Binomial edge ideals of small depth

Abstract Let G be a graph on [ n ] and J G be the binomial edge ideal of G in the polynomial ring S = K [ x 1 , … , x n , y 1 , … , y n ] . In this paper we investigate some topological properties of a poset associated to the minimal primary decomposition of J G . We show that this poset admits some specific subposets which are contractible. This in turn, provides some interesting algebraic consequences. In particular, we characterize all graphs G for which depth S / J G = 4 .