On the Barycentric Labeling of Certain Graphs

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]). Figure 1: An example for -zero sum magic graph. 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