Weakly Closed Graphs and F-Purity of Binomial Edge Ideals

Herzog, Hibi, Hreindattir et al. introduced the class of closed graphs, and they proved that the binomial edge ideal J(G) of a graph G has quadratic Grobner 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/J(G) of the polynomial ring S = K[x(1), . . ., x(n), y(1), . . . ,y(n),] with K a field and n = vertical bar V (G)vertical bar is F-pure if G is weakly closed. This fact is a generalization of Ohtani's theorem.