Depth and extremal Betti number of binomial edge ideals

Abstract

Let G be a simple graph on the vertex set [n] and let JG be the corresponding binomial edge ideal. Let G=v∗H be the cone of v on H. In this article, we compute all the Betti numbers of JG in terms of the Betti numbers of JH and as a consequence, we get the Betti diagram of wheel graph. Also, we study Cohen–Macaulay defect of S/JG in terms of Cohen–Macaulay defect of SH/JH and using this we construct a graph with Cohen–Macaulay defect q for any q≥1. We obtain the depth of binomial edge ideal of join of graphs. Also, we prove that for any pair (r,b) of positive integers with 1≤b<r, there exists a connected graph G such that reg(S/JG)=r and the number of extremal Betti numbers of S/JG is b.

Binomial edge ideals of small depth

“…Therefore, by Theorem 5.2, we have depth S ′ /J G ′ ≥ 4, where S ′ = K[x i , y i : i ∈ V (G ′ )]. This, together with [17,Theorem 4.3] and [17,Theorem 4.4], imply the result.…”

“…For example, consider G to be the graph depicted in Figure 3. Then we have depth S/J G = 4, by [17,Theorem 4.4]. Proof.…”

“…The study of algebraic properties as well as numerical invariants of binomial edge ideals has attracted a considerable attention in the meantime, see e.g. [1,3,5,8,9,10,14,16,17,18,20,21,22].…”

“…Also, in [8] the authors showed that depth S/J G = n + 1, for a connected block graph G. Later in [15] the authors computed the depth of a wider class of graphs which are called generalized block graphs. In [17], a nice formula was given for the depth of the join product of two graphs G 1 and G 2 . Roughly speaking, the join product of two graphs G 1 and G 2 , denoted by G 1 * G 2 , is the graph which is obtained from the union of G 1 and G 2 by joining all the vertices of G 1 to vertices of G 2 , (the precise definition is given in Section 2).…”

On the depth of binomial edge ideals of graphs

“…Next assume that G = H * 3K 1 , for some graph H. If H is a complete graph, then the result follows by [14,Theorem 3.9]. If H is not complete, then it follows from [14,Theorem 4.3] and [14,Theorem 4.4] that depth S/J G ≤ 5, as desired.…”