The Electronic Journal of Combinatorics· 2025Q1
Lattice Structure for Orientations of Graphs
- 84citations
- Q1SCImago
- 2025year
Short summary
The set of graph orientations with identical flow-differences around closed loops can be structured as a distributive lattice.
AI-generated from the title and abstract; the full text is not read.
Key points
- Identifies a distributive lattice structure for graph orientations with constant flow-differences.
- Generalizes partial orderings from alternating sign matrices.
- Extends to degree-constrained factors of bipartite planar graphs.
- Provides lattice structures for plane partitions, domino tilings, and spanning trees of planar graphs.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. The Electronic Journal of Combinatorics, 2025 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics