Research Article Open Access
On Super Edge-Magicness of Double Star
Stalin Kumar S*
Department of Mathematics, the American College, Madurai, Tamilnadu, India
*Corresponding author: Stalin Kumar S, Department of Mathematics, the American College, Madurai, Tamilnadu, India, 625 002, Tel: +919944035481; E-mail: @
Received: July 25, 2018; Accepted: August 07, 2018; Published: August 13, 2018
Citation: Stalin Kumar S (2018) On Super Edge-Magicness of Double Star. J Comp Sci Appl Inform Technol. 3(2): 1-3. DOI: 10.15226/2474-9257/3/2/00133
Abstract
For a graph $G$ , a bijection $f:V\left(G\right)\cup E\left(G\right)\to \left\{1,2,3,\dots ,p+q\right\}$ is called an edge-magic labeling of G if there exists a constant $s$ such that $f\left(u\right)+f\left(uv\right)=s$ , for any edge . $f$ is said to be super edge-magic if $f\left(V\left(G\right)\right)=\left\{1,2,3,\dots ,p\right\}.$ After investigating the super edge-magicness of double stat ${S}_{m,n}$ , a partial solution to a research problem (“For what $m$ and $n$ does ${S}_{m,n}$ have an edge magic labeling?, stated by Marr and Wallis in [6]”) was given in this paper

Keywords: Graph labeling; Edge-magic graphs; Super edge-magic graphs;
AMS Mathematics Subject Classification:
05C78.
Introduction
In this paper we consider only finite and simple undirected graphs. The vertex and edge sets of a graph G are denoted by V(G) and E(G) respectively and we let |V(G)| = p and |E(G)| = q. For graph theoretic notations, we follow [1,2]. A labeling of a graph G is a mapping that carries a set of graph elements, usually vertices and/or edges into a set of numbers, usually integers. Many kinds of labeling have been studied and an excellent survey of graph labeling can be found in [4].

In 1998, Enomoto, et al. [3] introduced the concept of super edge-magic graphs. In 2005, Sugeng and Xie, [7] constructed some super edge-magic total graphs. The usage of the word “super” was introduced in [4]. A edge-magic total labeling is a bijection f from to the integers {1,2,…,p+q} with the property that for every edge f(u) + f(v) + f(uv) = s for some constant s, such a labeling is (V )-super if f(V(G)) = {1,2,…,p}. A graph G is called edge-magic (resp. super edge-magic) if there exists an edge-magic (resp. super edge-magic) labeling of G.

Recently, Marimuthu and Balakrishnan, [5], introduced the notion of super edge-magic graceful graphs to solve some kind of network problems. A (p, q) graph G with p vertices and q edges is edge magic graceful if there exists a bijection f : such that , a constant for any edge uv of G. G is said to be super edge-magic graceful if f(V(G)) = {1,2,…,p}.

The double star has two adjacent central vertices x and y. There are m leaves(vertices) adjacent to x and n leaves(vertices) ${y}_{1},{y}_{2},\dots ,{y}_{n}$ adjacent to y. A (super) edgemagic total labeling of this graph can be specified by the list One solution for ${S}_{2,2}$ to be edge-magic is ({8,11},2,5, {4,10}) with s = 16. One solution for ${S}_{2,2}$ to be super edge-magic is ({6,3},2,5, {4,1}) with s = 16.

Some existing results found in [6]

i. All Paths have edge-magic total labeling.
ii. A Cycle ${C}_{n}$ is super edge-magic if and only if n is odd.
iii. The Complete bipartite graph is super edge-magic if and only if $n=1$ or $m=1.$

In [6], Marr and Wallis quoted a research problem “For what m and n does have an edge magic labeling?”. In this paper, we prove the following

i. is super edge-magic
ii. ${S}_{n,n}$ is super edge-magic if n is even.
Main Result
In this section, we prove two main theorems.

Theorem 4.1: The double star is super edge-magic.

Proof: Let has two adjacent central vertices x and y with n vertices ${x}_{1},{x}_{2},\dots ,{x}_{n}$ adjacent to $x$ and vertices ${y}_{1},{y}_{2},\dots ,{y}_{n+2}$ adjacent to $y$ . Clearly and . Define a total labeling $f:V\left(G\right)\cup E\left(G\right)\to \left\{1,2,\dots ,p+q\right\}$ by and for all

Now the edges of G can be labeled as shown in Table 1 and Table 2.
Table 1: The edge label of ${S}_{n,n+2}$ whose $n+2$ edges are labeled
 f ${y}_{1}$ ${y}_{2}$ ... ${y}_{i}$ ... ${y}_{n+1}$ ${y}_{n+2}$ y $4n+7$ $4n+5$ $4n-\left(2i-9\right)$ $2n+7$ $2n+5$
Table 2: The edge label of ${S}_{n,n+2}$ whose edges are labeled
 f ${y}_{1}$ ${y}_{2}$ ... ${y}_{i}$ ... ${y}_{n+1}$ ${y}_{n+2}$ x $4n+4$ $4n+2$ $4n-\left(2i-6\right)$ $2n+8$ $2n+6$
And . It is easily seen that f is super edge-magic labeling with magic constant . Hence the graph ${S}_{n,n+2}$ is super edge-magic. (Figure 1)
Figure 1: Super edge-magic labeling of ${S}_{3,5}$ with magic constant $s=24$
Theorem 4.2: The double star is super edge-magic if n is even.

Proof: Let $G\cong {S}_{n,n}$ has two adjacent central vertices x and y with n vertices

${x}_{1},{x}_{2},\dots ,{x}_{n}$ adjacent to and $n$ vertices ${y}_{1},{y}_{2},\dots ,{y}_{n}$ adjacent to $y$. Clearly and . Define a total labeling by
Now the edges of G can be labeled as shown in Table 3 and Table 4.
Table 3: The edge label of ${S}_{n,n}$ whose first edges are labeled
 $f$ $y$ ${y}_{1}$ $\left(3n+3\right)+\left(\frac{n}{2}\right)$ ${y}_{2}$ $\left(3n+3\right)+\left(\frac{n}{2}-1\right)$ ... ... ${y}_{j}$ $\left(3n+3\right)+\left(\frac{n}{2}+1-j\right)$ ... ... ${y}_{\frac{n}{2}}$ $\left(3n+3\right)+1$ ${y}_{\frac{n}{2}+1}$ $\left(2n+2\right)+\left(\frac{n}{2}\right)$ ... ... ${y}_{k}$ $\left(2n+2\right)+\left(n+1-k\right)$ ... ... ${y}_{n-1}$ $\left(2n+2\right)+2$ ${y}_{n}$ $\left(2n+2\right)+1$
Table 4: The edge label of ${S}_{n,n}$ whose second $n$ edges are labeled
 $f$ $x$ ${x}_{1}$ $\left(\frac{7n}{2}+3\right)+\left(\frac{n}{2}\right)$ ${x}_{2}$ $\left(\frac{7n}{2}+3\right)+\left(\frac{n}{2}-1\right)$ ... ... ${x}_{j}$ $\left(\frac{7n}{2}+3\right)+\left(\frac{n}{2}+1-j\right)$ ... ... ${x}_{\frac{n}{2}}$ $\left(\frac{7n}{2}+3\right)+1$ ${x}_{\frac{n}{2}+1}$ $\left(\frac{5n}{2}+2\right)+\left(\frac{n}{2}\right)$ ... ... ${x}_{k}$ $\left(\frac{5n}{2}+2\right)+\left(n+1-k\right)$ ... ... ${x}_{n-1}$ $\left(\frac{5n}{2}+2\right)+2$ ${x}_{n}$ $\left(\frac{5n}{2}+2\right)+1$
It is easily seen that $f$ is super edge-magic labeling with magic constant . Hence the graph is super edge-magic. (Figure 2)
Figure 2: Super edge-magic labeling of ${S}_{4,4}$ with magic constant $s=24$
Conclusion
In this paper, we had given two different solutions to a Research Problem given by Marr and Wallis in [6].
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