Koszul Binomial Edge Ideals

TL;DR

This paper characterizes exactly which graphs produce Koszul binomial edge ideals, showing they must be strongly chordal and claw-free, thus resolving a longstanding open problem in algebraic graph theory.

Contribution

It provides a complete graph-theoretic characterization of graphs whose binomial edge ideals are Koszul, linking algebraic properties to combinatorial graph structures.

Findings

A graph's binomial edge ideal is Koszul if and only if the graph is strongly chordal and claw-free.

The result unifies and extends previous partial characterizations.

It confirms the conjecture for a broad class of graphs.

Abstract

As the binomial edge ideal of a graph is always generated by homogeneous quadratic polynomials corresponding to the edges of the graph, the question of when a binomial edge ideal defines a Koszul algebra has been studied by many authors ever since the class of ideals was first defined. Several partial results are known, including a characterization of those binomial edge ideals that possess a quadratic Gr\"obner basis. However, a complete characterization of the graphs determining Koszul binomial edge ideals has remained elusive. Inspired by our recent work characterizing when the graded M\"obius algebras of graphic matroids are Koszul, we answer the question once and for all by proving that a graph defines a Koszul binomial edge ideal if and only if it is strongly chordal and claw-free.