ULD-Lattices and Delta-Bonds

TL;DR

This paper characterizes upper locally distributive lattices using edge colorings of cover graphs and applies this to prove distributive lattice structures in various combinatorial contexts, including Delta-bonds and chip-firing games.

Contribution

It introduces a new characterization of ULD-lattices via edge colorings and demonstrates its utility in establishing lattice structures in combinatorial objects.

Findings

Characterization of ULD-lattices through edge colorings.

Distributive lattice structure on Delta-bonds with circular flow-difference.

Application to various combinatorial structures like c-orientations and chip-firing games.

Abstract

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colorings of their cover graphs. 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 Delta-bonds with invariant circular flow-difference. This instance generalizes several previously studied lattice structures, in particular, c-orientations (Propp), alpha-orientations of planar graphs (Felsner, resp. de Mendez) and planar flows (Khuller, Naor and Klein). The characterization also applies to other instances, e.g. to chip-firing games.