SOME 4-TOTAL MEAN CORDIAL GRAPHS DERIVED FROM WHEEL

In this paper we investigate the 4-total mean cordial labeling behaviour of helm, closed helm, flower graph, sunflower graph, gear graph, subdivision of wheel, web graph.


INTRODUCTION
Graphs in this paper are finite, simple and undirected. k-total mean cordial labeling of graphs have been introduced in [3] and they investigate the 4-total mean cordial labeling behaviour of path, cycle, star, bistar, wheel, subdivision of star, subdivision of bistar, subdivision of comb, subdivision of crown, subdivision of doublecomb, subdivision of jellyfish, subdivision of ladder, subdivision of triangular snake in [3,4]. In this paper, we investigate the 4-total mean cordial labeling behaviour of helm, closed helm, flower graph, sunflower graph, gear graph, subdivision of wheel, web graph. Terms are not defined here follow from Harary [2] and Gallian [1].

PRELIMINARIES
Definition 3.1. The graph W n = C n + K 1 is called a wheel, where C n is the cycle u 1 u 2 . . . u n u 1 , Definition 3.5. The sunflower graph SF n is obtained by taking a wheel W n = C n + K 1 where C n is the cycle u 1 u 2 . . . u n u 1 , V (K 1 ) = {u} and new vertices v 1 ,v 2 ,. . .,v n where v i is join by the vertices u i u i+1 (mod n).
A 4-total mean cordial labeling is given in Table 2 n Theorem 4.2. The closed helm CH n is 4-total mean cordial for all values of n.
Proof. Take the vertex set and edge set of CH n as in definition 3.3. Clearly, |V (CH n )| + Assign the label 0 to the central vertex u.
Finally we assign the label 0 to the vertex v 4r+3 .
Let n = 4r + 1, r ∈ N. Assign the labels to the vertices u i ,v i (1 ≤ i ≤ 4r) as in case 1. Next we assign the labels 3,0 respectively to the vertices u 4r+1 ,v 4r+1 .
A 4-total mean cordial labeling is given in Table 6 n u u 1 u 2 u 3 v 1 v 2 v 3 n = 3 2 0 0 3 0 1 3  TABLE 6 Theorem 4.4. The sunflower graph SF n is 4-total mean cordial, for all n.
Proof. Take the vertex set and edge set of SF n as in definition 3.5. Clearly that |V (SF n )| + |E (SF n )| = 6n + 1. Assign the label 0 to the central vertex u.
Case 1. n is even.
We consider the vertices u 1 ,u 2 ,. . .,u n . Assign the label 0 to the n 2 vertices u 1 ,u 2 ,. . .,u n 2 . Then we assign the label 1 to the n 2 vertices un+2  Theorem 4.5. The Gear graph G n is 4-total mean cordial for every n.
Proof. Take the vertex set and edge set of the wheel W n as in definition 3.1. Let v i be the vertex which subdivide the edge u i u i+1 (1 ≤ i ≤ n − 1) and v n be the vertex which subdivide the edge u n u 1 . Clearly |V (G n )| + |E (G n )| = 5n + 1.
Assign the label 2 to the central vertex u.
The Table 8, establish that this vertex labeling f is a 4-total mean cordial labeling of gear G n .
A 4-total mean cordial labeling for this case is given in Table 9 Value n u u 1 u 2 u 3 v 1 v 2 v 3 n = 3 2 0 1 3 0 0 3  TABLE 9 Theorem 4.6. The subdivision of the wheel W n , S (W n ) is 4-total mean cordial for all values of n.
Finally we assign the label 1 to the vertex y 4r .
This vertex labeling f is a 4-total mean cordial labeling follows from the Tabel 10 Case 5. n = 3.
A 4-total mean cordial labeling for this case is given in Table 11 Theorem 4.7. The web graph W b n is 4-total mean cordial, for all n.
Assign the label 2 to the central vertex u.
We now consider the vertices u 1 ,u 2 ,. . .,u n . Assign the label 0 to the n vertices u 1 ,u 2 ,. . .,u n . Then we consider the vertices v 1 ,v 2 ,. . .,v n . We now assign the label 2 to the n vertices v 1 ,v 2 ,. . .,v n . We now move to the vertices x 1 ,x 2 ,. . .,x n . Finally we assign the label 3 to the n vertices x 1 ,x 2 ,. . .,x n .