Generalized hexagonal systems with each hexagon being resonant (Q1186165)

A hexagonal system (HS) is a finite 2-connected planar graph in which each interior face is a regular hexagon of side length 1, and a generalized hexagonal system (GHS) is a graph obtained by deleting some interior vertices and interior edges from a HS. An edge of a GHS is said to be not fixed if it belongs to some but not all perfect matchings of the GHS. The main result of the paper is the following: A GHS has no fixed edge if and only if the boundaries of its infinite face and nonhexagon faces are alternating cycles. Moreover, if a GHS has no fixed edge then every face of it is an alternating cycle. Here a cycle is called alternating if it is alternating with respect to some perfect matching.