Deficiency of forests

Abstract An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) onto the integers {1,2,…,n + m} with the property that there exists an integer constant c such that λ(x) + λ(y) + λ(xy) = c for any xy ∈ E(G). It is called super edge-magic total labeling if λ (V(G)) = {1,2,…,n}. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edge-magic total labeling, called super edge-magic deficiency of a graph G, is denoted by μs(G) [4]. If such vertices do not exist, then deficiency of G will be + ∞. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].


Basic definitions, notations and preliminary results
Let G D .V; E/ be a finite, simple, undirected graph having jV .G/j D n and jE.G/j D m, where V .G/ and E.G/ denote the vertex set and edge set, respectively. A general orientation for graph theoretic concepts can be seen in [10]. A labeling (or valuation) of a graph is a map that carries graph elements to numbers (usually to positive integers). A labeling that uses the vertex set only (or the edge set only), is known as vertex labeling (or the edge labeling). If the domain of the labeling includes all vertices and edges, then such a labeling is called total labeling. Cordial, graceful, harmonious and anti-magic are few types of labeling. A bijective labeling is called an edge-magic total if it satisfies the following property, given any edge xy 2 E.G/, .x/ C .y/ C .xy/ D c; (1) for some constant c. In other words, an edge-magic total labeling of a graph G is a bijective map from V .G/[E.G/ onto the integers f1; 2; ; : : : ; n C mg satisfying (1). The constant c is known as the magic constant and a graph that admits an edge-magic total labeling is called an edge-magic total graph. In [8,9], Kotzig and Rosa have given the origin of edge-magic total labeling of graphs. Recently, Enomoto et al. [1] brought in the name, super edge-magic labeling in the sense of Kotzig and Rosa, with the additional property that the vertices receive the smallest labels. In [1] they put forward the following conjecture: . "Every tree is super edge-magic total." In this paper we are focused on super edge-magic total labeling. A number of classification problems on edge-magic total graphs have been extensively investigated. For further details see recent survey of graph labelings [6]. Kotzig and Rosa in [9] show that there exists an edge-magic total graph H for any graph G such that H Š G [ nK 1 for some non-negative integer n. This verity provides the base for the concept of edge-magic total deficiency of a graph G [9], denoted by .G/, which is the minimum non-negative integer n such that G [ nK 1 is edge-magic total i.e., .G/ D mi nfn 0 W G [ nK 1 i s edge-magic totalg: In the same paper Kotzig and Rosa provide an upper bound of the edge-magic deficiency of a graph G having order n, .G/ Ä F nC2 2 n 1 2 n.n 1/; where F n denotes the n-th Fibonacci number. The super edge-magic deficiency of a graph G, denoted by s .G/ [4], is mathematically expressed as if It is easy to see that .G/ Ä s .G/. In [2], Figueroa-Centeno et al. conjectured, "Every forest with two components has the super edge-magic deficiency at most 1". Moreover, in the same paper they showed that s .P m [ K 1;n / is 1 if m D 2 and n is odd or m D 3 and n 6 Á 0.mod 3/ and 0 otherwise. In [7], S. Javed et al. gave the upper bound of deficiencies of disjoint union of graphs consisting of comb, generalized comb and star. In this paper, we frequently use the following two Lemmas.

Lemma 1.2 ([5])
. "A graph G with n-vertices and m-edges is super edge-magic total if and only if there exists a bijection W V .G/ ! f1; 2; : : : ; ng such that the set S D f .x/ C .y/jxy 2 E.G/g consists of m consecutive integers. In such a case, extends to a super edge-magic total labeling of G." The above condition is often easier to use than the original one. The following lemma was found first in [1].

Lemma 1.3 ([1]
). "If a graph G with n vertices and m edges is super edge-magic total then m Ä 2n 3."

Definition 1.4.
A comb is a graph derived from the path P n W u 1 ; u 2 ; : : : ; u n ; n 3, by adding n 1 new edges u iC1 w i I 1 Ä i Ä n 1 and this is denoted by C b n .

Definition 1.5.
A two-sided generalized comb, denoted by C b 2 n;m , consists of the vertex set, V .C b 2 n;m / D fu i;j I 1 Ä i Ä n; 1 Ä j Ä mg [ fu 0; mC1 2 g and the edge set, i.e., C b 2 n;m is deduced from n paths P i;m W u i;1 ; u i;2 ; : : : ; u i;m I u i;j u i;j C1 2 E.C b 2 n;m /I 1 Ä i Ä nI 1 Ä  The graph obtained by C b 2 n;m by deleting the set of vertices fu i;j I 1 Ä i Ä n; mC3 2 Ä j Ä mg and their adjacent edges is referred to as generalized comb, denoted by C b n .l; l; : : : ; l " ƒ‚ … n times /. The labeling of C b n .l; l; : : : ; l " ƒ‚ … n times / is discussed in [7]. Definition 1.6. A bistar on n vertices, denoted by BS.p; q/I p; q 1; p C q C 2 D n, is obtained from two stars K 1;p and K 1;q by joining their central vertices by an edge.
In this paper we formulate the super edge-magic total labeling of two sided generalized comb. Moreover, we determine an upper bound for super edge-magic total deficiency of forests containing comb, bistar and 2-sided generalized comb.

Super edge-magic deficiencies of forests of combs and bistar
In this section, we will provide precise value for super edge-magic deficiency of some specific number of copies of the comb C b n , we will also give an upper bound for super edge-magic deficiency for disjoint union of bistar BS.k; k/ and C b n with some restrictions on the parameters k and n.
Theorem 2.1. For n-odd, n 3, m-even and m Á 2.mod 4/, the graph G Š mC b n is super edge magic total.
Proof. Consider the graph G Š mC b n . Then jV .G/j D m.2n 1/ and jE.G/j D m.2n 2/, where Define a labeling f W V .G/ ! f1; 2; : : : ; m.2n 1/g as follows: The labeling f gives the following set of consecutive integers f 4mn mC6 4 ; 4mn mC10 4 ; : : : ; 12mn 9mC2 4 g that appears as the weights of the edges in the graph. g.u k n / D where the labeling f is defined in In the next Theorem we will compute an upper bound for the edge-magic deficiency of a forest consisting of bistar BS.k; k/ and comb C b n .

Theorem 2.3.
For k 2, consider the graph G Š BS.k; k/ [ C b n . Then 1. The graph G is super edge-magic total for n k C 2 and k-odd. 2. s .G/ Ä 1 for n k C 3 and k-even.
Proof. Consider the graph G Š BS.k; k/ [ C b n . We have which give jV .G/j D 2.n C k/ C 1 and E.G/ D 2.n C k/ 1.

1.
Define a labeling f W V .G/ ! f1; 2; : : : ; jV .G/jg in the following way: The set of edge weights formed under the labeling f consists of the following consecutive integers fn C k C 3; n C k C 4; : : : ; 3.n C k/ C 1g.

Consider the graph H
Define a labeling g W V .H / ! f1; 2; : : : ; 2.n C k C 1/g as follows: g.u/ D 2 The labeling g gives the following set of consecutive integers fn C k C 5; n C k C 6; : : : ; 3.n C k C 1/g as the edge weights. Proof. Consider the graph G Š C b 2 n;m . Then jV .G/j D mn C 1 and jE.G/j D mn, where

Super edge-magic total labeling of two-sided comb
To show that G is super edge-magic total, we will define a labeling f W V .G/ ! f1; 2; : : : ; mn C 1g as follows: if i Á 0.mod 2/ and j Á 1.mod 2/ for 2 Ä i Ä n and 1 Ä j Ä m.
The set of edge weights given by the labeling f consists of the following mn consecutive integers fd mn 2 e C 3; d mn 2 e C 4; : : : ; d 3mn 2 e C 2g. if i Á 0.mod 2/ and j Á 0.mod 2/ for 2 Ä i Ä n and 2 Ä j Ä m 1.
The set of edge weights formed under the labeling f is fb mn 2 c C 3; b mn 2 c C 4; : : : ; b 3mn 2 c C 2g. Proof. Define the graph G Š 2C b 2 n;m in the following way:

Super edge-magic deficiency of copies of two-sided comb
Then jV .G/j D 2.mn C 1/ and jE.G/j D 2mn. . .
The set of edge weights under the labeling g is fmn C 6; mn C 7; : : : ; 3mn C 5g.

Concluding remarks
In [3], Figueroa-Centeno et al. discovers that if a graph is super edge-magic, then an odd number of copies of the graph is also super edge-magic. In this paper, we extend this concept for an even number of copies of comb, so the result in [3] significantly generalizes our results. It is also shown that the two-sided generalized comb, denoted by C b 2 n;m is super edge-magic total. Moreover we have found upper bounds for the super edge-magic deficiency of forests mC b n , C b n [ BS.k; k/ and 2C b 2 n;m for different values of the parameters k; m and n. In this context we formulate some open problems: