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`Journal of Applied MathematicsVolume 2015, Article ID 320616, 7 pageshttp://dx.doi.org/10.1155/2015/320616`
Research Article

On -Vertex-Antimagic Edge Labeling of Regular Graphs

1Department of Applied Mathematics and Informatics, Technical University, Letná 9, 04200 Košice, Slovakia
2Department of Applied Mathematics, Tunghai University, Taichung 40704, Taiwan
3Department of Applied Mathematics, National Chung Hsing University, Taichung 402, Taiwan

Received 1 April 2015; Accepted 26 May 2015

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

An -vertex-antimagic edge labeling (or an -VAE labeling, for short) of is a bijective mapping from the edge set of a graph to the set of integers with the property that the vertex-weights form an arithmetic sequence starting from and having common difference , where and are two positive integers, and the vertex-weight is the sum of the labels of all edges incident to the vertex. A graph is called -antimagic if it admits an -VAE labeling. In this paper, we investigate the existence of -VAE labeling for disconnected 3-regular graphs. Also, we define and study a new concept -vertex-antimagic edge deficiency, as an extension of -VAE labeling, for measuring how close a graph is away from being an -antimagic graph. Furthermore, the -VAE deficiency of Hamiltonian regular graphs of even degree is completely determined. More open problems are mentioned in the concluding remarks.

1. Background and Introduction

All graphs in this paper are finite simple, undirected, and possibly disconnected, but without any isolated vertex or any isolated edge. For graph theoretic notations, we follow [1]. Over the past few decades, many kinds of graph labelings have been studied intensively, and an excellent survey of graph labeling can be found in Gallian’s paper [2].

Hartsfield and Ringel in [3] introduced the concept of an antimagic labeling. In their terminology, a -graph with vertices and edges is called antimagic if its edges are labeled with labels in such a way that all vertex-weights are pairwise distinct, where a vertex-weight of vertex is the sum of labels of all edges incident with . Hartsfield and Ringel [3] pointed out that antimagic graphs include paths , , cycles, wheels, and complete graphs , . They conjecture that every connected graph, except , is antimagic. Alon et al. [4] used several probabilistic tools and some techniques from analytic number theory to show that this conjecture is true for all graphs having minimum degree .

In 1993, Bodendiek and Walther [5] investigated antimagic labelings with certain restriction placed on the vertex-weights. They defined the concept of an -vertex-antimagic edge labeling as follows.

Definition 1. An -vertex-antimagic edge labeling (or an -VAE labeling for short) of a -graph is a bijective mapping from the edge set of a graph to the set of integers with the property that the vertex-weights form an arithmetic sequence starting from and having common difference , where and are two positive integers. The vertex-weight of the vertex is the sum of the labels of all edges incident with the vertex under the mapping . A graph is called -antimagic if it admits an -VAE labeling.

Bodendiek and Walther in [6, 7] proved that the Herschel graph is not -VAE and obtained both positive and negative results about -VAE labelings for various cases of graphs called parachutes . They investigated -VAE labelings for paths, cycles, and complete graphs in [8]. Characterization of all -antimagic graphs of the prism when is even is given in [9]. In [10, 11] the -VAE labelings for antiprisms have been investigated. It is proved in [12] that the generalized Petersen graph has an -VAE labeling if and only if is even, , , and . Nicholas et al. [13] obtained several results about -VAE labelings for caterpillars, unicyclic graphs, and complete bipartite graphs.

On the other hand, in 2002, MacDougall et al. [14] introduced the concept of vertex magic total labeling as follows. If is a finite simple undirected graph with vertices and edges, then a vertex magic total labeling is a bijection from to the integers with the property that, for every , the sum is a constant. Note that, for regular graphs, the vertex magic total labeling is equivalent to the -VAE labeling, while the vertices (edges) are assigned the smallest labels. More recently, Wang and Zhang [15] verified the existence of -VAE labeling for particular classes of 3-regular graph , where contains a 1-factor and a 2-factor which consists of two 2-regular subgraphs with equal size. These results generalize and contain previous known examples such as Generalized Petersen Graphs.

The following theorem was proved in [16] by Ivančo and Semaničová, which guarantees the existence of the -VAE-ness by adding an arbitrary even factor to an arbitrary -antimagic graph.

Theorem 2 (see [16]). Let the graph admit an -VAE labeling, and let be any -factor over . Then, still admits an -VAE labeling for some .

Therefore in order to study the -antimagicness of a general regular graph, based upon Theorem 2 and the fact, pointed out by Petersen [17], that any regular graph of even degree has a 2-factorization, it is natural to explore the -antimagicness of 2-regular graphs and 3-regular graphs, respectively. In this paper, we in particular investigate the existence of -VAE labeling for disconnected 3-regular graphs and also define a new concept -VAE deficiency, as an extension of -VAE labeling, for studying those (regular) graphs not admitting an -VAE labeling. The -VAE deficiency is a parameter to measure how close a graph is from being -antimagic. We notice that the method employed here is also valid for those graphs with multiple edges and loops. More examples and open problems will be provided in Section 5.

2. Preliminary Results

Suppose is a -graph with vertices and edges. The following are necessary conditions for graphs to admit an -VAE labeling and -VAE labeling in particular.

Lemma 3. If a -graph is -antimagic, then .

Proof. Consider that the total sum of all vertex-weights in -graph is by two-way counting, and it implies that .

By Lemma 3 and for , in case , then . Note that is a positive integer, which implies that is odd. Furthermore, in the case , one has , which implies that is odd. Therefore, we have the following properties for -VAE-ness.

Corollary 4. Let be a -graph. If is -antimagic, then is odd.

Corollary 5. Trees of even order are not -antimagic. In particular, a path of even order is not -antimagic.

Corollary 6. Let be a -graph. If is -antimagic, then is odd.

Corollary 7. The even cycle is not -antimagic.

Proposition 8. Let be an -regular graph of order and let admit an -VAE labeling. Then, we have the following: (1)If is odd, then .(2)If is even, then .

Proof . Let have edges. Then, by hand-shaking lemma, we have that . Since admits an -VAE labeling, by Lemma 3, we have . Combining these equations, one has that , which must be an integer. Therefore, it follows that if is odd, then , and if is even, then .

3. -VAE for 3-Regular Graphs

In this section, we study the -VAE labeling of disjoint union of 3-regular graphs. The main result is the following.

Theorem 9. Let be an -antimagic 3-regular graph having a perfect matching. Then, the disjoint union of arbitrary number of copies of , that is, , , is also a -antimagic graph.

Proof. Let be a 3-regular graph with vertices having a perfect matching.
Suppose that admits an -VAE labeling such that As contains a perfect matching, we can divide the edges of into two subsets and , such that where the subset consists of all edges belonging to the perfect matching.
For every vertex in , we denote by symbol the corresponding vertex in the th copy of in . Analogously, let denote the corresponding edge in the th copy of in .
Define the labeling for the edges of in the following way: Let . We consider two cases.
Case 1. If the number is assigned by the labeling to an edge from , then the corresponding edges in the copies in will receive labels under the labeling :Case 2. If the number is assigned by the labeling to an edge from , then the corresponding edges in the copies in will have the following labels under the labeling :It is easy to see that the edge labels in are not overlapping; thus, the labeling is a bijective function which assigns the integers to the edges of ; thus, is an edge labeling.
As is a 3-regular graph with a perfect matching, then every vertex in is incident to exactly one edge from and exactly two edges from . For a given vertex , let be the three edges incident to the vertex , where means that the edge belongs to the subset , .
For the vertex-weight of , we have As we get that the vertex-weights under the labeling in the components areThe reader can easily verify that the vertex-weights are distinct and consecutive: This means that has a -VAE labeling.

It is possible to extend the result from the previous theorem also for the disjoint union of arbitrary 3-regular graphs having a perfect matching that satisfy certain additional conditions.

We will follow the notation used in the proof of Theorem 9.

Theorem 10. Let be an -antimagic 3-regular graph of order having a perfect matching, . Let the set of all edge labels belonging to the perfect matching under the -VAE labeling of a graph be the same for every , .
Then, the disjoint union is also a -antimagic graph.

Proof. Let , , be a 3-regular graph of order having a perfect matching, , and note that is not necessarily isomorphic to for .
For each , , there exists an -VAE labeling such that the set of all edge labels belonging to the perfect matching under the -VAE labeling of a graph is the same for every graph . This means thatWe define the labeling for the edges of in the following way: Let . We consider two cases.
Case 1. If the number is assigned by the labeling to an edge from , then the corresponding edge in will receive the following label under the labeling :Case 2. If the number is assigned by the labeling to an edge from , then the corresponding edge in will receive this label under the labeling :It is easy to see that the edge labels in are not overlapping; thus, the labeling is a bijective function which assigns the integer to the edges of ; thus, is an edge labeling.
Moreover, analogously as in the proof of Theorem 9, we get that the vertex-weights form an arithmetic sequence with a difference 1. This produces the desired result.

4. -VAE Deficiency of Even Regular Graphs

We start this section by defining a new concept -vertex-antimagic edge deficiency for a -graph as follows. It is a parameter to study furthermore those graphs which do not admit any -VAE labeling and to measure how close they are away from being an -antimagic graph.

Definition 11. The -vertex-antimagic edge deficiency (or -VAE deficiency for short) is defined as the such that the edge labeling is -VAE. The -VAE deficiency of the graph is denoted by . Note that if is -antimagic and if no such exists for the graph to be -antimagic.

In the following, we determine completely the -VAE deficiency of paths and cycles. Also, as a corollary, the -VAE deficiency of Hamiltonian regular graphs of even degree is obtained. First, we have the following two lemmas giving -VAE labelings for odd cycles and paths; see also [8].

Lemma 12. The cycle is -antimagic for .

Proof. Given a notation for with and , we give an edge labeling for with and it implies the vertex-weight at as follows: Hence, is -antimagic.

Lemma 13. The path is -antimagic for .

Proof. Given a notation for with and , we give an edge labeling for such that and it implies the vertex-weight at as follows:Hence, is -antimagic.

Here, we have a general observation that every graph of order , where , can not be made -antimagic.

Lemma 14. Let be a graph of order , where . Then, .

Proof. Let have edges and vertices. Assign labels to edges of graph and suppose the existence of an -VAE labeling with the associated vertex-weights . Consider the sum of all vertex-weights: which is also equal to . Note that and are both even, but is odd, a contradiction.

In the following lemmas, we are dealing with the -VAE deficiency of cycles and paths.

Lemma 15. Consider for .

Proof. First, we find the missing value in the set of edge labels . Note that and it implies and thus . Suppose that , and then one has since . Therefore, must be or , and it implies that the missing value must be or . We show that for both missing values there exist -VAE labelings.
Let the vertex set and the edge set of be and .
Case 1. If the missing value is , then the -VAE labeling can be defined as follows: Case 2. If the missing value is , then, for example, consider the following edge labeling: Then, we find that the vertex-weights form the set for Case 1 and the set for Case 2. Hence, as required.

Lemma 16. Consider for .

Proof. First, we suppose that and we want to find the missing value in the set of edge labels . Then, and it implies that and thus . Suppose that , and then , since . Therefore, must be or and it implies that the missing value is or . As in the proof of the previous lemma we show that for both cases it is possible to find required -VAE labelings.
We denote the vertices and the edges of such that and .
Case 1. If the missing value is , then we define the edge labeling in the following way: Case 2. If the missing value is , then we define the labeling such that Then, we obtain that the vertex-weights attain the values from the sets for Case 1 and for Case 2. Thus, .

To summarize, we have the following -VAE deficiency for paths and cycles .

Theorem 17. Let . Then,

Theorem 18. Let . Then,

Therefore, as a corollary of Theorem 18, we immediately have the following result.

Theorem 19. Let be a -regular, , Hamiltonian graph of order . Then,

Proof. By Theorem 18, Lemma 12, Theorem 2, and also Petersen’s 2-factorization of regular graphs of even degree, we get that a Hamiltonian even regular graph of odd order is -antimagic. On the other hand, by Lemma 15 and Theorem 2, we get that the Hamiltonian even regular graphs of order have the -VAE deficiency equal to 1. Finally, by Lemma 14, we complete the proof.

As a corollary, we have the following example of -VAE deficiency for the Cartesian product of two cycles . Note that always contains a Hamiltonian cycle.

Corollary 20. Let . Then,

Note that similarly one may have the formula for -VAE deficiency of the higher dimensional toroidal grids, that is, the Cartesian product of cycles , for each .

5. Concluding Remarks and Further Studies

Notice that in 2009 Holden et al. [18] raised a conjecture for the existence of -VAE labeling of 2-regular graphs as follows: a 2-regular graph of odd order possesses an -VAE labeling if and only if it is not one of , , or . Note that the terminology of the labeling they made is the strong vertex magic total labeling, which is exactly equivalent to the -VAE labeling. Therefore, it is natural to ask for the following.

Problem 1. Determine the -VAE deficiency for , , and .

Moreover, as a generalization of the above result, we obtain in the last section the -VAE deficiency of a Hamiltonian regular graph of even degree, and we are concerned with the following situation. Note that Swaminathan and Jeyanthi [19] pointed out the following: is -antimagic if and only if are odd. Therefore, it is natural to ask for the following.

Problem 2. Determine the -VAE deficiency for the 2-regular graph .

If Problem 2 is answered, then the -VAE deficiency for an arbitrary regular graph of even degree containing a 2-factor is answered. More generally, we have the following.

Problem 3. Determine the -VAE deficiency for a general 2-regular graph.

If Problem 3 is answered, then the -VAE deficiency for an arbitrary regular graph of even degree is answered. As for 3-regular graphs and general odd regular graphs, we ask for the following.

Problem 4. Determine the -VAE deficiency for 3-regular Generalized Petersen Graphs and Mbius Ladder Graphs .

Problem 5. Determine the -VAE deficiency for a general 3-regular graph.

Problem 6. Determine the -VAE deficiency for a general odd regular graph.

It is not hard to check that does not admit any -VAE labeling. However, with the aid of computer programs, we have found that admits an -VAE labeling for . This leads to the following conjecture.

Conjecture 21. , the disjoint union of copies of , admits the -VAE labeling for .

Conflict of Interests

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

Acknowledgment

Tao-Ming Wang’s research is supported partially by the Grant no. MOST 103-2115-M-029-001 from the Ministry of Science and Technology of Taiwan.

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