On plane bipartite graphs without fixed edges

An edge of a graph HH with a perfect matching is a fixed edge if it either belongs to none or to all of the perfect matchings of HH. It is shown that a connected plane bipartite graph has no fixed edges if and only if the boundary of every face is an alternating cycle. Moreover, a polyhex fragment has no fixed edges if and only if the boundaries of its infinite face and the non-hexagonal finite faces are alternating cycles. These results extend results on generalized hexagonal systems from [F. Zhang, M. Zheng, Generalized hexagonal systems with each hexagon being resonant, Discrete Appl. Math. 36 (1992) 67–73].