Weakly closed graphs and F-purity of binomial edge ideals

Abstract

Herzog-Hibi-Hreindóttir-Kahle-Rauh introduced the class of closed graph and they proved that the binomial edge ideal $J(G)$ of a graph $G$ has quadratic Gröbner bases if $G$ is closed. In this paper, we introduce the class of weakly closed graph as a generalization of the closed graph and prove that the quotient ring $S/J(G)$ is $F$-pure if $G$ is weakly closed. This fact is a generalization of Ohtani's theorem.

12 pages, Major Revision(The organization of this paper has changed: Section1: Definition of the weakly closed graph and discussion about between the closed graph and the weakly closed graph, Section2: $F$-purity of binomial edge ideals associated with weakly closed graphs, Section3: Classification of all connected weakly closed graphs with 5 and 6 vertex. ) , References added

References in corpus (5)

• Generalized Binomial Edge Ideals
• Cohen-Macaulay binomial edge ideals of small deviation
• On approximately Cohen-Macaulay binomial edge ideal
• Extremal Betti numbers of some classes of binomial edge ideals
• Koszulness of binomial edge ideals

Cited by in corpus (2)

• On the binomial edge ideals of proper interval graphs
• On the Equality of Symbolic and Ordinary Powers of Binomial Edge Ideals