SUPER EDGE MAGIC TOTAL LABELING ON UNICYCLIC GRAPHS

Let G = (V,E) be a simple, connected and undirected graph with v vertices and e edges. An edge magic total labeling is a bijection f from V∪E to a set of integers {1,2,...,v+e} such that if xy is an edge of G then the weight of edge f(x)+f(y)+f(xy) = k, for some integer constant k. A super edge magic total labeling is an edge magic total labeling f which f(V) = {1, · · · , v}. In this paper we construct new super edge magic total labeling of special classes of unicyclic that we construct from a super edge magic total labeling of odd cycles. Our construction uses embedding process of odd cycles, which is labeled by edge magic total labeling to grid, and uses edge transformation to obtain interesting new classes of super edge magic total unicyclic graphs.


Introduction
Let G = (V,E) be a simple and finite undirected graph with v vertices and e edges.Let f be a bijection from V ∪ E to the set {1, 2, • • • , v + e}.f is called an edge magic total labeling if for any edge xy in G the weight of edge f(x) +f(y) +f(xy) = k, for some constant integer k.A super edge magic total labeling is an edge magic total labeling f which f(V ) = {1, • • • , v}.A graph that has a (super) edge magic total labeling is called a (super) edge magic total graph.
Sedláček introduced the concept of magic labeling in 1963 [1].However the notion of super edge-magic labeling has been used since it was defined by Enomoto et al. in [2].We can obtain complete results on edge magic total labeling in Gallian's survey [3].
According to attack the famous conjecture that said all trees are super edge magic total graphs [2,4], Bača, Lin and Muntaner-Batle [5] and Ngurah, Baskoro and Simanjuntak [6] in separate paper proved the following theorem.
Theorem 1 [5,6]: All path-like trees are super edge magic total graph.Sugeng and Silaban used the idea of path-like tree for constructed a subclass of tree, they called it caterpillarlike tree, which has super edge magic total labeling [7] Theorem 2 [7]: All caterpillar-like trees are super edge magic total graph.The super edge magic total labeling of odd cycle has been constructed by Enomoto et al. [2].It is also known that even cycles do not have super edge magic total labeling.If we consider the labeling of odd cycle, then the cycle can be decomposed to two paths P n-1 and P 2 .The super edge magic total labeling of cycle can be constructed by edge magic total labeling of path P n-1 , which one of the end vertex has 1 as its labels and then adjust the labels of all edges.Figure 1 shows a super edge magic total labeling of C 15 .

Embedding on grid
Let C n , n ≥ 4, be the cycle with can be drawn as a subgraph of the two dimensional grid.If n is even then there is no problem to make the embedding, but the even cycle is not a super edge magic total graph.However if n is odd then we have one edge of cycle C n that is represented as a free line in the grid.
Consider the ordered set of subpaths S 1 , S 2 , . . ., S t which are maximal straight segments in the embedding and it has the property that the end of S j is the beginning of S j+1 for any j = 1, 2, . . ., t − 1.
Suppose that S j = P 2 for some j, ), and distance of u and v is one for some vertex u of S j−1 and vertex v of S j+1 .The distance of u 0 and u in S j−1 is equal to the distance of v 0 and v in S j+1 .An edge transformation is replacing the edge u 0 v 0 by a new edge uv.A unicyclic C of order n is called cycle-like when it can be obtained from some embedding of C n in the two dimensional grid by a set of sequentially edge transformations.Since the labels of vertices of C n is similar with label of vertices of P n , then the edge transformations preserve the super edge magic total labeling property of a new graph.

Results
The following Lemma is needed to prove the main theorem.
Lemma 1 [8] : We generalize the idea of path-like tree by embedding the cycle to grid and using an edge transformation to construct a new subclass of unicyclic graph that we called cycle-like unicyclics.Figure 2 shows the edge transformation results on cycle with 15 vertices and the example of cycle-like unicyclic graph can be seen in Figure 3.The graph in Figure 3 is actually the same with the graph in Figure 2 after the edge transformations.
Theorem 3: All cycle-like unicyclics are super edge magic total graph.

Proof:
Let C be a cycle-like unycyclic order n which is a result of the embedding cycle C n on grid.Let V(C n ) = { x 1 , x 2 , …, x n } and k be a magic constant.Consider U o = C n , U 1 , U 2 ,…, U p as a sequence of unycyclic after applying some edge transformations sequentially.
Define a labeling on vertices and edges of C n as follows: Using Lemma 1, we can conclude that U j is a super edge magic total graph, where j=0,1,2,…,p.Thus All cycle-like unicyclics are super edge magic total graph.
Let G be a corona tail on every vertex.Figure 4 shows the super edge magic total labeling on 3 15 . If we ignore all the tails and look at the labeling on the cycle it self, then we can do the similar process with caterpillar-like tree as in [8] and we can have the following Theorem.This methods only works if the number of tails on every vertex are the same ( or equal to r).
In Figure 5 From Lemma 1, we obtain a super edge magic total labeling for the corona since the edge weight W f ={f(x)+f(y), xy ∈E} form a consecutive integers.
Let f be a super edge magic total labeling of a graph V 1 = CP n with a constant magic k.Suppose that graphs V j+1, j ∈{1,2,…,p}, is a result from the process of edge transformation by changing edge (u,v) of V i with edge (u',v').

C15 -SEMT (Corona) -Variation Figure 6. Example of super edge magic total labeling on corona-like unicyclics
For both cases we obtain ( ) Using Lemma 1, we can conclude that V j+1 is a super edge magic total graph, where j=0,1,2,…,p.Thus All corona-like unicyclics are super edge magic total graph.

Conclusion
In this paper we construct and prove two new subclass of unicyclic graphs, cycle-like unicyclic graphs and corona-like unicyclic graphs, to be a super edge magic total graph by using edge transformation on edges of embedded cycle in a two dimensional grid.If we have quite long tails from every vertex, we can do the same process (edge transformation) and treat the tails as a path, then we can have more exciting graphs.However, this graph is still categorized as unicyclic graph.

Figure 1 .Figure 2 .Figure 3 .
Figure 1.Super edge magic total labeling for C 15 graph G with v vertices and e edges is super edge magic graph if and only if there exists a bijective function f : V (G) ∪E(G) →{1, 2, . . .,v+e} such that the set S = {f(x) + f(y)| xy ∈ E(G)} consists of e consecutive integers.In such a case, f extends to a super edge magic total labeling of G with magic constant c = v+ e + s where s = min(S) and S = {f(x) + f(y)| xy ∈ E(G) ={c − (v + 1), c − (v + 2), ..., c − (v +e)}.By using Lemma 1, we only need to label all the vertices with 1,2,…,v and show that the set S = {f(x) + f(y)| xy ∈ E(G)} consists of e consecutive integers to prove that a bijection is a super edge magic total labeling.

Theorem 4 :
below, we can see the labeling on cycle of All corona-like unicyclics are super edge magic total graph.Proof:Let G be a corona with c i as vertices in the cycle, i = 1, ..., n and r as number tails on everyC i .Let V(CP n )={c 1 ,c 2 ,…,c n } and E(CP n )={c i c i+1 : i=1,…,n-1}∪{c n c 1 }.Let v ij be leaves of the tails on c i , i=1,2,…,n, j=1,…,r.Label the vertices as follows.Label the leaves vertices as follows. even