Parity binomial edge ideals

TL;DR

This paper introduces parity binomial edge ideals, explores their algebraic properties, and relates their minimal primes to graph walk combinatorics, revealing differences from traditional binomial edge ideals.

Contribution

It defines parity binomial edge ideals, analyzes their radicality, minimal primes, and decompositions, highlighting their unique algebraic and combinatorial features.

Findings

Parity binomial edge ideals are radical iff the graph is bipartite or the field characteristic is not two.

Minimal primes encode even and odd walk structures in graphs.

A mesoprimary decomposition is established, which is primary in characteristic two.

Abstract

Parity binomial edge ideals of simple undirected graphs are introduced. Unlike binomial edge ideals, they do not have square-free Gr\"obner bases and are radical if only if the graph is bipartite or the characteristic of the ground field is not two. The minimal primes are determined and shown to encode combinatorics of even and odd walks in the graph. A mesoprimary decomposition is determined and shown to be a primary decomposition in characteristic two.