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Lattice structure for orientations of graphs

James Propp

TL;DR

This work shows that the set of graph orientations with a fixed circulation $c$, relative to a chosen accessibility class, forms a distributive lattice under pushing-down moves, unifying combinatorial and statistical-mechanical constructions. The author introduces height-functions to prove the lattice structure and extends the framework dually to $d$-factors on planar graphs and to spanning trees, yielding analogous distributive lattices via face-twists and swinging-down moves. The results connect to classical models such as alternating-sign matrices, lozenge and domino tilings, and provide dual interpretations through graph duality and arborescences, highlighting broad applicability to tilings, matchings, and network flows. Beyond structural insights, the paper discusses algorithmic implications, sampling, and phase-diagram perspectives, illustrating the unifying power of the distributive-lattice viewpoint across discrete geometry and statistical mechanics.

Abstract

In 1986, Oliver Pretzel studied the set of orientations of a connected finite graph $G$ and showed that any two such orientations having the same flow-difference around all closed loops can be obtained from one another by a succession of local moves of a simple type. Here I show that the set of orientations of $G$ having the same flow-differences around all closed loops can be given the structure of a distributive lattice. When the graph is drawn on the plane, a dual version of the construction puts a distributive lattice structure on the set of orientations of $G$ having the same indegrees at all vertices. In both settings, adjacent lattice-elements are related by simple local moves. This construction unifies earlier, similar constructions in combinatorics and statistical mechanics. It also gives rise to an interesting lattice structure on spanning trees. This article is an updated version of a preprint originally distributed in 1993.

Lattice structure for orientations of graphs

TL;DR

This work shows that the set of graph orientations with a fixed circulation , relative to a chosen accessibility class, forms a distributive lattice under pushing-down moves, unifying combinatorial and statistical-mechanical constructions. The author introduces height-functions to prove the lattice structure and extends the framework dually to -factors on planar graphs and to spanning trees, yielding analogous distributive lattices via face-twists and swinging-down moves. The results connect to classical models such as alternating-sign matrices, lozenge and domino tilings, and provide dual interpretations through graph duality and arborescences, highlighting broad applicability to tilings, matchings, and network flows. Beyond structural insights, the paper discusses algorithmic implications, sampling, and phase-diagram perspectives, illustrating the unifying power of the distributive-lattice viewpoint across discrete geometry and statistical mechanics.

Abstract

In 1986, Oliver Pretzel studied the set of orientations of a connected finite graph and showed that any two such orientations having the same flow-difference around all closed loops can be obtained from one another by a succession of local moves of a simple type. Here I show that the set of orientations of having the same flow-differences around all closed loops can be given the structure of a distributive lattice. When the graph is drawn on the plane, a dual version of the construction puts a distributive lattice structure on the set of orientations of having the same indegrees at all vertices. In both settings, adjacent lattice-elements are related by simple local moves. This construction unifies earlier, similar constructions in combinatorics and statistical mechanics. It also gives rise to an interesting lattice structure on spanning trees. This article is an updated version of a preprint originally distributed in 1993.

Paper Structure

This paper contains 5 sections, 25 theorems, 10 equations, 20 figures.

Table of Contents

  1. Statement of results.
  2. Orientations of graphs.
  3. Matchings of Bipartite Graphs
  4. Spanning Trees
  5. Earlier and later related work

Key Result

Theorem 1

Let ${\cal R}$ be the (non-empty) set of orientations of a finite connected graph $G$ that have a fixed circulation $c$, and let $A^*$ be an accessibility class of $G$. If we say that one $c$-orientation $R$ covers another $c$-orientation $S$ exactly when $S$ is obtained from $R$ by pushing down at

Figures (20)

  • Figure 1: Two examples of pushing down.
  • Figure 2: Orientations of a rooted 4-cycle.
  • Figure 3: A six-sided face.
  • Figure 4: Twisting down a 1-factor.
  • Figure 5: A swinging-down move.
  • ...and 15 more figures

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • proof
  • Definition 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • ...and 50 more