Koszulness of binomial edge ideals

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}).