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Byline: Martin Baca and Ayesha Shabbir

ABSTRACT: A toroidal polyhex is a cubic bipartite graph embedded on the torus such that each face is a hexagon. A graph G(V , E) of order p and size q is called (a,d)-edge-antimagi c total if there exists a bijection f : V (G) - E(G) {1, 2, ..., p q} such that the edge-weights, w(uv) f (u) f (v) f (uv), uv E(G) , form an arithmetic sequence with the first term a and common difference d . Such a graph G is called super if the smallest possible labels appear on the verti ces. In this paper we study such labelings f o r toroidal polyhexes and give a characterization for super (a,d)-edge-antimagicness of toroidal polyhexes.

Key Words: ful lerene, toroidal polyhex, super edge-antimagic total labeling.


The discovery of the fullerene molecules and nan otubes has stimulated much interests in other possibilities for carbons. Classical fullerene is an all- carbon molecule in which the atoms are arranged on a pseudospherical framework made up entirely of pentagons and hexa g on s. Its molecular graph is a finite trivalent graph embedded on the surface of a sphere with only hexagonal and (exactly 12) pentagonal faces. Deza et al. [7] considered fullerene's extension to other closed surfaces and showed that only four surfaces are possible, namely sphere, torus, Klein bottle and pro jective plane. Unlike spherical fullerenes, toroidal and Klein bottle's fullerenes have been regarded as tessellations of entire hexagons on their surfaces since they must contain no pentagons, see [7] and [13].

A toroidal polyhex (toroidal fullerene) is a cubic bipartite graph embedded on the torus such that each face is a hexagon. Note that the torus is a closed surface that can carry graph of toroidal polyhex such that all its vertices have degree 3 and all faces of the embedding are hexagons, see Figure 1 and also [20].

Some features of toroidal polyhexes with chemical relevance were discussed in [11] and [12]. For example, a systematic coding and classification scheme were given for the enumeration of isomers of toroidal polyhexes, the calculation of the spectrum and the count for spanning trees. There have been a few work in the enumeration of perfect matchings of toroidal polyhexes by applying various techniques, such as transfer-matrix [14,16] and permanent of the adjacency matrix [5]. k-resonance of toroidal polyhexes have been studied in [21,22,23].

An (a,d)-edge-antimagic total labeling ((a,d)-EAT for short) of G is the total labeling with the property that the edge-weights form an arithmetic sequence starting from a and having common difference d, where a greater than 0 and d [greater than or equal to] 0 are two given integers. Definition of (a,d)-EAT labeling was introduced in [17] as a natural extension of magic valuation which also known as edge-magic labeling defined in [15]. An (a, d)-EAT labeling is called super if the smallest possible labels appear on the vertices. For more information on edge-magic and super edge-magic labelings, please see [8, 9, 19]. A graph that admits an (a,d)-EAT labeling or a super (a,d)-EAT labeling is called an (a,d)-EAT graph or super (a,d)-EAT graph, respectively.

Sugeng et al. in [18] described some ways to construct the super (a,d)-EAT labelings of the caterpillars for d [?] {0, 1, 2, 3}. Super (a,d)-edge-antimagic properties of certain classes of graphs are discussed in [2,4] and their relationship to mean labelings are described in [1].

The existence of super edge-antimagicness for disconnected graphs is investigated in [3,6,10]. In this paper we give a characterization for super (a,d)-edge-antimagicness of toroidal polyhexes.

2. Preliminaries for toroidal polyhex Let L be a regular hexagonal lattice and n P m be an mxn quadrilateral section (with m hexagons on the top and bottom sides and n hexagons on the lateral sides, n is even) cut from the regular hexagonal lattice L. First identify two lateral sides of Pnm to form a cylinder, and finally identify the top and bottom sides of nPm at their corresponding points, see Figure 2. From this we get a toroidal polyhex n m with mn hexagons.

The graph lying in the interior of the quadrilateral section nPm has a proper 2-coloring. The vertices incident with a downward vertical edge and with two upwardly oblique edges can be colored, say black, and the other vertices, say white. Such a 2-coloring is a proper 2-coloring of nm i.e., the end-vertices of each edge



The research for this article was supported by Slovak VEGA Grant 1/0130/12 and Higher Education Commission Pakistan Grant HEC(FD)/2007/555.


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Author:Baca, Martin; Shabbir, Ayesha
Publication:Science International
Article Type:Report
Geographic Code:9PAKI
Date:Sep 30, 2012

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