{n}{2}-j+1\leq i\leq k^{*}-j+1 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-k^{*}+\\+j+i-1 , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}-j+2 \leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}}x_{\frac{n}{2}+j-1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ { \ \ \ n-k^{*}\leq i\leq \frac{n}{2} }\\ \varphi(x_{k^{*}+1}x_{k^{*}+j})-\frac{n}{2}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \frac{n}{2}+1\leq i\leq k^{*}+1 }\\ \varphi(x_{n-j+1}x_{n})-k^{*}+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { k^{*}+2\leq i\leq n-j }. \end{array} \right. \end{align*}$

$\begin{align*} \varphi(x_{i}x_{i+n-k^{*}-1}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-\frac{n}{2}+2}x_{\frac{n}{2}})-n+k^{*}+\\+i+1 , \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq k^{*}-\frac{n}{2}+1 }\\ \varphi(x_{2k^{*}-n+3}x_{k^{*}+1})-\frac{n}{2}+i , \ \ \ \ \ \ \ \ \ { \ \ \ \ k^{*}-\frac{n}{2}+2\leq i\leq 2k^{*}-n+2 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}-3})-2k^{*}+\\+n+i-2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ 2k^{*}-n+3\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}}x_{\frac{3n}{2}-k^{*}-2})-n+k^{*}+\\+i+1 , \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq \frac{n}{2} }\\ \varphi(x_{k^{*}+1}x_{n-1})-\frac{n}{2}+i , \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \frac{n}{2}+1\leq i\leq k^{*}+1 }. \end{array} \right. \end{align*}$

For $n-k^{*}\leq j \leq \frac{n}{2}-1$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{\frac{n}{2}-j+1}x_{\frac{n}{2}})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq \frac{n}{2}-j }\\ \varphi(x_{k^{*}-j+2}x_{k^{*}+1})-\frac{n}{2}+j+i , \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \frac{n}{2}-j+1\leq i\leq k^{*}-j+1 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-k^{*}+\\+j+i-1 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k^{*}-j+2\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}}x_{\frac{n}{2}+j-1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ { n-k^{*}\leq i\leq \frac{n}{2} }\\ \varphi(x_{k^{*}+1}x_{k^{*}+j})-\frac{n}{2}+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \frac{n}{2}+1\leq i\leq n-j }. \end{array} \right. \end{align*}$

For $\frac{n}{2}\leq j\leq k^{*}$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \left\{ \begin{array}{l} \varphi(x_{k^{*}-j+2}x_{k^{*}+1})+i , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ 1\leq i\leq k^{*}-j+1 }\\ \varphi(x_{n-k^{*}-1}x_{n-k^{*}+j-2})-k^{*}+j+i-1 , \ \ \ \ \ \ \ \ \ \ \ { k^{*}-j+2\leq i\leq n-k^{*}-1 }\\ \varphi(x_{\frac{n}{2}}x_{\frac{n}{2}+j-1})-n+k^{*}+i+1 , \ \ \ \ \ \ \ \ \ \ \ \ \ { \ \ \ \ \ \ \ \ \ n-k^{*}\leq i\leq \ n-j }. \end{array} \right. \end{align*}$

For $k^{*}+1\leq j\leq r$ ,

$\begin{align*} \varphi(x_{i}x_{i+j}) = \varphi(x_{n-j+1}x_{n})+i, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ { 1\leq i\leq n-j }. \end{align*}$

As the pervious Subcase 2.3.1.2, the edge set can be divided into ten mutually separated subsets as follows:

● $A_{1} = \{x_{i}x_{i+j}:1\leq j \leq n-k^{*}-2, 1\leq j \leq n-k^{*}-j-1 \}$ : The set of all edges which have endpoints tagged with $0$ ,

● $A_{2} = \{x_{i}x_{i+j}:1\leq j\leq \frac{2k^{*}-n+2}{2}, n-k^{*}-j\leq i\leq n-k^{*}-1\}\bigcup\{x_{i}x_{i+j}:\frac{2k^{*}-n+4}{2}\leq j\leq n-k^{*}-1, n-k^{*}-j\leq i\leq \frac{n}{2}-j\}\bigcup\{x_{i}x_{i+j}:n-k^{*}\leq j\leq \frac{n}{2}-1, 1\leq i\leq \frac{n}{2}-j\}$ : The set of all edges which have endpoints tagged with $0$ and $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor$ ,

● $A_{3} = \{x_{i}x_{i+j}:1\leq j \leq \frac{2k^{*}-n}{2}, n-k^{*}\leq i\leq\frac{n}{2}-j\}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(k^{*}-2)(k^{*}-3)}{4}\rfloor$ ,

● $A_{4} = \{x_{i}x_{i+j}:1\leq j \leq \frac{2k^{*}-n+2}{2}, \frac{n+2}{2}-j\leq i\leq\frac{n}{2} \}\bigcup\{x_{i}x_{i+j}:\frac{2k^{*}-n+4}{2}\leq j \leq 2k^{*}-n+1, n-k^{*}\leq i\leq k^{*}-j+1 \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor$ and $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+2)}{4}\rfloor$ ,

● $A_{6} = \{x_{i}x_{i+j}:\frac{2k^{*}-n+4}{2}\leq j \leq 2k^{*}-n+2, k^{*}-j+2\leq i\leq \frac{n}{2} \}\bigcup\{x_{i}x_{i+j}:2k^{*}-n+3\leq j \leq k^{*}, n-k^{*}\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor$ and $k$ ,

● $A_{7} = \{x_{i}x_{i+j}:1\leq j \leq \frac{2k^{*}-n+2}{2}, k^{*}-j+2\leq i\leq k^{*}+1 \}\bigcup\{x_{i}x_{i+j}:\frac{2k^{*}-n+4}{2} \leq j \leq n-k^{*}-1, \frac{n}{2}+1\leq i\leq k^{*}+1 \}\bigcup \{n-k^{*}\leq j \leq \frac{n}{2}-1, \frac{n}{2}+1\leq i \leq n-j \}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+2)}{4}\rfloor$ and $k$ ,

● $A_{8} = \{x_{i}x_{i+j}:\frac{2k^{*}-n+4}{2}\leq j\leq 2k^{*}-n+2, \frac{n}{2}-j+1\leq i \leq n-k^{*}-1 \} \bigcup\{x_{i}x_{i+j}:2k^{*}-n+3\leq j\leq \frac{n}{2}, \frac{n}{2}+1-j\leq i \leq k^{*}-j+1\}\bigcup\{x_{i}x_{i+j}:\frac{n}{2}+1\leq j\leq k^{*}, 1\leq i \leq k^{*}-j+1 \}$ : The set of all edges which have endpoints tagged with $0$ and $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+2)}{4}\rfloor$ ,

● $A_{9} = \{x_{i}x_{i+j}:1\leq j \leq \frac{(2k^{*}-n)}{2}, \frac{n+3}{2}\leq i\leq k^{*}+j-1\}$ : The set of all edges which have endpoints tagged with $2\lfloor\frac{(n-k^{*}-1)(n-k^{*}-2)}{4}\rfloor+2\lfloor\frac{(n-k^{*}-1)(2k^{*}-n+2)}{4}\rfloor$ ,

● $A_{10} = \{x_{i}x_{i+j}:1\leq j\leq n-k^{*}-2, k^{*}+2\leq i\leq n-j \}$ : The set of all edges which have endpoints tagged with $k$ .

One can easily check that under the total $k$ -labeling $\varphi$ the edge (edges):

1. from the set $A_{1}$ , get the weights from the successive numbers $\{1, 2, ..., \frac{(n-k^{*}-1)(n-k^{*}-2)}{2}\}$ ,

2. from the set $A_{2}$ , get the weights from the successive numbers $\{\frac{(n-k^{*}-1)(n-k^{*}-2)}{2}+1, ..., \frac{k^{*}(n-k^{*}-1)}{2}\}$ ,

3. from the set $A_{3}$ , get the weights from the successive numbers $\{\frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}+1, ..., \frac{4k^{*}(n-k^{*}-1)+(2k^{*}-n+2)(3n-2k^{*}-4)}{8}\}$ ,

4. from the set $A_{4}$ , get the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{n(4k^{*}-n+2)}{8}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(2k^{*}-n+2)(3n-2k^{*}-4)}{8}\}$ ,

5. from the set $A_{5}$ , admit the weights from the successive numbers\\ $\{\frac{4k^{*}(n-k^{*}-1)+(2k^{*}-n+2)(3n-2k^{*}-4)}{8}+1, ..., \frac{r(2n-r-1)}{2}-\frac{n(4k^{*}-n+2)}{8}\}$ ,

6. from the set $A_{6}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{n(4k^{*}-n+2)}{8}\}$ ,

7. from the set $A_{7}$ , obtain the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{k^{*}(n-k^{*}-1)}{2}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(n-k^{*}-2)}{2}\}$ ,

8. from the set $A_{8}$ , obtain the weights from the successive numbers $\{\frac{k^{*}(n-k^{*}-1)}{2}+1, ..., \frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}\}$ ,

9. from the set $A_{9}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(2k^{*}-n+2)(3n-2k^{*}-4)}{8}+1, ..., \frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(3k^{*}-n+2)}{2}\}$ ,

10. from the set $A_{10}$ , admit the weights from the successive numbers $\{\frac{r(2n-r-1)}{2}-\frac{(n-k^{*}-1)(n-k^{*}-2)}{2}+1, ..., \frac{r(2n-r-1)}{2}\}$ .

By direct computation we can verify that in the both subcases all vertices are labeled by even numbers and the edge weights of the edges are different numbers from the set $\{1, 2, 3, ..., \frac{r(2n-r-1)}{2}\}$ . Hence we prove that in all cases vertices are even numbers and the labels of edges are less than or equal to $k = \lceil\frac{r(2n-r-1)}{6}\rceil+1$ where $\frac{r(2n-r-1)}{2}\equiv 2, 3$ (mod $6$ ) and less than or equal to $k = \lceil\frac{r(2n-r-1)}{6}\rceil$ where $\frac{r(2n-r-1)}{2}\not\equiv 2, 3$ (mod $6$ ). Thus the edge irregular reflexive strength of the $r$ -th power of the path $P^{r}_{n}$ :

${res}\left(P_{n}^{r}\right)= \begin{cases}\left\lceil\frac{r(2 n-r-1)}{6}\right\rceil, & \text { if } \frac{r(2 n-r-1)}{2} \not \equiv 2,3(\bmod 6) \\ \left\lceil\frac{r(2 n-r-1)}{6}\right\rceil+1, & \text { if } \frac{r(2 n-r-1)}{2} \equiv 2,3(\bmod 6) .\end{cases}$

3. Conclusions

In this paper we discussed the edge irregular reflexive k-labeling of the $r$ -th power of the path $P_{n}$ , where $r\geq2$ , $n\geq r+4$ . Also, we computed the precise value of the reflexive edge strength of $P^{r}_{n}$ , $r\geq2$ , $n\geq r+4$ . In the future, we would like to calculate the reflexive edge strength, res, for r-th power of other graphs.

Conflict of interest

The authors declare that they have no competing interest.

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