Bounds for the regularity of product of edge ideals

TL;DR

This paper establishes bounds and exact formulas for the Castelnuovo-Mumford regularity of products of edge ideals, advancing understanding of their algebraic properties in combinatorial commutative algebra.

Contribution

It provides general bounds and explicit regularity formulas for products of edge ideals, including cases with nested ideals and specific graph structures.

Findings

Derived upper and lower bounds for regularity of product ideals.

Computed exact regularity for products of certain classes of edge ideals.

Provided explicit formulas for regularity when multiple ideals are nested and the last is from a complete graph.

Abstract

Let $I$ and $J$ be edge ideals in a polynomial ring $R = \mathbb{K}[x_1,\ldots,x_n]$ with $I \subseteq J$. In this paper, we obtain a general upper and lower bound for the Castelnuovo-Mumford regularity of $IJ$ in terms of certain invariants associated with $I$ and $J$. Using these results, we explicitly compute the regularity of $IJ$ for several classes of edge ideals. Let $J_1,\ldots,J_d$ be edge ideals in a polynomial ring $R$ with $J_1 \subseteq \cdots \subseteq J_d$. Finally, we compute the precise expression for the regularity of $J_1 J_2\cdots J_d$ when $d \in \{3,4\}$ and $J_d$ is the edge ideal of complete graph.