Regularity of powers of (parity) binomial edge ideals

TL;DR

This paper derives exact formulas for the Castelnuovo-Mumford regularity of powers of certain ideals, specifically parity binomial edge ideals of connected graphs, with a notable exception involving odd cycles.

Contribution

It provides explicit formulas for the regularity of powers of parity binomial edge ideals, extending understanding of their algebraic properties.

Findings

Formulas for regularity of powers of almost complete intersection ideals.

Explicit regularity formulas for parity binomial edge ideals of connected graphs.

An exception identified for graphs formed by connecting disjoint odd cycles.

Abstract

In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an almost complete intersection ideal $I$ which is generated by a homogeneous $d$-sequence. As applications, when $I$ is an almost complete intersection, taking the form of the (parity) binomial edge ideal of a connected graph, we can describe explicitly formulas for $\reg(I^t)$ for $t\ge 2$. The only exception is when $I$ is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.