Tree-Antimagicness of Disconnected Graphs
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.
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 . These labelings are the generalization of the edge-magic and super edge-magic labelings that were introduced by Kotzig and Rosa  and Enomoto et al. , respectively. For further information about (super) edge-magic labelings, one can see [4–8]. Gutiérrez and Lladó  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  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.  and also Salman et al.  proved that certain families of trees are path-supermagic. Ngurah et al.  proved that chains, wheels, triangles, ladders, and grids are cycle-supermagic.
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  as a natural extension of the edge-magic labeling defined by Kotzig and Rosa in  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  calls such labeling magic and describes magic (-antimagic) labelings of type for wheels, friendship graphs, and prisms. In , 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 , the following results are proved on path-antimagicness of cycles and paths.
Proposition 4 (see ). Let and be positive integers. The cycle is a super --antimagic for every .
Proposition 5 (see ). Let and be positive integers. The path is a super --antimagic for every .
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 .
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.
The work was supported by Slovak VEGA Grant 1/0056/15.
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