E-cordial and product e-cordial labeling for the extended duplicate graph of splitting graph of path

Based on the works of Yilmaz and Cahit on E-cordial labeling, we prove the existence of E-cordial labeling, total E-cordial labeling, product E-cordial labeling and total product E-cordial labeling for the extended duplicate graph of splitting graph of path Pm.


Introduction
One of the most famous and productive labeling of graph theory is cordial labeling. This labeling was introduced by Cahit in the year 1987. For an extensive survey on graph labeling, we refer to Gallian [5]. E.Sampthkumar [3,4] introduced the concept of duplicate graph and splitting graph. In 1997, Yilmaz and Cahit have introduced a weaker version of edge-graceful called E-cordial [8]. B.Selvam, K.Thirusangu, and P.P. Ulaganathan have introduced the idea of extended duplicate graph of twig graphs and they proved that the EDG of twig graphs is E-cordial and product E-cordial [1,9]. E. Bala and K.Thirusangu have studied on E-Cordial labeling for Competition graph [2]. R. Avudainayaki,et.al., have proved that the Prime Cordial and Signed Product Cordial Labeling for the Extended Duplicate Graph of Arrow Graph [7]. Vaidya and Barasara proposed edge product cordial labeling [10]. P. Lawrence Rozario Raj and S. Koilraj have proved that cordial labeling for the splitting graph of some standard graphs [6].

Preliminaries
First, we will give brief summary of definitions which are useful for the present investigations.
Definition 1: each vertex v of a graph G, take a new vertex v. Join v to all the vertices of G adjacent to v. The graph Spl(G) thus obtained is called splitting graph of G.
is bijective and the edge set E1 of DG is defined as the edge ab is in E if and only if both ab and ab are edges in E1.
Theorem 2 : For m 2 , EDG(Spl(Pm)) is total E-cordial for m is even. Proof: In theorem 1, 2m vertices are allotted '0' and 2m vertices are allotted the label '1'. When m is odd, 3m-2 edges labeled with '0' and 3m-3 edges labeled with '1' and when m is even, 3m-2 edges labeled with '1' and 3m-3 edges labeled with '0'. In both cases, we see that the number of vertices and edges labeled with '1' is 2m + 3m-3 = 5m-3 and the number of vertices and edges labeled with '0' is 2m+3m-2 = 5m-2 differ by one and satisfies the required condition. Thus EDG (Spl(Pm)), m  2 admits total E-cordial labeling for m is even.

Product e-cordial
Here, we prove the existence of product E-cordial and total product E-cordial for EDG(Spl(Pm)) , m  2.
Theorem 3 : For m  2 , EDG(Spl(Pm)) is product E-cordial. Proof: Let Spl(Pm), m  2 be a splitting path graph. Let EDG(Spl(Pm)) ,m  2 be a extended duplicate graph of splitting path graph.

Conclusion
We have shown that EDG (Spl(Pm)) , m 2 is E-cordial, total E-cordial, product E-cordial and total product E-cordial. In future it would be interesting to extend the different type of graphs and its possible labeling for EDG (Spl(Pm)).