Hexagonal systems with forcing edges

An edge of a hexagonal system H is said to be forcing if it belongs to exactly one perfect matching of H. Using the concept of Z-transformation of hexagonal system, we give a characterization for the hexagonal systems with forcing edges and determine all forcing edges is such systems. We also give the generating function of all hexagonal systems with forcing edges. A hexaoonal system, also called hexanimal, polyhex or benzenoid system [2-4], is a finite connected planar graph without cut vertices in which every interior region is surrounded by a regular hexagon of side length 1. Recently, three books have been published on this kind of systems. The concept of forcing edges was first defined in [7], which related to some chemical and physical problems, see [8-10], namely, the innate degree of freedom of ;t-electron couplings and long-range order for spin pairing. We first give the following definition. Definition 1. Let H be a hexagonal system with perfect matchings. An edge e is called a forcing edge if e is contained in exactly one perfect matching of H. In this paper we attempt to give a complete characterization for the hexagonal systems with forcing edges. Hence, we confine our considerations to hexagonal systems with perfect matchings. First, let us recall the concept of Z-transformation graphs of perfect matchings of hexagonal systems and its properties. Definition 2. The Z-transformation graph Z ( H ) of a hexagonal H is the graph where the vertices are the perfect matchings of H and where two perfect matchings M1 and Supported by NFSC. * Corresponding author. 0012-365X/95/$09.50 © 1995--Elsevier Science B,V. All rights reserved SSDI 0012-365X(93)E0184-6 254 F. Zhang, X. Li / Discrete Mathematics 140 (1995) 253-263 M 2 a r e joined by an edge provided that their symmetric difference Mt A M2 is a hexagon of H. In [14] we showed that if H is a hexagonal system with perfect matchings, then Z(H) is connected and bipartite. Furthermore, the connectivity of Z(H) is equal to its minimum degree [13]. Some more notations are as follows. Let H be a hexagonal system, So be a hexagon of H, 0 be the center of So. Draw three straight lines through 0 such that every line perpendicularly intersects a group of parallel edges of H. In fact, we obtain six half lines denoted by OA t, OA2, OBI, OB2, OCt and 0C2, see Fig. 1. We call OArOBrOCi (i = 1, 2) a coordinate system with respect to (w.r.t.) So. Evidently, a coordinate system OArOBrOCi w.r.t. So divides the plane into three areas A~OB~, BiOCi, C~OAv Let OA-OB-OC be a coordinate system w.r.t So. For a point W lying in some areas, say AOB, we define the coordinates of W to be the lengths of O ~ and O I, VB, see Fig. 2, and denote them by W(OA) and I,V(OB), respectively. The characteristic graph T(H) of a hexagonal system H is defined to be the graph whose vertex set is in 1-1 correspondence with the set of the hexagons of H, and where two vertices are joined by an edge providing the corresponding hexagons have a common edge in H. In fact, there is a natural way to draw the graph T(H) of H. We can always let the vertices of T(H) be the centers of hexagons of H, two centers Oi and Oj being joined by an edge if the corresponding hexagons have a common edge, see Fig. 3. Note that the 'characteristic graph T(H)' is usually called the 'inner dual of H'. Obviously, T(H) is a planar graph. We define the perimeter of T(H) to be the boundary of the exterior face, i.e., a closed walk in which each cut edge of T(H) is traversed twice, see Fig. 3.