Binomial edge ideals of weakly closed graphs

TL;DR

This paper characterizes weakly-closed graphs through their binomial edge ideals, linking graph properties with algebraic structures and extending known theorems about F-purity and minimal primes.

Contribution

It introduces a novel characterization of weakly-closed graphs via Knutson ideals and extends existing results on F-purity and minimal primes of binomial edge ideals.

Findings

Weakly-closed graphs are characterized by their binomial edge ideals being Knutson ideals.

Re-proves Matsuda's theorem on F-purity for weakly closed graphs.

Provides a detailed description of minimal primes of binomial edge and Knutson ideals.

Abstract

Closed graphs have been characterized by Herzog et al. as the graphs whose binomial edge ideals have a quadratic Gr\"obner basis with respect to a diagonal term order. In this paper, we focus on a generalization of closed graphs, namely weakly-closed graphs (or co-comparability graphs). Build on some results about Knutson ideals of generic matrices, we characterize weakly closed graphs as the only graphs whose binomial edge ideals are Knutson ideals for a certain polynomial $f$. In doing so, we re-prove Matsuda's theorem about the F-purity of binomial edge ideals of weakly closed graphs in positive characteristic and we extend it to generalized binomial edge ideals. Furthermore, we give a characterization of weakly closed graphs in terms of the minimal primes of their binomial edge ideals and we characterize all minimal primes of Knutson ideals for this choice of $f$.