New Constructions of Edge Bimagic Graphs from Magic Graphs

An edge magic total labeling of a graph G(V,E) with p vertices and q edges is a bijection f from the set of vertices and edges to such that for every edge uv in E, f(u) + f(uv) + f(v) is a constant k. If there exist two constants k1 and k2 such that the above sum is either k1 or k2, it is said to be an edge bimagic total labeling. A total edge magic (edge bimagic) graph is called a super edge magic (super edge bimagic) if f(V(G)) = . In this paper we define super edge edge-magic labeling and exhibit some interesting constructions related to Edge bimagic total labeling. 1, 2, ,   p q


Introduction
A labeling of a graph G is an assignment f of labels to either the vertices or the edges or both subject to certain conditions.Labeled graphs are becoming an increasingly useful family of Mathematical Models from a broad range of applications.Graph labeling was first introduced in the late 1960's.A useful survey on graph labeling by J. A. Gallian (2010) can be found in [1].All graphs considered here are finite, simple and undirected.We follow the notation and terminology of [2].In most applications labels are positive (or nonnegative) integers, though in general real numbers could be used.A (p, q)-graph with p vertices and q edges is called total edge magic if there is a bijection such that there exists a constant k for any edge uv in E, The original concept of total edge-magic graph is due to Kotzig and Rosa [3].They called it magic graph.A total edge-magic graph is called a super edge-magic if )) {1,2, , } f V G p   .Wallis [4] called super edge-magic as strongly edgemagic.An Edge antimagic total labeling of a graph with p vertices and q edges is a bijection from the set of edges to such that the sums of the label of the edge and incident vertices are pairwise distinct.It becomes interesting when we arrive with magic type labeling summing to exactly two distinct constants say or Edge bimagic totally labeling was introduced by J. Baskar Babujee [5] and studied in [6] as (1,1) edge with p vertices and q edges is called total edge bimagic if there exists a bijection such that for any edge  we have two constants 1 and 2 with  .Super edge-bimagic labeling for path, star-1, 1, , n nn are proved in [7].Super edge-bimagic labeling for cycles, Wheel graph, Fan graph, Gear graph, Maximal Planar class-Pl n : , , ( 1), are proved in [8][9][10].In this paper we define super edge edge-magic and exhibit some interesting constructions related to Edge bimagic total labeling.For our convenience, we state total edge-magic as edge-magic total labeling throughout the paper.

Main Results
On renaming Super edge-magic as Super vertex edgemagic it motivates us to define super edge edge-magic labeling in graphs.

Definition 2.1 A graph
with p vertices and q edges is called total edge magic if there is a bijection function for any edge uv in E we have a constant k with Theorems 2.3 If G has super edge edge-magic total labeling then, admits edge bimagic total labeling.Proof: Let G(p, q) be super edge edge-magic graph with the bijective function is the maximum value.Consider the path P n with vertex set { : We superimpose one of the pendent vertex of the path P n say x 1 on the vertex of G. Now we define the new graph called with vertex set and Consider the bijective function defined by for all From our construction of new graph , , ( ) ) Since the graph G is super edge edge-magic with common count implies that for all Now we have to prove that the remaining edges in the set have the common count k 2 . .
For the edge wx 2 ,

is total edge bimagic for any arbitrary super edge edge-magic Graph G. ô n G K
Proof: Let G(p,q) be super edge edge-magic graph with the bijective function is the maximum value.Consider the star K 1,n with vertex set and edge set 0 { :1 }.

i x x i n  
We superimpose the vertex x 0 of the star K 1,n graph on the vertex wV of G. Now we define the new graph called Since the graph G is super edge edge-magic with common count k 1 , implies that all Now we have to prove that the remaining edges joining w and .

uv E 
: For any edge wx i , i i g w g wx g x p q p q i p q n i p q n k has two common count k 1 and k 2 .Hence has edge bimagic total labeling.6 If G has super edge edge-magic total labeling then, admits edge bimagic total labeling.

1,
Proof: Let G(p,q) be super edge edge-magic graph with the bijective function and edge set 0 1 We superimpose the vertex x 0 of the Fan F 1,n graph on the vertex of G. Now we define the new graph called with vertex set and Consider the bijective function Since the graph G is super edge edge-magic with common count k 1 , implies that i i i i g w g wx g x p q p q n i p q

G G
Proof: Let G 1 be the super edge edge-magic then there exist the bijective function Let G 2 be the super vertex edge-magic then there exist the bijective function  be the vertex whose label is the maximum value 1 1 p q  and 1 2 x V  be the vertex with label 1.We superimpose the vertex x 1 of G 2 graph on the vertex 1 w V  .
Now we define the new graph called with vertex set V V and Consider the bijective function 1 1

( ) for all ( ). h uv p q g uv uv E G      
For the edges in G 1, we have 1 Since magic labeling is preserved in a graph if all the vertices and edges are increased by any constants, for the edges in G 2 , we have .
So has two common count k 1 and k 3 .Hence admits edge bimagic total labeling.
Theorem 2.9 If G has super edge edge-magic total labeling then, G + K 1 admits edge bimagic total labeling.Proof: Let G(p,q) be super edge edge-magic.Then there exist a bijective function function such that with vertex set and Consider the bijective function defined as follows, i i g x g xv g v p q p q i p q i p q k Thus we have 1 has two common count k 1 and k 2 .Hence has edge bimagic total labeling.

Concluding Remarks
Theorem 2.8 shows that 1 2 admit edge bimagic total labeling if G 1 has super edge edge-magic labeling and G 2 has super vertex edge-magic labeling.Further investigation can be done to obtain the conditions at which 1 2 admits edge bimagic total labeling for any two arbitrary total magic graphs.

Acknowledgements
The referee is gratefully acknowledged for their suggestions that improved the manuscript.
for all uv E. Now we have to prove that the remaining edges in the set { : have the common count k 2 .
k 1 and k 2 .
Let k 1 be the constant edge count of an

2
Since there are p vertices in the graph G, Since the graph G is super edge edge-magic with common count k 1 , implies that Now we have to prove that the remaining p edges joining V and x have the common count k 2 .For any edge xv i ,