On Square Divisor Cordial Graphs

Abstract The square divisor cordial labeling is a variant of cordial labeling and divisor cordial labeling. Here we prove that the graphs like flower , n Fl bistar , n n B , restricted square graph of , n n B , shadow graph of , n n B as well as splitting graphs of star 1,n K and bistar , n n B are square divisor cordial graphs. Moreover we show that the degree splitting graphs of , n n B and n P admit square divisor cordial labeling.


Introduction
We begin with simple, finite, connected and undirected graph ( ( ), ( )) G V G E G with p vertices and q edges.For standard terminology and notations related to graph theory we refer to Gross and Yellen [1] while for any concept of number theory we refer to Burton [2].We will provide brief summary of definitions and other information which are prerequisites for the present investigations.Definition 1.1 If the vertices are assigned values subject to certain condition(s) then it is known as graph labeling.
The graph labeling is described as a frontier between number theory and structure of graphs by Beineke and Hegde [3].There are enormous applications of graph labeling in various fields including computer science and communication networks.Yegnanaryanan and Vaidhyanathan [4] described applications of edge balanced graph labeling, edge magic labeling and (1,1) edge magic graph.For a dynamic survey of various graph labeling problems along with an extensive bibliography we refer to Gallian [5].Definition 1.2 A mapping : ( ) {0,1} f V G is called binary vertex labeling of G and () fvis called the label of the vertex v of G under f .
The concept of cordial labeling was introduced by Cahit [6].This concept has been explored by many researchers and various labeling schemes are also introduced with minor variations in cordial theme.Product cordial labeling, total product cordial labeling and prime cordial labeling and divisor cordial labeling are among mention a few.The present work is focused on square divisor cordial labeling.The concept of prime cordial labeling was introduced by Sundaram et al. [7] and in the same paper they have investigated several results on prime cordial labeling.Vaidya and Vihol [8,9] as well as Vaidya and Shah [10,11] have proved many results on prime cordial labeling.
Varatharajan et al. [12] have introduced a new concept called divisor cordial labeling by combining the divisibility of numbers and the concept of Cordial labeling.Definition 1.6 Let ( ( ), ( )) G V G E G be a simple graph and : {1, 2,... | ( ) |} f V G be a bijection.For each edge uv , assign the label 1 if ( ) | ( ) A graph with a divisor cordial labeling is called a divisor cordial graph.
In the same paper [12] they have proved that path, cycle, wheel, star, 2,n K and 3,n K are divisor cordial graphs while n K is not divisor cordial for 7 n . Same authors in [13] have discussed divisor cordial labeling of full binary tree as well as some star related graphs.Vaidya and Shah [14] have proved that some star and bistar related graphs are divisor cordial graph.Same authors [15] have shown that helm, flower and gear graphs admit divisor cordial labeling.Moreover the graphs obtained from switching of a vertex in various graphs are proved to be divisor cordial.
Motivated by the concept of divisor cordial labeling, Murugesan et al. [16] have introduced the concept of square divisor cordial labeling and many graphs are proved to be square divisor cordial graphs.Hence n Fl is a square divisor cordial graph for each n .

E G n
We define vertex labeling : ( ) {1, 2, , 2 2} f V G n as follows.Let 1 p be the highest prime number < 2n+2.B is shown in Fig. 3.
p be the highest prime number, 2 p be the second highest prime number and 3 p be the third highest prime number such that 3 ( 1); 1 ( ) 2 ; 1 ( ) 4( 1), ( ) 4 , For the vertices 3 4  , , , , n v v v v be the pendant vertices and v be the apex vertex of 1, n K and , , , , , n u u u u u are added vertices corresponding to

Concluding Remarks
The square divisor cordial labeling is a labeling with the blend of cordial and divisor cordial labelings.As all the graphs do not admit square divisor cordial labeling, it is very interesting to find out graph or graph families which are square divisor cordial graphs.Here we contribute some new results and many graphs are proved to be square divisor cordial graphs.

Notation 1 . 3
If for an edge e uv , the induced edge labeling * : ( ) {0,1} f of vertices of having label under ( ) number of vertices of having label under * In view of the above labeling pattern we have, (

4 :
of the above labeling pattern we have, Square divisor cordial labeling of the graph 8,8

Illustration 2 . 8 :
In view of the above defined labeling pattern we have, nn DB is a square divisor cordial graph.Square divisor cordial labeling of the graph 2 5,5 () DB is shown in Fig.4.

Illustration 2 . 10 :
In view of the above defined labeling pattern we have, 22 Square divisor cordial labeling of the graph

Illustration 2 . 12 :
Square divisor cordial labeling of the graph

Illustration 2 . 16 :
Square divisor cordial labeling of the graph 7 () DS P is shown in Fig.

Fig. 8
Fig. 8 Fl is the graph obtained from a helm n H by joining each pendant vertex to the apex of the helm.It contains three types of vertices: an apex of degree 2n , n vertices of degree 4 and n vertices of degree 2. Flower graph n Fl is a square divisor cordial graph for each n .
n Definition 1.10 For a simple connected graph G the square of graph G is denoted by 2 G and defined as the graph with the same vertex set as of G and two vertices are adjacent in 2G if they are at a distance 1 or 2 apart in G .