Parity Binomial Edge Ideals with Pure Resolutions

TL;DR

This paper characterizes graphs whose parity binomial edge ideals have pure resolutions, showing they are either complete bipartite graphs or unions of paths and odd cycles, thus linking graph structure to algebraic properties.

Contribution

It provides a complete characterization of graphs with pure parity binomial edge ideal resolutions, a novel connection between graph theory and algebraic geometry.

Findings

Pure resolutions occur only for complete bipartite graphs, disjoint unions of paths, or odd cycles.

The minimal free resolution is pure if and only if the graph belongs to these classes.

This characterizes the algebraic structure of parity binomial edge ideals based on graph properties.

Abstract

We provide a characterisation of all graphs whose parity binomial edge ideals have pure resolutions. In particular, we show that the minimal free resolution of a parity binomial edge ideal is pure if and only if the corresponding graph is a complete bipartite graph, or a disjoint union of paths and odd cycles.