Cordial, Total Cordial, Edge Cordial, Total Edge Cordial Labeling of Some Box Type Fractal Graphs

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1 International Journal of Algebra and Statistics Volume 1: 2(2012), Published by Modern Science Publishers Available at: Cordial, Total Cordial, Edge Cordial, Total Edge Cordial Labeling of Some Box Type Fractal Graphs A. A. Sathakathulla a, Muhammad Akram a, P. G. Rajeswari a a Information Technology Department, Higher College of Technology, Muscat, Oman. (Received: 10 July 2012; Accepted: 2 August 2012) Abstract. This paper deals with the concept of self-similarity fractals of three types of box type fractal with existence of cordial and Edge cordial labeling. A square graph is considered as base for square fractals which leads to construction of some box type fractals. For our study each iteration and the generalized form are considered as graph. Eventually each graph is checked with cordial, edge cordial and total cordial, total edge cordial labeling. 1. Introduction A Graph G =< V, E, ψ > consists of a non empty set V called the set of nodes (points, vertices) of the graph, E is said to be the set of edges (may be empty) of the graph and ψ is the mapping from the set of edges E to a set of ordered or unordered pair of elements of V. It would be convenient to write a graph G as < V, E > or simply as G. A graph labeling is an assignment of integers to the vertices or edges, or both subject to certain conditions. Many types of labeling like harmonious, graceful, etc. are used by various researchers [1 3] in practice. A graph G with q edges is harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f (x) + f (y) (modq), the resulting edge labels are distinct. A graph G with q edges is graceful if f is an injection from the vertices of G to the set f : V {0, 1,..., q} such that, when each edge xy is assigned the label f (x) f (y), the resulting edge labels are distinct. Eventually after the introduction of the concept of cordial labeling by (Cahit I.,1987, p ) many researchers have investigated graph families or graphs which admit cordial labeling with minor variations in cordial theme like product cordial labeling, total product cordial labeling and prime cordial labeling (Harary F., 1972). The brief summary of definitions which are useful for the present investigations are given below. Definition 1.1. If the vertices of the graph are assigned values subject to certain conditions then it is known as graph labeling. For a dynamic survey on graph labeling we refer to (Gallian J., 2009). A detailed study on variety of applications of graph labeling is reported in (Bloom G. S., 1977, p ) Mathematics Subject Classification. 28A80, 05C78. Keywords. Fractals; cordial; Edge cordial labeling; square graphs; Vicsek fractal. address: makram_69@yahoo.com (Muhammad Akram)

2 A. A. Sathakathulla et al. / Int. J. of Algebra and Statistics 1 (2012), Definition 1.2. Let G be a graph. A mapping f : E(G) {0, 1} is called a binary egde labeling of G and f (e).it is called the label of the edge e of G under f. For an edge e = uv, the induced edge labeling f : E(G) {0, 1} is given by f (e) = f (u) f (v). Let v f (0) and v f (1) be the number of vertices of G having labels 0 and 1 respectively under f while e f (0), e f (1) be the number of edges having labels 0 and 1 respectively under f. Definition 1.3. A binary vertex labeling of a graph G is called a cordial labeling if e f (0) e f (1) 1. A graph G is cordial if it admits cordial labeling. v f (0) v f (1) 1and Definition 1.4. Let G be a graph with two or more vertices then the total graph T(G)of a graph G is the graph whose vertex set is V(G) E(G) and two vertices are adjacent whenever they are either adjacent or incident in G. Definition 1.5. A binary edge labeling of a graph G is called a edge cordial labeling if v f (0) v f (1) 1and e f (0) e f (1) 1. A graph G is edge cordial if it admits cordial labeling Definition 1.6. Yilmaz and Cahit [9] introduced edge-cordial labeling as a binary edge labeling f : E(G) {0, 1}, with the induced vertex labeling given by f (v) = uv E f (uv)(mod2) for each v V, such that e f (0) e f (1) 1 and v f (0) v f (1) 1, where e f (i)and v f (i), i = 0, 1 denote the number of edges and vertices labeled with 0 and 1 respectively. Definition 1.7. As an extension of the above, we define a total edge-cordial labeling of a graph G with vertex set V and edge set E as an edge-cordial labeling such that number of vertices and edges labeled with 0 and the number of vertices and edges labeled with 1 differ by at most 1 (i.e). A graph with a total edge-cordial labeling is called a total edge-cordial graph. For our study, we have considered three types of fractals namely Vicsek fractals two types and the third one is similar to Greek cross fractal but has the same base as of Vicsek fractals. The existence of cordial, total cordial, edge cordial, and total edge cordial are discussed. 2. Main results A fractal [6] on all scales is an object or quantity that displays self-similarity in a somewhat technical sense. The object need not exhibit exactly the same structure at all scales, but the same type of structures must appear on all scales Vicsek fractals (saltire form) In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal. The Box Fractal can be formed using the cluster fractal generation method or generator iteration. The basic square is decomposed into nine smaller squares in the 3-by-3 grid. The four squares at the corners are left and the middle squares are the other squares being removed. The process is repeated recursively for each of the five remaining subsquares. The Vicsek fractal is the set obtained at the limit of this procedure. It has applications in compact antennas, particularly in cellular phones. Figure 2.1. Construction of Vicsek fractal (saltire form)

3 A. A. Sathakathulla et al. / Int. J. of Algebra and Statistics 1 (2012), Figure 2.2. Cordial labeling of Vicsek fractal (saltire form) The pattern of labeling of vertices are easily understandable by seeing the above figure(fig 2.2). Vertically all vertices in column wise labeled with 0 s and 1 s alternatively. Hence, if one column is labeled with 0 s then the next column, all vertices are labeled with 1 s. It is clearly observed that all vertical edges are having label 0 s are denoted by a tick mark ( ) and correspondingly all horizontal edges are labeled with 1 sdenoted by a cross mark(x). Hence it satisfies existence of cordial labeling. In each iteration, the similar fashion of labeling can be followed to check the existence of cordial labeling that it satisfies the condition of total cordial labeling also. Hence, the above fractal holds good both cordial and total cordial labeling. The iteration wise number of edges and vertices are provided in the Table 2.1. All values hold good for the said definitions. Figure 2.3. Edge Cordial labeling of Vicsek fractal (saltire form) The pattern of labeling of vertices are easily understandable by seeing the below figure(fig 2.3). Initially a single square is labeled with two zeros and two ones adjacent to each other and it satisfies edge cordial labeling definition. Then for second iteration, to satisfy condition for vertices, edges are labeled with zeros and once as given in the figure. Hence for the third iteration the copy of second iteration is retained at two portions and for the remaining portion the labels are done in accordance to satisfy the definition.the same pattern of labeling can be extended further. The vertices are labeled with label 0 s denoted by a tick

4 A. A. Sathakathulla et al. / Int. J. of Algebra and Statistics 1 (2012), mark( ) and correspondingly 1 sdenoted by a cross mark(x). In each iteration, it satisfies the condition of edge cordial labeling, total cordial labeling. The results are provided in the following table to hold hood. Table 2.1 Iterations No.ofSquares Vertices Edges 1 1 v f (0) = v f (1) = 2 e f (0) = e f (1) = v f (0) = v f (1) = 8 e f (0) = e f (1) = v f (0) = v f (1) = 38 e f (0) = e f (1) = v f (0) = v f (1) = 188 e f (0) = e f (1) = 250 n 5 (n 1) v f (0) = v f (1) = 2 + (3(5(n 1) 1)) 2 e f (0) = e f (1) = 2(5 (n 1) ) 2.2. Vicsek fractals (cross form) Construction of Vicsek fractal (cross form) is also starts as similar to saltire form with single square using the cluster fractal generation method.the basic square is decomposed into nine smaller squares in the 3-by-3 grid. The four squares at the corners are removed and the middle squares are left. The process is repeated recursively for each of the five remaining subsquares. The Vicsek fractal is the set obtained at the limit of this procedure Figure 2.4. Construction of Vicsek fractal (cross form) The pattern of labeling of vertices are easily understandable by seeing the above figure(fig 2.5). Similar to previous saltire form, vertically all vertices in column wise labeled with 0 s and 1 s alternatively. Hence, if one column is labeled with 0 s then the next column all vertices are labeled with 1 s. It is clearly observed that all vertical edges are having label 0 s denoted by a tick mark( ) and correspondingly all horizontal edges are labeled with 1 sdenoted by a cross mark(x). Hence it satisfies existence of cordial labeling. In each iteration, the similar fashion of labeling can be followed to check the existence of cordial labeling that satisfies the condition of total cordial labeling. Hence, the above fractal holds good both cordial and total cordial labeling. The iteration wise no of edges and vertices are provided in the Table 2.2. All values hold good for the said definitions.

5 A. A. Sathakathulla et al. / Int. J. of Algebra and Statistics 1 (2012), Figure 2.5. Cordial labeling of Vicsek fractal (cross form) The pattern of labeling of edges are easily understandable by seeing the above figure(fig 2.6). Initially a single square is labeled with two zeros and two ones adjacent to each other and it satisfies edge cordial labeling definition. Then for second iteration, to satisfy condition for vertices, edges are labeled with zeros and once as given in the figure. Hence for the third iteration the copy of second iteration is retained at two portions and for the remaining portion the labels are done in accordance to satisfy the definition. The same pattern of labeling can be extended further. The vertices are labeled with label 0 s denoted by a tick mark( ) and correspondingly 1 s denoted by a cross mark(x). In each iteration, it satisfies the condition of edge cordial labeling, total edge cordial labeling. The results provided in the following table are holds good. Figure 2.6. Edge labeling of Vicsek fractal (cross form)

6 A. A. Sathakathulla et al. / Int. J. of Algebra and Statistics 1 (2012), Table 2.2 Iterations No. of Squares Vertices Edges 1 1 v f (0) = v f (1) = 2 e f (0) = e f (1) = v f (0) = v f (1) = 6 e f (0) = e f (1) = v f (0) = v f (1) = 26 e f (0) = e f (1) = v f (0) = v f (1) = 126 e f (0) = e f (1) = v f (0) = v f (1) = 626 e f (0) = e f (1) = 938 n 5 n 1 v f (0) = v f (1) = 5 n e f (0) = e f (1) = 2 + 3(5n 1 1) Box Fractal (Type-2) Construction of Box fractal is also starts with a single square in the first iteration. Then each side of the square is added with another square, it leads to plus type five squares. Similarly each side is added again with square and the same fashion is extended. This same construction can be done through cluster fractal generation method by removing some squares in order. Figure 2.7 Construction of Box fractal (Type -2) The pattern of the labeling is easily understandable by seeing the figure (Fig 2.8). Each alternate row is labeled with 1 s and 0 s. Correspondingly the edges are labeled with 1 s and 0 s as per definition. The edges with label 0 s denoted by a tick mark( ) and correspondingly 1 s denoted by a cross mark(x). For every iteration same pattern is extended. The results are provided in the following Table 2.3 hold good for cordial labeling and total cordial labeling. Figure 2.8. Cordial labeling of Box fractal (Type -2) The pattern of labeling of edges are easily understandable by seeing the below figure(fig 2.9). As previous, initially a single square is labeled with two zeros and two ones adjacent to each other and it satisfies edge

7 A. A. Sathakathulla et al. / Int. J. of Algebra and Statistics 1 (2012), cordial labeling definition. Then for second iteration, to satisfy condition for vertices, edges are labeled with zeros and ones as given in the figure. Similarly, it is extended for third iteration and so on.. The vertices are labeled with 0 s denoted by a tick mark( ) and correspondingly 1 s denoted by a cross mark(x). In each iteration, it satisfies the condition of edge cordial labeling, total edge cordial labeling. The results are provided in the following table holds good. Figure 2.9. Edge Cordial labeling of Box fractal(type -2) Table 2.3 Iterations No. of Squares Vertices Edges 1 1 v f (0) = v f (1) = 2 e f (0) = e f (1) = v f (0) = v f (1) = 6 e f (0) = e f (1) = v f (0) = v f (1) = 12 e f (0) = e f (1) = v f (0) = v f (1) = 20 e f (0) = e f (1) = v f (0) = v f (1) = 30 e f (0) = e f (1) = 50 n n 2 + (n 1) 2 v f (0) = v f (1) = n(n + 1) e f (0) = e f (1) = 2n 2 3. Conclusion In this paper we have constructed three types of box fractal graphs. For each type of graph, cordial labeling total cordial labeling, Edge cordial labeling and Total edge cordial labeling are applicable in each iterations. It is checked and proved for iteration wise and results are provided as detailed in the tables. Hence, it is concluded that the said three types of box fractal graphs are cordial, total cordial, Edge cordial and total edge cordial. References [1] G. S. Bloom and S. W. Golomb, Applications of numbered undirected graphs, Proc of IEEE, 65(4): , [2] I. Cahit, Cordial Graphs: A weaker version of graceful and harmonious Graphs, Ars Combinatoria, 23: , [3] J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 16, DS 6, [4] F. Harary, Graph Theory, Massachusetts, Addison Wesley, 1972.

8 A. A. Sathakathulla et al. / Int. J. of Algebra and Statistics 1 (2012), [5] G. James, Chaos, Vintage Publishers, [6] M. Barnsley, Fractals Everywhere, Academic Press Inc., [7] M. Seoud and A. E. I. Abdel Maqsoud, On cordial and balanced labelings of graphs, J. Egyptian Math. Soc., 7: , [8] R. Devaney and L. Keen, eds., Chaos and Fractals: The Mathematics Behind the Computer Graphics, American Mathematical Society, Providence, RI, [9] M. Sundaram, R. Ponraj and S. Somasundram, Prime Cordial Labeling of graphs, J. Indian Acad. Math.,27(2): ,2005. [10] R. Venkatesan, A.A. Sathakathulla and A. Saibulla, On Edge-Cordial and Total Edge-CordialLabeling of Path, Cycle and Koch Snowflake Graphs, paper presented in the international conference on ICETMCA, Dec 16-18, 2010.