Cohen-Macaulay binomial edge ideals

Abstract

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.

On the binomial edge ideals of block graphs

“…Moreover, for m ≥ n, we show that reg J G = reg(in < (J G )) = n, where n is the number of vertices of the graph G. When m < n, then we provide an upper bound for the regularity of in < (J G ) and, therefore, for the regularity of J G as well. Our results generalize the ones obtained in the papers [3,8,12] for classical binomial edge ideals associated with (generalized) block graphs.…”

“…By applying Depth lemma to our exact sequence (6), and taking into account equations (7), (8) and (9), we get…”

On the Extremal Betti Numbers of Binomial Edge Ideals of Block Graphs

“…Since f (G) = f (H), (7) implies that β n−1,n−1+i(G)+1 (S/J G ) = f (G) − 1, and together with (6) it follows that β n−1,n−1+i(G)+1 (S/J G ) is an extremal Betti number. Now we prove the assertions regarding in < (J G ).…”

“…The following exact sequence [7] is used for our induction step. By the proof of [7, Theorem 1.1] we know that proj dim S/J G = proj dim S/J G ′ = n − 1, proj dim S/((x i , y i ) + J H ) = n, and proj dim S/((x i , y i )+J G ′′ ) = n−q, where q+1 is the number of connected components of G ′′ .…”

“…Thus it is natural to test the above conjectures for binomial edge ideals. A positive answer to this conjecture was given in [7] for Cohen-Macaulay binomial edge ideals of PI graphs (proper interval graphs). In that case actually all the graded Betti numbers of the binomial edge ideal and its initial ideal coincide.…”