Distributive Lattices from Graphs

Several instances of distributive lattices on graph structures are known. This includes c-orientations (Propp), α-orientations of planar graphs (Felsner/de Mendez) planar flows (Khuller, Naor and Klein) as well as some more special instances, e.g., spanning trees of a planar graph, matchings of planar bipartite graphs and Schnyder woods. We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their Hasse diagrams. In many instances where a set of combinatorial objects carries the order structure of a lattice this characterization yields a slick proof of distributivity or UL-distributivity. This is exemplified by proving a distributive lattice structure on the ∆-bonds of a graph. All the previously known instances of distributive lattices from graphs turn out to be special ∆-bond lattices. Let a D-polytope be a polytope that is closed under componentwise max and min, i.e., the points of the polytope are an infinite distributive lattice. A characterization of D-polytopes reveals that each D-polytope has an underlying graph model. The associated graph models have two descriptions either edge based or vertex based.