On the Integer-Magic Spectra of Honeycomb Graphs

For a positive integer k, a graph G (V, E) is £k-magic if there exists a function, namely, a labeling, I : E(G) -+ £k such that the induced vertex set labeling [+ : V(G) £k, where [+(v) is the sum of the labels of the edges incident with a vertex v is a constant map. The set of all positive integer k such that G is k-magic is denoted by IM(G). We call this set the integer-magic spectrum of G. In this paper, we investigate the integer-magic spectra of the honeycomb graphs. We show that besides N, there are only two types of integer­ magic spectra of honeycomb graphs. Introduction Let A be an abelian group written additivel y and A" A {o}. A labeling = is a map 1 from E(G) to Z;'. Given a labeling on the edge set of G, we can induce a vertex set labeling 1+ by adding all the labels of the edges incident with a vertex v, that is, l+(v) = '2)l(u,v): (u,v) E E(G)}. CONGRESS US NUMERANTIUM 193 (2008), pp. 49-65 A graph G is known as A-magic if there is a labeling I : E( G) ---> A' such that for each vertex v, the sum of the labels of the edges incident with v are all equal to the same constant; i.e., l+(v) = c for some fixed c in A. We call (G, l) an A-magic graph. In general, a graph G may admit more than one labeling to become an A-magic graph. We denote the class of all graphs (either simple or multiple graphs) by Gph and the class of all abelian groups by Ab. For an abelian group A in Ab, we also denote the class of all A-magic graphs byAMGp. The Z-magic graphs are considered in Stanley [26, 27]. He pointed out in [26] that the theory of magic labelings can be put into the more general context of linear homogeneous diophantine equations. When the group is Zk, we shall refer the Zk magic graph as k-magic. The k-magic graphs are studied by [14], [16] and [17]. A-magic graphs are also considered by Doob in [2],[3] and [4] where A is an abelian group. Given a graph G, the problem of deciding whether G admits a magic labeling is equivalent to the problem of deciding whether a set of linear homogeneous Diophantine equation has a solution. (See [23].) At present, no general efficient algorithm is known for finding magic labelings of a graph for a given abelian group. The original concept of A-magic graph is due to J. Sedlacek [24] and [25], who defines it to be a graph with real-valued edge labeling such that (i) distinct edges have distinct nonnegative labels, and (ii) the sum of the labels of the edges incident to a particular vertex is the same for all vertices. In this paper we use N to denote the set of natural numbers, {I, 2, 3, ... }. A graph G equipped with a magic labeling I : E(G) ---> N is called N-magic. It is well-known that a graph G is N-magic if and only if each edge of G is contained in a I-factor (a perfect matching) or a {I, 2}-factor. (See [9].) Readers shall refer to [5],[6],[7],[8],[12],[14],[24],[25] for N-magic graphs. Note that the Z-magic condition is weaker than N-magic condition. Figure 1 shows a graph which is Z-magic but not N-magic. Figure 1: Z-magic but not N-magic. 50 For convenience, we name Z-magic as I-magic. For a graph G, the set of all positive integer k such that G is k-magic is denoted by IM(G). We call IM(G) the integer-magic spectrum of G. In [16], Lee, Sun, Wen and the first author investigate these sets for general graphs . It is shown in [17] that all the grid graph Pm X Pn has IM(Pm x Pn) = N {2}, except P2 x P2 which is IM(P2 x P2) = N. In fact, a more general result is obtained for polyominoes from square lattices in [22]. A polyomino is a finite connected union of cells on a lattice. Polyomi­ noes are sprung from recreative mathematics domains, from physics such as Ising models, and from polymer of chemistry. In this paper, we consider polyominoes with hexagonal cells, (See Figure 2,) namely, a honeycomb graph. They are called poly hexes in Harary and Read in [5]. These graphs have been studied by chemists as models of the molecular structure of or­ ganic compounds build up entirely from benzene rings. We will investigate the integer-magic spectra of honeycomb graphs. Figure 2: Polyominoes with hexagonal cells The Honeycomb graphs have been studied from various directions. We put some papers in the reference, even they do not relate to the magic spectrum we studied here. For example, the magic valuation considered in [4] and [13] does not relate to our concept. Papers [15],[16],[18] and[21] deal with more general concept of k-magic graphs. 2 Honeycomb graphs whose dual graphs are trees. Let H be a honeycomb graph with only one cell. Since H is eulerian with even number of edges, by a result in [21], its integer-magic spectrum is N. Given a honeycomb graph G, we can associate it with a graph Dual(G) where the vertices of Dual(G) is the set of all hexagonal components and two components A and B are adjacent if they share a common edge. The graph Dual(G) is called the interior dual graph of the honeycomb graph G. If we replace the cell of the polyomino by a vertex at the center and join the vertex to its neighbor vertex, then we obtain a dual graph of