Lattice structure for orientations of graphs

James Propp

Published 2002-09-01, updated 2020-05-26Version 2

Earlier researchers have studied the set of orientations of a connected finite graph $G$, and have shown 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. The construction generalizes partial orderings that arise in the study of alternating sign matrices. It also gives rise to lattices for the set of degree-constrained factors of a bipartite planar graph; as special cases, one obtains lattices that arise in the study of plane partitions and domino tilings. Lastly, the theory gives a lattice structure to the set of spanning trees of a planar graph.