On Generalizations of Cycles and Chordality to Hypergraphs from an Algebraic Viewpoint

TL;DR

This paper explores how concepts of cycles and chordality in hypergraphs can be understood through algebraic properties like linear resolutions and linear quotients, proposing unified definitions and relationships.

Contribution

It introduces algebraic interpretations of hypergraph cycles and chordality, and establishes conditions linking these notions to algebraic properties such as linear quotients and stability.

Findings

Chordality relates to linear resolutions in algebraic terms.

Certain hypergraph classes are proven to be chordal under algebraic conditions.

Unified algebraic definitions for cycles and chordality in hypergraphs are proposed.

Abstract

In this paper, we study the notion of chordality and cycles in hypergraphs from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. However, there is no unified definition for cycle or chordality in hypergraphs in the literature, so we consider several generalizations of these notions and study their algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of hypergraphs. Also we show that if $\mathcal{C}$ is a hypergraph such that $\langle \mathcal{C} \rangle$ is a vertex decomposable simplicial complex or $I(\bar{\mathcal{C}})$ is squarefree stable, then $\mathcal{C}$ is chordal according to one of the most promising definitions.