4-TOTAL DIFFERENCE CORDIAL LABELING OF CORONA OF SNAKE GRAPHS WITH K1

Let G be a graph. Let f : V (G) → {0,1,2, . . . ,k − 1} be a map where k ∈ N and k > 1. For each edge uv, assign the label | f (u)− f (v)|. f is called k-total difference cordial labeling of G if ∣ ∣td f (i)− td f ( j) ∣ ∣ ≤ 1, i, j ∈{0,1,2, . . . ,k−1}where td f (x) denotes the total number of vertices and the edges labeled with x. A graph with admits a k-total difference cordial labeling is called k-total difference cordial graphs. In this paper we investigate the 4-total difference cordial labeling behaviour of corona of snake graphs with K1.


INTRODUCTION
All graphs in this paper are finite, simple and undirecte. The k-total difference cordial graph was introduced in [3]. In [3,4], 3-total difference cordial labeling behaviour of path, complete graph, comb, armed crown, crown, wheel, star etc have been investigated . Also 4-total difference cordial labeling of path, star , bistar, comb, crown, P n ∪ K 1,n , S(P n ∪ K 1,n ), P n ∪ B n,n etc.,have been invetigated [5]. In this paper we investigate 4-total difference of cordial labeling of Corona of triangular snake and quadrilateral snake graphs with K 1 . Definition 2.2. The Triangular snake T n is obtained from the path P n : u 1 u 2 . . . u n with V (T n ) =

PRELIMINARIES
Definition 2.3. The Quadrilateral snake Q n is obtained from the path P n : Definition 2.4. The The Alternate triangular snake of A(T n ) is obtained from the path P n : u 1 u 2 . . . u n by joining u i and u i+1 (alternatively) to the vertex v i . That is every alternate edge of a path is replaced by C 3 .
Definition 2.5. Let G 1 , G 2 respectively be p 1 , q 1 ,p 2 , q 2 graphs. The corona of G 1 with G 2 ,G 1 ⊙ G 2 is the graph is obtained by taking one copy of G 1 and p 1 copies of G 2 and joining the i th vertex of G 1 with an edge to every vertex in the i th copy of G 2 .

MAIN RESULTS
Theorem 3.1. The corona of triangular snake T n with K 1 , T n ⊙ K 1 is 4-total difference cordial.
Next assign the labels 2, 3, 2 and 1 to the vertices x 3 , x 4 , x 5 and x 6 . Assign the labels 2, 3, 2 and 1 to the next four vertices x 7 , x 8 , x 9 and x 10 . Proceeding in this way until we reach the vertex x n−1 . Clearly the vertex x n−1 receive the label 2 when n ≡ 1, 3 (mod 4) and 3 or 1 according as n ≡ 0 (mod 4) or n ≡ 2 (mod 3).
Next assign the labels 1, 3, 3 and 3 to the vertices y 3 , y 4 , y 5 and y 6 . Assign the labels 1, 3, 3 and 3 to the next four vertices y 7 , y 8 , y 9 and y 10 . Proceeding like this until we reach the vertex y n . Clearly the vertex y n receive the label 3 or 1 when n ≡ 0, 1, 2 (mod 4) or n ≡ 3 (mod 4).
Case 2. n ≤ 3. Table 1 gives a 4-total difference cordial labeling for this case.
The table 2 shows that this vertex labeling is a 4-total difference cordial labeling.   Theorem 3.2. The corona of quadrilateral snake Q n with K 1 , Q n ⊙ K 1 is 4-total difference cordial.
Proof. Take the vertex set and edge set of Q n as in definition 2.2. Let x i be the pendent vertices adjacent to v i and z i be the pendent vertices adjacent to Assign the label 3 to the all the path vertices u 1 u 2 . . . u n . Next assign the labels 3, 3, 1 and 1 to the vertices v 1 , v 2 , v 3 and v 4 . Assign the labels 3, 3, 1 and 1 to the vertices v 5 , v 6 , v 7 and v 8 .
Continue in this pattern until we reach the vertex v n−1 . Clearly the vertex v n−1 receive the label 3 or 1 according as n ≡ 1, 2 (mod 4) or n ≡ 0, 3 (mod 4).
We now consider the vertices w i . Assign the labels 3, 3, 1 and 2 to the vertices w 1 , w 2 , w 3 and w 4 . Next assign the labels 3, 3, 1 and 2 to the vertices w 5 , w 6 , w 7 and w 8 . Proceeding like this until we reach the vertex w n−1 . Clearly the vertex w n−1 receive the label 3 when n ≡ 1, 2 (mod 4) and 1 or 2 when n ≡ 0, 3 (mod 4).
Now we consider the vertices x i . Assign the labels 1, 1, 2 and 3 to the vertices x 1 , x 2 , x 3 and x 4 . Next assign the labels 1, 1, 2 and 3 to the vertices x 5 , x 6 , x 7 and x 8 . Proceeding like this until we reach the vertex x n−1 . Clearly the vertex x n−1 receive the label 1 when n ≡ 1, 2 (mod 4) and 2 or 3 when n ≡ 3, 0 (mod 4).
We now move to the vertices z i . Assign the labels 1, 1, 3 and 3 to the vertices z 1 , z 2 , z 3 and z 4 . Next assign the labels 1, 1, 3 and 3 to the vertices z 5 , z 6 , z 7 and z 8 . Proceeding like this until we reach the vertex z n−1 . Clearly the vertex z n−1 receive the labels 1 or 3 according as n ≡ 1, 2 (mod 4) or n ≡ 3, 0 (mod 4).
Next we move to the pendent vertices of path. Fix the label 1 to the vertex y i . Assign the labels 1, 1, 3 and 3 to the vertices y 2 , y 3 , y 4 and y 5 . Next assign the labels 1, 1, 3 and 3 to the vertices y 6 , y 7 , y 8 and y 9 . Proceeding like this until we reach the vertex y n . Clearly the vertex y n receive the label 1 or 3 according as n ≡ 2, 3 (mod 4) or n ≡ 0, 1 (mod 4).
The table 3 shows that this vertex labeling is a 4-total difference cordial labeling.   and v 2 . Fix the label 1 to the vertices x 1 , x 2 , y 1 and y 2 . Next assign the labels 2, 3, 2 and 1 to the vertices x 3 , x 4 , x 5 and x 6 . Assign the labels 2, 3, 2 and 1 to the next four vertices x 7 , x 8 , x 9 and x 10 . Continue in this pattern until we reach the vertex x n 2 . Clearly the vertex x n 2 receive the label 2 when n ≡ 1, 3 (mod 4) and 3 or 1 according as n ≡ 0 (mod 4) or n ≡ 2 (mod 3).
We now consider the vertices v i . Assign the labels 1, 2, 1 and 3 to the vertices v 3 , v 4 , v 5 and v 6 . Similarly assign the labels 1, 2, 1 and 3 to the next four vertices v 7 , v 8 , v 9 and v 10 . Continue in this pattern until we reach the vertex v n 2 . Clearly the vertex v n 2 receive the label 1 when n ≡ 1, 3 (mod 4) and 2 or 3 according as n ≡ 2 (mod 4) or n ≡ 0 (mod 3).
The table 4 shows that this vertex labeling is a 4-total difference cordial labeling.
n ≡ 0 (mod 8) 13n  TABLE 4 Case 2. The edge u 1 u 2 lies in a triangle and the edge u n−2 u n−1 lies in a triangle.In this case n is odd.
Clearly removal of the edge u n−1 u n is the graph as in case(i). Assign the label to the vertices 2 ) as in case (i). Finally assign the labels 3 and 1 respect to the vertices u n and v n .
The table 5 shows that this vertex labeling is a 4-total difference cordial labeling.  Case 3. The edge u 2 u 3 lies in a triangle and the edge u n−2 u n−1 lies in a triangle.
Obviously removal of the edge u 1 u 2 as in case(ii). Assign the label to the vertices u i (2 ≤ i ≤ n) and v i (2 ≤ i ≤ n−2 2 ) as in case (i). Next assign the labels 3 and 1 respect to the vertices u 1 and v 1 .
The table 6 shows that this vertex labeling is a 4-total difference cordial labeling.  Theorem 3.4. The corona of alternate quadrilateral snake A(Q n ) with K 1 , A(Q n ) ⊙ K 1 is 4-total difference cordial.
Proof. Take the vertex set and edge set of A(Q n ) as in definition 2.3.
Case 1. The edge u 1 u 2 lies in a Quadrilateral and the edge u n−1 u n lies in a Quadrilateral.
be the pendent vertices adjacent to w i (1 ≤ i ≤ n) and y i (1 ≤ i ≤ n) be the pendent vertices adjacent to u i 1 ≤ i ≤ n. Clearly n is even. In this case |V (A(Q n )) ⊙ K 1 | + |E(A(Q n ))| = 17n−11 2 . Assign the label 3 to the path vertices u 1 u 2 . . . u n . Next fix the label 3 to the vertices v 1 and w 1 . Fix the label 1 to the vertices x 1 , z 1 and y i (1 ≤ i ≤ n). Next assign the labels 1, 3, 3 and 3 to the vertices x 2 , x 3 , x 4 and x 5 . Assign the labels 1, 3, 3 and 3 to the next four vertices x 6 , x 7 , x 8 and x 9 . Continue in this pattern until we reach the vertex x n 2 . Clearly the vertex x n 2 receive the label 1 when n ≡ 2 (mod 4) and 3 when n ≡ 0, 1, 3 (mod 4).
We now consider the vertices z i (1 ≤ i ≤ n 2 ). Assign the labels 1, 2, 3 and 2 to the vertices z 2 , z 3 , z 4 and z 5 . Similarly assign the labels 1, 2, 3 and 2 to the next four vertices z 6 , z 7 , z 8 and z 9 . Continue in this pattern until we reach the vertex z n 2 . Clearly the vertex z n 2 receive the label 2 when n ≡ 1, 3 (mod 4) and 1 or 3 according as n ≡ 0, 2 (mod 4).

Consider the vertices
. Assign the label 3 to the vertices v 1 , v 2 , v n 2 . Next assign the labels 3, 1, 2 and 1 to the vertices w 2 , w 3 , w 4 and w 5 . Assign the labels 3, 1, 2 and 1 to the next four vertices w 6 , w 7 , w 8 and w 9 . Continue in this way until we reach the vertex w n 2 . Clearly the vertex w n 2 receive the label 1 when n ≡ 1, 3 (mod 4) and 3 or 2 when n ≡ 0, 2 (mod 4).
The table 7 shows that this vertex labeling is a 4-total difference cordial labeling.  TABLE 8 Case 3. The edge u 2 u 3 lies in a Quadrilatral and the edge u n−2 u n−1 lies in a Quadrilatral.
Obviously removal of the edge u 1 u 2 as in case(ii). Assign the label to the vertices u i (2 ≤ i ≤ n) and v i (2 ≤ i ≤ n − 1) and w i (1 ≤ i ≤ n 2 ) as in case (i). Next assign the labels 3 and 1 respect to the vertices u 1 and v 1 .
The table 9 shows that this vertex labeling is a 4-total difference cordial labeling.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.