Non-matchable distributive lattices

Based on an acyclic orientation of the Z -transformation graph, a finite distributive lattice (FDL for short) M ( G ) is established on the set of all 1-factors of a plane (weakly) elementary bipartite graph G . For an FDL L , if there exists a plane bipartite graph G such that L is isomorphic to M ( G ) , then L is called a matchable FDL. A natural question arises: Is every FDL a matchable FDL? In this paper we give a negative answer to this question. Further, we obtain a series of non-matchable FDLs by characterizing sub-structures of matchable FDLs with cut-elements.

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