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Introduction and Preliminaries
Let G be finite, simple and undirected graph. Let VG be the vertex set and EG be the edge set of graph 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 νh(x);
(2) The number of edges labeled by x is eh(x);
(3) νh(x,y) = νh(x) − νh(y);
(4) eh(x,y) = eh(x) − eh(y);
(5) s(x) = νh(x) + eh(x);
Now we will define different types of cordial labeling.
Definition 1.
A vertex labeling function h : VG → {0,1} can induce an edge labeling h* : EG → {0,1} as for each edge uν, h* (uν) = |h(u) − h(ν)| if it satisfies |νh(1,0)| ≤ 1 and |eh(1,0)| ≤ 1 holds then it is called cordial. A graph G is cordial if it admits a cordial labeling as proved in [2].
A vertex labeling function h : VG → {0,1} can induce an edge labeling h* : EG → {0,1} as for each edge uν, h* (uν) = h(u)h(ν) if it satisfies |νh(1,0)| ≤ 1 and |eh(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 : VG → {0,1,2,...,k − 1} such that k is an integer 2 ≤ k ≤ |EG| 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 : EG → {0,1} can induce a vertex labeling h* : VG → {0,1} such that h* (ν) = h(e1)h(e2)...h(en) for edge e1,e2,...,en that are incident to ν. If it satisfies |νh(1,0)| ≤ 1 and |eh(1,0)| ≤ 1 holds then it is called edge product cordial(EPC) labeling. A graph G is EPC if it admits a EPC labeling as proved in [10].
A edge labeling is a function h : EG → {0,1,2,...,k − 1} can induce a vertex labeling h* : VG → {0,1,2,..., k − 1} such that h* (ν) = h(e1)h(e2)...h(en)(modk) for edge e1,e2,...,en that are incident to 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 : EG → {0,1,2,...,k − 1} such that k is an integer 2 ≤ k ≤ |EG|. For each vertex ν assign the label by h(e1)h(e2)...h(en)(modk) such that e1,e2,...,en are edges incident to 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.
Definition 2.
The web Wbn is the graph obtained by joining the pendant vertices of a helm Hn to form a cycle and then adding a pendant edge to each of the vertices of outer cycle. An example of web graph Wb4 are given in Figure 1.
Fig. 1
Web graph Wb4
The helm Hn is the graph obtained from a wheel Wn by attaching a pendent edge at each vertex of the n-cycle. An example of helm graph H5 are given in Figure 2.
Fig. 2
Helm graph H5
A gear graph Gn is obtained from the wheel graph Wn by adding a vertex between every pair of adjacent vertices of the n-cycle. An example of gear graph G6 are given in Figure 3.
Fig. 3
Gear graph G6
A ladder graph Ln of order n is a planer undirected graph with 2n vertices and n + 2(n − 1) edges. An example of ladder graph L6 are given in Figure 4.
Fig. 4
Ladder graph L6
Now we will define corona of two graphs.
Definition 3.
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.
3-TEPC labeling of Web graph and Helm graph
In this section we will study 3-TEPC labeling of web graph and helm graph
3-TEPC labeling of Web graph
Theorem 1.
Let G be a Web graph Wbn of n + 1 vertices then G admits 3-TEPC labeling.
Proof. : Let VG = {u,νx,wx, 1 ≤ x ≤ n} and EG = {uxνx, 1 ≤ x ≤ n} ∪ {νxνx+1, 1 ≤ x ≤ n − 1} ∪ {νxwx, 1 ≤ x ≤ n} ∪ {uxux+1, 1 ≤ x ≤ n − 1}. We consider three cases as follows:
Case 1
Let n ≡ 0 (mod 3) which implies n = 3t, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as
In this case we have s(0) = 7t + 3, s(1) = s(2) = 7t. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence h is a 3-TEPC labeling as elaborated in Figure 6.
Fig. 6
3-TEPC labeling of W7
Case 3
Let n ≡ 2 (mod 3) which implies n = 3t + 2, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as
In this case we have s(0) = 7t + 4, s(1) = s(2) = 7t + 5. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence h is a 3-TEPC labeling as elaborated in Figure 7.
Fig. 7
3-TEPC labeling of W8
3-TEPC labeling of Helm graph
Theorem 2.
Let G be a Helm graph Hn of n + 1 vertices then G admits 3-TEPC labeling.
Proof. Let VG = {u,νx,wx, 1 ≤ x ≤ n} and EG = {uνx, 1 ≤ x ≤ n} ∪ {νxνx+1, 1 ≤ x ≤ n − 1} ∪ {νxwx, 1 ≤ x ≤ n}. We consider three cases as follows:
Case 1
Let n ≡ 0 (mod 3) which implies n = 3t, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as h(uνx) = 1, for 1 ≤ x ≤ 3t.
In this case we have s(0) = 5t + 1, s(1) = s(2) = 5t. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence h is a 3-TEPC labeling as elaborated in Figure 8.
Fig. 8
3-TEPC labeling of H9
Case 2
Let n ≡ 1 (mod 3) which implies n = 3t + 1, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as
In this case we have s(0) = s(1) = s(2) = 5t + 1. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence h is a 3-TEPC labeling as elaborated in Figure 9.
Fig. 9
3-TEPC labeling of H10
Case 3
Let n ≡ 2 (mod 3) which implies n = 3t + 2, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as
In this case we have s(0) = 5t + 3, s(1) = s(2) = 5t + 4. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence h is a 3-TEPC labeling as elaborated in Figure 10.
Fig. 10
3-TEPC labeling of H11
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-TEPC Labeling of Gear graph
Theorem 3.
Let G be the Gear graph Gn, then G is 3-TEPC.
Proof. Let VG = {u,νx, 1 ≤ x ≤ n} and EG = {uν2x−1, 1 ≤ x ≤ n} ∪ {νxνx+1, 1 ≤ x ≤ n − 1}. We consider three cases as follows:
Case 1
Let n ≡ 0( mod 3) which implies n = 3t, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as
In this case we have s(0) = 5t + 1,s(1) = s(2) = 5t. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence h is a 3-TEPC labeling as elaborated in Figure 11.
Fig. 11
3-TEPC labeling of G6
Case 2
Let n ≡ 1 (mod 3) which implies n = 3t + 1, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as
In this case we have s(0) = 5t + 3,s(1) = s(2) = 5t + 4. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence h is a 3-TEPC labeling as elaborated in Figure 13.
Fig. 13
3-TEPC labeling of G8
3-TEPC of Corona Ln ⊙ 2K1
Theorem 4.
Let G be a corona of Ln ⊙ 2K1 then G admits 3-TEPC labeling.
Proof. Let VG = {ux,νx,wx,w′x, 1 ≤ x ≤ n} and EG = {uxνx, 1 ≤ x ≤ n} ∪ {νxνx+1, 1 ≤ x ≤ n − 1} ∪ {uxux+1, 1 ≤ x ≤ n − 1} ∪ {uxwx, 1 ≤ x ≤ n} ∪ {νxw′x, 1 ≤ x ≤ n}. we consider three cases as follows:
Case 1
Let n ≡ 0( mod 3) then n = 3t, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as h(νxk1) = 2, for 1 ≤ x ≤ 3t.
In this case we have s(0) = 9t, s(1) = s(2) = 9t − 1. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence g is a 3-TEPC labeling as elaborated in Figure 14.
Fig. 14
3-TEPC labeling of graph L6 ⊙ 2K1
Case 2
Let n ≡ 1( mod 3) then n = 3t + 1, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as
In this case we have s(0) = 9t + 3, s(1) = s(2) = 9t + 2. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence g is a 3-TEPC labeling as elaborated in Figure 15.
Fig. 15
3-TEPC labeling of graph L7 ⊙ 2K1
Case 3
Let n ≡ 2( mod 3) then n = 3t + 2, for some integer t ≥ 1. We define the edge labeling h : EG → {0,1,2} as
In this case we have s(0) = 9t + 6, s(1) = s(2) = 9t + 5. Therefore |s(x) − s(y)| ≤ 1 for 0 ≤ x < y ≤ 2. Hence g is a 3-TEPC labeling as elaborated in Figure 16.
