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ULD-Lattices and Δ-Bonds

Published online by Cambridge University Press:  01 September 2009

STEFAN FELSNER  and
KOLJA B. KNAUER
STEFAN FELSNER
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Germany (e-mail: felsner@math.tu-berlin.de, knauer@math.tu-berlin.de)
KOLJA B. KNAUER
Affiliation:
Institut für Mathematik, Technische Universität Berlin, Germany (e-mail: felsner@math.tu-berlin.de, knauer@math.tu-berlin.de)
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Abstract

We provide a characterization of upper locally distributive lattices (ULD-lattices) in terms of edge colourings 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 Δ-bonds with invariant circular flow-difference. This instance generalizes several previously studied lattice structures, in particular, c-orientations (Propp), α-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.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 5: New Directions in Algorithms, Combinatorics and Optimization , September 2009 , pp. 707 - 724
Copyright
Copyright © Cambridge University Press 2009

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