Decomposition theorem on matchable distributive lattices

Abstract

A distributive lattice structure $\mathbf{M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then $\mathbf{M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.

19 pages, 7 figures

References in corpus (1)

• Lattice structure for orientations of graphs

Cited by in corpus (3)

• Outerplane bipartite graphs with isomorphic resonance graphs
• Structure and enumeration results of matchable Lucas cubes
• A class of non-matchable distributive lattices