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3-total edge product cordial labeling of rhombic grid
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3-total edge product cordial labeling of rhombic grid Aisha Javed a , Muhammad Kamran Jamil b , ∗ a Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan b Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, 14 Ali Road,
Lahore, Pakistan Received 12 September 2017; received in revised form 14 December 2017; accepted 15 December 2017 Available online xxxx
Abstract For a simple graph G = (V (G), E(G)), this paper deals with the existence of an edge labeling χ : E(G) → {0, 1, . . . , k − 1}, 2 ≤ k ≤ |E(G)|, which induces a vertex labeling χ ∗ : V (G) → {0, 1, . . . , k − 1} in such a way that for each vertex v, assigns the label χ (e1 ) · χ (e2 ) · · · · · χ (en )( (mod k)), where e1 , e2 , . . . , en are the edges incident to the vertex v. The labeling χ is called a k-total edge product cordial labeling of G if |(eχ (i) + vχ ∗ (i)) − (eχ ( j) + vχ ∗ ( j))| ≤ 1 for every i, j, 0 ≤ i < j ≤ k − 1, where eχ (i) and vχ ∗ (i) are the number of edges and vertices with χ (e) = i and χ ∗ (e) = i, respectively. In this paper, we examine the existence of such labeling for rhombic grid graph. c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Cordial labeling; 3-total edge product cordial labeling; Rhombic grid graph
1. Introduction Let G = (V (G), E(G)) be a simple, finite and connected graph, where V (G) and E(G) represent the vertex set and edge set, respectively. For basic notations and definitions from graph theory see [1]. Any mapping that sends some set of graph elements to a set of numbers or colors is called a labeling of that graph. A labeling in which we only label vertices (respectively, edges) is known as vertex (respectively, edge) labeling. If we label both vertices and edges, then the labeling is called a total labeling. Labeled graphs are becoming an increasingly useful family of mathematical models for a broad range of applications. A vertex labeling χ : V (G) → {0, 1} induces and edge labeling χ ∗ : E(G) → {0, 1} defined as χ ∗ (uv) = |χ (u) − χ(v)|. For χ and i ∈ {0, 1}, a vertex u is an i-vertex if χ(u) = i and an edge e is an i-edge if χ ∗ (e) = i. Denote the numbers of 0-vertices, 1-vertices, 0-edges and 1-edges of G under χ and χ ∗ by vχ (0), vχ (1), eχ ∗ (0) and eχ ∗ (1), respectively. A vertex labeling χ is called cordial if |vχ (0) − vχ (1)| ≤ 1 and |eχ ∗ (0) − eχ ∗ (1)| ≤ 1. Cahit [2] Peer review under responsibility of Kalasalingam University.
∗ Corresponding author.
E-mail addresses: [email protected] (A. Javed), [email protected] (M.K. Jamil). https://doi.org/10.1016/j.akcej.2017.12.002 c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Fig. 1. A rhombic grid graph.
introduced the notation of cordial labeling as a weaker version of graceful labeling. A lot of work on cordial labeling have been done. For history and recent result, we refer [3–10]. A binary vertex labeling χ : V (G) → {0, 1} with induced edge labeling χ ∗ : E(G) → {0, 1} defined as ∗ χ (uv) = χ(u)χ(v) is called a product cordial labeling if |vχ (0) − vχ (1)| ≤ 1 and |eχ ∗ (0) − eχ ∗ (1)| ≤ 1. Sundaram et al. [11] and Barasara and Vaidya [12,13] introduced the edge product and total edge product cordial labeling with some variation in cordial definition. Let k be an integer, 2 ≤ k ≤ |E(G)|. An edge labeling χ : E(G) → {0, 1, 2, . . . , k − 1} with induced vertex labeling χ ∗ : V (G) → {0, 1, 2, . . . , k − 1} defined as χ ∗ (v) = χ (e1 ) · χ(e2 ) · · · · · χ(el )( (mod k)), where e1 , e2 , . . . , el are the edges incident to the vertex v, is called a k-total edge product cordial labeling of G if |(eχ (i) + vχ ∗ (i)) − (eχ ( j) + vχ ∗ ( j))| ≤ 1 for every i, j, 0 ≤ i < j ≤ k − 1. Azaizeh et al. [14] gave the concept of the k-total edge product cordial labeling. A graph G with a k-total edge product cordial labeling is called a k-total edge product cordial graph. For further study on k-total edge product cordial graphs, see [13–15]. The square grid graph is an important class of graphs, which can be used in the design of local area networks [16]. The square grid graph has been studied extensively in recent years, and results on the square grid graph can be found in [17–19]. The rhombic grid graph is another grid graph. A rhombic grid graph is shown in Fig. 1, and n and m represents the number of rhombuses in a row and a column, respectively. This structure contains 2mn + m + n vertices and 4mn edges. In this paper, we showed that the rhombic grid graph admits a 3-total edge product cordial labeling. 2. Main results The rhombic grid graph is denoted by Rnm where n denotes the number of rhombuses in a row and m denotes the number of rhombuses in a column. The total number of vertices and number of edges in a rhombic grid graph Rnm is 6mn + m + n. In this paper, we will use the notations open edge that means the edge with only one end vertex and isolated edge that means the edge without both end vertices. We will use two operations for gluing the segments, the symbol ⊕v represents an operation gluing two segments in the vertical direction and the symbol ⊕h represents an operation of gluing two segments in the horizontal direction. Theorem 1. The graph R1m is 3-total edge product cordial for m ≥ 1. Proof. For m = 1, 2, 3 the corresponding 3-total edge product cordial labelings of R1m are illustrated in Fig. 2. Table 1 shows multiplicity of numbers 0,1 and 2 used in R1m for m = 1, 2, 3. Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Table 1 Multiplicity of 0s, 1s and 2s used in R1m for m = 1, 2, 3. R1m
eχ (0) + vχ ∗ (0)
eχ (1) + vχ ∗ (1)
eχ (2) + vχ ∗ (2)
m=1 m=2 m=3
3 5 7
2 5 8
3 5 7
Fig. 2. The 3-total edge product cordial labelings of R1m , m = 1, 2, 3.
Fig. 3. The 3-total edge product cordial labeling of the segment A31 .
Fig. 3 shows the A31 segment and its labeling. Note that this labeling has properties that open edges are labeled with 1 and every number of 0,1 and 2 is used exactly 7 times. If m = 3t, t ≥ 1, then to obtain the graph R1m , we glue (t − 1) segments A31 together in the vertical direction. Since the open edges in the segment A31 are labeled with number 1 it follows that by gluing these segments we do not change the vertex labels in the segment A31 ⊕v A31 ⊕v · · · ⊕v A31 = (t − 1)A31 . We glue (t − 1)A31 vertically to R13 to obtain R1m , i.e. ⎡ ⎤ (t − 1)A31 ⎢ ⎥ R1m = ⎣ ⊕v ⎦. R13 One can easily notice that R1m contains number of 0 and 2 exactly 7t times and the number 1 exactly 7t + 1 times. Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Table 2 Multiplicity of 0s, 1s and 2s used in R2m for m = 2, 3. R2m
eχ (0) + vχ ∗ (0)
eχ (1) + vχ ∗ (1)
eχ (2) + vχ ∗ (2)
m=2 m=3
9 14
9 13
10 14
Fig. 4. The 3-total edge product cordial labelings of R2m , m = 2, 3.
If m = 3t + 1, t ≥ 1, then to obtain the graph R1m , we glue t segments A31 together and the segment R11 in the vertical direction. Thus, we obtain ⎤ ⎡ t A31 ⎥ ⎢ R1m = ⎣ ⊕v ⎦ . R11 One can easily notice that R1m contains numbers 0 and 2 exactly 7t + 3 times while 1 is used exactly 7t + 2 times. If m = 3t + 2, t ≥ 1, then to obtain the graph R1m , we glue t segments A31 together and the segment R12 in the vertical direction. Then, we obtain ⎤ ⎡ t A31 ⎥ ⎢ R1m = ⎣ ⊕v ⎦ . R12 One can easily notice that R1m contains numbers 0,1 and 2 exactly 7t + 5 times. Theorem 2. The graph R2m is 3-total edge product cordial for m ≥ 1. Proof. The graph R12 is isomorphic to R21 , and its 3-total edge product cordial labeling is illustrated in Fig. 2. For m = 2, 3 the corresponding 3-total edge product cordial labelings of R2m are illustrated in Fig. 4. Table 2 shows the number of 0,1 and 2 used in R2m for m = 2, 3. The segment A32 and its labeling is shown in Fig. 5. Note that this labeling has properties that open edges are labeled with 1 and every number 0,1 and 2 is used exactly 13 times. If m = 3t, t ≥ 1, then to obtain the graph R2m , we glue (t − 1) segments A32 together and the segment R12 in the vertical direction. Hence, we obtain ⎡ ⎤ (t − 1)A32 ⎢ ⎥ R2m = ⎣ ⊕v ⎦. 3 R2 One can easily notice that R2m contains numbers of 0 and 2 exactly 13t + 1 times and the number 1 exactly 13t times. Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Fig. 5. The 3-total edge product labeling of A32 . Table 3 Multiplicity of 0s, 1s and 2s used in R33 . R3m
eχ (0) + vχ ∗ (0)
eχ (1) + vχ ∗ (1)
eχ (2) + vχ ∗ (2)
m=3
20
20
20
If m = 3t + 1, t ≥ 1, then to obtain the graph R2m , we glue t segments A32 together in the vertical direction. Since the open edges in the segment A32 are labeled with number 1 it follows that by gluing these segments we do not change the vertex labels in the segment A32 ⊕v A32 ⊕v · · · ⊕v A32 = t A32 . Then we glue t A32 vertically to R21 to obtain t A32 ⊕v R21 = R2m , i.e. ⎡ ⎤ t A32 ⎢ ⎥ R2m = ⎣ ⊕v ⎦ . R21 One can easily notice that the R2m contains numbers 0,1 and 2 exactly 13t + 5 times. If m = 3t + 2, t ≥ 1, then to obtain the graph R2m , we glue t segments A32 together and the segment R22 in the vertical direction. Then we obtain ⎡ ⎤ t A32 ⎢ ⎥ R2m = ⎣ ⊕v ⎦ . R22 One can easily notice that R2m contains numbers 0 and 1 exactly 13t + 9 times and 2 exactly 13t + 10 times. Theorem 3. The graph R3m is 3-total edge product cordial for m ≥ 1. Proof. As the graph R31 is isomorphic to R13 and the graph R32 is isomorphic to R23 we have that they are 3-total edge product cordial. The 3-total edge product cordial labelings of these graphs are illustrated in Figs. 2 and 4, respectively. A 3-total edge product cordial labeling of R33 is illustrated in Fig. 6. Table 3 shows the number of 0,1 and 2 used in R3m for m = 3. Fig. 7 illustrate the segment A33 and its labeling. Note that this labeling has properties that open edges are labeled with 1 and every number 0,1 and 2 is used exactly 19 times. If m = 3t, t ≥ 1, then to obtain the graph R3m , we glue (t − 1) segments A33 together and the segment R31 in the vertical direction. Since the open edges in the segment A33 are labeled with number 1 it follows that by gluing these segments we do not change the vertex labels in the segment A33 ⊕v A33 ⊕v · · · ⊕v A33 = (t −1)A33 . Then we glue (t −1)A33 Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Fig. 6. The 3-total edge product cordial labeling of R33 .
Fig. 7. The 3-total edge product cordial labeling of A33 .
vertically to R33 to obtain (t − 1)A33 ⊕v R33 = R3m , i.e. ⎡
⎤ (t − 1)A33 ⎥ ⎢ R3m = ⎣ ⊕v ⎦. 3 R3 One can easily notice that R3m contains numbers of 0,1 and 2 exactly 19t + 1 times. If m = 3t + 1, t ≥ 1, then to obtain the graph R3m , we glue t segments A33 together and the segment R31 in the vertical direction. Then we obtain ⎡ ⎤ t A33 ⎢ ⎥ R3m = ⎣ ⊕v ⎦ . R31 One can easily notice that R3m contains numbers 0 and 2 exactly 19t + 7 times while 1 is exactly 19t + 8 times. Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Fig. 8. The 3-total edge product cordial labeling of B33 .
If m = 3t + 2, t ≥ 1, then to obtain the graph R3m , we glue t segments A33 together and the segment R32 in the vertical direction. This gives ⎤ ⎡ t A33 ⎥ ⎢ R3m = ⎣ ⊕v ⎦ . R32 One can easily notice that R3m contains numbers 0 and 2 exactly 19t + 14 times and 1 exactly 19t + 13 times. Theorem 4. The graph Rnm is 3-total edge product cordial for m, n ≥ 1. Proof. For obtaining a 3-total edge product cordial labeling of Rnm , we need a new labeled segment B33 . The segment B33 has ten open edges and one isolated edge. Fig. 8 illustrates the segment B33 and its labeling. Note that this labeling has properties that open edges and isolated edge are labeled with 1 and every number 0,1 and 2 is used exactly 18 times. To obtain the 3-total edge product cordial labeling of Rnm for m = 1, 2, 3 and for all n we used Theorems 1–3. − → − → − → Moreover, we will use the segments A33 , A31 and A32 obtained by rotation of the segments A31 , A32 and A33 in a clockwise direction through the angle 90◦ about its center. Depending upon the values of m and n we have the following cases: Case 1. When m = 3t, t ≥ 1. First we glue the segment B33 vertically t −1 times. Since the open edges in the segment B33 are labeled with number 1 it follows that by gluing these segments we do not change the vertex labels in the segment B33 ⊕v B33 ⊕v · · · ⊕v B33 = − → (n − 1)B33 . Then we glue in the vertical direction the segment A33 to the segment (n − 1)B33 to obtain the segment A. Hence ⎡ ⎤ (t − 1)B33 ⎥ ⎢ A = ⎣⊕v ⎦. − →3 A3 Note that open edges of the segment have label 1 and each number 0,1 and 2 has multiplicity 18t + 1. Subcase i. When n = 3s, s ≥ 1. We glue the segment A horizontally s − 1 times and moreover, we glue horizontally the segment R3m and we obtain [ ] Rnm = R3m ⊕h (s − 1)A . Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Subcase ii. When n = 3s + 1, s ≥ 1. The 3-total edge product cordial labeling of rhombic grid Rnm is obtained as [ ] Rnm = R1m ⊕h s A . Subcase iii. When n = 3s + 2, s ≥ 1. We glue the segment A horizontally s times and moreover, we glue horizontally the segment R2m hence [ ] Rnm = R2m ⊕h s A . Case 2. When m = 3t + 1, t ≥ 1. − → The segment B is obtained by gluing vertically the segment A31 with vertically t times of the segment B33 , i.e. ⎡− →3 ⎤ A ⎢ 1 ⎥ B = ⎣⊕v ⎦ . t B33 Note that open edges of the segment have label 1 and each number 0,1 and 2 has multiplicity 18t + 7. Subcase i. When n = 3s, s ≥ 1. The rhombic grid Rnm is obtained as [ ] Rnm = R3m ⊕h (s − 1)B . Subcase ii. When n = 3s + 1, s ≥ 1. The 3-total edge product cordial labeling of rhombic grid Rnm can be obtained as [ ] Rnm = R1m ⊕h s B . Subcase iii. When n = 3s + 2, s ≥ 1. We glue the segment A3t+1 horizontally s times and moreover, we glue horizontally the segment R2m and we obtain 3 [ ] Rnm = R2m ⊕h s B . Case 3. When m = 3t + 2, t ≥ 1. − → The segment C is obtained by gluing vertically the segment A32 with vertically t times of the segment B33 , i.e. ⎡− →3 ⎤ A ⎢ 2 ⎥ C = ⎣⊕v ⎦ . t B33 Note that open edges of the segment has label 1 and each number 0,1 and 2 have multiplicity 18t + 13. Subcase i. When n = 3s, s ≥ 1. To obtain the rhombic grid, we glue the segment A3t+2 horizontally s − 1 times and then we glue the resulting 3 segment horizontally to R3m . Hence [ ] Rnm = R3m ⊕h (s − 1)C . Subcase ii. When n = 3s + 1, s ≥ 1. The 3-total edge product cordial labeling of rhombic grid Rnm can be obtained as [ ] Rnm = R1m ⊕h sC . Subcase iii. When n = 3s + 2, s ≥ 1. The segment Rnm is obtained by gluing the segment A3t+2 horizontally s times and the resulting segment is glued 3 horizontally to R2m and we obtain [ ] Rnm = R2m ⊕h sC . Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Table 4 Multiplicity of 0s, 1s and 2s used in the rhombic grid Rnm for m, n ≥ 1. Case
Grid Rnm
eχ (0) + vχ ∗ (0)
1.
m = 3t, t ≥ 1
d = 18st + s + t
i. ii. iii.
n = 3s, s ≥ 1 n = 3s + 1, s ≥ 1 n = 3s + 2, s ≥ 1
d d + 6t d + 12t + 1
2.
m = 3t + 1, t ≥ 1
d = 18st + 7s + t
i. ii. iii.
n = 3s, s ≥ 1 n = 3s + 1, s ≥ 1 n = 3s + 2, s ≥ 1
d d + 6t + 3 d + 12t + 5
3.
m = 3t + 2, t ≥ 1
d = 18st + 13s + t
i. ii. iii.
n = 3s, s ≥ 1 n = 3s + 1, s ≥ 1 n = 3s + 2, s ≥ 1
d +1 d + 6t + 5 d + 12t + 9
eχ (1) + vχ ∗ (1)
eχ (2) + vχ ∗ (2)
d d + 6t + 1 d + 12t
d d + 6t d + 12t + 1
d +1 d + 6t + 2 d + 12t + 5
d d + 6t + 3 d + 12t + 5
d d + 6t + 5 d + 12t + 9
d +1 d + 6t + 5 d + 12t + 10
All possible cases for obtaining the grid Rmn for m, n ≥ 1, are described in Table 4, where it is shown how many times the numbers 0,1 and 2 are used as edge and vertex labels. One can see that the resulting grid Rmn in each previous cases satisfies the property of having a 3-total edge product cordial labeling. 3. Future work The rhombic grid graph can be embed on cylinder and torus. So, in future with the help of this work one can find the 3-total edge product cordial labeling of rhombic grid graph than can embed on cylinder and torus. Acknowledgments The authors would like to thank the reviewers for their valuable comments and suggestions to improve the quality of the paper. References [1] D.B. West, Introduction To Graph Theory, second ed., Prentice-Hall, New Jersey, USA, 2003. [2] I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin. 23 (1987) 201–207. [3] A. Ahmad, M. Baˇca, M. Naseem, A. Semaniˇcová, On 3-total edge product cordial labeling of honeycomb, AKCE Int. J. Graphs Combin. 14 (2017) 149–157. [4] I. Cahit, On cordial and 3-equitable labelings of graphs, Util. Math. 37 (1990) 189–198. [5] M. Hovey, A-cordial graphs, Discrete Math. 93 (1991) 183–194. [6] Y.S. Ho, S.M. Lee, S.C. Shee, Cordial labelings of the Cartesian product and composition of graphs, Ars Combin. 29 (1990) 169–180. [7] Y.S. Ho, S.C. Shee, The cordiality of one-point union of n copies of a graph, Discrete Math. 117 (1993) 225–243. [8] W.W. Kirchherr, On the cordiality of some specific graphs, Ars Combin. 31 (1991) 127–137. [9] D. Kuo, G.J. Chang, Y.H.H. Kwong, Cordial labeling of m K n , Discrete Math. 169 (1997) 121–131. [10] S.M. Lee, A. Liu, A construction of cordial graphs from smaller cordial graphs, Ars Combin. 32 (1991) 209–214. [11] M. Sundaram, R. Ponraj, S. Somasundaram, Product cordial labeling of graphs, Bull. Pure Appl. Sci. Sect. E Math. Stat. 23 (2004) 155–163. [12] S.K. Vaidya, C.M. Barasara, Edge product cordial labeling of graphs, J. Math. Comput. Sci. 2 (5) (2012) 1436–1450. [13] S.K. Vaidya, C.M. Barasara, Total edge product cordial labeling of graphs, Malay. J. Mat. 3 (1) (2013) 55–63. [14] A. Azaizeh, R. Hasni, A. Ahmad, G.C. Lau, 3-total edge product cordial labeling of graphs, Far East J. Math. Sci. 96 (2) (2015) 193–209. [15] J. Ivanˇco, On k-total edge product cordial graphs, Aust. J. Combin. 67 (3) (2017) 476–486. [16] J.C. Bermond, F. Comellas, D.F. Hsu, Distributed loop computer networks: a survey, J. Parallel Distrib. Comput. 24 (1995) 2–10. [17] P. Andersen, C. Grigorious, M. Miller, Minimum weight resolving sets of grid graphs, arXiv:1409.4510v1 [math.CO]. [18] M. Baˇca, Y. Lin, M. Miller, Antimagic labelings of grids, Util. Math. 72 (2007) 65–75. [19] M.J. Lee, W.H. Tsai, C. Lin, Super (a,1)-cycle-antimagic labeling of the grid, Ars Combin. 112 (2013) 3–12.
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3-total edge product cordial labeling of rhombic grid Aisha Javed a , Muhammad Kamran Jamil b , ∗ a Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan b Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, 14 Ali Road,
Lahore, Pakistan Received 12 September 2017; received in revised form 14 December 2017; accepted 15 December 2017 Available online xxxx
Abstract For a simple graph G = (V (G), E(G)), this paper deals with the existence of an edge labeling χ : E(G) → {0, 1, . . . , k − 1}, 2 ≤ k ≤ |E(G)|, which induces a vertex labeling χ ∗ : V (G) → {0, 1, . . . , k − 1} in such a way that for each vertex v, assigns the label χ (e1 ) · χ (e2 ) · · · · · χ (en )( (mod k)), where e1 , e2 , . . . , en are the edges incident to the vertex v. The labeling χ is called a k-total edge product cordial labeling of G if |(eχ (i) + vχ ∗ (i)) − (eχ ( j) + vχ ∗ ( j))| ≤ 1 for every i, j, 0 ≤ i < j ≤ k − 1, where eχ (i) and vχ ∗ (i) are the number of edges and vertices with χ (e) = i and χ ∗ (e) = i, respectively. In this paper, we examine the existence of such labeling for rhombic grid graph. c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Cordial labeling; 3-total edge product cordial labeling; Rhombic grid graph
1. Introduction Let G = (V (G), E(G)) be a simple, finite and connected graph, where V (G) and E(G) represent the vertex set and edge set, respectively. For basic notations and definitions from graph theory see [1]. Any mapping that sends some set of graph elements to a set of numbers or colors is called a labeling of that graph. A labeling in which we only label vertices (respectively, edges) is known as vertex (respectively, edge) labeling. If we label both vertices and edges, then the labeling is called a total labeling. Labeled graphs are becoming an increasingly useful family of mathematical models for a broad range of applications. A vertex labeling χ : V (G) → {0, 1} induces and edge labeling χ ∗ : E(G) → {0, 1} defined as χ ∗ (uv) = |χ (u) − χ(v)|. For χ and i ∈ {0, 1}, a vertex u is an i-vertex if χ(u) = i and an edge e is an i-edge if χ ∗ (e) = i. Denote the numbers of 0-vertices, 1-vertices, 0-edges and 1-edges of G under χ and χ ∗ by vχ (0), vχ (1), eχ ∗ (0) and eχ ∗ (1), respectively. A vertex labeling χ is called cordial if |vχ (0) − vχ (1)| ≤ 1 and |eχ ∗ (0) − eχ ∗ (1)| ≤ 1. Cahit [2] Peer review under responsibility of Kalasalingam University.
∗ Corresponding author.
E-mail addresses: [email protected] (A. Javed), [email protected] (M.K. Jamil). https://doi.org/10.1016/j.akcej.2017.12.002 c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Fig. 1. A rhombic grid graph.
introduced the notation of cordial labeling as a weaker version of graceful labeling. A lot of work on cordial labeling have been done. For history and recent result, we refer [3–10]. A binary vertex labeling χ : V (G) → {0, 1} with induced edge labeling χ ∗ : E(G) → {0, 1} defined as ∗ χ (uv) = χ(u)χ(v) is called a product cordial labeling if |vχ (0) − vχ (1)| ≤ 1 and |eχ ∗ (0) − eχ ∗ (1)| ≤ 1. Sundaram et al. [11] and Barasara and Vaidya [12,13] introduced the edge product and total edge product cordial labeling with some variation in cordial definition. Let k be an integer, 2 ≤ k ≤ |E(G)|. An edge labeling χ : E(G) → {0, 1, 2, . . . , k − 1} with induced vertex labeling χ ∗ : V (G) → {0, 1, 2, . . . , k − 1} defined as χ ∗ (v) = χ (e1 ) · χ(e2 ) · · · · · χ(el )( (mod k)), where e1 , e2 , . . . , el are the edges incident to the vertex v, is called a k-total edge product cordial labeling of G if |(eχ (i) + vχ ∗ (i)) − (eχ ( j) + vχ ∗ ( j))| ≤ 1 for every i, j, 0 ≤ i < j ≤ k − 1. Azaizeh et al. [14] gave the concept of the k-total edge product cordial labeling. A graph G with a k-total edge product cordial labeling is called a k-total edge product cordial graph. For further study on k-total edge product cordial graphs, see [13–15]. The square grid graph is an important class of graphs, which can be used in the design of local area networks [16]. The square grid graph has been studied extensively in recent years, and results on the square grid graph can be found in [17–19]. The rhombic grid graph is another grid graph. A rhombic grid graph is shown in Fig. 1, and n and m represents the number of rhombuses in a row and a column, respectively. This structure contains 2mn + m + n vertices and 4mn edges. In this paper, we showed that the rhombic grid graph admits a 3-total edge product cordial labeling. 2. Main results The rhombic grid graph is denoted by Rnm where n denotes the number of rhombuses in a row and m denotes the number of rhombuses in a column. The total number of vertices and number of edges in a rhombic grid graph Rnm is 6mn + m + n. In this paper, we will use the notations open edge that means the edge with only one end vertex and isolated edge that means the edge without both end vertices. We will use two operations for gluing the segments, the symbol ⊕v represents an operation gluing two segments in the vertical direction and the symbol ⊕h represents an operation of gluing two segments in the horizontal direction. Theorem 1. The graph R1m is 3-total edge product cordial for m ≥ 1. Proof. For m = 1, 2, 3 the corresponding 3-total edge product cordial labelings of R1m are illustrated in Fig. 2. Table 1 shows multiplicity of numbers 0,1 and 2 used in R1m for m = 1, 2, 3. Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Table 1 Multiplicity of 0s, 1s and 2s used in R1m for m = 1, 2, 3. R1m
eχ (0) + vχ ∗ (0)
eχ (1) + vχ ∗ (1)
eχ (2) + vχ ∗ (2)
m=1 m=2 m=3
3 5 7
2 5 8
3 5 7
Fig. 2. The 3-total edge product cordial labelings of R1m , m = 1, 2, 3.
Fig. 3. The 3-total edge product cordial labeling of the segment A31 .
Fig. 3 shows the A31 segment and its labeling. Note that this labeling has properties that open edges are labeled with 1 and every number of 0,1 and 2 is used exactly 7 times. If m = 3t, t ≥ 1, then to obtain the graph R1m , we glue (t − 1) segments A31 together in the vertical direction. Since the open edges in the segment A31 are labeled with number 1 it follows that by gluing these segments we do not change the vertex labels in the segment A31 ⊕v A31 ⊕v · · · ⊕v A31 = (t − 1)A31 . We glue (t − 1)A31 vertically to R13 to obtain R1m , i.e. ⎡ ⎤ (t − 1)A31 ⎢ ⎥ R1m = ⎣ ⊕v ⎦. R13 One can easily notice that R1m contains number of 0 and 2 exactly 7t times and the number 1 exactly 7t + 1 times. Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Table 2 Multiplicity of 0s, 1s and 2s used in R2m for m = 2, 3. R2m
eχ (0) + vχ ∗ (0)
eχ (1) + vχ ∗ (1)
eχ (2) + vχ ∗ (2)
m=2 m=3
9 14
9 13
10 14
Fig. 4. The 3-total edge product cordial labelings of R2m , m = 2, 3.
If m = 3t + 1, t ≥ 1, then to obtain the graph R1m , we glue t segments A31 together and the segment R11 in the vertical direction. Thus, we obtain ⎤ ⎡ t A31 ⎥ ⎢ R1m = ⎣ ⊕v ⎦ . R11 One can easily notice that R1m contains numbers 0 and 2 exactly 7t + 3 times while 1 is used exactly 7t + 2 times. If m = 3t + 2, t ≥ 1, then to obtain the graph R1m , we glue t segments A31 together and the segment R12 in the vertical direction. Then, we obtain ⎤ ⎡ t A31 ⎥ ⎢ R1m = ⎣ ⊕v ⎦ . R12 One can easily notice that R1m contains numbers 0,1 and 2 exactly 7t + 5 times. Theorem 2. The graph R2m is 3-total edge product cordial for m ≥ 1. Proof. The graph R12 is isomorphic to R21 , and its 3-total edge product cordial labeling is illustrated in Fig. 2. For m = 2, 3 the corresponding 3-total edge product cordial labelings of R2m are illustrated in Fig. 4. Table 2 shows the number of 0,1 and 2 used in R2m for m = 2, 3. The segment A32 and its labeling is shown in Fig. 5. Note that this labeling has properties that open edges are labeled with 1 and every number 0,1 and 2 is used exactly 13 times. If m = 3t, t ≥ 1, then to obtain the graph R2m , we glue (t − 1) segments A32 together and the segment R12 in the vertical direction. Hence, we obtain ⎡ ⎤ (t − 1)A32 ⎢ ⎥ R2m = ⎣ ⊕v ⎦. 3 R2 One can easily notice that R2m contains numbers of 0 and 2 exactly 13t + 1 times and the number 1 exactly 13t times. Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Fig. 5. The 3-total edge product labeling of A32 . Table 3 Multiplicity of 0s, 1s and 2s used in R33 . R3m
eχ (0) + vχ ∗ (0)
eχ (1) + vχ ∗ (1)
eχ (2) + vχ ∗ (2)
m=3
20
20
20
If m = 3t + 1, t ≥ 1, then to obtain the graph R2m , we glue t segments A32 together in the vertical direction. Since the open edges in the segment A32 are labeled with number 1 it follows that by gluing these segments we do not change the vertex labels in the segment A32 ⊕v A32 ⊕v · · · ⊕v A32 = t A32 . Then we glue t A32 vertically to R21 to obtain t A32 ⊕v R21 = R2m , i.e. ⎡ ⎤ t A32 ⎢ ⎥ R2m = ⎣ ⊕v ⎦ . R21 One can easily notice that the R2m contains numbers 0,1 and 2 exactly 13t + 5 times. If m = 3t + 2, t ≥ 1, then to obtain the graph R2m , we glue t segments A32 together and the segment R22 in the vertical direction. Then we obtain ⎡ ⎤ t A32 ⎢ ⎥ R2m = ⎣ ⊕v ⎦ . R22 One can easily notice that R2m contains numbers 0 and 1 exactly 13t + 9 times and 2 exactly 13t + 10 times. Theorem 3. The graph R3m is 3-total edge product cordial for m ≥ 1. Proof. As the graph R31 is isomorphic to R13 and the graph R32 is isomorphic to R23 we have that they are 3-total edge product cordial. The 3-total edge product cordial labelings of these graphs are illustrated in Figs. 2 and 4, respectively. A 3-total edge product cordial labeling of R33 is illustrated in Fig. 6. Table 3 shows the number of 0,1 and 2 used in R3m for m = 3. Fig. 7 illustrate the segment A33 and its labeling. Note that this labeling has properties that open edges are labeled with 1 and every number 0,1 and 2 is used exactly 19 times. If m = 3t, t ≥ 1, then to obtain the graph R3m , we glue (t − 1) segments A33 together and the segment R31 in the vertical direction. Since the open edges in the segment A33 are labeled with number 1 it follows that by gluing these segments we do not change the vertex labels in the segment A33 ⊕v A33 ⊕v · · · ⊕v A33 = (t −1)A33 . Then we glue (t −1)A33 Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Fig. 6. The 3-total edge product cordial labeling of R33 .
Fig. 7. The 3-total edge product cordial labeling of A33 .
vertically to R33 to obtain (t − 1)A33 ⊕v R33 = R3m , i.e. ⎡
⎤ (t − 1)A33 ⎥ ⎢ R3m = ⎣ ⊕v ⎦. 3 R3 One can easily notice that R3m contains numbers of 0,1 and 2 exactly 19t + 1 times. If m = 3t + 1, t ≥ 1, then to obtain the graph R3m , we glue t segments A33 together and the segment R31 in the vertical direction. Then we obtain ⎡ ⎤ t A33 ⎢ ⎥ R3m = ⎣ ⊕v ⎦ . R31 One can easily notice that R3m contains numbers 0 and 2 exactly 19t + 7 times while 1 is exactly 19t + 8 times. Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Fig. 8. The 3-total edge product cordial labeling of B33 .
If m = 3t + 2, t ≥ 1, then to obtain the graph R3m , we glue t segments A33 together and the segment R32 in the vertical direction. This gives ⎤ ⎡ t A33 ⎥ ⎢ R3m = ⎣ ⊕v ⎦ . R32 One can easily notice that R3m contains numbers 0 and 2 exactly 19t + 14 times and 1 exactly 19t + 13 times. Theorem 4. The graph Rnm is 3-total edge product cordial for m, n ≥ 1. Proof. For obtaining a 3-total edge product cordial labeling of Rnm , we need a new labeled segment B33 . The segment B33 has ten open edges and one isolated edge. Fig. 8 illustrates the segment B33 and its labeling. Note that this labeling has properties that open edges and isolated edge are labeled with 1 and every number 0,1 and 2 is used exactly 18 times. To obtain the 3-total edge product cordial labeling of Rnm for m = 1, 2, 3 and for all n we used Theorems 1–3. − → − → − → Moreover, we will use the segments A33 , A31 and A32 obtained by rotation of the segments A31 , A32 and A33 in a clockwise direction through the angle 90◦ about its center. Depending upon the values of m and n we have the following cases: Case 1. When m = 3t, t ≥ 1. First we glue the segment B33 vertically t −1 times. Since the open edges in the segment B33 are labeled with number 1 it follows that by gluing these segments we do not change the vertex labels in the segment B33 ⊕v B33 ⊕v · · · ⊕v B33 = − → (n − 1)B33 . Then we glue in the vertical direction the segment A33 to the segment (n − 1)B33 to obtain the segment A. Hence ⎡ ⎤ (t − 1)B33 ⎥ ⎢ A = ⎣⊕v ⎦. − →3 A3 Note that open edges of the segment have label 1 and each number 0,1 and 2 has multiplicity 18t + 1. Subcase i. When n = 3s, s ≥ 1. We glue the segment A horizontally s − 1 times and moreover, we glue horizontally the segment R3m and we obtain [ ] Rnm = R3m ⊕h (s − 1)A . Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Subcase ii. When n = 3s + 1, s ≥ 1. The 3-total edge product cordial labeling of rhombic grid Rnm is obtained as [ ] Rnm = R1m ⊕h s A . Subcase iii. When n = 3s + 2, s ≥ 1. We glue the segment A horizontally s times and moreover, we glue horizontally the segment R2m hence [ ] Rnm = R2m ⊕h s A . Case 2. When m = 3t + 1, t ≥ 1. − → The segment B is obtained by gluing vertically the segment A31 with vertically t times of the segment B33 , i.e. ⎡− →3 ⎤ A ⎢ 1 ⎥ B = ⎣⊕v ⎦ . t B33 Note that open edges of the segment have label 1 and each number 0,1 and 2 has multiplicity 18t + 7. Subcase i. When n = 3s, s ≥ 1. The rhombic grid Rnm is obtained as [ ] Rnm = R3m ⊕h (s − 1)B . Subcase ii. When n = 3s + 1, s ≥ 1. The 3-total edge product cordial labeling of rhombic grid Rnm can be obtained as [ ] Rnm = R1m ⊕h s B . Subcase iii. When n = 3s + 2, s ≥ 1. We glue the segment A3t+1 horizontally s times and moreover, we glue horizontally the segment R2m and we obtain 3 [ ] Rnm = R2m ⊕h s B . Case 3. When m = 3t + 2, t ≥ 1. − → The segment C is obtained by gluing vertically the segment A32 with vertically t times of the segment B33 , i.e. ⎡− →3 ⎤ A ⎢ 2 ⎥ C = ⎣⊕v ⎦ . t B33 Note that open edges of the segment has label 1 and each number 0,1 and 2 have multiplicity 18t + 13. Subcase i. When n = 3s, s ≥ 1. To obtain the rhombic grid, we glue the segment A3t+2 horizontally s − 1 times and then we glue the resulting 3 segment horizontally to R3m . Hence [ ] Rnm = R3m ⊕h (s − 1)C . Subcase ii. When n = 3s + 1, s ≥ 1. The 3-total edge product cordial labeling of rhombic grid Rnm can be obtained as [ ] Rnm = R1m ⊕h sC . Subcase iii. When n = 3s + 2, s ≥ 1. The segment Rnm is obtained by gluing the segment A3t+2 horizontally s times and the resulting segment is glued 3 horizontally to R2m and we obtain [ ] Rnm = R2m ⊕h sC . Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.
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Table 4 Multiplicity of 0s, 1s and 2s used in the rhombic grid Rnm for m, n ≥ 1. Case
Grid Rnm
eχ (0) + vχ ∗ (0)
1.
m = 3t, t ≥ 1
d = 18st + s + t
i. ii. iii.
n = 3s, s ≥ 1 n = 3s + 1, s ≥ 1 n = 3s + 2, s ≥ 1
d d + 6t d + 12t + 1
2.
m = 3t + 1, t ≥ 1
d = 18st + 7s + t
i. ii. iii.
n = 3s, s ≥ 1 n = 3s + 1, s ≥ 1 n = 3s + 2, s ≥ 1
d d + 6t + 3 d + 12t + 5
3.
m = 3t + 2, t ≥ 1
d = 18st + 13s + t
i. ii. iii.
n = 3s, s ≥ 1 n = 3s + 1, s ≥ 1 n = 3s + 2, s ≥ 1
d +1 d + 6t + 5 d + 12t + 9
eχ (1) + vχ ∗ (1)
eχ (2) + vχ ∗ (2)
d d + 6t + 1 d + 12t
d d + 6t d + 12t + 1
d +1 d + 6t + 2 d + 12t + 5
d d + 6t + 3 d + 12t + 5
d d + 6t + 5 d + 12t + 9
d +1 d + 6t + 5 d + 12t + 10
All possible cases for obtaining the grid Rmn for m, n ≥ 1, are described in Table 4, where it is shown how many times the numbers 0,1 and 2 are used as edge and vertex labels. One can see that the resulting grid Rmn in each previous cases satisfies the property of having a 3-total edge product cordial labeling. 3. Future work The rhombic grid graph can be embed on cylinder and torus. So, in future with the help of this work one can find the 3-total edge product cordial labeling of rhombic grid graph than can embed on cylinder and torus. Acknowledgments The authors would like to thank the reviewers for their valuable comments and suggestions to improve the quality of the paper. References [1] D.B. West, Introduction To Graph Theory, second ed., Prentice-Hall, New Jersey, USA, 2003. [2] I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin. 23 (1987) 201–207. [3] A. Ahmad, M. Baˇca, M. Naseem, A. Semaniˇcová, On 3-total edge product cordial labeling of honeycomb, AKCE Int. J. Graphs Combin. 14 (2017) 149–157. [4] I. Cahit, On cordial and 3-equitable labelings of graphs, Util. Math. 37 (1990) 189–198. [5] M. Hovey, A-cordial graphs, Discrete Math. 93 (1991) 183–194. [6] Y.S. Ho, S.M. Lee, S.C. Shee, Cordial labelings of the Cartesian product and composition of graphs, Ars Combin. 29 (1990) 169–180. [7] Y.S. Ho, S.C. Shee, The cordiality of one-point union of n copies of a graph, Discrete Math. 117 (1993) 225–243. [8] W.W. Kirchherr, On the cordiality of some specific graphs, Ars Combin. 31 (1991) 127–137. [9] D. Kuo, G.J. Chang, Y.H.H. Kwong, Cordial labeling of m K n , Discrete Math. 169 (1997) 121–131. [10] S.M. Lee, A. Liu, A construction of cordial graphs from smaller cordial graphs, Ars Combin. 32 (1991) 209–214. [11] M. Sundaram, R. Ponraj, S. Somasundaram, Product cordial labeling of graphs, Bull. Pure Appl. Sci. Sect. E Math. Stat. 23 (2004) 155–163. [12] S.K. Vaidya, C.M. Barasara, Edge product cordial labeling of graphs, J. Math. Comput. Sci. 2 (5) (2012) 1436–1450. [13] S.K. Vaidya, C.M. Barasara, Total edge product cordial labeling of graphs, Malay. J. Mat. 3 (1) (2013) 55–63. [14] A. Azaizeh, R. Hasni, A. Ahmad, G.C. Lau, 3-total edge product cordial labeling of graphs, Far East J. Math. Sci. 96 (2) (2015) 193–209. [15] J. Ivanˇco, On k-total edge product cordial graphs, Aust. J. Combin. 67 (3) (2017) 476–486. [16] J.C. Bermond, F. Comellas, D.F. Hsu, Distributed loop computer networks: a survey, J. Parallel Distrib. Comput. 24 (1995) 2–10. [17] P. Andersen, C. Grigorious, M. Miller, Minimum weight resolving sets of grid graphs, arXiv:1409.4510v1 [math.CO]. [18] M. Baˇca, Y. Lin, M. Miller, Antimagic labelings of grids, Util. Math. 72 (2007) 65–75. [19] M.J. Lee, W.H. Tsai, C. Lin, Super (a,1)-cycle-antimagic labeling of the grid, Ars Combin. 112 (2013) 3–12.
Please cite this article in press as: A. Javed, M.K. Jamil, 3-total edge product cordial labeling of rhombic grid, AKCE International Journal of Graphs and Combinatorics (2018), https://doi.org/10.1016/j.akcej.2017.12.002.