Koszul Binomial Edge Ideals
Adam LaClair, Matthew Mastroeni, Jason McCullough, Irena Peeva
TL;DR
The paper resolves when a binomial edge ideal $J_G$ yields a Koszul algebra by proving $J_G$ is Koszul if and only if $G$ is strongly chordal and claw-free (equivalently chordal, claw-free, and tent-free). The authors develop a combinatorial framework using strong elimination orders and inductive arguments, and contrast Koszulness with the existence of a quadratic Gröbner basis (the latter characterized by closed graphs). They establish obstructions via explicit non-Koszul examples (cycles $C_n$ for $n\ge4$, claw, tent) and derive a precise structural characterization for graphs with small clique number, showing how nets serve as the boundary between Koszul and non-Koszul cases. This completes a long-standing question and clarifies how graph structure governs Koszul properties in binomial edge ideals, with implications for related graph-algebra constructions and their homological behavior.
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 Gröbner 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 Möbius 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.