Super Edge-magic Labeling of Graphs: Deficiency and Maximality

A graph G of order p and size q is called super edge-magic if there exists a bijective function f from V(G) U E(G) to {1, 2, 3, ..., p+q} such that f(x) + f(xy) + f(y) is a constant for every edge $xy \in E(G)$ and f(V(G)) = {1, 2, 3, ..., p}. The super edge-magic deficiency of a graph G is either the smallest nonnegative integer n such that G U nK_1 is super edge-magic or +~ if there exists no such integer n. In this paper, we study the super edge-magic deficiency of join product graphs. We found a lower bound of the super edge-magic deficiency of join product of any connected graph with isolated vertices and a better upper bound of the super edge-magic deficiency of join product of super edge-magic graphs with isolated vertices. Also, we provide constructions of some maximal graphs, ie. super edge-magic graphs with maximal number of edges.


Introduction
Let G be a finite and simple graph of order |V (G)| and size |E(G)|. A graph G is edgemagic if there exists a bijective function f : V (G) ∪ E(G) → {1, 2, 3, · · · , |V (G)| + |E(G)|} such that f (x) + f (xy) + f (y) = k is a constant, called the magic constant of f , for every edge xy of G. In such a case f is called an edge-magic labeling of G. The concepts of edgemagic graphs and edge-magic labelings introduced by Kotzig and Rosa in 1970 [11]. Motivated by the concept of edge-magic labelings, Enomoto, Lladó, Nakamigawa, and Ringel [5] introduced the concepts of super edge-magic labelings and super edge-magic graphs as follows: A super edge-magic labeling of a graph G is an edge-magic labeling f of G with the extra condition that f (V (G)) = {1, 2, 3, · · · , |V (G)|}. A graph having a super edge-magic labeling is a super edgemagic graph. In [6], Figueroa-Centeno et al. provided a useful characterization of a super edgemagic graph that we state in the following lemma. In light of above result, it suffices to exhibit the vertex labeling of a super edge-magic graph. The next lemma proved by Enomoto et al. [5] gives sufficient condition for un-existence of super edge-magic labeling of a graph.
In [11], Kotzig and Rosa proved that for every graph G there exists a nonnegative integer n such that G ∪ nK 1 is edge-magic. This fact motivated them to introduced the concept of edge-magic deficiency of a graph. The edge-magic deficiency of a graph G, µ(G), is the smallest nonnegative integer n such that G ∪ nK 1 is an edge-magic graph. Motivated by Kotzig and Rosa's concept of edge-magic deficiency, Figueroa-Centeno et al. [7] introduced the concept of super edge-magic deficiency of a graph. The super edge-magic deficiency of a graph G, µ s (G), is either the smallest nonnegative integer n such that G ∪ nK 1 is a super edge-magic graph or +∞ if there exists no such n.
Some papers concerning on the super edge-magic deficiency of graphs have been published. The super edge-magic deficiency of cycles, complete graphs, complete bipartite graphs K 2,m , disjoin union of cycles, and some forest with two components can be found in [7,8]. The super edge-magic deficiency of some classes of chain graphs, wheels, fans, double fans, and disjoint union of complete bipartite graphs can be found in [13,14]. The super edge-magic deficiency of complete bipartite graphs K m,n and disjoin union of stars are investigated in [10]. The super edgemagic deficiency of some classes of unicyclic graphs are studied in [1,2] . The latest developments in these and other types of graph labelings can be found in [9].

Super edge-magic deficiency of join product graphs
Join product of two graphs is their graph union with additional edges that connect all vertices of the first graph to each vertex of the second graph. The join product of graphs G and H is denoted by G + H. Hence, and Ngurah and Simanjuntak [12] studied super edge-magic deficiency of the join product of a path, a star, and a cycle, respectively, with isolated vertices. We gave the lower and upper bounds of super edge-magic deficiency of P n + mK 1 , K 1,n + mK 1 , and C n + mK 1 , respectively. We also gave the following result.
In 2008, Ngurah et al. [14] studied super edge-magic deficiency of P n + 2K 1 . They have the following result and conjectured that for any odd integer n, µ s (P n + 2K 1 ) = n−1 2 .
In this section, we study super edge-magic deficiency of join product of any connected graphs with isolated vertices. Notice that the only connected graphs of order 1 and 2 are K 1 and K 2 , respectively. It is well known that K 1 + mK 1 and K 2 + mK 1 are super edge-magic for any integers m ≥ 1. In other words, for any integers m ≥ 1, Here, we study super edge-magic deficiency of G + mK 1 where G is any connected graph with |V (G)| ≥ 3. Our first result provides a lower bound of its super edge-magic deficiency.
is not a super edge-magic graph. Hence, we have the desire result.
In [12], Ngurah and Simanjuntak also showed the finiteness of the super edge-magic deficiency of the join product of any super edge-magic graph with isolated vertices. We proved the following result.

Theorem 2.2. [12] Let G be a super edge-magic graph of order p with a super edge-magic labeling
In the next theorem, we provide a better upper bound than the one in Theorem 2.2.
Next, define an injective function g of H as follows.
Next, we consider two cases depending on the value of m.

Constructions of maximal graphs
As a consequence of Lemma 1.2, maximal number of edges of a super edge-magic graph G is 2|V (G)| − 3. A super edge-magic graph G which satisfy |E(G)| = 2|V (G)| − 3 is called maximal. In this section, we seek for some maximal graphs. First, we study the maximality of graphs obtained from chain graphs by adding or deleting an edge of the chain graphs. A chain graph with blocks B 1 , B 2 , B 3 , · · · , B k , denoted by C[B 1 , B 2 , · · · , B k ], is defined as a graph such that for every i, B i and B i+1 have a common vertex in such a way that the block-cut-vertex graph is a path. The concept of a chain graph was firstly introduced by Barrientos in 2002 [3]. If B i = H, for every i, then C[B 1 , B 2 , · · · , B k ] denoted by kH-path. Hence, kK 4 -path is a chain graph with k blocks where each block is identical and isomorphic to the complete graph K 4 . If c 1 , c 2 , . . . , c k−1 are the consecutive cut vertices of For every integer k ≥ 3, let H = kK 4 -path be a graph with vertex set and edge set as follows.
As we can see, H is a graph of order 3k + 1 and size 6k. Ngurah et al. [13] proved that µ s (H) = 1.
The following vertex labeling f is an alternative vertex labeling that extends to a super edge-magic labeling of H ∪ K 1 .
From now on f refers to this vertex labeling.
It is clear that by removing one edge from H, the resulting graph has number of edges satisfying the maximal condition. We shall study two of such graphs. Let j be an integer where For case j = 1, For case j = 2, 3, . . . , ⌈ k 2 ⌉, It can be checked that g is a bijection from V (H [j] ) to {1, 2, 3, . . . , 3k+1} and for {3, 4, 5, . . . , 6k + 1}. By Lemma 1.1, g can be extended to a super edge-magic labeling of H [j] . Also, it is easy to verify that for 1 ≤ j ≤ ⌈ k 2 ⌉, g(x j ) + g(y j ) = g(c j ) + g(c j+1 ). Hence, g can be extended to a super edge-magic labeling of H It is a routine procedure to verify that h can be extended to a super edge-magic labeling of H − {x 1 c 1 } and H − {x 1 c 2 }.
Next, we consider graphs obtaining from ladders L m = P m × P 2 , m ≥ 2, by adding a diagonal in each rectangle of L m . We denote such a graph by G[m]. Hence, G[m] can be defined as a graph with the vertex and edge sets as follows.
can be extended to a super edge-magic labeling of G[m] with magic constant 6m.
Our results presented in Theorems 3.1, 3.2, 3.3, 3.4, and 3.5, give impression that graphs satisfy the maximal condition are super edge-magic. However, it is not the case. As an example P n + K 1 , which satisfies the maximal condition, is super edge-magic if and only if 1 ≤ n ≤ 6 (see [7]).