Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 504251, 4 pages
http://dx.doi.org/10.1155/2015/504251

Research Article

Tree-Antimagicness of Disconnected Graphs

Martin Bača,¹ Zuzana Kimáková,¹ Andrea Semaničová-Feňovčíková,¹ and Muhammad Awais Umar²

¹Department of Applied Mathematics and Informatics, Technical University, 042 00 Košice, Slovakia
²Abdus Salam International Center of Mathematics, Information Technology University, Lahore, Pakistan

Received 10 October 2014; Accepted 5 January 2015

Academic Editor: Qing-Wen Wang

Copyright © 2015 Martin Bača et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A simple graph admits an -covering if every edge in belongs to a subgraph of isomorphic to . The graph is said to be (, )--antimagic if there exists a bijection from the vertex set and the edge set onto the set of integers such that, for all subgraphs of isomorphic to , the sum of labels of all vertices and edges belonging to constitute the arithmetic progression with the initial term and the common difference . is said to be a super (, )--antimagic if the smallest possible labels appear on the vertices. In this paper, we study super tree-antimagic total labelings of disjoint union of graphs.

1. Introduction

We consider finite undirected graphs without loops and multiple edges. The vertex and edge set of a graph are denoted by and , respectively. An edge-covering of is a family of subgraphs such that each edge of belongs to at least one of the subgraphs , . In this case we say that admits an -(edge) covering. If every subgraph is isomorphic to given graph , then the graph admits an -covering. A bijective function which we call a total labeling and the associated -weight of subgraph is . An --antimagic total labeling of graph admitting an -covering is a total labeling with the property that, for all subgraphs isomorphic to , the -weights form an arithmetic progression , where and are two integers and is the number of all subgraphs of isomorphic to . Such a labeling is called a super if the smallest possible labels appear on the vertices. A graph that admits a (super) --antimagic total labeling is called a (super) --antimagic. For , it is called -magic and -supermagic, respectively.

The -(super)magic labelings were first studied by Gutiérrez and Lladó [1]. These labelings are the generalization of the edge-magic and super edge-magic labelings that were introduced by Kotzig and Rosa [2] and Enomoto et al. [3], respectively. For further information about (super) edge-magic labelings, one can see [4–8]. Gutiérrez and Lladó [1] proved that certain classes of connected graphs are -(super)magic, such as the star and the complete bipartite graphs are -supermagic for some . They also proved that the cycle is -supermagic for any such that . Lladó and Moragas [9] studied the cycle-(super)magic behavior of several classes of connected graphs. They proved that wheels, windmills, books, and prisms are -magic for some . Maryati et al. [10] and also Salman et al. [11] proved that certain families of trees are path-supermagic. Ngurah et al. [12] proved that chains, wheels, triangles, ladders, and grids are cycle-supermagic.

The --antimagic total labeling is introduced by Inayah et al. [13]. In [14], the super --antimagic labelings are investigated for some shackles of connected graph .

When is isomorphic to , a super --antimagic total labeling is also called super -edge-antimagic total. The notion of -edge-antimagic total labeling was introduced by Simanjuntak et al. in [15] as a natural extension of the edge-magic labeling defined by Kotzig and Rosa in [2] as magic valuation.

The (super) --antimagic total labeling is related to a super -antimagic labeling of type of a plane graph. A labeling of type , that is, a total labeling, of a plane graph is said to be -antimagic if for every positive integer the set of weights of all -sided faces is for some integers and , where is the number of the -sided faces. Note that there are allowed different sets for different . The weight of a face under a labeling of type is the sum of labels of all the edges and vertices surrounding that face. If , then Lih [16] calls such labeling magic and describes magic (-antimagic) labelings of type for wheels, friendship graphs, and prisms. In [17], Ahmad et al. investigate the existence of super -antimagic labelings of type for disjoint union of plane graphs for several values of difference .

In this paper we mainly investigate the existence of super -tree-antimagic total labelings for disconnected graphs. We concentrate on the following problem: if graph admits a (super) -tree-antimagic total labeling, does the disjoint union of copies of the graph , denoted by , admit a (super) -tree-antimagic total labeling as well?

2. Super -Tree-Antimagic Total Labeling

In this section, we will study the super -tree-antimagicness for disconnected graphs. The main result is the following.

Theorem 1. Let be positive integers. Let with a -covering be a super --antimagic graph of order and size , , where is a tree and every tree , , is isomorphic to . Then the disjoint union is also a super --antimagic graph.

Proof. Let and be positive integers and let be a tree of order . Let , , be a graph with vertices and edges that admits a -covering. Note that is not necessarily isomorphic to for . Assume that every , , has a super --antimagic total labeling ; thus

is the set of the corresponding -weights, where every tree , , is isomorphic to .
Define the labeling for the vertices and edges of in the following way:

First we will show that the vertices of under the labeling , use integers from up to ; that is, if , then the vertices of successively attain values , if , then the vertices of successively assume values , the values of vertices of are equal successively to , and if , then the vertices of successively assume values .
Second we can see that the edges of under the labeling use integers from up to . It means that if , then the edges of successively assume values , if , then the edges of successively attain values , , the values of edges of are equal successively to , , , and if , then the edges of successively attain values , .
It is not difficult to see that the labeling is a bijective function which assigns the integer to the vertices and edges of ; thus is a total labeling.
For the weight of every subgraph isomorphic to the tree under the labeling , we have

As every , , , is isomorphic to the tree , it holds

Thus for the -weights we get

According to (1), we get that the -weights in components are the following:

 if , then the -weights in are , , and ; if , then the -weights in are , , and ; if , then the -weights in are , , and ;  if , then the -weights in successively attain values , , , and  ; if , then the -weights in successively assume values , , , and  .

It is easy to see that the set of all -weights in consists of distinct and consecutive integers:

Thus the graph is a super --antimagic.

Immediately from the previous theorem we get that arbitrary number of copies of a super --antimagic graph is a super --antimagic.

Corollary 2. Let be a super --antimagic graph, where is a tree. Then the disjoint union of arbitrary number of copies of , that is, , , also admits a super --antimagic total labeling.

Moreover, for copies of a graph which is --antimagic but is not super, we can derive the following result.

Theorem 3. Let be an --antimagic graph of order and size , where is a tree. Then , , is also a --antimagic graph.

Proof. Let be a tree of order and let be an --antimagic graph of order and size with corresponding labeling and the corresponding -weights .
For every vertex in , we denote by symbol the corresponding vertex in the th copy of in . Analogously, let be the edge corresponding to the edge in the th copy of in .
For , we define labeling of as follows:

Let . We consider two cases.
Case 1. If the number is assigned by the labeling to a vertex of , then the corresponding vertices , , in the copies in will receive labels

Case 2. If the number is assigned by the labeling to an edge of , then the corresponding edges , , in the copies in will have labels

We can see that the vertex labels and edge labels in are not overlapping, and the maximum used label is . Thus is a total labeling.
Analogously as in the proof of Theorem 1 we get that the weight of every subgraph , , , under the labeling attains the value

and the set of all -weights in successively attain consecutive values . Thus the resulting labeling is a --antimagic total labeling.

3. Disjoint Union of Certain Families of Graphs

In [18], the following results are proved on path-antimagicness of cycles and paths.

Proposition 4 (see [18]). Let and be positive integers. The cycle is a super --antimagic for every .

Proposition 5 (see [18]). Let and be positive integers. The path is a super --antimagic for every .

In light of Propositions 4 and 5 and Corollary 2 we immediately obtain the following.

Corollary 6. Let , , and be positive integers. Then copy of a cycle , that is, the graph , is a super --antimagic for every .

Corollary 7. Let , , and be positive integers. Then copy of a path , that is, the graph , is a super --antimagic for every .

4. Conclusion

In this paper we have shown that the disjoint union of multiple copies of a (super) -tree-antimagic graph is also a (super) -tree-antimagic. It is a natural question whether the similar result holds also for other differences and other -antimagic graphs. For further investigation we propose the following open problems.

Open Problem 1. Let be a (super) --antimagic graph, where is a tree different from . For the graph , determine if there is a (super) --antimagic total labeling, for and all .

Open Problem 2. Let be a (super) --antimagic graph. For the graph , determine if there is a (super) --antimagic total labeling, for certain values of and all .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work was supported by Slovak VEGA Grant 1/0056/15.

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