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

Let G be a simple graph with vertex set V(G) and edge set E(G). An edge labeling δ: E(G)⟶ 0, 1, . . . , p − 1 􏼈 􏼉, where p is an integer, 1≤p≤ |E(G)|, induces a vertex labeling δ∗: V(H)⟶ 0, 1, . . . , p − 1 􏼈 􏼉 defined by δ∗(v) � δ(e1)δ(e2) · δ(en)(modp), where e1, e2, . . . , en are edges incident to v. +e labeling δ is said to be p-total edge product cordial (TEPC) labeling of G if |eδ(i) + vδ∗(i) − (eδ(j) + vδ∗(j))|≤ 1 for every i, j, 0≤ i≤ j≤p − 1, where eδ(i) 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.


Introduction and Definitions
Let G be a simple, finite, and connected graph with the vertex set V(G) and edge set E(G). 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 V(G) ∪ E(G), 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.
Let P n 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 P m with P n . e graph of rectangular grid is denoted by L(m, n) and has n and m squares in each row and column respectively. It is easy to observe that rectangular grid L(m, n) has mn vertices and mn − m − n + 1 edges. e stellation of L(m, n) is obtained by adding a vertex in each face of L(m, n) and then joining this vertex to each vertex of the respective face. We denote the stellation of L(m, n) by G m n , as shown in Figure 1. In this paper, we show that the graph G m n admits 3-TEPC labeling.

Main Results
Let m, n ≥ 1 and G m n be stellation of rectangular grid containing m rows and n columns. Observe that G m n has 2mn + m + n + 1 vertices and 6mn + m + n edges. We use the notations G 1 ⊕ v G 2 for gluing the graph G 1 with G 2 vertically. Similarly, G 1 ⊕ h G 2 represent gluing G 1 with G 2 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 → .
Proof. e 3-TEPC labeling of G 1 1 and G 2 1 is shown in Figure 2. Similarly, the 3-TEPC labeling of G 3 1 and the labeled segment S 3 1 are shown in Figure 3. e segment S 3 1 has the property that open edges are assigned labeled 1. Hence, if we glue the segment S 3 1 with itself vertically, then it will not change the vertex labels in S 3 1 ⊕ v S 3 1 : � 2S 3 1 . Observe that the labels 0, 1, and 2 are used 10 times in the segment S 3 1 . Table 1 shows the multiplicity of numbers 0, 1, and 2, respectively, used in the labeled graph G m 1 for m � 1, 2, 3.
To construct labeled graph G m 1 , we will use the labeled segments S 3 1 . First, glue r − 1 copies of labeled segment Finally, glue vertically the label segment G 3 1 to the open edges of (r − 1)S 3 1 to get labeled graph G m 1 , that is, In the labeled graph G m 1 , the multiplicity of 0, 1, and 2 is 10r Journal of Mathematics To construct the labeled graph G m 1 , we glue r copies of the labeled segment S 3 1 and then finally glue G 1 1 vertically. at is, In the labeled graph G m 1 , 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 graph G m 1 by gluing r times the labeled segment S 3 1 and finally gluing G 2 1 in vertical direction. at is, In the labeled graph G m 1 , the multiplicity of 0 is 10r + 7, whereas the multiplicity of 1 and 2 is 10r Proof. Observe that the graphs G 2 1 and G 1 2 are isomorphic and the 3-total edge cordial labeling of G 2 1 is given in Figure 2. erefore, G 1 2 is 3-TEPC. e 3-total edge product cordial labeling of the graphs G 2 2 and G 3 2 is given in Figures 4  and 5, respectively. Table 2 shows the multiplicity of numbers 0, 1, and 2 used in G 2 2 and G 3 2 . Figure 6 depicts the labeled segment S 3 2 , which has the property that open edges are assigned labeled 1 and each number 0, 1, and 2 is used 18 times.
To construct labeled graph G m 2 , we will use the labeled segments S 3 2 . First, we glue r − 1 copies of labeled segment S 3 2 vertically, that is, Since the open edges of S 3 2 are labeled with 1, therefore, this gluing process does not change the label of other vertices of (r − 1)S 3 2 . Finally, we glue vertically the label segment G 3 2 to the open edges of (r − 1)S 3 2 to get labeled graph G m 2 . at is, In the labeled graph G m 2 , 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 graph G m 2 , we glue r copies of the labeled segment S 3 2 and then finally glue G 1 2 vertically. at is, In the labeled graph G m 2 , the multiplicity of 0 is 18r + 7 whereas the multiplicity of 1 and 2 is 18r + 8. Case (iii): m � 3r + 2, r ≥ 1.
e labeled graph G m 2 can be obtained by gluing r times the labeled segment S 3 2 and then gluing G 2 2 in vertical direction. at is, 19 20 20 In the labeled graph G m 2 , the multiplicity of 0 is 18r + 13, whereas the multiplicity of 1 and 2 is 18r + 14. Proof. Observe that the graphs G 3 1 and G 1 3 are isomorphic. Similarly, the graphs G 3 2 and G 2 3 are also isomorphic. e 3-TEPC labeling of G 3 1 and G 3 2 are given in Figures 3 and 5, respectively. e 3-TEPC labeling of G 3 3 is shown in Figure 7. In the labeled graph G 3 3 , the multiplicity of 0 is 29, whereas the multiplicity of 1 and 2 is 28. Figure 8 shows the labeled segment S 3 3 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 graph G m 3 , we use the labeled segment S 3 3 . First, glue r − 1 copies of labeled segment S 3 3 vertically, that is, Since the open edges of S 3 3 are labeled with 1, therefore, this gluing process does not change the label of other vertices of (r − 1)S 3 3 . Finally, glue vertically the label segment G 3 3 to the open edges of (r − 1)S 3 2 to get labeled graph G m 3 . at is, In the labeled graph G m 3 , 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 graph G m 3 , we glue r copies of the labeled segment S 3 3 vertically and then finally glue G 1 3 vertically. at is, In the labeled graph G m 3 , the multiplicity of 0, 1, and 2 is 26r + 11. Case (iii): m � 3r + 2, r ≥ 1. We obtain the labeled graph G m 3 by gluing r times the labeled segment S 3 3 vertically and then finally glue G 2 3 in vertical direction. at is, In the labeled graph G m 3 , the multiplicity of 0 is 26r + 19, whereas the multiplicity of 1 and 2 is 26r + 20.  e graph G m n is 3-TEPC for m, n ≥ 1.
Proof. To construct the labeled graph of G m n and to examine its 3-TEPC labeling, we introduced a new segment R 3 3 . is segment has 17 open edges which are labeled with the number 1 and multiplicity of 0, 1, and 2 is 24. e labeled segment R 3 3 is shown in Figure 9. Case 1: m � 3r, r ≥ 1. First, we glue the segment R 3 3 vertically r − 1 times, that is, 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 − 1)R 3 3 . Finally, we glue the segment S 3 3 → in the vertical direction. is gives a new segment X and is defined as Note that the labels of open edges of X are 1 and multiplicity of each number 0, 1, and 2 is 24r + 2. Subcase 1: n � 3s, s ≥ 1. First, we glue s − 1 times the segment X horizontally and finally glue the labeled segment G m 3 horizontally to obtain the labeled graph G m n . at is, Subcase 2: n � 3s + 1, s ≥ 1. First, we glue s times the segment X horizontally and finally glue the labeled segment G m 1 horizontally with sX to obtain the labeled graph G m n . at is, Subcase 3: n � 3s + 2, s ≥ 1. First, we glue s times the segment X horizontally and finally glue the labeled segment G m 2 horizontally with sX to obtain the labeled graph G m n . at is, Case 2: when m � 3r + 1, r ≥ 1. First, we glue the segment R 3 3 vertically r times, that is, 3 3 . en, we glue the segment S 3 1 → in the vertical direction. is gives us a new segment Y defined as Note that the labels of open edges of Y are 1 and multiplicity of each number 0, 1, and 2 is 24r + 10. Subcase 1: n � 3s, s ≥ 1.
First, we glue s − 1 times the segment Y horizontally and finally glue the labeled segment G m 3 horizontally with (s − 1)Y to obtain the labeled graph G m n . at is, Subcase 2: n � 3s + 1, s ≥ 1. First, we glue s times the segment Y horizontally and finally glue the labeled segment G m 1 horizontally with sY to obtain the labeled graph G m n . at is, Subcase 3: n � 3s + 2, s ≥ 1. First, we glue s times the segment Y horizontally and finally glue the labeled segment G m 2 horizontally with sY to obtain the labeled graph G m n . at is, Case 3: m � 3r + 2, r ≥ 1. First, we glue the segment R 3 3 vertically r times, that is, 3 3 . en, we glue in the vertical direction of the segment S 3 2 → . is gives us a new segment Z defined as Note that the labels of open edges of Z are 1 and multiplicity of each number 0, 1, and 2 is 24r + 18. Subcase 1: n � 3s, s ≥ 1. First, we glue s − 1 times the segment Y horizontally and finally glue the labeled segment G m 3 horizontally with (s − 1)Z to obtain the labeled graph G m n . at is, Journal of Mathematics 5 Subcase 2: n � 3s + 1, s ≥ 1. First, we glue s times the segment Y horizontally and finally glue the labeled segment G m 1 horizontally with sZ to obtain the labeled graph G m n . at is, Subcase 3: n � 3s + 2, s ≥ 1.
First, we glue s times the segment Z horizontally and finally glue the labeled segment G m 2 horizontally with sZ to obtain the labeled graph G m n . at is, e multiplicity of the numbers 0, 1, and 2 in the graph G m n for m, n ≥ 1 is shown in Table 3.

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

Data Availability
No data were used to support the findings of the study.