Cordial and 3-Equitable Labeling for Some Shell Related Graphs

We present here cordial and 3-equitable labeling for the graphs obtained by joining apex vertices of two shells to a new vertex. We extend these results for k copies of shells.


I. INTRODUCTION
W E begin with simple, finite and undirected graph G = (V, E). In the present work W n = C n + K 1 (n ≥ 3) denotes the wheel and in W n vertices correspond to C n are called rim vertices and vertex which corresponds to K 1 is called an apex vertex. For all other terminology and notations we follow Harary [7]. We will give brief summary of definitions which are useful for the present investigations. (2) n then G =< W (1) n : W (2) n > is the graph obtained by joining apex vertices of wheels to a new vertex x. Note that G has 2n + 3 vertices and 4n + 2 edges.

Definition
1.2 Consider k copies of wheels namely W (1) n , W (2) n , W According to Hegde [8] most interesting graph labeling problems have following three important characteristics.
1) a set of numbers from which the labels are chosen; 2) a rule that assigns a value to each edge; 3) a condition that these values must satisfy. The recent survey on graph labeling can be found in Gallian [6]. Vast amount of literature is available on different types of graph labeling. According to Beineke and Hegde [2] graph labeling serves as a frontier between number theory and structure of graphs.
Labeled good radar type codes and convolution codes with optimal autocorrelation properties. Labeled graph plays vital role in the study of X-Ray crystallography, communication network and to determine optimal circuit layouts. A detailed study on variety of applications of graph labeling is carried out in Bloom and Golomb [3].
For an edge e = uv, the induced edge labeling be the number of vertices of G having labels 0 and 1 respectively under f and let e f (0),e f (1) be the number of edges having labels 0 and 1 respectively under f * .
The concept of cordial labeling was introduced by Cahit [4].
Many researchers have studied cordiality of graphs. e.g.Cahit [4] proved that tree is cordial. In the same paper he proved that K n is cordial if and only if n ≤ 3. Ho et al. [9] proved that unicyclic graph is cordial unless it is C 4k+2 while Andar et al. [1] have discussed cordiality of multiple shells. Vaidya et al. [10], [11], [12] have also discussed the cordiality of various graphs.
The concept of 3-equitable labeling was introduced by Cahit [5] and in the same paper he proved that Eulerian graphs with number of edges congruent to 3(mod6) are not 3-equitable. Youssef [17] proved that W n is 3-equitable for all n ≥ 4. Several results on 3-equitable labeling for some wheel related graphs in the context of vertex duplication are reported in Vaidya et al. [13].
In the present investigations we prove that graphs < W (1) n : W (2) n > and < W (1) n : W
The labeling pattern defined above covers all possible arrangement of vertices. The graph G satisfies the vertex condition v f (0) + 1 = v f (1) and edge condition e f (0) = e f (1). i.e. G admits cordial labeling.

>.
Here n = 6. The cordial labeling is as shown in Figure 1.
In this case we define labeling function f for first k − 1 wheels as Fig For i = 1, 2, . . . n and j = 1, 2, .
To define labeling function f for k th copy of wheel we consider following subcases The labeling pattern defined above exhaust all the possibilities and in each one the graph G under consideration satisfies the conditions |v f (0) − v f (1)| ≤ 1 and |e f (0) − e f (1)| ≤ 1 as shown in Table 1. i.e. G admits cordial labeling.
(In Table 1  >. Here n = 7 and k = 4 i.e k is even. The cordial labeling is as shown in Figure 2. >. Here n = 5 i.e n ≡ 1(mod4) and k = 3 i.e k is odd. The cordial labeling is as shown in Figure 3.
3 , . . . v (2) n be the rim vertices W  Case 1: n ≡ 0(mod6) In this case we define labeling f as: In this case we define labeling f as: In this case we define labeling f as: Case 4: n ≡ 3(mod6) Subcase 1: n = 3 In this case we define labeling f as: 3 ) = f (x) = 2; Case 5: n ≡ 4(mod6) In this case we define labeling f as: In this case we define labeling f as: The labeling pattern defined above covers all the possible arrangement of vertices and in each case the resulting labeling satisfies the conditions |v Table 2. i.e. G admits 3-equitable labeling.
(In Table 2 n = 6a + b and a ∈ N ∪ {0}) Let us understand the labeling pattern defined in Theorem 2.5 by means of following Illustration 2.6.
Here n = 5 i.e n ≡ 5(mod6). The corresponding 3-equitable labeling is shown in Figure 4. Case 1: For n ≡ 0(mod6). In this case we define labeling function f as follows Subcase 1: For k ≡ 0(mod3). For For remaining vertices take j = k − 1 and label them as in subcase 1.
For remaining vertices take j = k − 2 and label them as in subcase 1. Case 2: For n ≡ 1(mod6). In this case we define labeling function f as follows For remaining vertices take j = k − 1 and label them as in subcase 1.
For remaining vertices take j = k − 2 and label them as in subcase 1.
For remaining vertices take j = k − 2 and label them as in subcase 1.
For remaining vertices take j = k − 2 and label them as in subcase 1.
The labeling pattern defined above covers all possible arrangement of vertices. In each case, the graph G under consideration satisfies the conditions |v f (i) − v f (j)| ≤ 1 and |e f (i) − e f (j)| ≤ 1 for all 0 ≤ i, j ≤ 2 as shown in Table 3. i.e. G admits 3-equitable labeling.
(In Table 3 n = 6a + b and k = 3c + d where a ∈ N ∪ {0},c ∈ N ) The labeling pattern defined above is demonstrated by means of following Illustration 2.8. >. Here n = 6 and k = 4. The corresponding 3-equitable labeling is as shown in Figure 5.

III. CONCLUDING REMARKS
Cordial and 3-equitable labeling of some star and shell related graphs are reported in Vaidya et al. [14], [15] while the present work corresponds to cordial and 3-equitable labeling of some wheel related graphs. Here we provide cordial and 3-equitable labeling for the larger graphs constructed from the standard graph.

Further scope of research
• Similar investigations can be carried out in the context of different graph labeling techniques and for various standard graphs. • Cordial labeling in the context of arbitrary super subdivision of graphs is discussed in Vaidya and Kanani [16].
In likeway all the results reported here can be discussed in the context of arbitrary super subdivision of graphs.