Abstract
Let be a simple graph with vertex set and edge set . An edge labeling , where is an integer, , induces a vertex labeling defined by , where are edges incident to . The labeling is said to be -total edge product cordial (TEPC) labeling of if for every , , where and are numbers of edges and vertices labeled with integer , respectively. In this paper, we have proved that the stellation of square grid graph admits a 3-total edge product cordial labeling.
1. Introduction and Definitions
Let be a simple, finite, and connected graph with the vertex set and edge set . For basic notions related to graph theory, we refer the reader to the book by West [1]. A graph labeling 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 is called vertex (edge) labeling. If the domain is , then 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 be a vertex labeling which induces edge labeling defined by . . The labeling is said to be cordial if and , where and denote the number of vertices and number of edges labeled with integer , 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 induces an edge labeling defined by which is called product cordial labeling if and , where and represent the number of vertices that are labeled 0 and 1, respectively. While and 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].
Let be an integer. An edge labeling induces a vertex labeling defined by , where are edges incident to . The labeling is said to be -TEPC labeling of if for every , , where and are numbers of edges and vertices labeled with integer , respectively. Azaizeh et al. [14] introduced the concept of -TEPC labeling. A graph that admits a -TEPC labeling is called a -TEPC graph. Baca et al. [15] investigated the 3-TEPC labeling of carbon nanotube networks. Ahmad et al. [16] showed that the grid graph admits a 3-TEPC labeling. Ahmad et al. [3] proved that the hexagonal grid admits 3-TEPC labeling. Javed and Jamil [17] proved that the Rhombic grid is 3-TEPC for .
Let denote a path graph on vertices. A rectangular grid is an lattice graph and is obtained by taking the Cartesian product of with . The graph of rectangular grid is denoted by and has and squares in each row and column respectively. It is easy to observe that rectangular grid has vertices and edges. The stellation of is obtained by adding a vertex in each face of and then joining this vertex to each vertex of the respective face. We denote the stellation of by , as shown in Figure 1. In this paper, we show that the graph admits 3-TEPC labeling.
2. Main Results
Let and be stellation of rectangular grid containing rows and columns. Observe that has vertices and edges. We use the notations for gluing the graph with vertically. Similarly, represent gluing with horizontally. If we have a labeled segment or labeled graph and we rotate it by 90 degree in clockwise direction, then we will denote it by .
Theorem 1. For , the graph is 3-TEPC.
Proof. The 3-TEPC labeling of and is shown in Figure 2. Similarly, the 3-TEPC labeling of and the labeled segment are shown in Figure 3. The segment has the property that open edges are assigned labeled 1. Hence, if we glue the segment with itself vertically, then it will not change the vertex labels in . Observe that the labels 0, 1, and 2 are used 10 times in the segment . Table 1 shows the multiplicity of numbers 0, 1, and 2, respectively, used in the labeled graph for . Case (i). m = 3r, . To construct labeled graph , we will use the labeled segments . First, glue copies of labeled segment vertically that is : = . Finally, glue vertically the label segment to the open edges of to get labeled graph , that is, In the labeled graph , the multiplicity of 0, 1, and 2 is exactly. Case (ii): m = 3r + 1, . To construct the labeled graph , we glue copies of the labeled segment and then finally glue vertically. That is, In the labeled graph , the multiplicity of 0 is , whereas the multiplicity of 1 and 2 is . Case (iii): m = 3r + 2, . We obtain the labeled graph by gluing times the labeled segment and finally gluing in vertical direction. That is, In the labeled graph , the multiplicity of 0 is , whereas the multiplicity of 1 and 2 is .
Theorem 2. For m , the graph is 3-total edge product cordial.
Proof. Observe that the graphs and are isomorphic and the 3-total edge cordial labeling of is given in Figure 2. Therefore, is 3-TEPC. The 3-total edge product cordial labeling of the graphs and is given in Figures 4 and 5, respectively. Table 2 shows the multiplicity of numbers 0, 1, and 2 used in and .
Figure 6 depicts the labeled segment , 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, . To construct labeled graph , we will use the labeled segments . First, we glue copies of labeled segment vertically, that is, . Since the open edges of are labeled with 1, therefore, this gluing process does not change the label of other vertices of . Finally, we glue vertically the label segment to the open edges of to get labeled graph . That is, In the labeled graph , the multiplicity of 0 is , whereas the multiplicity of 1 and 2 is . Case (ii): m = 3r + 1, . To construct the labeled graph , we glue copies of the labeled segment and then finally glue vertically. That is, In the labeled graph , the multiplicity of 0 is whereas the multiplicity of 1 and 2 is . Case (iii): m = 3r + 2, . The labeled graph can be obtained by gluing times the labeled segment and then gluing in vertical direction. That is, In the labeled graph , the multiplicity of 0 is , whereas the multiplicity of 1 and 2 is .
Theorem 3. The graph is 3-TEPC for .
Proof. Observe that the graphs and are isomorphic. Similarly, the graphs and are also isomorphic. The 3-TEPC labeling of and are given in Figures 3 and 5, respectively. The 3-TEPC labeling of is shown in Figure 7. In the labeled graph , the multiplicity of 0 is 29, whereas the multiplicity of 1 and 2 is 28.
Figure 8 shows the labeled segment 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, . To construct labeled graph , we use the labeled segment . First, glue copies of labeled segment vertically, that is, : = . Since the open edges of are labeled with 1, therefore, this gluing process does not change the label of other vertices of . Finally, glue vertically the label segment to the open edges of to get labeled graph . That is, In the labeled graph , the multiplicity of 0 is , whereas the multiplicity of 1 and 2 is . Case (ii): m = 3r + 1, . To construct the labeled graph , we glue copies of the labeled segment vertically and then finally glue vertically. That is, In the labeled graph , the multiplicity of 0, 1, and 2 is . Case (iii): m = 3r + 2, . We obtain the labeled graph by gluing times the labeled segment vertically and then finally glue in vertical direction. That is, In the labeled graph , the multiplicity of 0 is , whereas the multiplicity of 1 and 2 is .
Theorem 4. The graph is 3-TEPC for .
Proof. To construct the labeled graph of and to examine its 3-TEPC labeling, we introduced a new segment . 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 is shown in Figure 9. Case 1: . First, we glue the segment vertically 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 . Finally, we glue the segment in the vertical direction. This gives a new segment and is defined as Note that the labels of open edges of are 1 and multiplicity of each number 0, 1, and 2 is . Subcase 1: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally to obtain the labeled graph . That is, Subcase 2: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally with to obtain the labeled graph . That is, Subcase 3: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally with to obtain the labeled graph . That is, Case 2: when . First, we glue the segment vertically times, that is, = . Then, we glue the segment in the vertical direction. This gives us a new segment defined as Note that the labels of open edges of are 1 and multiplicity of each number 0, 1, and 2 is . Subcase 1: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally with to obtain the labeled graph . That is, Subcase 2: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally with to obtain the labeled graph . That is, Subcase 3: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally with to obtain the labeled graph . That is, Case 3: . First, we glue the segment vertically times, that is, = . Then, we glue in the vertical direction of the segment . This gives us a new segment defined as Note that the labels of open edges of are 1 and multiplicity of each number 0, 1, and 2 is . Subcase 1: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally with to obtain the labeled graph . That is, Subcase 2: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally with to obtain the labeled graph . That is, Subcase 3: . First, we glue times the segment horizontally and finally glue the labeled segment horizontally with to obtain the labeled graph . That is, The multiplicity of the numbers 0, 1, and 2 in the graph for is shown in Table 3.
3. Conclusion
In this paper, we constructed 3-TEPC labeling for the stellation of square grid graph . For every and every , we proved that is 3-TEPC.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.