Abstract. We study the depth of classes of binomial edge ideals and classify all closed graphs whose binomial edge ideal is Cohen-Macaulay.Binomial edge ideals were introduced in [5]. They appear independently, and at about the same time, also in [6]. In simple terms, a binomial edge ideal is just an ideal generated by an arbitrary collection of 2-minors of a 2 × n-matrix whose entries are all indeterminates. Thus, the generators of such an ideal are of the form f ij = x i y j − x j y i , with i < j. It is then natural to associate with such an ideal the graph G on the vertex set [n] for which {i, j} is an edge if and only if f ij belongs to our ideal. This explains the naming for this type of ideals. The binomial edge ideal of graph G is denoted by J G . In [5], the relevance of this class of ideals for algebraic statistics is explained.The goal of this paper is to characterize Cohen-Macaulay binomial edge ideals for simple graphs with vertex set [n]. Similar to ordinary edge ideals, which were introduced by Villarreal [7], a general classification of CohenMacaulay binomial edge ideals seems to be hopeless. Thus, we have to restrict our attention to special classes of graphs. In Section 1, we first consider the class of chordal graphs with the property that any two maximal cliques of it intersect in at most one vertex. These graphs include, of course, all forests. We show in Theorem 1.1 that for these graphs we have depth S/J G = n + c, where n is the number of vertices of G and c is the number of connected components of G. As an application we show that the binomial edge ideal of a forest is Cohen-Macaulay if and only if each of its connected components is a path graph, and this is the case if and only if S/J G is a complete intersection.

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In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Gröbner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals.2

We introduce the concept of t-spread monomials and t-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the t-fold stretching operator. It is shown that t-spread strongly stable ideals are componentwise linear. Their height, their graded Betti numbers and their generic initial ideal are determined. We also consider the toric rings whose generators come from t-spread principal Borel ideals.2010 Mathematics Subject Classification. 05E40, 13C14, 13D02.

We study the regularity of binomial edge ideals. For a closed graph G we show that the regularity of the binomial edge ideal JG coincides with the regularity of in lex (JG) and can be expressed in terms of the combinatorial data of G. In addition, we give positive answers to Matsuda‐Murai conjecture for some classes of graphs.