## Abstract

In this paper, we study 3–total edge product cordial (3–TEPC) labeling which is a variant of edge product cordial labeling. We discuss Web, Helm, Ladder and Gear graphs in this context of 3–TEPC labeling. We also discuss 3–TEPC labeling of some particular examples with corona graph.

## 1 Introduction and Preliminaries

Let *G* be finite, simple and undirected graph. Let *V _{G}* be the vertex set and

*E*be the edge set of graph

_{G}*G*. We follow the standard notations and terminology of graph theory as in [15]. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition (s). If the domain of mapping is the set of vertices (or edges) then the labeling is called a vertex labeling (or an edge labeling) proved as in [6].

To define some different types of cordial labelings *h* we have the following notations:

(1) The number of vertices labeled by

*x*is*ν*(_{h}*x*);(2) The number of edges labeled by

*x*is*e*(_{h}*x*);(3)

*ν*(_{h}*x,y*) =*ν*(_{h}*x*) −*ν*(_{h}*y*);(4)

*e*(_{h}*x,y*) =*e*(_{h}*x*) −*e*(_{h}*y*);(5)

*s*(*x*) =*ν*(_{h}*x*) +*e*(_{h}*x*);

Now we will define different types of cordial labeling.

A vertex labeling function

*h*:*V*→ {0,1} can induce an edge labeling_{G}*h** :*E*→ {0,1} as for each edge_{G}*uν*,*h** (*uν*) = |*h*(*u*) −*h*(*ν*)| if it satisfies |*ν*(1,0)| ≤ 1 and |_{h}*e*(1,0)| ≤ 1 holds then it is called cordial. A graph_{h}*G*is cordial if it admits a cordial labeling as proved in [2].A vertex labeling function

*h*:*V*→ {0,1} can induce an edge labeling_{G}*h** :*E*→ {0,1} as for each edge_{G}*uν*,*h** (*uν*) =*h*(*u*)*h*(*ν*) if it satisfies |*ν*_{h}(1,0)| ≤ 1 and |*e*_{h}(1,0)| ≤ 1 holds then it is called product cordial labeling. A graph*G*is product cordial if it admits a product cordial labeling as proved in [8].For a graph

*G*a function*h*:*V*→ {0,1,2,...,_{G}*k*− 1} such that*k*is an integer 2 ≤*k*≤ |*E*_{G}| gives label to each edge*uν*with*h*(*u*)*h*(*ν*). If |*s*(*x*) −*s*(*y*)| ≤ 1 for*x,y ∈*{0,1,...,*k*− 1} holds then it is called total product cordial(TPC) labeling. A graph*G*is TPC if it admits a TPC labeling as proved in [9].A edge labeling is a function

*h*:*E*→ {0,1} can induce a vertex labeling_{G}*h** :*V*→ {0,1} such that_{G}*h** (*ν*) =*h*(*e*_{1})*h*(*e*_{2})*...h*(*e*) for edge_{n}*e*_{1},*e*_{2},...,*e*that are incident to_{n}*ν*. If it satisfies |*ν*(1,0)| ≤ 1 and |_{h}*e*(1,0)| ≤ 1 holds then it is called edge product cordial(EPC) labeling. A graph_{h}*G*is EPC if it admits a EPC labeling as proved in [10].A edge labeling is a function

*h*:*E*→ {0,1,2,...,_{G}*k*− 1} can induce a vertex labeling*h** :*V*→ {0,1,2,...,_{G}*k*− 1} such that*h** (*ν*) =*h*(*e*_{1})*h*(*e*_{2})...*h*(*e*)(_{n}*modk*) for edge*e*_{1},*e*_{2},...,*e*that are incident to_{n}*ν*. If |*s*(*x*)*−s*(*y*)| ≤ 1 for*x,y*∈ {0,1,...,*k*− 1} holds then it is called k-edge product cordial(k-EPC) labeling. A graph*G*is k-EPC if it admits a k-EPC labeling as proved in [10].For a graph

*G*a function*h*:*E*→ {0,1,2,...,_{G}*k*− 1} such that*k*is an integer 2 ≤*k*≤ |*E*_{G}|. For each vertex*ν*assign the label by*h*(*e*_{1})*h*(*e*_{2})...*h*(*e*)(_{n}*modk*) such that*e*_{1},*e*_{2},...,*e*are edges incident to_{n}*ν*. If |*s*(*x*) −*s*(*y*)| ≤ 1 for*x,y*∈ {0,1,...,*k*− 1} holds then it is called k-total edge product cordial(k-TEPC) labeling. A graph*G*is k-TEPC if it admits a k-TEPC labeling as proved in [1, 11].

Now we will define different family of graphs.

The web

*Wb*is the graph obtained by joining the pendant vertices of a helm_{n}*H*to form a cycle and then adding a pendant edge to each of the vertices of outer cycle. An example of web graph_{n}*Wb*_{4}are given in Figure 1.The helm

*H*is the graph obtained from a wheel_{n}*W*by attaching a pendent edge at each vertex of the_{n}*n*-cycle. An example of helm graph*H*_{5}are given in Figure 2.Fig. 2 Helm graph

*H*_{5}Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

A gear graph

*G*is obtained from the wheel graph_{n}*W*by adding a vertex between every pair of adjacent vertices of the_{n}*n*-cycle. An example of gear graph*G*_{6}are given in Figure 3.Fig. 3 Gear graph

*G*_{6}Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

A ladder graph

*L*of order_{n}*n*is a planer undirected graph with 2*n*vertices and*n*+ 2(*n −*1) edges. An example of ladder graph*L*_{6}are given in Figure 4.Fig. 4 Ladder graph

*L*_{6}Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Now we will define corona of two graphs.

The corona *G* ⊙ *H* of two graphs *G* with *n* vertices and a graph *H* with *n* copies can be obtained by connecting *i*-th vertex of *G* to each vertex of *i*-th copy of *H* with an edge.

Azaizeh at al in [1] introduced the concept of 3-TEPC labeling and they proved many results on this newly concept. They discussed 3-TEPC labeling of Path, Circle and Star graphs. Madiha at el in [7] has discussed 3-TEPC labeling of Dutch Windmill and copies of *n*-cycle graphs. This paper is structured as follows. In Section 2, we discussed 3-total edge product cordial (3-TEPC) labeling of web graph and helm graph. 3-TEPC labeling of gear graph and a class of corona graph are discussed in Section 3.

## 2 3-TEPC labeling of Web graph and Helm graph

In this section we will study 3-TEPC labeling of web graph and helm graph

### 2.1 3-TEPC labeling of Web graph

*Let G be a Web graph Wb _{n} of n* + 1

*vertices then G admits*3

*-TEPC labeling.*

*Proof.* : Let *V _{G}* = {

*u,ν*, 1 ≤

_{x},w_{x}*x*≤

*n*} and

*E*= {

_{G}*u*, 1 ≤

_{x}ν_{x}*x*≤

*n*} ∪ {

*ν*

_{x}*ν*

_{x+1}, 1 ≤

*x*≤

*n*− 1} ∪ {

*ν*, 1 ≤

_{x}w_{x}*x*≤

*n*} ∪ {

*u*

_{x}*u*

_{x+1}, 1 ≤

*x*≤

*n*− 1}. We consider three cases as follows:

Let *n* ≡ 0 (mod 3) which implies *n* = 3*t*, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = *s*(1) = *s*(2) = 7*t.* Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 5.

3-TEPC labeling of *W*_{9}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Let *n* ≡ 1 (mod 3) which implies *n* = 3*t* + 1, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = 7*t* + 3, *s*(1) = *s*(2) = 7*t.* Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 6.

3-TEPC labeling of *W*_{7}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Let *n* ≡ 2 (mod 3) which implies *n* = 3*t* + 2, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = 7*t* + 4, *s*(1) = *s*(2) = 7*t* + 5. Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 7.

3-TEPC labeling of *W*_{8}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

### 2.2 3-TEPC labeling of Helm graph

*Let G be a Helm graph H _{n} of n* + 1

*vertices then G admits*3

*-TEPC labeling.*

*Proof.* Let *V _{G}* = {

*u,ν*, 1 ≤

_{x},w_{x}*x*≤

*n*} and

*E*= {

_{G}*uν*, 1 ≤

_{x}*x*≤

*n*} ∪ {

*ν*

_{x}ν_{x+1}, 1 ≤

*x*≤

*n*− 1} ∪ {

*ν*, 1 ≤

_{x}w_{x}*x*≤

*n*}. We consider three cases as follows:

Let *n* ≡ 0 (mod 3) which implies *n* = 3*t*, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

*h*(

*uν*) = 1, for 1 ≤

_{x}*x*≤ 3

*t*.

In this case we have *s*(0) = 5*t* + 1, *s*(1) = *s*(2) = 5*t.* Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 8.

3-TEPC labeling of *H*_{9}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Let *n* ≡ 1 (mod 3) which implies *n* = 3*t* + 1, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = *s*(1) = *s*(2) = 5*t* + 1. Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 9.

3-TEPC labeling of *H*_{10}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Let *n* ≡ 2 (mod 3) which implies *n* = 3*t* + 2, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = 5*t* + 3, *s*(1) = *s*(2) = 5*t* + 4. Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 10.

3-TEPC labeling of *H*_{11}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

## 3 3–TEPC Labeling of Gear graph and some classes of corona graph

In this section we will study 3-TEPC labeling of Gear graph and a class of corona graph.

### 3.1 3-TEPC Labeling of Gear graph

*Let G be the Gear graph G _{n}, then G is* 3

*-TEPC.*

*Proof.* Let *V _{G}* = {

*u,ν*, 1 ≤

_{x}*x*≤

*n*} and

*E*= {

_{G}*uν*

_{2x−1}, 1 ≤

*x*≤

*n*} ∪ {

*ν*

_{x}ν_{x+1}, 1 ≤

*x*≤

*n*− 1}. We consider three cases as follows:

Let *n* ≡ 0( mod 3) which implies *n* = 3*t*, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = 5*t* + 1,*s*(1) = *s*(2) = 5*t.* Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 11.

3-TEPC labeling of *G*_{6}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Let *n* ≡ 1 (mod 3) which implies *n* = 3*t* + 1, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

So we have *s*(0) = *s*(1) = *s*(2) = 5*t* + 2. Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 12.

3-TEPC labeling of *G*_{7}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Let *n* ≡ 2( mod 3) which implies *n* = 3*t* + 2, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = 5*t* + 3,*s*(1) = *s*(2) = 5*t* + 4. Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *h* is a 3-TEPC labeling as elaborated in Figure 13.

3-TEPC labeling of *G*_{8}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

### 3.2 3-TEPC of Corona *L*_{n} ⊙ 2*K*_{1}

_{n}

*Let G be a corona of L _{n} ⊙* 2

*K*

_{1}

*then G admits*3

*-TEPC labeling.*

*Proof.* Let *V _{G}* = {

*u*, 1 ≤

_{x},ν_{x},w_{x},w^{′}_{x}*x*≤

*n*} and

*E*= {

_{G}*u*, 1 ≤

_{x}ν_{x}*x*≤

*n*} ∪ {

*ν*

_{x}ν_{x+1}, 1 ≤

*x*≤

*n*− 1} ∪ {

*u*

_{x}u_{x+1}, 1 ≤

*x*≤

*n*− 1} ∪ {

*u*, 1 ≤

_{x}w_{x}*x*≤

*n*} ∪ {

*ν*, 1 ≤

_{x}w^{′}_{x}*x*≤

*n*}. we consider three cases as follows:

Let *n* ≡ 0( mod 3) then *n* = 3*t*, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

*h*(

*ν*

_{x}k_{1}) = 2, for 1 ≤

*x*≤ 3

*t*.

In this case we have *s*(0) = 9*t, s*(1) = *s*(2) = 9*t* − 1. Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *g* is a 3-TEPC labeling as elaborated in Figure 14.

3-TEPC labeling of graph *L*_{6} ⊙ 2*K*_{1}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Let *n* ≡ 1( mod 3) then *n* = 3*t* + 1, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = 9*t* + 3,* s*(1) = *s*(2) = 9*t* + 2. Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *g* is a 3-TEPC labeling as elaborated in Figure 15.

3-TEPC labeling of graph *L*_{7} ⊙ 2*K*_{1}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

Let *n* ≡ 2( mod 3) then *n* = 3*t* + 2, for some integer *t* ≥ 1. We define the edge labeling *h* : *E _{G}* → {0,1,2} as

In this case we have *s*(0) = 9*t* + 6, *s*(1) = *s*(2) = 9*t* + 5. Therefore |*s*(*x*) − *s*(*y*)| ≤ 1 for 0 ≤ *x* < *y* ≤ 2. Hence *g* is a 3-TEPC labeling as elaborated in Figure 16.

3-TEPC labeling of graph *L*_{8} ⊙ 2*K*_{1}

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00005

**Communicated by:** Wei Gao

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