Abstract
Let be an abelian group. A graph is called -magic if there exists edge labeling such that the induced vertex set labeling , defined by , where the sum is over all edges in , is a constant map. A graph is -barycentric-magic (or has -barycentric labeling) if is -magic and also satisfies for all and for some vertex adjacent to . In this paper we consider some graphs and characterize all for which is -barycentric-magic.
1. Introduction
Let be a finite, simple, and undirected graph. Labeling for a graph is a map that takes graph elements to numbers (usually positive or nonnegative integers). Let be an abelian group (written additively). The graph is called -magic if there exists labeling such that, for each vertex , the sum of values of all edges incident with , denoted by , is a constant; that is, , for some . When this constant is , is said to be -zero-sum magic. The integer-magic spectrum of a graph is the set .
As an example, Figure 1 shows a graph which is -zero sum magic, for every group (see [1]).
Let us state some easy lemmas (or observations). They are straightforward to verify and can be found in [2].
Lemma 1. A graph is -magic if and only if every vertex of is of the same parity.
Lemma 2. An Eulerian graph with even size is -magic.
Lemma 3. If is a subgroup of and graph is -magic, then is -magic.
Various authors have introduced labeling that generalizes the idea of magic square. Kotzig and Rosa [3] defined a magic labeling to be total labeling on the vertices and edges in which the labels are the integers from to . The sum of labels on an edge and its two endpoints is constant. In 1996 Ringel and Llado [4] redefined this type of labeling as edge-magic. Also, Enomoto et al. [5] have introduced the name super edge-magic for magic labeling in the sense of Kotzig and Rosa, with the added property that the vertices receive the smaller labels, . Lee et al. [6] defined the concept of -edge magic graphs and studied it for certain graphs (see, e.g., [7]). Recently authors in [8] defined a new kind of group magicness graphs.
Here we recall the following definition.
Definition 4 (see [8]). If there exists labeling for a graph , whose induced vertex set labeling is a constant map and for all the sum also satisfies for some vertex adjacent to is said to be -barycentric-magic.
Note that the motivation of Definition 4 is the following definition of -barycentric sequence which was introduced in [9] and has already been used in graph labeling problems, specially in Ramsey theory [9–11].
Definition 5. Let be elements of an abelian group . This sequence is -barycentric if there exists such that . The element is called a barycenter.
Example 6. Figure 2 shows a graph with and which is -barycentric-magic. Note that in this example the sum is (mod 4) for all .
The study of the barycentric-magic graphs is motivated by the relationship between different types of magic labeling and the behavior of the sums of sequences in abelian groups. That is, -magic labeling of a graph is equivalent to sequences of nonzero elements of with the same sum; -zero-sum magic labeling is equivalent to zero-sum sequences in .
Note that -barycentric-magic labeling is equivalent to sequences of nonzero elements of that contain one element which is the “average” of its terms, called barycentric sequence.
Similar to the definition of integer magic spectrum of -magic graphs we state the definition of barycenter-magic spectrum of graph.
Definition 7 (see [8]). For a given graph the set of all positive integers for which is -barycentric-magic is called the barycenter-magic spectrum of and is denoted by .
In this paper, we consider specific graphs and characterize all for which is -barycentric-magic.
2. Barycentric-Magic Labeling of Certain Graphs
In this section, we characterize all for which is -barycentric-magic. First we consider some complete bipartite graphs.
First we state the following theorem.
Theorem 8 (see [8]). The complete bipartite graph is not -barycentric-magic for any .
We generalize the previous theorem.
Theorem 9. For , is -barycentric-magic if and only if .
Proof. Let be the set of vertices of . For each , the edges incidents to must have the same label. Suppose that and . Then (mod m). We consider the two following cases.
Case 1 ( is odd). In this case, the condition (mod ) implies that all edges have the same label, say . By the condition we have (mod ) and this is impossible when . If , then using gives a barycentric-magic labeling.
Case 2 ( is even). Here we give two different approaches. In this case, the condition (mod ) implies that there are at most two different labels and , such that (mod ). Now label as follows: for and for , for some . Then, the edges incidents to must be labeled in the same way. This labeling is barycentric-magic if and only if
or
Without loss of generality, we consider only the first relation. The condition (mod ) is satisfied only when . So suppose that . Choose , and and since is even we get
Therefore with this labeling is -barycentric-magic.
Now we state the second approach.
Since (mod ), if , then , a contradiction. Conversely, let . This is also a -barycentric-magic labeling for .
Example 10. Consider the graph and the group . Let be the set of vertices of . In this case, since choose and and . The labeling of is as follows: for and ; for and ; then (mod 10) for ; (mod 10) for ; (mod 10) for . Then with this labeling we get barycentric-magic labeling.
Here we consider friendship graphs.
Let be any positive integer and Dutch-Windmill, or friendship graph with vertices and edges. In other words, the friendship graph is a graph that can be constructed by coalescence copies of the cycle graph of length with a common vertex. The friendship theorem of Erdös et al. [12] states that graphs with the property that every two vertices have exactly one neighbour in common are exactly the friendship graphs. Figure 3 shows some examples of friendship graphs.
Theorem 11. Friendship graphs are -barycentric-magic graphs for any .
Proof. Consider friendship graph and suppose that and for . We label all the edges of with . Since (mod ) (e.g., put ), the graphs are -barycentric-magic.
Here we consider another families of graphs denoted by (see Figure 4).
Theorem 12. The graphs are not -barycentric-magic for every .
Proof. Suppose that the set of vertices of is , where and and , , and . One can consider some cases to prove that there is no barycentric-magic labeling for . Here we state two cases. We label the graph as follows.
Case 1. Label all edges of with . In this case (mod ) or (mod ) which is not true.
Case 2. We label as follows: for each and for each and for each and and ). From we have (mod ) or (), but is the label of edges , so this labeling is not barycentric-magic.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to express their gratitude to the referee for careful reading and helpful comments.