A class of hypergraphs that generalizes chordal graphs

TL;DR

This paper introduces a new class of hypergraphs called chordal hypergraphs, extending the concept of chordal graphs, and demonstrates that their associated hypergraph algebras have linear resolutions, generalizing known properties of graph algebras.

Contribution

The paper defines a new class of hypergraphs called chordal hypergraphs and extends existing algebraic results to these hypergraphs, including linear resolutions of their algebras.

Findings

Generalization of chordal graphs to hypergraphs.

Hypergraph algebras of generalized chordal hypergraphs have linear resolutions.

Introduction of $d$-flag complexes as a higher-dimensional analogue.

Abstract

In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given in \cite{VT}, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. In \cite{F1}, Fr\"oberg shows that the chordal graphs corresponds to graph algebras, $R/I(\mc{G})$, with linear resolutions. We extend Fr\"oberg's method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The definitions we give, yield a natural higher dimensional version of the well known flag property of simplicial complexes. We obtain what we call $d$-flag complexes.