Z-transformation graphs of perfect matchings of hexagonal systems

Let H be a hexagonal system. The Z-transformation graph Z(H) is the graph where the vertices are the perfect matchings of H and where two perfect matchings are joined by an edge provided their symmetric difference is a hexagon of H (Z. Fu-ji et al., 1988). In this paper we prove that Z(H) has a Hamilton path if H is a catacondensed hexagonal system. A hexagonal system [ll], also called honeycomb system or hexanimal (see, eg. [lo]) is a finite connected plane graph with no cut-vertices, in which every interior region is surrounded by a regular hexagon of side length 1. Hexagonal systems are of chemical significance since a hexagonal system with perfect matchings is the skeleton of a benzenoid hydrocarbon molecule [9]. Recall that a perfect matching of a graph G is a set of disjoint edges of G covering all the vertices of G. In the following discussion we confine our considerations to those hexagonal systems with at least one perfect matching. Let H be a hexagonal system. The Z-transformation graph Z(H) [3,4] is the graph where the vertices are the perfect matchings of H and where two perfect matchings Ml and M2 are joined by an edge provided their symmetric difference Ml A MZ, i.e. (M, u M,) (M, n M,), is a hexagon of H. Z-transformation graphs have some interesting properties. Z(H) is either a path or a bipartite graph with girth 4, and the connectivity of Z(H) is equal to the minimum degree of the vertices of Z(H) [3,4]. Furthermore, Z(H) has at most two vertices of degree one [3]. The construction feature for the class of hexagonal systems whose Z-transformation graphs have at least one vertex of degree one was reported in [S]. Z-transformation graphs are useful *Corresponding author 0166-218X/97/$17.00