Regularity of parity binomial edge ideals

TL;DR

This paper investigates the algebraic properties of parity binomial edge ideals of graphs, providing bounds on their regularity, and classifying graphs with specific regularity and resolution properties.

Contribution

It introduces a lower bound for the regularity of parity binomial edge ideals and classifies graphs based on their ideal's regularity and resolution purity.

Findings

Established a lower bound for regularity of parity binomial edge ideals.

Classified graphs with parity binomial edge ideals of regularity 3.

Identified graphs with pure resolutions of their parity binomial edge ideals.

Abstract

Let $G$ be a simple graph on $n$ vertices and $\mathcal{I}_G$ denotes parity binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n].$ We obtain a lower bound for the regularity of parity binomial edge ideals of graphs. We then classify all graphs whose parity binomial edge ideals have regularity $3$. We classify graphs whose parity binomial edge ideals have pure resolution.