Edge Pair Sum Labeling

An injective map f : E(G) → is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function defined by is one – one, where denotes the set of edges in G that are incident with a vertex v and is either of the form or ⋃ according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that path , cycle , triangular snake, , are edge pair sum graphs.


Introduction
We consider only finite, simple, planar and undirected graphs.A graph G(p,q) has the vertex-set V(G) and the edge-set E(G) with |V(G)|=p and |E(G)| = q.A vertex labeling f of a graph G is an assigned of labels to the vertices of G that induces a label for each edge xy depending on the vertex labels.An edge labeling f of a graph G is an assigned of labels to the edges of G that induces a label for each vertex v depending on the labels of the edges incident on it.Terms and terminology are used in the sense of Harary [1].
Ponraj and Parthipan [2] introduced the concept of pair sum labeling.An injective map ƒ: V (G) → {±1, ±2,…,±p} is said to be a pair sum labeling of a graph G(p,q) if the induced edge function ƒ e :E(G) → Z -{0} defined by ƒ e (uv) =ƒ(u)+ƒ(v) is one-one and ƒ e (E(G)) is either of the form {±k 1 , ±k 2 ,…, ± } or {±k 1 , ±k 2 ,…, ± } { } according as q is even or odd.A graph with a pair sum labeling is called a pair sum graph.The pair sum behavior of graphs like complete graph, path, bistar, cycle, all trees of order ≤8, and ≤9 and some more standard graphs are investigated in refs.[3][4][5][6].Motivated by Ponraj and Parthipan [2], we define a new labeling called an edge pair sum labeling analogous to pair sum labeling.Let G(p, q) be a graph.An injective map f : E(G) → is said to be an edge pair sum labeling if the induced vertex function defined by is oneone where denotes the set of edges in G that are incident with a vertex v and is either of the form or ⋃ according as p is even or odd.A graph with an edge pair sum labeling is called an edge pair sum graph.
We use the following definitions in the subsequent sequel.

Definition 1.1
The union of two graphs and is the graph

Definition 1.2
The corona is the graph obtained by taking one copy of G and n copies of H and joins the ith vertex of G with an edge to every vertex in the ith copy of H where .

Main results
Theorem 2.1: Every path is an edge pair sum graph for n 3.
Proof: Let and be the vertex set and edge set of respectively, where , 1 We consider the following four cases: Define the labeling by , The induced vertex labeling are Hence, f is an edge pair sum labeling of .Case (ii) n = 4. Define the labeling by , The induced vertex labelings are .Hence, f is an edge pair sum labeling of .Case (iii) n is even.Take n = 2k, k ≥ 3.

Define the labeling by
The induced vertex labelings are , , for , , , , and for .

Define the labeling by
The induced vertex labelings are , for , , and for .
From the above argument we get 1)⋃6.Hence, f is an edge pair sum labeling of . Theorem 2.2: Every cycle is an edge pair sum graph.
Proof : Let and be the vertex set and the edge set of , where , and .
Define the edge labeling by considering the following three cases.The induced vertex labelings are as follows: For , for = -(2i + 1), .
From the above labeling, we get .Hence, f is an edge pair sum labeling.Case (iv) n is odd.Take n = 2m + 1, .
Subcase (i) for for .
The induced vertex labelings are , for , , for , .
From the above arguments, we get Hence, f is an edge pair sum labeling.
1,± ⋃−4. Theorem 2.3: The star graph is an edge pair sum graph if and only if n is even.