Weakly Closed Graphs and F-Purity of Binomial Edge Ideals

Herzog,Hibi,Hreinddttir et al.introduced the class of closed graphs,and they proved that the binomial edge ideal JG of a graph G has quadratic GrSbner bases if G is closed.In this paper,we introduce the class of weakly closed graphs as a generalization of the closed graph,and we prove that the quotient ring S/JG of the polynomial ring S=K[x1,...,xn,y1,...,yn]with K a field and n=|V(G)|is F-pure if G is weakly closed.This fact is a generalization of Ohtani's theorem.