Volume: 02, October 2013, Pages: 243-247 International Journal of Computing Algorithm Integrated Intelligent Research (IIR) 243 ON E-CORDIAL LABELINGS OF COMPETITION GRAPH E.Bala, K.Thirusangu Department of mathematics, S.I.V.E.T. College, Gowrivakkam, Chennai–73 kthirusangu@gmail.com Abstract In 1968, Joel E. Cohen has introduced the notion of Competition graph in connection with a problem of ecology. The competition graph of a diagram D denoted by C(D) has the same vertex set of D and there is an edge between the vertices x and y if and only if there exists a vertex z in D such that (x,z) and (y,z) are arcs of D. In this paper we present an algorithm and prove the existence of graph labelings such as E-cordial, total E-cordial, product E-cordial, total product E-cordial labelings for the Competition graph of the Cayley digraphs associated with 2 –generated 2-groups. Key words: Graph labelings, Cayley digraphs, Competition graph, 2-generated 2-groups. 1. Introduction In 1878, Cayley constructed a graph with a generating set which is now popularly known as Cayley graphs. A directed graph or digraph is a finite set of points called vertices and a set of arrows called arcs connecting some vertices. The Cayley graphs and Cayley digraphs are excellent models for interconnection networks [ 6 ]. Many well- known interconnection networks are Cayley digraphs. For example hypercube, butterfly, and cube-connected cycle’s networks are Cayley graphs. The Cayley digraph of a group provides a method of visualizing the group and its properties. The properties such as commutativity and the multiplication table of a group can be recovered from a Cayley digraph. In [2], it is proved that the Competition graph of Cayley digraph associated with diheadral group D n admits Z 3 magic, Cordial, total cordial, E-cordial, total E-cordial, Product cordial, total product cordial, Product E-cordial and total product E- cordial labelings. In this paper we prove the existence of graph labelings such as E-cordial, total E-cordial, Product E-cordial and total product E-cordial for the competition graph of the Cayley digraphs associated with 2- generated 2-groups. 2. Preliminaries In this section we give the basic notion relevant to this paper. Definition 2.1 A (p,q)-digraph G = (V,E) is defined by a set V of vertices such that |V|=p and a set E of arcs or directed edges with |E|=q. The set E is a subset of elements (u,v) of V × V. The out-degree (or in-degree of a vertex u of a digraph G is the number of arcs (u,v)(or(v,u)) of G and is denoted by d + (u) (or d - (u)). A digraph is said to be regular if d + (u) = d - (v) for every vertex u of G. Definition 2.2 Let G be a finite group and S be a generating subset of G. The Cayley digraph Cay(G,S) is the digraph whose vertices are the elements of G, and there is an edge from g to gs whenever g ∈ G and s ∈ S. If S = S -1 then there is an edge from g to gs if and only if there is an arc from gs to g Definition 2.3 A group G is said to be a 2- group if o(G) = p m for m ≥ 1. It is said to be 2- generated if the minimal generating set of G has exactly two elements. Throughout this paper we take o(G) = p m = n and p =2 Definition 2.4 The structure of the Cayley digraph Cay ( G , (α,β)) for the 2-group is defined as follows. Using definition 2.2, the Cayley digraph for the 2-group, Cay ( G , (α,β)) has n vertices and 2n arcs. Let us denote the vertex set of Cay ( G , (α,β)) as V ={v 1 , v 2 , v 3 ….v n }. Define the arc set as Cay ( G , (α,β)) as E(E α , E β ) where E α = {(v, αv)| v∈V} and E β = {(v, βv)| v∈V}. Denote the arcs in E α as {gα (v i ) | v i ∈V} and E β as {g β (v i ) | v i ∈V}. Clearly each vertex in Cay (G, (α, β)) has