Total edge irregularity strength for special types of square snake graphs

Abstract

One of the extremely useful branches in graph theory is the labeling of a graph. Graph labeling plays a vital role in many fields such as database management, astronomy, coding theory, X-ray crystallography, communication network addressing and radar. A labeling of a connected simple graph $G\left( {V,E} \right)$ is a map that assign each element in $G$ with a positive integer number. An edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling is a map $\beta :V\left( G \right) \cup E\left( G \right) \to \left\{ 1,2,3, \ldots ,\lambda^{\!\!\!\!\!-} \right\}$ such that $W_\beta \left( h \right) \ne W_\beta \left( z \right)$ where $W_\beta \left( h \right)$ and $W_\beta \left( z \right)$ are weights for any two distinct edges. In this case, $G$ has total edge irregularity strength (TEIS) if $\lambda^{\!\!\!\!\!-}$ is minimum. In this paper, a new family of graphs called square snake graphs is defined and denoted by $C_{4,n}$. Moreover, we define some related graphs of square snake graphs named double square snake graph $D\left( {C_{4,n} } \right)$, triple square snake graph $T\left( {C_{4,n} } \right)$ and $m$-multiple square snake graph $M_m \left( {C_{4,n} } \right)$. Finally, we determine TEIS for square snake graphs, double square snake graph, triple square snake graph and $m$-multiple square snake graph, which have many applications in coding theory and physics.

1 Introduction

For a simple, connected and undirected graph $G\left( {V,E} \right)$, Bača et al. (2007) have introduced the notion of an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling $\beta :V\left( G \right) \cup E\left( G \right) \to \left\{ 1,2,3, \ldots ,\lambda^{\!\!\!\!\!-} \right\}$ to be a labeling of edges and vertices in such a way that any two edges $pq$ and $p^* q^*$ of a graph $G$ have distinct weights, i.e. $w_\beta \left( {pq} \right) \ne w_\beta \left( {p^* q^* } \right)$ where $w_\beta \left( {pq} \right) = \beta \left( {pq} \right) + \beta \left( p \right) + \beta \left( q \right)$. The minimum $\lambda^{\!\!\!\!\!-}$ for which $G\left( {V,E} \right)$ has an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling is named TEIS and denoted by $tes\left( G \right).$ Also, They have estimated the bounds of TEIS of $tes\left( G \right)$ for a graph $G$ in the following inequality

$$ {\text{tes}}\left( G \right) \ge \max \left\{ {\left\lceil {\frac{{\left| {E\left( G \right)} \right| + 2}}{3}} \right\rceil , \left\lceil {\frac{\Delta G + 1}{2}} \right\rceil } \right\} $$

(1)

where $\Delta G$ is maximum degree of vertices of a graph $G.$

In Ivanĉo and Jendroî (2006) Ivanĉo and Jendroî introduced the conjecture which gives a TEIS for any graph $G$, except $K_5$, in the form

$$ {\text{tes}}\left( G \right) = \max \left\{ {\left\lceil {\frac{\Delta G + 1}{2}} \right\rceil , \left\lceil {\frac{{\left| {E\left( G \right)} \right| + 2}}{3}} \right\rceil } \right\}. $$

(2)

In addition, they have determined TEIS for a tree $T$ as

$$ {\text{tes}}\left( T \right) = \max \left\{ {\left\lceil {\frac{k + 1}{3}} \right\rceil , \left\lceil {\frac{\Delta G + 1}{2}} \right\rceil } \right\}. $$

(3)

The conjecture of Ivanĉo and Jendroî has been verified for a heptagonal snake graph, uniform theta snake graphs, quintet snake graph, a polar grid graph and special families of graphs in Salama (2022, 2021, 2019, 2020a) and Salama and Abo Elanin (2021), for cylindrical accordion graph and spiral accordion graph in Siddiqui et al. (2017), for complete bipartite graphs and complete graphs in Jendroî et al. (2007), for corona product of a path with certain graphs in Salman and Baskoro (2008), for the categorical product of two paths $P_n \times P_m$ in Ahmad and Bača (2014), for centralized uniform theta graphs in Putra and Susanti (2018), for zigzag graphs in Ahmad et al. (2012), for sunlet graph and the line of sunlet graph in Salama (2020b), for hexagonal grid graphs in Al-Mushayt et al. (2012), for a wheel graph, a fan graph, a triangular Book graph and a friendship graph in Tilukay et al. (2015), for generalized prism in Bača and Siddiqui (2014), for disjoint union of isomorphic copies of generalized Petersen graph in Naeem et al. (2017) and for subdivision of star graph in Siddiqui (2012), for more details see Majerski and Przybylo (2014), Rajasingh and Arockiamary (2015), Jeyanthi and Sudha (2015), Tarawneh et al. (2021), Yang et al. (2018), Ahmad et al. (2016, 2015), Jegan et al. (2022), Nurdini et al. (2020), Mary et al. (2018), Ratnasari et al. (2019) and Arockiamary and Vistthra (2022).

Motivated by the previous results, in this paper, we define some related graph of square snake graphs named double square snake graph $D\left( {C_{4,n} } \right)$, triple square snake graph $T\left( {C_{4,n} } \right)$ and m-multiple square snake graph Finally, we determine the exact value of the total edge irregularity strength (TEIS) for square snake graphs, double square snake graph, triple square snake graph and m-multiple square snake graph, in the form:

$$ {\text{tes}}\left( {C_{4,n} } \right) = \left\lceil {\frac{4n + 2}{3}} \right\rceil , $$

(4)

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil , $$

(5)

$$ {\text{tes}}\left( {T\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{12n + 2}{3}} \right\rceil , $$

(6)

$$ {\text{tes}}\left( {M_m \left( {C_{4,n} } \right) } \right) = \left\lceil {\frac{4mn + 2}{3}} \right\rceil . $$

(7)

Which have many applications in coding theory and physics.

2 Main results

The table of notations and acronyms is used in this paper (Table 1).

Table 1 Notations and acronyms used in this paper

In this section we define a new family of graphs and determine the exact value of TEIS for it which are used in scanner and bar code as follows:

Let the vertices $x_1 , x_2 , x_3 , \ldots ,x_n$ be $n$ computer scanners jointed with $n$ automatic scanners. Let the vertices $c_1 , c_2 , c_3 , \ldots ,c_{n + 1}$ be representing $n + 1$ automatic scanner and item details, connected with $n$ computer scanner and $n$ alarm. Now, we have two options, one is an alarm system barcode-neutralized system, other is a barcode-neutralized system that is combined with an exit option (Fig. 1).

Fig. 1

Scanner and bar code

Definition 1

A square snake graph $C_{4,n}$ is a graph obtained by replacing every edge of a path $P_n$ by a square $C_4$, as shown in Fig. 2.

Fig. 2

Square snake graph $C_{4,n}$

In the following the flow chart of the steps which follow through our work,

In the following theorem, we calculate the TEIS of the square snake graph $C_{4,n}$ by using inequality (1):

Theorem 1

For a square snake graph $C_{4,n}$ a TEIS is given by:

$$ {\text{tes}}\left( {C_{4,n} } \right) = \left\lceil {\frac{4n + 2}{3}} \right\rceil . $$

Proof

For a square snake graph, we find $\left| {E\left( {C_{4,n} } \right)} \right| = 4n$ and $\Delta \left( {C_{4,n} } \right) = 4$, then from (1) we have

$$ {\text{tes}}\left( {C_{4,n} } \right) \ge \left\lceil {\frac{4n + 2}{3}} \right\rceil . $$

(8)

To prove the inverse inequality, we will show that there exists an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling for $C_{4,n}$. Consider $\lambda^{\!\!\!\!\!-} = \left\lceil {\frac{4n + 2}{3}} \right\rceil$ and $\beta :V\left( {C_{4,n} } \right) \cup E\left( {C_{4,n} } \right) \to \left\{1,2,3, \ldots ,\lambda^{\!\!\!\!\!-} \right\}$ is a total $\lambda^{\!\!\!\!\!-}$-labeling which is defined in the following cases as:

Case 1: In this case, we will discuss an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling when the remainder of the division of $4n + 2$ by 3 is 0 or 1 (Fig. 3).

Fig. 3

The flow chart of the steps of our work

β is defined as:

$$ \begin{aligned} \beta \left( {c_i } \right) & = \left\{ {\begin{array}{*{20}l} i \hfill & {\quad {\text{for}}\,1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{2} } \hfill \\ \lambda^{\!\!\!\!\!-} \hfill & {\quad {\text{for}}\,\frac{\lambda^{\!\!\!\!\!-}}{2} + 1 \le i \le n + 1} \hfill \\ \end{array} } \right., \\ \beta \left( {x_i } \right) & = \beta \left( {y_i } \right) = \left\{ {\begin{array}{*{20}l} i \hfill & {\quad {\text{for}}\, 1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{2} - 1 } \hfill \\ \lambda^{\!\!\!\!\!-} \hfill & {\quad {\text{for}}\,\frac{\lambda^{\!\!\!\!\!-}}{2} \le i \le n} \hfill \\ \end{array} } \right., \\ \beta \left( {c_i x_i } \right) & = \left\{ {\begin{array}{*{20}l} {2i - 1} \hfill & {\quad {\text{for}}\,1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{2} - 1 } \hfill \\ {\frac{\lambda^{\!\!\!\!\!-}}{2} - 1} \hfill & { \quad {\text{for}}\,i = \frac{\lambda^{\!\!\!\!\!-}}{2} } \hfill \\ {4i - 2\lambda^{\!\!\!\!\!-} - 1} \hfill & {\quad {\text{for}}\, \frac{\lambda^{\!\!\!\!\!-}}{2} + 1 \le i \le n} \hfill \\ \end{array} ,} \right. \\ \beta \left( {c_{i + 1} x_i } \right) & = \left\{ {\begin{array}{*{20}l} {2i} \hfill & {\quad {\text{for}}\, 1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{2} - 1 } \hfill \\ {4i - 2\lambda^{\!\!\!\!\!-} + 1} \hfill & {\quad {\text{for}}\, \frac{\lambda^{\!\!\!\!\!-}}{2} \le i \le n} \hfill \\ \end{array} } \right., \\ \beta \left( {c_i y_i } \right) & = \beta \left( {c_i x_i } \right) + 1, \\ \beta \left( {c_{i + 1} y_i } \right) & = \beta \left( {c_{i + 1} x_i } \right) + 1. \\ \end{aligned} $$

It is clear that $\lambda^{\!\!\!\!\!-}$ is the greatest label. In addition, the weights of edges of $C_{4,n}$ are given by:

$$ \begin{aligned} w_\beta \left( {c_i x_i } \right) & = 4i - 1\quad {\text{for}}\, 1 \le i \le n, \\ w_\beta \left( {c_{i + 1} x_i } \right) & = 4i + 1\quad {\text{for}}\,1 \le i \le n, \\ w_\beta \left( {c_i y_i } \right) & = 4i\quad {\text{for}}\,1 \le i \le n, \\ w_\beta \left( {c_{i + 1} y_i } \right) & = 4i + 2\quad {\text{for}}\,1 \le i \le n. \\ \end{aligned} $$

Obviously, the edges weights are distinct. Therefore, $\beta$ is an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling. Hence.

$$ {\text{tes}}\left( {C_{4,n} } \right) = \left\lceil {\frac{4n + 2}{3}} \right\rceil . $$

Case 2: In this case, we will discuss an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling when the remainder of the division of $4n + 2$ by 3 is 2.

Define $\beta$ as:

$$ \begin{aligned} \beta \left( {c_i } \right) & = \left\{ {\begin{array}{*{20}l} i \hfill & {\quad {\text{for}}\,1 \le i \le \left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil } \hfill \\ \lambda^{\!\!\!\!\!-} \hfill & {\quad {\text{for}}\, \left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil + 1 \le i \le n + 1} \hfill \\ \end{array} } \right., \\ \beta \left( {x_i } \right) & = \beta \left( {y_i } \right) = \left\{ {\begin{array}{*{20}l} i \hfill & {\quad {\text{for}}\,1 \le i \le \left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil - 1 } \hfill \\ \lambda^{\!\!\!\!\!-} \hfill & {\quad {\text{for}}\,\left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil \le i \le n} \hfill \\ \end{array} } \right., \\ \beta \left( {c_i x_i } \right) & = \left\{ {\begin{array}{*{20}l} {2i - 1} \hfill & {\quad {\text{for}}\,1 \le i \le \left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil - 1 } \hfill \\ {\frac{\lambda^{\!\!\!\!\!-}}{2}} \hfill & { \quad {\text{for}}\, i = \left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil } \hfill \\ {4i - 4\frac{\lambda^{\!\!\!\!\!-}}{2} + 1} \hfill & { \quad {\text{for}}\,\left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil \frac{\lambda^{\!\!\!\!\!-}}{2} + 1 \le i \le n} \hfill \\ \end{array} } \right., \\ \beta \left( {c_{i + 1} x_i } \right) & = \left\{ {\begin{array}{*{20}l} {2i} \hfill & { \quad {\text{for}}\,1 \le i \le \left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil - 1 } \hfill \\ {4i - 4\frac{\lambda^{\!\!\!\!\!-}}{2} + 3} \hfill & { \quad {\text{for}}\,\left\lceil {\frac{\lambda^{\!\!\!\!\!-}}{2}} \right\rceil \le i \le n} \hfill \\ \end{array} } \right., \\ \beta \left( {c_i y_i } \right) & = \beta \left( {c_i x_i } \right) + 1, \\ \beta \left( {c_{i + 1} y_i } \right) & = \beta \left( {c_{i + 1} x_i } \right) + 1. \\ \end{aligned} $$

Clearly, $\lambda^{\!\!\!\!\!-}$ is the most label of edges and vertices. The weights of the edges are given as follows:

$$ \begin{aligned} w_\beta \left( {c_i x_i } \right) & = \left\{ {\begin{array}{*{20}l} {4i - 1} \hfill & {\quad {\text{for}}\,~1 \le i \le \left\lceil {\frac{\lambda^{\!\!\!\!\!-} }{2}} \right\rceil ~ - 1~} \hfill \\ {2\left\lceil {\frac{\lambda^{\!\!\!\!\!-} }{2}} \right\rceil + \lambda^{\!\!\!\!\!-} } \hfill & {\quad {\text{for}}\,~i = \left\lceil {\frac{\lambda^{\!\!\!\!\!-} }{2}} \right\rceil ~} \hfill \\ {4i - 4\left\lceil {\frac{\lambda^{\!\!\!\!\!-} }{2}} \right\rceil + 2\lambda^{\!\!\!\!\!-} + 1~} \hfill & {\quad {\text{for}}\,~\left\lceil {\frac{\lambda^{\!\!\!\!\!-} }{2}} \right\rceil + 1 \le i \le n} \hfill \\ \end{array} } \right., \\ w_\beta \left( {c_{i + 1} x_i } \right) & = \left\{ {\begin{array}{*{20}l} {4i + 1} \hfill & {\quad {\text{for}}\,~1 \le i \le \left\lceil {\frac{\lambda^{\!\!\!\!\!-} }{2}} \right\rceil ~ - 1~} \hfill \\ {4i - 4\left\lceil {\frac{\lambda^{\!\!\!\!\!-} }{2}} \right\rceil + 2\lambda^{\!\!\!\!\!-} - 1} \hfill & {\quad {\text{for}}\,~\left\lceil {\frac{\lambda^{\!\!\!\!\!-} }{2}} \right\rceil \le i \le n} \hfill \\ \end{array} } \right., \\ w_\beta \left( {c_i y_i } \right) & = w_\beta \left( {c_i x_i } \right) + 1, \\ w_\beta \left( {c_{i + 1} y_i } \right) & = w_\beta \left( {c_{i + 1} x_i } \right) + 1, \\ \end{aligned} $$

From the above equations we can deduce that the weights of edges are distinct. Then $\beta$ is an edge irregular itotal $\lambda^{\!\!\!\!\!-}$-labeling. Hence,

$$ {\text{tes}}\left( {C_{4,n} } \right) = \left\lceil {\frac{4n + 2}{3}} \right\rceil . $$

Definition 2

A double square snake graph $D\left( {C_{4,n} } \right)$ consists of two square graphs that have a common path $P_n$, see Fig. 4.

Fig. 4

A double square snake graph $D\left( {C_{4,n} } \right)$

Lemma 1

For a double square snake graph $D\left( {C_{4,n} } \right)$, where $3 \le n \le 11$. We have

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

Proof

As $\left| {E\left( {D\left( {C_{4,n} } \right)} \right)} \right| = 8n$, $3 \le n \le 11$, then (1) becomes

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) \ge \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

(9)

Our aim is proving equality, so we need only to show that there exists an edge irregular total $\hbar$-labeling for $D\left( {C_{4,n} } \right)$ where $3 \le n \le 11$ with $\hbar = \left\lceil {\frac{8n + 2}{3}} \right\rceil$. Let $\hbar = \left\lceil {\frac{8n + 2}{3}} \right\rceil$ and a total $\hbar$-labeling $\beta :V\left( {D\left( {C_{4,n} } \right)} \right) \cup E\left( {D\left( {C_{4,n} } \right)} \right) \to \left\{ {1,2, \ldots ,\hbar} \right\}$ is defined as:

$$ \begin{aligned} \beta \left( {c_i } \right) & = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {3i - 3 } \hfill & {\quad {\text{for}}\,2 \le i \le n } \hfill \\ \hbar \hfill & {\quad {\text{for}}\,i = n + 1} \hfill \\ \end{array} } \right. , \\ \beta \left( {x_{1,i} } \right) & = \left\{ {\begin{array}{*{20}l} {3i - 2} \hfill & {\quad {\text{for}}\,1 \le i \le n\quad {\text{if}}\,i \in \left\{ {3, \ldots ,7} \right\} } \hfill \\ {3i - 2 } \hfill & {\quad {\text{for}}\,1 \le i \le n - 1\quad {\text{if}}\, i \in \left\{ {8, \ldots ,11} \right\}} \hfill \\ \hbar \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\, i \in \left\{ {8, \ldots ,11} \right\}} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{2,i} } \right) & = \left\{ {\begin{array}{*{20}l} {3i - 1 } \hfill & {\quad {\text{for}}\,1 \le i \le n\quad {\text{if}}\, i \in \left\{ {3, \ldots ,7} \right\}} \hfill \\ {3i - 1 } \hfill & {\quad {\text{for}}\,1 \le i \le n - 1\quad {\text{if}}\, i \in \left\{ {8, \ldots ,11} \right\}} \hfill \\ \hbar \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\, i \in \left\{ {8, \ldots ,11} \right\}} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{1,i} } \right) & = \left\{ {\begin{array}{*{20}l} {3i - 2 } \hfill & {\quad {\text{for}}\,1 \le i \le n\quad {\text{if}}\, i \in \left\{ {3, \ldots ,7} \right\} } \hfill \\ {3i - 2 } \hfill & {\quad {\text{for}}\,1 \le i \le n - 1\quad {\text{if}}\, i \in \left\{ {8, \ldots ,11} \right\}} \hfill \\ \hbar \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i \in \left\{ {8, \ldots ,11} \right\}} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{2,i} } \right) & = \left\{ {\begin{array}{*{20}l} {3i - 1 } \hfill & {\quad {\text{for}}\,1 \le i \le n\quad {\text{if}}\, i \in \left\{ {3, \ldots ,7} \right\} } \hfill \\ {3i - 1 } \hfill & {\quad {\text{for}}\,1 \le i \le n - 1\quad {\text{if}}\, i \in \left\{ {8, \ldots ,11} \right\}} \hfill \\ \hbar \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\,i \in \left\{ {8, \ldots ,11} \right\}} \hfill \\ \end{array} } \right., \\ \beta \left( {c_i x_{1,i} } \right) & = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad {\text{for}}\,i = 1 } \hfill \\ {2i} \hfill & {\quad {\text{for}}\,2 \le i \le n - 1 } \hfill \\ {2n} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\,i \in \left\{ {3, \ldots ,10} \right\}} \hfill \\ {2n + 2} \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\, i = 11} \hfill \\ \end{array} } \right., \\ \beta \left( {c_i y_{1,i} } \right) & = \beta \left( {c_i x_{1,i} } \right) + 1, \\ \beta \left( {c_i x_{2,i} } \right) & = \left\{ {\begin{array}{*{20}l} 2 \hfill & {\quad {\text{for}}\,i = 1 } \hfill \\ {2i + 1} \hfill & {\quad {\text{for}}\, 2 \le i \le n - 1 } \hfill \\ {2n + 1} \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\, i \in \left\{ {3, \ldots ,7} \right\}} \hfill \\ {2n + 2} \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\, i \in \left\{ {8,9,10} \right\}} \hfill \\ {2n + 4} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i = 11} \hfill \\ \end{array} } \right., \\ \beta \left( {c_i y_{2,i} } \right) & = \beta \left( {c_i x_{2,i} } \right) + 1, \\ \beta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {2i + 1} \hfill & {\quad {\text{for}}\,1 \le i \le n - 1 } \hfill \\ {2n + 1} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i \in \left\{ {3,4} \right\} } \hfill \\ {2n + 2} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i \in \left\{ {5,6,7} \right\} } \hfill \\ {2n + 3} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i \in \left\{ {8,9,10} \right\} } \hfill \\ {2n + 5} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i = 11} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{1,i} c_{i + 1} } \right) & = \beta \left( {x_{1,i} c_{i + 1} } \right) + 1, \\ \beta \left( {x_{2,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {2i + 2} \hfill & {\quad {\text{for}}\,1 \le i \le n - 1 } \hfill \\ {2n + 2} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i \in \left\{ {3,4} \right\} } \hfill \\ {2n + 3} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i \in \left\{ {5,6,7} \right\} } \hfill \\ {2n + 5} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i \in \left\{ {8,9,10} \right\} } \hfill \\ {2n + 7} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\, i = 11} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{2,i} c_{i + 1} } \right) & = \beta \left( {x_{2,i} c_{i + 1} } \right) + 1. \\ \end{aligned} $$

From the previous equations we deduce that, $\hbar$ is the greatest label of vertices and edges. We get the weights of edges as follows:

$$ \begin{aligned} W_\beta \left( {c_i x_{1,i} } \right) & = \left\{ {\left| {\begin{array}{*{20}l} 3 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {8i - 5 } \hfill & {\quad {\text{for}}\, 2 \le i \le n - 1 } \hfill \\ {5n + \hbar - 3 } \hfill & {\quad {\text{for}}\, i = n \quad {\text{if}}\,i \in \left\{ {3, \ldots ,10} \right\} } \hfill \\ {5n + \hbar - 1} \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\,i = 11} \hfill \\ \end{array} } \right.} \right., \\ W_\beta \left( {c_i x_{2,i} } \right) & = \left\{ {\begin{array}{*{20}l} 5 \hfill & {\quad {\text{for}}\,i = 1 } \hfill \\ {8i - 3} \hfill & {\quad {\text{for}}\,2 \le i \le n - 1 } \hfill \\ {8n - 3} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\,i \in \left\{ {3, \ldots ,7} \right\}} \hfill \\ {5n + \hbar - 1} \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\,i \in \left\{ {8,9,10} \right\}} \hfill \\ {5n + \hbar + 1} \hfill & {\quad {\text{for}}\, i = n\quad {\text{if}}\,i = 11} \hfill \\ \end{array} } \right., \\ W_\beta \left( {c_i y_{1,i} } \right) & = W_\beta \left( {c_i x_{1,i} } \right) + 1, \\ W_\beta \left( {c_i y_{2,i} } \right) & = W_\beta \left( {c_i x_{2,i} } \right) + 1, \\ W_\beta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {8i - 1} \hfill & {\quad {\text{for}}\,1 \le i \le n - 1 } \hfill \\ {5n + \hbar - 1 } \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\,i \in \left\{ {3,4} \right\} } \hfill \\ {5n + \hbar } \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\,i \in \left\{ {5,6,7} \right\} } \hfill \\ {2n + 2\hbar + 3} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\,i \in \left\{ {8,9,10} \right\} } \hfill \\ {2n + 2\hbar + 5} \hfill & {\quad {\text{for}}\,i = n\quad {\text{if}}\,i = 11 } \hfill \\ \end{array} } \right., \\ W_\beta \left( {x_{2,i} c_{i + 1} } \right) & = W_\beta \left( {x_{1,i} c_{i + 1} } \right) + 2, \\ W_\beta \left( {y_{1,i} c_{i + 1} } \right) & = W_\beta \left( {x_{1,i} c_{i + 1} } \right) + 1, \\ W_\beta \left( {y_{2,i} c_{i + 1} } \right) & = W_\beta \left( {x_{1,i} c_{i + 1} } \right) + 3. \\ \end{aligned} $$

From the above equations of weights of edges we find that they are distinct. Hence, $\theta$ is an edge irregular total $\hbar$-labeling. Thus

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

Theorem 2

If $D\left( {C_{4,n} } \right)$ is a double square snake graph, where $n \ge 12$. Then

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

Proof

Since $\left| {E\left( {D\left( {C_{4,n} } \right)} \right)} \right| = 8n$ and $\Delta \left( {D\left( {C_{4,n} } \right)} \right) = 8$. Then from inequality (1) we have

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) \ge \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

In the following, we define an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling to get the upper pound. Consider a total $\lambda^{\!\!\!\!\!-}$-labeling is a map $\theta :V\left( {D\left( {C_{4,n} } \right)} \right) \cup E\left( {D\left( {C_{4,n} } \right)} \right) \to \left\{ 1,2, \ldots , \lambda^{\!\!\!\!\!-} \right\},{ }\lambda^{\!\!\!\!\!-} = \left\lceil {\frac{8n + 2}{3}} \right\rceil$.

Throughout the proof we take the following notations:

• I is the condition:

  $$ 1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3} \quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right) \quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right) $$

• II is the condition:

  $$ 1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3} + 1\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right) $$

• III is the condition:

  $$ \frac{\lambda^{\!\!\!\!\!-}}{3} + 1 \le i \le n\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right)\quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right) $$

• IV is the condition:

  $$ \frac{\lambda^{\!\!\!\!\!-}}{3} + 2 \le i \le n\quad {\text{if}}\,\lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right) $$

• V is the condition:

  $$ 2 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3}\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right)\quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right) $$

• VI is the condition:

  $$ 2 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3} + 1\quad {\text{if}}\,\lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right) $$

Now, we define $\theta$ in the following three cases as:

Case 1: In this case, we will discuss an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling when the remainder of the division of $8n + 2$ by 3 is 0.

$\theta$ is defined as:

$$\begin{aligned} \theta \left( {c_i } \right) & = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad {\text{for}}\,~i = 1~} \hfill \\ {3i - 3~} \hfill & {\quad {\text{for}}\,~2 \le i \le \frac{\lambda^{\!\!\!\!\!-} }{3} + 1~} \hfill \\ {~\lambda^{\!\!\!\!\!-} } \hfill & {\quad {\text{for}}\,~\frac{\lambda^{\!\!\!\!\!-} }{3} + 2 \le i \le n + 1} \hfill \\ \end{array} } \right. \\ \theta \left( {x_{1,i} } \right) & = \theta \left( {y_{1,i} } \right) = \left\{ {\begin{array}{*{20}l} {3i - 2} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \right.} \hfill \\ \lambda^{\!\!\!\!\!-} \hfill & {\quad {\text{for}}~\left\{ {\begin{array}{*{20}c} {{\text{III}}} \\ {{\text{IV}}} \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ \theta \left( {x_{2,i} } \right) & = \theta \left( {y_{2,i} } \right)\left\{ {\begin{array}{*{20}l} {3i - 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \right.} \hfill \\ \lambda^{\!\!\!\!\!-} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {{\text{III}}} \\ {{\text{IV}}} \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ \theta \left( {c_i x_{1,i} } \right) & = ~\left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad {\text{for}}\,~i = 1~} \hfill \\ {2i} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {\text{V}} \\ {{\text{VI}}} \\ \end{array} } \right.~} \hfill \\ {~\lambda^{\!\!\!\!\!-} - 8\left( {n + 1 - i} \right) - 7} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {{\text{III}}} \\ {{\text{IV}}} \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ \theta \left( {c_i x_{2,i} } \right) & = ~\left\{ {\begin{array}{*{20}l} 2 \hfill & {\quad {\text{for}}\,~i = 1~} \hfill \\ {2i + 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {\text{V}} \\ {{\text{VI}}} \\ \end{array} } \right.~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 8\left( {n + 1 - i} \right) - 5} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {{\text{III}}} \\ {{\text{IV}}} \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ \theta \left( {c_i y_{1,i} } \right) & = ~\theta \left( {c_i x_{1,i} } \right) + 1 \\ \theta \left( {c_i y_{2,i} } \right) & = \theta \left( {c_i x_{2,i} } \right) + 1 \\ \theta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {2i + 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {\text{I}} \\ {{\text{II}}} \\ \end{array} } \right.} \hfill \\ {\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 3} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {{\text{III}}} \\ {{\text{IV}}} \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ \theta \left( {x_{2,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {2i + 2} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {\text{I}} \\ {{\text{II}}} \\ \end{array} } \right.} \hfill \\ {\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}c} {{\text{III}}} \\ {{\text{IV}}} \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ \theta \left( {y_{1,i} c_{i + 1} } \right) & = \theta \left( {x_{1,i} c_{i + 1} } \right) + 1 \\ \theta \left( {y_{2,i} c_{i + 1} } \right) & = \theta \left( {x_{2,i} c_{i + 1} } \right) + 1. \\ \end{aligned}$$

From the previous equations, we can say that $\lambda^{\!\!\!\!\!-}$ is the greatest label. Now, the weights of the edge of $D\left( {C_{4,n} } \right)$ are given by:

$$ \begin{aligned} W_\theta \left( {c_i x_{1,i} } \right) & = \left\{ {\begin{array}{*{20}l} 3 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {8i - 5} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{V}} \hfill \\ {{\text{VI}}} \hfill \\ \end{array} } \right.} \hfill \\ { 3\lambda^{\!\!\!\!\!-} - 8\left( {n + 1 - i} \right) + 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ W_\theta \left( {c_i x_{2,i} } \right) & = W_\theta \left( {c_i x_{1,i} } \right) + 2 \\ W_\theta \left( {c_i y_{1,i} } \right) & = W_\theta \left( {c_i x_{1,i} } \right) + 1 \\ W_\theta \left( {c_i y_{2,i} } \right) & = W_\theta \left( {c_i x_{1,i} } \right) + 3 \\ W_\theta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {8i - 4} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \right.} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) + 5 } \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ W_\theta \left( {x_{2,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 2 \\ W_\theta \left( {y_{1,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 1 \\ W_\theta \left( {y_{2,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 3 \\ \end{aligned} $$

Therefore, $\theta$ is an edge I irregular itotal $\lambda^{\!\!\!\!\!-}$-labeling. Hence

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

Case 2: In this case, we will discuss an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling when the remainder of the division of $8n + 2$ by 3 is 1.

$\theta$ is defined for vertices like case 1, but for edges is given by:

$$ \begin{aligned} \theta \left( {c_i x_{1,i} } \right) & = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {2i} \hfill & {\quad {\text{for}}\,\left\{ {\begin{array}{*{20}l} {\text{V}} \hfill \\ {{\text{VI}}} \hfill \\ \end{array} } \right. } \hfill \\ { \lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 9} \hfill & {\quad {\text{for}}\,\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} ,} \right. \\ \theta \left( {c_i x_{2,i} } \right) & = \left\{ {\begin{array}{*{20}l} 2 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {2i + 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{V}} \hfill \\ {{\text{VI}}} \hfill \\ \end{array} } \right. } \hfill \\ { \lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 7 } \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right., \\ \theta \left( {c_i y_{1,i} } \right) & = \theta \left( {c_i x_{1,i} } \right) + 1, \\ \theta \left( {c_i y_{2,i} } \right) & = \theta \left( {c_i x_{2,i} } \right) + 1, \\ \theta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {2i + 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \right.} \hfill \\ {\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 5} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} ,} \right. \\ \theta \left( {x_{2,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {2i + 2} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \right.} \hfill \\ {\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 3} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right., \\ \theta \left( {y_{1,i} c_{i + 1} } \right) & = \theta \left( {x_{1,i} c_{i + 1} } \right) + 1, \\ \theta \left( {y_{2,i} c_{i + 1} } \right) & = \theta \left( {x_{2,i} c_{i + 1} } \right) + 1. \\ \end{aligned} $$

From the previous equations, we can say that $\lambda^{\!\!\!\!\!-}$ is the greatest upper bound. Now, the weights of edge of $D(C_{4,n} )$ are given by:

$$ \begin{aligned} W_\theta \left( {c_i x_{1,i} } \right) = & \left\{ {\begin{array}{*{20}l} 3 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {8i - 5 } \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{V}} \hfill \\ {{\text{VI}}} \hfill \\ \end{array} } \right.} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 8\left( {n + 1 - i} \right) - 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ W_\theta \left( {c_i x_{2,i} } \right) = & W_\theta \left( {c_i x_{1,i} } \right) + 2 \\ W_\theta \left( {c_i y_{1,i} } \right) & = W_\theta \left( {c_i x_{1,i} } \right) + 1 \\ W_\theta \left( {c_i y_{2,i} } \right) & = W_\theta \left( {c_i x_{1,i} } \right) + 3 \\ W_\theta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {8i - 4} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \right.} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) + 3 } \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array} } \right. \\ W_\theta \left( {x_{2,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 2 \\ W_\theta \left( {y_{1,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 1 \\ W_\theta \left( {y_{2,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 3 \\ \end{aligned} $$

Therefore, $\theta$ is an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling and

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

Case 3: In this case, we will discuss an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling when the remainder of the division of $8n + 2$ by 3 is 2.

The function $\theta$ of vertices $c_i , x_{1,i} , x_{2,i} ,y_{1,i}$ and $y_{2,i}$ is the same like case 1. For edges, it is given by:

$$ \begin{aligned} \theta \left( {c_i x_{1,i} } \right) & = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {2i} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{V}} \hfill \\ {{\text{VI}}} \hfill \\ \end{array} } \right. } \hfill \\ { \lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 8} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right. } \hfill \\ \end{array} } \right., \\ \theta \left( {c_i x_{2,i} } \right) & = \left\{ {\begin{array}{*{20}l} 2 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {2i + 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{V}} \hfill \\ {{\text{VI}}} \hfill \\ \end{array} } \right. } \hfill \\ {\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 6} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right. } \hfill \\ \end{array} } \right., \\ \theta \left( {c_i y_{1,i} } \right) = & \theta \left( {c_i x_{1,i} } \right) + 1, \\ \theta \left( {c_i y_{2,i} } \right) & = \theta \left( {c_i x_{2,i} } \right) + 1, \\ \theta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {2i + 1} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \right. } \hfill \\ {\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 4} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right. } \hfill \\ \end{array} } \right., \\ \theta \left( {x_{2,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {2i + 2} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \right. } \hfill \\ {\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) - 2} \hfill & {\quad {\text{for}}\left\{ {\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \right. } \hfill \\ \end{array} } \right., \\ \theta \left( {y_{1,i} c_{i + 1} } \right) & = \theta \left( {x_{2,i} c_{i + 1} } \right) + 1, \\ \theta \left( {y_{2,i} c_{i + 1} } \right) & = \theta \left( {x_{2,i} c_{i + 1} } \right) + 1. \\ \end{aligned} $$

From the previous equations we can say that $\lambda^{\!\!\!\!\!-}$ is the greatest upper bound. Now, the weights of edge of $D(C_{4,n} )$ are given by:

$$ \begin{aligned} W_\theta \left( {c_i x_{1,i} } \right) & = \left\{ {\begin{array}{*{20}l} 3 \hfill & {\quad {\text{for}}\, i = 1 } \hfill \\ {8i - 5} \hfill & {\quad {\text{for}}\begin{array}{*{20}l} {\text{V}} \hfill \\ {{\text{VI}}} \hfill \\ \end{array} } \hfill \\ { 3\lambda^{\!\!\!\!\!-} - 8\left( {n + 1 - i} \right) } \hfill & {\quad {\text{for}}\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \hfill \\ \end{array} } \right. , \\ W_\theta \left( {c_i x_{2,i} } \right) & = W_\theta \left( {c_i x_{1,i} } \right) + 2 \\ W_\theta \left( {c_i y_{1,i} } \right) & = W_\theta \left( {c_i x_{1,i} } \right) + 1 \\ W_\theta \left( {c_i y_{2,i} } \right) & = W_\theta \left( {c_i x_{1,i} } \right) + 3 \\ W_\theta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {8i - 4} \hfill & {\quad {\text{for}}\begin{array}{*{20}l} {\text{I}} \hfill \\ {{\text{II}}} \hfill \\ \end{array} } \hfill \\ {3\lambda^{\!\!\!\!\!-} - 8\left( {n - i} \right) + 4 } \hfill & {\quad {\text{for}}\begin{array}{*{20}l} {{\text{III}}} \hfill \\ {{\text{IV}}} \hfill \\ \end{array} } \hfill \\ \end{array} } \right. \\ W_\theta \left( {x_{2,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 2 \\ W_\theta \left( {y_{1,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 1 \\ W_\theta \left( {y_{2,i} c_{i + 1} } \right) & = W_\theta \left( {x_{1,i} c_{i + 1} } \right) + 3. \\ \end{aligned} $$

Therefore, $\theta$ is an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling and

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

Definition 3

A triple square snake graph $T\left( {C_{4,n} } \right)$ consists of three square graphs that have a common path $P_n$, as shown in Fig. 5.

Fig. 5

A triple square snake graph $T\left( {C_{4,n} } \right)$

Theorem 3

For a triple square snake graph $T\left( {C_{4,n} } \right)$ we have

$$ {\text{tes}}\left( {T\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{12n + 2}{3}} \right\rceil . $$

Proof

Since $\left| {E\left( {T\left( {C_{4,n} } \right)} \right))} \right| = 12n$ and $\Delta \left( {T\left( {C_{4,n} } \right)} \right) = 12$. Then from (1) we have

$$ {\text{tes}}\left( {T\left( {C_{4,n} } \right)} \right) \ge \left\lceil {\frac{12n + 2}{3}} \right\rceil . $$

To prove the inverse inequality, we show that $\lambda^{\!\!\!\!\!-}$-labeling is an edge irregular total for $T\left( {C_{4,n} } \right)$, where $\lambda^{\!\!\!\!\!-} = \left\lceil {\frac{12n + 2}{3}} \right\rceil$. Let $\beta :V\left( {T\left( {C_{4,n} } \right)} \right) \cup E\left( {T\left( {C_{4,n} } \right) } \right) \to \left\{ 1,2,3, \ldots ,\lambda^{\!\!\!\!\!-} \right\}$ be a total $\lambda^{\!\!\!\!\!-}$-labeling defined as:

$$ \begin{aligned} \beta \left( {c_i } \right) & = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad {\text{for}}\,~i = 1} \hfill \\ {4i - 4} \hfill & {\quad {\text{for}}\,~2 \le i \le n~} \hfill \\ \lambda^{\!\!\!\!\!-} \hfill & {\quad {\text{for}}\,~i = n + 1} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{1,~i} } \right) & = \left\{ {\begin{array}{*{20}l} {4i - 3} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 3} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{2,~i} } \right) & = \left\{ {\begin{array}{*{20}l} {4i - 2} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1} \hfill \\ {\lambda^{\!\!\!\!\!-} - 2} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} ,} \right. \\ \beta \left( {x_{3,~i} } \right) & = \left\{ {\begin{array}{*{20}l} {4i - 1} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 1} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{1,~i} } \right) & = \left\{ {\begin{array}{*{20}l} {4i - 3} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 3} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{2,~i} } \right) & = \left\{ {\begin{array}{*{20}l} {4i - 2} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 2} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{3,~i} } \right) & = \left\{ {\begin{array}{*{20}l} {4i - 1} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 1} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{1,~i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad {\text{for}}\,~i = 1} \hfill \\ {4i - 2} \hfill & {\quad {\text{for}}\,~2 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 4} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{2,~i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 2 \hfill & {\quad {\text{for}}\,~i = 1} \hfill \\ {4i - 1} \hfill & {\quad {\text{for}}\,~2 \le i \le n - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 3} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{3,~i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 3 \hfill & {\quad {\text{for}}\,~i = 1} \hfill \\ {4i} \hfill & {\quad {\text{for}}\,~2 \le i \le n - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 2} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{1,~i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 2 \hfill & {\quad {\text{for}}\,~i = 1} \hfill \\ {4i - 1} \hfill & {\quad {\text{for}}\,~2 \le i \le n - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 3} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{2,~i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 3 \hfill & {\quad {\text{for}}\,~i = 1} \hfill \\ {4i} \hfill & {\quad {\text{for}}\,~2 \le i \le n~ - 1} \hfill \\ {\lambda^{\!\!\!\!\!-} - 2} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{3,~i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 4 \hfill & {\quad {\text{for}}\,~i = 1} \hfill \\ {4i + 1} \hfill & {\quad {\text{for}}\,~2 \le i \le n - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 1} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{1,~i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {4i} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 3} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{2,~i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {4i + 1} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 2} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {x_{3,~i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {4i + 2} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 1} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{1,~i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {4i + 1} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 2} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{2,~i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {4i + 2} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ {\lambda^{\!\!\!\!\!-} - 1} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right., \\ \beta \left( {y_{3,~i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {4i + 3} \hfill & {\quad {\text{for}}\,~1 \le i \le n~ - 1~} \hfill \\ \lambda^{\!\!\!\!\!-} \hfill & {\quad {\text{for}}\,~i = n} \hfill \\ \end{array} } \right.. \\ \end{aligned} $$

One can check that edges and vertices labels are at most $\lambda^{\!\!\!\!\!-}$. For the edges weights under the labeling $\beta$ we have

$$ \begin{aligned} W_\beta \left( {x_{1,i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 3 \hfill & {\quad {\text{for}}\, i = 1} \hfill \\ {12i - 9} \hfill & {\quad {\text{for}}\, 2 \le i \le n - 1} \hfill \\ {2\lambda^{\!\!\!\!\!-} + 4n - 11} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {x_{2,i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 5 \hfill & {\quad {\text{for}}\, i = 1} \hfill \\ {12i - 7} \hfill & {\quad {\text{for}}\, 2 \le i \le n - 1} \hfill \\ {2\lambda^{\!\!\!\!\!-} + 4n - 9} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {x_{3,i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 7 \hfill & {\quad {\text{for}}\, i = 1} \hfill \\ {12i - 5} \hfill & {\quad {\text{for}}\, 2 \le i \le n - 1} \hfill \\ {2\lambda^{\!\!\!\!\!-} + 4n - 7} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {y_{1,i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 4 \hfill & {\quad {\text{for}}\, i = 1} \hfill \\ {12i - 8} \hfill & {\quad {\text{for}}\, 2 \le i \le n - 1} \hfill \\ {2\lambda^{\!\!\!\!\!-} + 4n - 10} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {y_{2,i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 6 \hfill & {\quad {\text{for}}\, i = 1} \hfill \\ {12i - 6} \hfill & {\quad {\text{for}}\, 2 \le i \le n - 1} \hfill \\ {2\lambda^{\!\!\!\!\!-} + 4n - 8} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {y_{3,i} c_i } \right) & = \left\{ {\begin{array}{*{20}l} 8 \hfill & {\quad {\text{for}}\, i = 1} \hfill \\ {12i - 4} \hfill & {\quad {\text{for}}\, 2 \le i \le n - 1} \hfill \\ {2\lambda^{\!\!\!\!\!-} + 4n - 6} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {x_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {12i - 3} \hfill & {\quad {\text{for}}\, 1 \le i \le n - 1} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 6} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {x_{2,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {12i - 1} \hfill & {\quad {\text{for}}\, 1 \le i \le n - 1} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 4} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {x_{3,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {12i + 1} \hfill & {\quad {\text{for}}\, 1 \le i \le n - 1} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 2} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {y_{1,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {12i - 2} \hfill & {\quad {\text{for}}\, 1 \le i \le n - 1} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 5} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {y_{2,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {12i} \hfill & {\quad {\text{for}}\, 1 \le i \le n - 1} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 3} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right., \\ W_\beta \left( {y_{3,i} c_{i + 1} } \right) & = \left\{ {\begin{array}{*{20}l} {12i + 2} \hfill & {\quad {\text{for}}\, 1 \le i \le n - 1} \hfill \\ {3\lambda^{\!\!\!\!\!-} - 1} \hfill & {\quad {\text{for}}\, i = n} \hfill \\ \end{array} } \right.. \\ \end{aligned} $$

We deduced from the previous equations that edges’ weights consist of different numbers. Hence, $\beta$ is an edge irregular total $\lambda^{\!\!\!\!\!-}$-labeling of $T\left( {C_{4,n} } \right)$ and

$$ {\text{tes}}\left( {T(C_{4,n} )} \right) = \left\lceil {\frac{12n + 2}{3}} \right\rceil = 4n + 1. $$

Definition 4

If we replaced every edge in the path P_n by m-multiple square graphs we have which called m-multiple square snake graph $M\left( {C_{4,n} } \right)$, see Fig. 6.

Fig. 6

An m-multiple square snake graph

Theorem 4

Let $M_m \left( {C_{4,n} } \right)$ be m-multiple square snake graph. Then

$$ {\text{tes}}\left( {M_m \left( {C_{4,n} } \right) } \right) = \left\lceil {\frac{4mn + 2}{3}} \right\rceil $$

(10)

where $M_1 \left( {C_{4,n} } \right) = C_{4,n}$, $M_2 \left( {C_{4,n} } \right) = D(C_{4,n} )$ and $M_3 \left( {C_{4,n} } \right) = T(C_{4,n} )$.

Proof

We will use a proof by induction. To prove the relation (10), for $m = 1$, we have by definition

$$ M_1 \left( {C_{4,n} } \right) = C_{4,n} , $$

And from Theorem 1, we find that

$$ {\text{tes}}\left( {C_{4,n} } \right) = \left\lceil {\frac{4n + 2}{3}} \right\rceil . $$

For $m = 2$ from Theorem 2, we get

$$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

On the other hand, we have

$$ M_2 \left( {C_{4,n} } \right) = D\left( {C_{4,n} } \right) $$

Therefore,

$$ {\text{tes}}\left( {M_2 \left( {C_{4,n} } \right)} \right) = tes\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil = \left\lceil {\frac{4n + 2}{3} + \frac{4n}{3}} \right\rceil $$

where $\left\lceil {\frac{4n + 2}{3}} \right\rceil$ is TEIS for $M_1 \left( {C_{4,n} } \right) = C_{4,n}$ from Theorem 1. Then the relation (10) is true for $m = 2$.

Assume that the relation (10) is true for $m = k$ i.e.

$$ {\text{tes}}\left( {M_k \left( {C_{4,n} } \right){ }} \right) = \left\lceil {\frac{4kn + 2}{3}} \right\rceil = \left\lceil {\frac{4(k - 1)n + 2}{3} + \frac{4n}{3}} \right\rceil $$

Now, we will prove that the relation (10) is true for $m = k + 1$ i.e.

$$ {\text{tes}}\left( {M_k \left( {C_{4,n} } \right){ }} \right) = \left\lceil {\frac{{4\left( {k + 1} \right)n + 2}}{3}} \right\rceil = \left\lceil {\frac{4kn + 2}{3} + \frac{4n}{3}} \right\rceil . $$

Since $\left\lceil {\frac{4kn + 2}{3}} \right\rceil$ is TEIS for $M_k \left( {C_{4,n} } \right)$. Then, the relation (10) is true for $m = k + 1$. Hence, $tes\left( {M_m \left( {C_{4,n} } \right){ }} \right) = \left\lceil {\frac{4mn + 2}{3}} \right\rceil$ for any $m$.

3 Conclusion

In this paper, a new family of graphs called square snake graph has been defined and denoted by $C_{4,n}$. After that, we deduced the exact value of TEISs for a square snake graph $C_{4,n}$, double square snake graph $D(C_{4,n} )$, triple square snake graph $T(C_{4,n} )$ and m-multiple square snake graph $M(C_{4,n} )$.

The main findings and major contributions of the present work are

• 1. Square snake graph was defined.

• 2. Related graphs of square snake graph, like double square snake graph, triple square snake graph and m-multiple square snake graph, were defined.

• 3. The exact values of TERS for a square snake graph $C_{4,n}$ was calculated in the form

  $$ {\text{tes}}\left( {C_{4,n} } \right) = \left\lceil {\frac{4n + 2}{3}} \right\rceil . $$

• 4. The exact values of TERS for double square snake graph $D\left( {C_{4,n} } \right)$ was deduced and given by

  $$ {\text{tes}}\left( {D\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{8n + 2}{3}} \right\rceil . $$

• 5. The exact values of TERS for triple square snake graph $T(C_{4,n} )$ was calculated and given in the form

  $$ {\text{tes}}\left( {T\left( {C_{4,n} } \right)} \right) = \left\lceil {\frac{12n + 2}{3}} \right\rceil . $$

• 6. The exact values of TERS for m-multiple square snake graph $M(C_{4,n} )$ was deduced in the form

  $$ {\text{tes}}\left( {M_m \left( {C_{4,n} } \right)~} \right) = \left\lceil {\frac{{4mn + 2}}{3}} \right\rceil.$$