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Journal of Applied Mathematics
Volume 2013, Article ID 896815, 6 pages
http://dx.doi.org/10.1155/2013/896815
Research Article

On Super -Edge-Antimagic Total Labeling of Special Types of Crown Graphs

1FAST-National University of Computer and Emerging Sciences, Peshawar 25000, Pakistan
2Department of Applied Mathematics and Informatics, Technical University, 042 00 Košice, Slovakia

Received 8 April 2013; Accepted 9 June 2013

Academic Editor: Zhihong Guan

Copyright © 2013 Himayat Ullah 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

For a graph , a bijection from is called -edge-antimagic total (-EAT) labeling of if the edge-weights , form an arithmetic progression starting from and having a common difference , where and are two fixed integers. An -EAT labeling is called super -EAT labeling if the vertices are labeled with the smallest possible numbers; that is, . In this paper, we study super -EAT labeling of cycles with some pendant edges attached to different vertices of the cycle.

1. Introduction

All graphs considered here are finite, undirected, and without loops and multiple edges. Let be a graph with the vertex set and the edge set . For a general reference of the graph theoretic notions, see [1, 2].

A labeling (or valuation) of a graph is a map that carries graph elements to numbers, usually to positive or nonnegative integers. In this paper, the domain of the map is the set of all vertices and all edges of a graph. Such type of labeling is called total labeling. Some labelings use the vertex-set only, or the edge-set only, and we will call them vertex labelings or edge labelings, respectively. The most complete recent survey of graph labelings can be seen in [3, 4].

A bijection is called -edge-antimagic vertex (-EAV) labeling of if the set of edge-weights of all edges in is equal to the set , where and are two fixed integers. The edge-weight of an edge under the vertex labeling is defined as the sum of the labels of its end vertices; that is, . A graph that admits -EAV labeling is called an -EAV graph.

A bijection is called -edge-antimagic total (-EAT) labeling of if the set of edge-weights of all edges in forms an arithmetic progression starting from with the difference , where and are two fixed integers. The edge-weight of an edge under the total labeling is defined as the sum of the edge label and the labels of its end vertices. This means that. Moreover, if the vertices are labeled with the smallest possible numbers, that is, , then the labeling is called super -edge-antimagic total (super -EAT). A graph that admits an -EAT labeling or a super -EAT labeling is called an -EAT graph or a super -EAT graph, respectively.

The super -EAT labelings are usually called super edge-magic; see [58]. Definition of super -EAT labeling was introduced by Simanjuntak et al. [9]. This labeling is a natural extension of the notions of edge-magic labeling; see [7, 8]. Many other researchers investigated different types of antimagic graphs. For example, see Bodendiek and Walther [10], Hartsfield and Ringel [11].

In [9], Simanjuntak et al. defined the concept of -EAV graphs and studied the properties of -EAV labeling and -EAT labeling and gave constructions of -EAT labelings for cycles and paths. Bača et al. [12] presented some relation between -EAT labeling and other labelings, namely, edge-magic vertex labeling and edge-magic total labeling.

In this paper, we study super -EAT labeling of the class of graphs that can be obtained from a cycle by attaching some pendant edges to different vertices of the cycle.

2. Basic Properties

Let us first recall the known upper bound for the parameter of super -EAT labeling. Assume that a graph has a super -EAT labeling , . The minimum possible edge-weight in the labeling is at least . Thus, . On the other hand, the maximum possible edge-weight is at most . So

Thus, we have the upper bound for the difference . In particular, from (2), it follows that, for any connected graph, where , the feasible value is no more than .

The next proposition, proved by Bača et al. [12], gives a method on how to extend an edge-antimagic vertex labeling to a super edge-antimagic total labeling.

Proposition 1 (see [12]). If a graph has an -EAV labeling, then (i)has a super -EAT labeling, and(ii) has a super -EAT labeling.

The following lemma will be useful to obtain a super edge-antimagic total labeling.

Lemma 2 (see [13]). Let be a sequence , even. Then, there exists a permutation of the elements of , such that .

Using this lemma, we obtain that, if is an -EAV graph with odd number of edges, then is also super -EAT.

3. Crowns

If has order , the corona of with , denoted by , is the graph obtained by taking one copy of and copies of and joining the th vertex of with an edge to every vertex in the th copy of . A cycle of order with an pendant edges attached at each vertex, that is, , is called an -crown with cycle of order . A -crown, or only crown, is a cycle with exactly one pendant edge attached at each vertex of the cycle. For the sake of brevity, we will refer to the crown with cycle of order simply as the crown if its cycle order is clear from the context. Note that a crown is also known in the literature as a sun graph. In this section, we will deal with the graphs related to -crown with cycle of order .

Silaban and Sugeng [14] showed that, if the -crown is -EAT, then . They also describe -EAT, labeling of the -crown for and . Note that the -EAT labeling of -crown presented in [14] is super -EAT. Moreover, they proved that, if , and are odd; there is no -EAT labeling for . They also proposed the following open problem.

Open Problem 1 (see [14]). Find if there is an -EAT labeling, for -crown graphs .

Figueroa-Centeno et al. [15] proved that the -crown graph has a super -EAT labeling.

Proposition 3 (see [15]). For every two integers and , the -crown is super -EAT.

According to inequality (2), we have that, if the crown is super -EAT, then . Immediately from Proposition 3 and the results proved in [14], we have that the crown is super -EAT and super -EAT for every positive integer . Moreover, for odd, the crown is not -EAT. In the following theorem, we prove that the crown is super -EAT for even. Thus, we partially give an answer to Open Problem 1.

Theorem 4. For every even positive integer , , the crown is super -EAT.

Proof. Let be the vertex set and be the edge set of ; see Figure 1.
We define a labeling , as follows:
It is easy to check that is a bijection. For the edge-weights under the labeling , we have Thus, the edge-weights are distinct number from the set . This means that is a super -EAT labeling of the crown .

896815.fig.001
Figure 1: The crown .

In [16], was proved the following.

Proposition 5 (see [16]). Let be a super -EAT graph. Then, the disjoint union of arbitrary number of copies of , that is, , , also admits a super -EAT labeling.

In [17], Figueroa-Centeno et al. proved the following.

Proposition 6 (see [17]). If is a (super) -EAT bipartite or tripartite graph and is odd, then is (super) -EAT.

For (super) -EAT labeling for disjoint union of copies of a graph, was shown the following.

Proposition 7 (see [18]). If is a (super) -EAT bipartite or tripartite graph and is odd, then is (super) -EAT.

Using the above mentioned results, we immediately obtain the following theorem.

Theorem 8. Let be an even positive integer, . Then, the disjoint union of arbitrary number of copies of the crown , that is, , , admits a super -EAT labeling.

Since the crown is either a bipartite graph, for even, or a tripartite graph, for odd, thus, we have the following.

Theorem 9. Let be a positive integer, . Then, the disjoint union of odd number of copies of the crown , that is, , , admits a super -EAT labeling for .

According to these results we are able to give partially positive answers to the open problems listed in [19].

Open Problem 2 (see [19]). For the graph , even, and , determine if there is a super -EAT labeling with .

Open Problem 3 (see [19]). For the graph , odd, and even, determine if there is a super -EAT labeling.

4. Graphs Related to Crown Graphs

Let us consider the graph obtained from a crown graph by deleting one pendant edge.

Theorem 10. For odd, , the graph obtained from a crown graph by removing a pendant edge is super -EAT for .

Proof. Let be a graph obtained from a crown graph by removing a pendant edge. Without loss of generality, we can assume that the removed edge is . Other edges and vertices we denote in the same manner as that of Theorem 4. Thus and . It follows from (2) that .
Define a vertex labeling as follows: It is easy to see that labeling is a bijection from the vertex set to the set . For the edge-weights under the labeling , we have Thus, the edge-weights are consecutive numbers . This means that is the -EAV labeling of . According to Proposition 1, the labeling can be extended to the super -EAT and the super -EAT labeling of . Moreover, as the size of is odd, , and according to Lemma 2, we have that is also super -EAT.

Note that we found some -EAV labelings for the graphs obtained from a crown graph by removing a pendant edge only for small size of , where is even. In Figure 2, there are depicted -EAV labelings of these graphs for . However, we propose the following conjecture.

896815.fig.002
Figure 2: The -EAV labelings for the graphs obtained from a crown graph by removing one pendant edge for . The edge-weights under the corresponding labelings are depicted in italic.

Conjecture 11. The graph obtained from a crown graph , , by removing a pendant edge is super -EAT for .

Now, we will deal with the graph obtained from a crown graph by removing two pendant edges at distance 1 and at distance 2.

First, consider the case when we remove two pendant edges at distance . Let us consider the graph with the -EAV labeling defined in the proof of Theorem 10. It is easy to see that the vertex is labeled with the maximal vertex label, . Also, the edge-weight of the edge is the maximal possible, . Thus, it is possible to remove the edge from the graph ; we denote the graph by , and for the labeling restricted to the graph , we denote it by . Clearly, is -EAV labeling of . According to Proposition 1 from the labeling , we obtain the super -EAT and the super -EAT labeling of . Note that, as the size of is even, , the labeling can not be extended to a super -EAT labeling of .

Theorem 12. For odd , , the graph obtained from a crown graph by removing two pendant edges at distance is super -EAT for .

Next, we show that, if we remove from a crown graph , , two pendant edges at distance , the resulting graph is super -EAT for .

Theorem 13. For odd , , the graph obtained from a crown graph by removing two pendant edges at distance is super -EAT for .

Proof. Let be a graph obtained from a crown graph by removing two pendant edges at distance . Let , , be the vertex set and , , be the edge set of . Thus, and . Using (2) gives .
Define a vertex labeling in the following way: It is not difficult to check that the labeling is a bijection from the vertex set of to the set and that the edge-weights under the labeling are consecutive numbers . Thus is the -EAV labeling of . According to Proposition 1, it is possible to extend the labeling to the super -EAT and the super -EAT labelings of .

Result in the following theorem is based on the Petersen Theorem.

Proposition 14 (Petersen Theorem). Let be a -regular graph. Then, there exists a -factor in .

Notice that, after removing edges of the -factor guaranteed by the Petersen Theorem, we have again an even regular graph. Thus, by induction, an even regular graph has a -factorization.

The construction in the following theorem allows to find a super -EAT labeling of any graph that arose from an even regular graph by adding even number of pendant edges to different vertices of the original graph. Notice that the construction does not require the graph to be connected.

Theorem 15. Let be a graph that arose from a -regular graph , , by adding pendant edges to different vertices of , . If is an even integer, then the graph is super -EAT.

Proof. Let be a graph that arose from a -regular graph , , by adding pendant edges to different vertices of , and let . Thus, , , and .
Let be an even integer. Let us denote the pendant edges of by symbols . We denote the vertices of such that and moreover We denote the remaining vertices of arbitrarily by the symbols .
By the Petersen Theorem, there exists a -factorization of . We denote the -factors by , . Without loss of generality, we can suppose that for all , and . Each factor is a collection of cycles. We order and orient the cycles arbitrarily such that the arcs form oriented cycles. Now, we denote by the symbol the unique outgoing arc that forms the vertex in the factor , . Note that each edge is denoted by two symbols.
We define a total labeling of in the following way:
It is easy to see that the vertices are labeled by the first integers, the edges by the next labels, and the edges of by consecutive integers starting at . Thus, is a bijection .
It is not difficult to verify that is a super -EAT labeling of . For the weights of the edges , , we have Thus, the edge-weights are the numbers For convenience, we denote by the unique vertex such that in , where . The weights of the edges in , , are for all , and . Since is a factor in it holds . Hence, we have that the set of the edge-weights in the factor is and thus, the set of all edge-weights in under the labeling is

We conclude the paper with the result that immediately follows from the previous theorem.

Corollary 16. Let be a graph obtained from a crown graph , , by deleting any pendant edges, . If is even, then is a super -EAT graph.

Acknowledgments

This research is partially supported by FAST-National University of Computer and Emerging Sciences, Peshawar, and Higher Education Commission of Pakistan. The research for this paper was also supported by Slovak VEGA Grant 1/0130/12.

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