# 3-Total Edge Product Cordial Labeling for Stellation of Square Grid Graph

Authors: Rizwan Ullah; Gul Rahmat; Muhammad Numan; Kraidi Anoh Yannick; Adnan Aslam
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetG be a simple graph with vertex set VG and edge set EG. An edge labeling δ:EG⟶0,1,…,p−1, where p is an integer, 1≤p≤EG, induces a vertex labeling δ∗:VH⟶0,1,…,p−1 defined by δ∗v=δe1δe2⋅δenmodp, where e1,e2,…,en are edges incident to v. The labeling δ is said to be p-total edge product cordial (TEPC) labeling of G if eδi+vδ∗i−eδj+vδ∗j≤1 for every i,j, 0≤i≤j≤p−1, where eδi and vδ∗i are numbers of edges and vertices labeled with integer i, respectively. In this paper, we have proved that the stellation of square grid graph admits a 3-total edge product cordial labeling.


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## Body

## 1. Introduction and Definitions

LetG be a simple, finite, and connected graph with the vertex set VG and edge set EG. For basic notions related to graph theory, we refer the reader to the book by West [1]. A graph labeling δ is a map that sends one of the graph element (vertex set or edge set or both) to set of numbers. If the domain is the vertex set (edge set), then δ is called vertex (edge) labeling. If the domain is VG∪EG, then δ is called total labeling. Graph labeling has a wide range of applications such as X-ray crystallography, coding theory, radar, astronomy, circuit design, network, and communication design.Letδ:VH⟶0,1 be a vertex labeling which induces edge labeling δ∗:EH⟶0,1 defined by δ∗xy=δx−δy. δ. The labeling δ is said to be cordial if vδ0−vδ1≤1 and eδ∗0−eδ∗1≤1, where vδi and eδ∗i denote the number of vertices and number of edges labeled with integer i, respectively. The concept of cordial labeling was first introduced by Cahit [2]. A considerable amount of work have been done on cordial labeling. For latest results, see [3–10]. A vertex labeling δ:VG⟶0,1 induces an edge labeling δ∗:EG⟶0,1 defined by δ∗xy=δxδy which is called product cordial labeling if vδ0−vδ1≤1 and eδ∗0−eδ∗1≤1, where vδ0 and vδ1 represent the number of vertices that are labeled 0 and 1, respectively. While eδ∗0 and eδ∗1 represent the number of edges labeled with 0 and 1, respectively. The concept named product cordial labeling was first presented by Sundaram et al. [11]. A variation in the cordial theme, namely, edge product cordial labeling and a TEPC labeling was introduced by Vaidya and Barasara [12, 13].Let2≤p≤EG be an integer. An edge labeling δ:EG⟶0,1,…,p−1 induces a vertex labeling δ∗:VH⟶0,1,…,p−1 defined by δ∗v=δe1δe2⋅δenmodp, where e1,e2,…,en are edges incident to v. The labeling δ is said to be p-TEPC labeling of G if eδi+vδ∗i−eδj+vδ∗j≤1 for every i,j, 0≤i≤j≤p−1, where eδi and vδ∗i are numbers of edges and vertices labeled with integer i, respectively. Azaizeh et al. [14] introduced the concept of p-TEPC labeling. A graph G that admits a p-TEPC labeling is called a p-TEPC graph. Baca et al. [15] investigated the 3-TEPC labeling of carbon nanotube networks. Ahmad et al. [16] showed that the grid graph Pm□Pn admits a 3-TEPC labeling. Ahmad et al. [3] proved that the hexagonal grid Hmn admits 3-TEPC labeling. Javed and Jamil [17] proved that the Rhombic grid Rmn is 3-TEPC for m,n≥1.LetPn denote a path graph on n vertices. A rectangular grid is an m×n lattice graph and is obtained by taking the Cartesian product of Pm with Pn. The graph of rectangular grid is denoted by Lm,n and has n and m squares in each row and column respectively. It is easy to observe that rectangular grid Lm,n has mn vertices and mn−m−n+1 edges. The stellation of Lm,n is obtained by adding a vertex in each face of Lm,n and then joining this vertex to each vertex of the respective face. We denote the stellation of Lm,n by Gnm, as shown in Figure 1. In this paper, we show that the graph Gnm admits 3-TEPC labeling.Figure 1
Stellation of square grid graphG73.


## 2. Main Results

Letm,n≥1 and Gnm be stellation of rectangular grid containing m rows and n columns. Observe that Gnm has 2mn+m+n+1 vertices and 6mn+m+n edges. We use the notations G1⊕vG2 for gluing the graph G1 with G2 vertically. Similarly, G1⊕hG2 represent gluing G1 with G2 horizontally. If we have a labeled segment or labeled graph H and we rotate it by 90 degree in clockwise direction, then we will denote it by H⟶.Theorem 1.
Form≥1, the graph G1m is 3-TEPC.Proof.
The 3-TEPC labeling ofG11 and G12 is shown in Figure 2. Similarly, the 3-TEPC labeling of G13 and the labeled segment S13 are shown in Figure 3. The segment S13 has the property that open edges are assigned labeled 1. Hence, if we glue the segment S13 with itself vertically, then it will not change the vertex labels in S13⊕vS13:=2S13. Observe that the labels 0, 1, and 2 are used 10 times in the segment S13. Table 1 shows the multiplicity of numbers 0, 1, and 2, respectively, used in the labeled graph G1m for m=1,2,3.Case (i).m = 3r, r≥1.To construct labeled graphG1m, we will use the labeled segments S13. First, glue r−1 copies of labeled segment S13 vertically that is S13⊕vS13⊕v⋯⊕vS13: = r−1S13. Finally, glue vertically the label segment G13 to the open edges of r−1S13 to get labeled graph G1m, that is,(1)G1m=r−1S13⊕vG13.In the labeled graphG1m, the multiplicity of 0, 1, and 2 is 10r+1 exactly.Case (ii):m = 3r + 1, r≥1.To construct the labeled graphG1m, we glue r copies of the labeled segment S13 and then finally glue G11 vertically. That is,(2)G1m=rS13⊕vG11.In the labeled graphG1m, the multiplicity of 0 is 10r+5, whereas the multiplicity of 1 and 2 is 10r+4.Case (iii):m = 3r + 2, r≥1.We obtain the labeled graphG1m by gluing r times the labeled segment S13 and finally gluing G12 in vertical direction. That is,(3)G1m=rS13⊕vG12.In the labeled graphG1m, the multiplicity of 0 is 10r+7, whereas the multiplicity of 1 and 2 is 10r+8.Figure 2
3-TEPC labelingG11 and G12.Figure 3
3-TEPC labelingG13 and label segment S13.Table 1
The multiplicity of 0, 1, and 2 used inG1m, for m=1,2,3.
G
1
m
e
δ
0 + vδ∗0
e
δ
1 + vδ∗1
e
δ
2 + vδ∗2
m
=
1
5
4
4
m
=
2
7
8
8
m
=
3
11
11
11Theorem 2.
Form ≥1, the graph G2m is 3-total edge product cordial.Proof.
Observe that the graphsG12 and G21 are isomorphic and the 3-total edge cordial labeling of G12 is given in Figure 2. Therefore, G21 is 3-TEPC. The 3-total edge product cordial labeling of the graphs G22 and G23 is given in Figures 4 and 5, respectively. Table 2 shows the multiplicity of numbers 0, 1, and 2 used in G22 and G23.
Figure6 depicts the labeled segment S23, which has the property that open edges are assigned labeled 1 and each number 0, 1, and 2 is used 18 times.Case (i):m = 3r, r≥1.To construct labeled graphG2m, we will use the labeled segments S23. First, we glue r−1 copies of labeled segment S23 vertically, that is, S23⊕vS23⊕v⋯⊕vS23:=r−1S23. Since the open edges of S23 are labeled with 1, therefore, this gluing process does not change the label of other vertices of r−1S23. Finally, we glue vertically the label segment G23 to the open edges of r−1S23 to get labeled graph G2m. That is,(4)G2m=r−1S23⊕vG23.In the labeled graphG2m, the multiplicity of 0 is 18r+1, whereas the multiplicity of 1 and 2 is 18r+2.Case (ii):m = 3r + 1, r≥1.To construct the labeled graphG2m, we glue r copies of the labeled segment S23 and then finally glue G21 vertically. That is,(5)G2m=rS23⊕vG21.In the labeled graphG2m, the multiplicity of 0 is 18r+7 whereas the multiplicity of 1 and 2 is 18r+8.Case (iii):m = 3r + 2, r≥1.The labeled graphG2m can be obtained by gluing r times the labeled segment S23 and then gluing G22 in vertical direction. That is,(6)G2m=rS23⊕vG22.In the labeled graphG2m, the multiplicity of 0 is 18r+13, whereas the multiplicity of 1 and 2 is 18r+14.Figure 4
3-TEPC labeling ofG22.Figure 5
3-TEPC labeling ofG23.Table 2
The multiplicity of 0, 1, and 2 inG2m, for m=2,3.
G
1
m
e
δ
0 + vδ∗0
e
δ
1 + vδ∗1
e
δ
2 + vδ∗2
m
=
2
13
14
14
m
=
3
19
20
20Figure 6
Label segmentS23.Theorem 3.
The graphG3m is 3-TEPC for m≥1.Proof.
Observe that the graphsG13 and G31 are isomorphic. Similarly, the graphs G23 and G32 are also isomorphic. The 3-TEPC labeling of G13 and G23 are given in Figures 3 and 5, respectively. The 3-TEPC labeling of G33 is shown in Figure 7. In the labeled graph G33, the multiplicity of 0 is 29, whereas the multiplicity of 1 and 2 is 28.
Figure8 shows the labeled segment S33 which has the property that open edges are assigned with label 1 and each number 0, 1, and 2 appears 26 times.Case (i):m = 3r, r≥1.To construct labeled graphG3m, we use the labeled segment S33. First, glue r−1 copies of labeled segment S33 vertically, that is, S33⊕vS33⊕v⋯⊕vS33: = r−1S33. Since the open edges of S33 are labeled with 1, therefore, this gluing process does not change the label of other vertices of r−1S33. Finally, glue vertically the label segment G33 to the open edges of r−1S23 to get labeled graph G3m. That is,(7)G3m=r−1S33⊕vG33.In the labeled graphG3m, the multiplicity of 0 is 26r+3, whereas the multiplicity of 1 and 2 is 26r+2.Case (ii):m = 3r + 1, r≥1.To construct the labeled graphG3m, we glue r copies of the labeled segment S33 vertically and then finally glue G31 vertically. That is,(8)G3m=rS33⊕vG31.In the labeled graphG3m, the multiplicity of 0, 1, and 2 is 26r+11.Case (iii):m = 3r + 2, r≥1.We obtain the labeled graphG3m by gluing r times the labeled segment S33 vertically and then finally glue G32 in vertical direction. That is,(9)G3m=rS33⊕vG32.In the labeled graphG3m, the multiplicity of 0 is 26r+19, whereas the multiplicity of 1 and 2 is 26r+20.Figure 7
3-TEPC labeling ofG33.Figure 8
Label segmentS33.Theorem 4.
The graphGnm is 3-TEPC for m,n≥1.Proof.
To construct the labeled graph ofGnm and to examine its 3-TEPC labeling, we introduced a new segment R33. This segment has 17 open edges which are labeled with the number 1 and multiplicity of 0, 1, and 2 is 24. The labeled segment R33 is shown in Figure 9.Case 1:m=3r,r≥1.First, we glue the segmentR33 vertically r−1 times, that is, R33⊕vR33⊕v⋯⊕vR33 = r−1R33. Since the open edges in the segment are labeled with number 1, it follows that gluing these segments do not change the vertex labels in the segment r−1R33. Finally, we glue the segment S33⟶ in the vertical direction. This gives a new segment X and is defined as(10)X=r−1R33⊕vS33⟶.Note that the labels of open edges ofX are 1 and multiplicity of each number 0, 1, and 2 is 24r+2.Subcase 1:n=3s,s≥1.First, we glues−1 times the segment X horizontally and finally glue the labeled segment G3m horizontally to obtain the labeled graph Gnm. That is,(11)Gnm=s−1X⊕hG3m.Subcase 2:n=3s+1,s≥1.First, we glues times the segment X horizontally and finally glue the labeled segment G1m horizontally with sX to obtain the labeled graph Gnm. That is,(12)Gnm=sX⊕hG1m.Subcase 3:n=3s+2,s≥1.First, we glues times the segment X horizontally and finally glue the labeled segment G2m horizontally with sX to obtain the labeled graph Gnm. That is,(13)Gnm=sX⊕hG2m.Case 2: whenm=3r+1,r≥1.First, we glue the segmentR33 vertically r times, that is, R33⊕vR33⊕v⋯⊕vR33 = rR33. Then, we glue the segment S13⟶ in the vertical direction. This gives us a new segment Y defined as(14)Y=rR33⊕vS13⟶.Note that the labels of open edges ofY are 1 and multiplicity of each number 0, 1, and 2 is 24r+10.Subcase 1:n=3s,s≥1.First, we glues−1 times the segment Y horizontally and finally glue the labeled segment G3m horizontally with s−1Y to obtain the labeled graph Gnm. That is,(15)Gnm=s−1Y⊕hG3m.Subcase 2:n=3s+1,s≥1.First, we glues times the segment Y horizontally and finally glue the labeled segment G1m horizontally with sY to obtain the labeled graph Gnm. That is,(16)Gnm=sY⊕hG1m.Subcase 3:n=3s+2,s≥1.First, we glues times the segment Y horizontally and finally glue the labeled segment G2m horizontally with sY to obtain the labeled graph Gnm. That is,(17)Gnm=sY⊕hG2m.Case 3:m=3r+2,r≥1.First, we glue the segmentR33 vertically r times, that is, R33⊕vR33⊕v⋯⊕vR33 = rR33. Then, we glue in the vertical direction of the segment S23⟶. This gives us a new segment Z defined as(18)Z=rR33⊕vS23⟶.Note that the labels of open edges ofZ are 1 and multiplicity of each number 0, 1, and 2 is 24r+18.Subcase 1:n=3s,s≥1.First, we glues−1 times the segment Y horizontally and finally glue the labeled segment G3m horizontally with s−1Z to obtain the labeled graph Gnm. That is,(19)Gnm=s−1Z⊕hG3m.Subcase 2:n=3s+1,s≥1.First, we glues times the segment Y horizontally and finally glue the labeled segment G1m horizontally with sZ to obtain the labeled graph Gnm. That is,(20)Gnm=sZ⊕hG1m.Subcase 3:n=3s+2,s≥1.First, we glues times the segment Z horizontally and finally glue the labeled segment G2m horizontally with sZ to obtain the labeled graph Gnm. That is,(21)Gnm=sZ⊕hG2m.The multiplicity of the numbers 0, 1, and 2 in the graphGnm for m,n≥1 is shown in Table 3.Figure 9
Label segmentR33.Table 3
The multiplicity of 0, 1, and 2 inGnm, for m,n≥1.
G
m
n
e
δ
0 + vδ∗0
e
δ
1 + vδ∗1
e
δ
2 + vδ∗2
m
=
3
r, n=3s
24
r
s
+
2
s
+
2
r
+
1
24
r
s
+
2
s
+
2
r
24
r
s
+
2
s
+
2
r
m
=
3
r, n=3s+1
24
r
s
+
2
s
+
10
r
+
1
24
r
s
+
2
s
+
10
r
+
1
24
r
s
+
2
s
+
10
r
+
1
m
=
3
r, n=3s+2
24
r
s
+
2
s
+
18
r
+
1
24
r
s
+
2
s
+
18
r
+
2
24
r
s
+
2
s
+
18
r
+
2
m
=
3
r
+
1, n=3s
24
r
s
+
10
s
+
2
r
+
1
24
r
s
+
10
s
+
2
r
+
1
24
r
s
+
10
s
+
2
r
+
1
m
=
3
r
+
1, n=3s+1
24
r
s
+
10
s
+
10
r
+
5
24
r
s
+
10
s
+
10
r
+
4
24
r
s
+
10
s
+
10
r
+
4
m
=
3
r
+
1, n=3s+2
24
r
s
+
10
s
+
18
r
+
7
24
r
s
+
10
s
+
18
r
+
8
24
r
s
+
10
s
+
18
r
+
8
m
=
3
r
+
2, n=3s
24
r
s
+
18
s
+
2
r
+
1
24
r
s
+
18
s
+
2
r
+
2
24
r
s
+
18
s
+
2
r
+
2
m
=
3
r
+
2, n=3s+1
24
r
s
+
18
s
+
10
r
+
8
24
r
s
+
18
s
+
10
r
+
8
24
r
s
+
18
s
+
10
r
+
8
m
=
3
r
+
2, n=3s+2
24
r
s
+
18
s
+
18
r
+
13
24
r
s
+
18
s
+
18
r
+
14
24
r
s
+
18
s
+
18
r
+
14


## 3. Conclusion

In this paper, we constructed 3-TEPC labeling for the stellation of square grid graphGnm. For every m≥1 and every n≥1, we proved that Gnm is 3-TEPC.



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