Z-transformation graphs of perfect matchings of hexagonal systems

Abstract

Let H be a hexagonal system. We define the Z-transformation graph Z(H) to be 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. We prove that Z(H) is a connected bipartite graph if H has at least one perfect matching. Furthermore,Z(H) is either an elementary chain or graph with girth 4; and Z(H) - Vm is 2-connected, where Vm is the set of monovalency vertices in Z(H). Finally, we give those hexagonal systems whose Z-transformation graphs are not 2-connected.