3 research outputs found

    Decomposition theorem on matchable distributive lattices

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    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

    A Min-Max Result on Catacondensed Benzenoid Graphs

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    The resonance graph of a benzenoid graph G has the 1-factors of G as vertices, two 1-factors being adjacent if their symmetric difference forms the edge set of a hexagon of G. It is proved that the smallest number of elementary cuts that cover a catacondensed benzenoid graph equals the dimension of a largest induced hypercube of its resonance graph
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