SOME RESULTS ON SUPER EDGE-MAGIC DEFICIENCY OF GRAPHS

An edge-magic total labeling of a graph G is a bijection f : V (G) ∪ E(G) → {1, 2, . . . , |V (G)| + |E(G)|}, where there exists a constant k such that f(u) + f(uv) + f(v) = k, for every edge uv ∈ E(G). Moreover, if the vertices are labeled with the numbers 1, 2, . . . , |V (G)| such a labeling is called a super edge-magic total labeling. The super edge-magic deficiency of a graph G, denoted by μs(G), is the minimum nonnegative integer n such that G ∪ nK1 has a super edge-magic total labeling or is defined to be ∞ if there exists no such n. In this paper we study the super edge-magic deficiencies of two types of snake graph and a prism graph Dn for n ≡ 0 (mod 4). We also give an exact value of super edge-magic deficiency for a ladder Pn × K2 with 1 pendant edge attached at each vertex of the ladder, for n odd, and an exact value of super edge-magic deficiency for a square of a path Pn for n ≥ 3.


Introduction
In this paper, we consider only finite, simple and undirected graphs. We denote the vertex set and edge set of a graph G by V (G) and E(G), respectively. Let |V (G)| = p and |E(G)| = q.
An edge-magic total labeling of a graph G is a bijection f : V (G) ∪ E(G) → {1, 2, . . . , p + q}, where there exists a constant k such that smallest possible labels, i.e., with the numbers 1, 2, . . . , p. A graph that admits a (super) edge-magic total labeling is called (super) edge-magic total.
The concept of edge-magic total labeling was given by Kotzig and Rosa [8]. Super edge-magic total labelings were originally defined by Enomoto et al. in [3]. However Acharya and Hegde had introduced in [1] the concept of strongly indexable graphs that is equivalent to the one of super edge-magic total labeling. Kotzig and Rosa [8] proved that for any graph G there exists an edge-magic graph H such that H ∼ = G ∪ nK 1 for some nonnegative integer n. This fact leads to the concept of edge-magic deficiency of a graph G, which is the minimum nonnegative integer n such that G ∪ nK 1 is edge-magic total and it is denoted by µ(G). In particular, µ(G) = min{n ≥ 0 : G ∪ nK 1 is edge-magic total}.
In the same paper, Kotzig and Rosa gave an upper bound for the edge-magic deficiency of a graph G with n vertices, where F n is the nth Fibonacci number. Motivated by Kotzig and Rosa's concept of edge-magic deficiency, Figueroa-Centeno, Ichishima and Muntaner-Batle [5] defined a similar concept for the super edge-magic total labelings. The super edge-magic deficiency of a graph G, denoted by µ s (G), is the minimum nonnegative integer n such that G ∪ nK 1 has a super edge-magic total labeling, or is defined to be ∞ if there exists no such n. More precisely, if It is easy to see that for every graph G it holds In [5,7] Figueroa-Centeno, Ichishima and Muntaner-Batle found the exact values of the super edge-magic deficiencies of several classes of graphs, such as cycles, complete graphs, 2-regular graphs and complete bipartite graphs K 2,m . They also proved that all forests have finite deficiency. In particular, they proved that In [10] Ngurah, Simanjuntak and Baskoro gave some upper bounds for the super edge-magic deficiency of fans, double fans and wheels. In In the same paper, they showed that , if m is a multiple of n + 1 or n is a multiple of m + 1, 1, otherwise.
They also conjectured that every forest with two components has super edge-magic deficiency less than or equal to 1. Baig, Baskoro and Semaničová-Feňovčíková [2] have determined the super edge magic deficiency of a star forest. Santhosh and Singh [11] studied the corona product of K 2 and C n and they showed that µ s (K 2 C n ) ≤ n−3 2 , for n odd at least 3.
In this paper we study the super edge-magic deficiencies for several classes of graphs. We give upper bounds for the super edge-magic deficiencies of two types of snake graph and for prism graph D n for n ≡ 0 (mod 4). We also give an exact value of super edge-magic deficiency for a ladder P n × K 2 with 1 pendant edge attached at each vertex of the ladder, for n odd, and an exact value of super edge-magic deficiency for a square of a path P n for every positive integer n, n ≥ 3.
To prove the results presented in this paper, we frequently use the following lemma.

Upper Bounds
In graph theory a block graph is a graph in which every bi-connected component (block) is a clique. Block graphs are sometimes erroneously said to be "Husimi trees", but that name more properly refers to cactus graphs, graphs in which every nontrivial bi-connected component is a cycle. In graph theory block graphs may be described as the intersection graphs of the blocks of arbitrary undirected graphs.
Let G be a graph and u and v are two fixed vertices in G. The G n -snake is a graph obtained from n copies of G by identifying the vertex corresponding to the vertex v in the ith copy of G with the vertex corresponding to the vertex u in the (i + 1)th copy of G, for i = 1, 2, . . . , n − 1. The wheel W k , k ≥ 3 is a graph obtained by joining every vertex of a cycle C k with a new vertex.
In the following theorem we will deal with the super edge-magic deficiency of W n 4 -snake. Let us denote the vertex set and the edge set of W n 4 -snake such that Theorem 2.1. The graph W n 4 -snake has super edge-magic deficiency at most 1.

Proof. Let us denote the vertices and edges of
The graph G has 4n + 2 vertices and 8n edges.
We define the vertex labeling f of G in the following way It is easy to see that the vertices of G are labeled with the numbers 1, 2, 3, . . . , 4n + 2 as the sets of vertex labels are Next we will count the edge sums of the edges in the blocks. For i = 1, 2, . . . , n it holds It means that the edge sums are consecutive integers 3, 4, . . . , 8n + 2. According to Lemma 1.1 the labeling f can be extended to a super edge-magic total labeling of G with magic constant 12n + 5.
A graph is called a cactus graph if every block is either a cycle or a complete graph K 2 . Next we will deal with a special type of a cactus graph called an alternate quadrilateral snake. An alternate quadrilateral snake A(C n 4 ) is obtained from a path x 1 x 2 . . . x n by joining the vertices x i , x i+1 , for every odd i, to new vertices y i , y i+1 , respectively and then joining y i and y i+1 . That is every alternate edge of the path is replaced by a cycle Proof. Let n be an even positive integer. Let us denote the vertex set and the edge ). We define the vertex labeling of the graph G in the following way The remaining n 2 numbers n + 2, n + 5, . . . , 5n 2 − 1 are used to label the isolated vertices v 1 , v 2 , . . . , v n 2 of the graph G arbitrary. It is easy to see that f is a bijection from the vertex set of G onto the set of integers 1, 2, . . . , 5n 2 . For the edge sums we have the following. The edge sum of the edges x i y i , y i y i+1 , x i x i+1 and y i+1 x i+1 , for i = 1, 3, . . . , n − 1, are f (x i+1 ) + f (y i+1 ) = n + 3(i + 1) 2 + (i + 1) = n + 5i − 1 2 + 3.
A prism graph D n , sometimes also called a circular ladder graph, is a graph corresponding to the skeleton of an n-prism. Prism graphs are both planar and polyhedral. An n-prism graph consist of 2n vertices and 3n edges, which is equivalent to generalized Petersen graph P n,1 . The n-prism is isomorphic to circulant graph Ci 2n (2, n) for odd n, and can be showed by rotating the inner cycle by 180 • , and its radius is equal to that of the outer cycle in the top embedding above. In addition, for odd n, D n is isomorphic to Ci 2n (4, n), Ci 2n (6, n), . . . , Ci 2n (n − 1, n). The prism D n is isomorphic to the Cartesian product C n × K 2 , where C n is the cycle on n vertices and K 2 is the complete graph of order 2. The prism graph D n is equivalent to the Cayley graph of the dihedral group D n , with respect to the generating set {x, x −1 , y}.
We denote the vertices and edges of D n such that The cardinality of the vertex set and the edge set of D n is 2n and 3n, respectively. In [4] Figueroa-Centeno, Ichishima and Muntaner-Batle proved that for n odd the graph D n is super edge-magic total. Ngurah and Baskoro [9] showed that for n even the prism D n is not edge-magic total. In the following theorem we are dealing with the case when n is divisible by 4.
Proof. Let n be a positive integer, n ≡ 0 (mod 4). Let us denote the isolated vertices of G ∼ = D n ∪ ( 3n 2 − 1)K 1 by the symbols v 1 , v 2 , . . . , v 3n We define the vertex labeling f of G in the following way. . . . . . .
. . . Hence the edge sums are the numbers 7n 4 + 1, 7n 4 + 2, . . . , 19n 4 . According to Lemma 1.1 the labeling f can be extended to the super edge-magic total labeling of G with the magic constant 33n 4 .

Exact Values
If G has order p, the corona of G with H, denoted by G H, is the graph obtained by taking one copy of G and p copies of H and joining the ith vertex of G with an edge to every vertex in the ith copy of H.
Let us consider the Cartesian product P n × K 2 , where P n is the path on n vertices and K 2 is the complete graph of order 2. This graph is also called a ladder. In this section we deal with the super edge-magic deficiency of a ladder P n × K 2 with 1 pendant edge attached at each vertex of P n × K 2 , i.e., the corona (P n × K 2 ) K 1 .
Theorem 3.1. For every odd positive integer n the graph (P n × K 2 ) K 1 is super edge-magic total, i.e., Proof. Let n be a positive odd integer. We denote the vertex set and the edge set of G ∼ = (P n × K 2 ) K 1 as follows The graph G is of order 4n and of size 5n − 2.
For n ≥ 5 we define the vertex labeling f of G such that   Figure 2. A super edge-magic total labeling of (P 3 × K 2 ) K 1 .
of some graphs or to find the upper bound of this parameter for several classes of graphs. In Theorem 2.3 we described the upper bound of the super edge-magic deficiency of prism D n for n ≡ 0 (mod 4). As it is known, see [4], that for n odd the prism D n is super edge-magic. To conclude the problem of finding the super edge-magic deficiency of prism D n also for n even, for further investigation we state the following open problem. Open Problem. Find the super edge-magic deficiency of prism D n , for n ≡ 2 (mod 4).