Abstract Let G be a plane elementary bipartite graph with more than two vertices. Then its resonance graph Z ( G ) is a median graph and the set M ( G ) of all perfect matchings of G with a specific partial order is a finite distributive lattice. In this paper, we prove that Z ( G ) is cube-free if and only if it can be obtained from an edge by a sequence of convex path expansions with respect to a reducible face decomposition of G . As a corollary, a structure characterization is provided for G whose Z ( G ) is cube-free. Furthermore, Z ( G ) is cube-free if and only if the Clar number of G is at most two, and sharp lower bounds on the number of perfect matchings of G can be expressed by the number of finite faces of G and the number of Clar formulas of G . It is known that a cube-free median graph is not necessarily planar. Using the lattice structure on M ( G ) , we show that Z ( G ) is cube-free if and only if Z ( G ) is planar if and only if M ( G ) is an irreducible sublattice of m × n . We raise a question on how to characterize irreducible sublattices of m × n that are M ( G ) .

中文翻译：

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无立方体共振图

摘要 设 G 是一个具有两个以上顶点的平面初等二分图。那么它的共振图 Z ( G ) 是一个中值图， G 具有特定偏序的所有完美匹配的集合 M ( G ) 是一个有限分配格。在本文中，我们证明了 Z ( G ) 是无立方体的，当且仅当它可以通过关于 G 的可约面分解的一系列凸路径展开从边缘获得。作为推论，为 G 提供了结构表征，其 Z ( G ) 是无立方体的。此外，当且仅当 G 的 Clar 数最多为 2 时，Z ( G ) 是无立方体的，并且 G 的完全匹配数的尖锐下界可以由 G 的有限面数和数表示G 的 Clar 公式。众所周知，无立方体的中值图不一定是平面的。使用 M ( G ) 上的晶格结构，我们证明 Z ( G ) 是无立方体的当且仅当 Z ( G ) 是平面的当且仅当 M ( G ) 是 m × n 的不可约子晶格。我们提出了一个问题，即如何刻画 m × n 的不可约子晶格，即 M ( G ) 。