PRODUCT CORDIAL LABELING OF DOUBLE PATH UNION OF C 3 RELATED GRAPHS

: In this paper we discuss The double path unions obtained on C 3 related graphs for product cordial labeling given by P m (2G’) where G’ = C 3 , FL(C 3 ), C 3+ , bull, tail(C 3 ,2P 2 ) etc .


Introduction
The graphs we consider are simple, finite, undirected and connected. For terminology and definitions we depend on Graph Theory by Harary [9], A dynamic survey of graph labeling by J.Gallian [8] and Douglas West. [11]. I.Cahit introduced the concept of cordial labeling [5]. There are variety of cordial labeling available in labeling of graphs. Sundaram, Ponraj, and Somasundaram [10] introduced the notion of product cordial labeling. A product cordial labeling of a graph G with vertex set V is a function f from V to {0,1} such that if each edge uv is assigned the label f(u)f(v), the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1, and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a product cordial labeling is called a product cordial graph. We use vf(0,1) = (a, b) to denote the number of vertices with label 1 are a in number and the number of vertices with label 0 are b in number. Similar notion on edges follows for ef(0,1) = (x, y).
A lot of work is done in this type of labeling so far. One interested in survey may refer Dynamic survey in Graph labeling by J. Gillian. We mention some part of it. Sundaram, Ponraj, and Somasundaram have shown that trees; unicyclic graphs of odd order; triangular snakes; dragons; helms; Pm∪Pn; Cm∪Pn; Pm∪K1,n; Wm∪Fn (Fn is the fan Pn+K1); K1,m∪K1,n; Wm∪ K1,n; Wm∪Pn; Wm∪Cn; the total graph of Pn (the total graph of Pn has vertex set V (Pn)∪E(Pn) with two vertices adjacent whenever they are neighbors in Pn); Cn if and only if n is odd; Cn (t) , the one-point union of t copies of Cn, provided t is even or both t and n are even; K2+mK1 if and only if m is odd; Cm∪Pn if and only if m+n is odd; Km,n∪Ps if s >mn; Cn+2∪K1,n; Kn∪Kn,(n−1)/2 when n is odd; Kn∪Kn−1,n/2 when n is even; and P2 n if and only if n is odd. They also prove that Km,n (m,n> 2), Pm ×Pn (m,n> 2) and wheels are not product cordial and if a (p,q)-graph is product cordial graph, then q 6 (p−1)(p + 1)/4 + 1.In this paper we show thatPm(2G') where G' = C3, FL(C3), C3 + , bull, tail(C3,2P2) are product cordial graphs.

Fusion of Vertex
Let G be a (p,q) graph. Let u≠v be two vertices of G. We replace them with single vertex w and all edges incident with u and that with v are made incident with w. If a loop is formed is deleted.
The new graph has p-1vertices and at least q-1 edges.If uϵG1 and vϵG2, where G1 is (p1,q1) and G2 is (p2,q2) graph. Take a new vertex w and all the edges incident to u and v are joined to w and vertices u and v are deleted. The new graph has p1+p2-1 vertices and q1 + q2 edges. Sometimes this is referred as u is identified with the concept is well elaborated in John Clark,Holton [6]

Crown Graph
It is CnOK2.At each vertex of cycle a n edge was attached. We develop the concept further to obtain crown for any graph. Thus crown (G) is a graph G OK2.It has a pendent edge attached to each of it's vertex. If G is a (p,q) graph then crown(G) has q+p edges and 2p vertices.

2.3.
Flag of a Graph G denoted by FL(G) is obtained by taking a graph G=G(p,q).At suitable vertex of G attach a pendent edge.It has p+1 vertices and q+1 edges.

2.4.
A bull Graph Bull (G) was initially defined for a C3-bull. It has a copy of G with an pendent edge each fused with any two adjacent vertices of G. For G is a (p,q) graph, bull(G) has p+2 vertices and q+2 edges.

2.5.
A Tail Graph (Also called as antenna graph) is obtained by fusing a path pk to some vertex of G. This is denoted by tail(G, Pk). If there are t number of tails of equal length say (k-1) then it is denoted by tail(G, tpk). If G is a (p,q) graph and a tail Pk is attached to it then tail(G, Pk) has p+k-1 vertices and q+k-1 edges4.

2.6.
Path union of G, i.e.Pm(G) is obtained by taking a path pm and take m copies of graph G . Then fuse a copy each of G at every vertex of path at given fixed point on G. It has mp vertices and mq +m-1 edges. Where G is a (p,q) graph.
Thus the graph is product cordial for all m. #. Theorem 3.4 Double path union on crown C3 + given by G =Pm(2C3 + ) is product cordial.