Results on the regularity of square-free monomial ideals

TL;DR

This paper generalizes bounds on the regularity of edge ideals from graphs to hypergraphs, introducing concepts like 2-collages and recursive formulas for vertex-decomposable hypergraphs, advancing understanding of algebraic invariants.

Contribution

It extends known graph regularity bounds to hypergraphs using 2-collages and provides recursive methods for vertex-decomposable hypergraphs, broadening the theoretical framework.

Findings

Regularity bounded by a multiple of the minimum 2-collage size

Recursive formula for vertex-decomposable hypergraphs

Bound on regularity via maximal packings of star subgraphs

Abstract

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the collage such that |E \ F| < 2. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.