On the depth of binomial edge ideals of graphs

Let G be a graph on the vertex set [n] and J(G) the associated binomial edge ideal in the polynomial ring S=K[x(1), ..., x(n),y(1), ..., y(n)]. In this paper, we investigate the depth of binomial edge ideals. More precisely, we first establish a combinatorial lower bound for the depth of S/J(G) based on some graphical invariants of G. Next, we combinatorially characterize all binomial edge ideals J(G) with depthS/J(G)=5. To achieve this goal, we associate a new poset M-G with the binomial edge ideal of G and then elaborate some topological properties of certain subposets of M-G in order to compute some local cohomology modules of S/J(G).