4 research outputs found

    Decomposition theorem on matchable distributive lattices

    Full text link
    A distributive lattice structure M(G){\mathbf M}(G) has been established on the set of perfect matchings of a plane bipartite graph GG. 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 GG is elementary, then M(G){\mathbf M}(G) is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice L\mathbf{L} is an MDL if and only if each factor in any cartesian product decomposition of L\mathbf{L} is an MDL. Two types of MDLs are presented: J(m×n)J(\mathbf{m}\times \mathbf{n}) and J(T)J(\mathbf{T}), where m×n\mathbf{m}\times \mathbf{n} denotes the cartesian product between mm-element chain and nn-element chain, and T\mathbf{T} is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

    Block graphs of z-transformation graphs of perfect matchings of plane elementary bipartite graphs

    No full text
    Let G be a connected plane bipartite graph. The Z-transformation graph Z(G) is a graph where the vertices are the perfect matchings of G and where two perfect matchings are joined by an edge provided their symmetric difference is the boundary of an interior face of G. For a plane elementary bipartite graph G it is shown that the block graph of Z-transformation graph Z(G) is a path. As an immediate consequence, we have that Z(G) has at most two vertices of degree one
    corecore