Koszulness of binomial edge ideals

TL;DR

This paper investigates the algebraic property of Koszulness for binomial edge ideals associated with a special class of graphs called closed graphs, providing criteria that do not depend on vertex labeling.

Contribution

It introduces criteria for the closedness of graphs independent of vertex labeling and links these to the Koszulness of their binomial edge ideals.

Findings

Criteria for graph closedness independent of labeling.

Conditions under which the quotient algebra is Koszul.

Characterization of binomial edge ideals for closed graphs.

Abstract

Let $G$ be a simple graph on the vertex set $V(G) = [n] = \{1,...,n\}$ and edge ideal $E(G)$. We consider the class of closed graphs. A closed graph is a simple graph satisfying the following property: for all edges $\{i, j\}$ and $\{k, \ell\}$ with $i < j$ and $k < \ell$ one has $\{j, \ell\}\in E(G)$ if $i = k$, and $\{i, k\}\in E(G)$ if $j = \ell$. We state some criteria for the closedness of a graph $G$ that do not depend necessarily from the labelling of its vertex set. Consequently, if $S = K[x_1,..., x_n, y_1,..., y_n]$ is a polynomial ring in $2n$ variables with coefficients in a field $K$, we obtain some criteria for the Koszulness of the quotient algebra $S /J_G$, where $J_G$ is the binomial edge ideal of $S$ i.e. the ideal generated by the binomials $f_{ij} = x_iy_j - x_jy_i$ such that $i<j$ and $\{i,j\}$ is an edge of $G$ (\cite{HH}).