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

Abstract

We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.

Square-free Gröbner degenerations

“…The above conjecture, in various forms, has been discussed in several occasions by Herzog and his collaborators. It appeared in print only recently in [CDG18b, Conjecture 1.7] and in the introduction of [HR18]. In this paper we solve positively Herzog's conjecture.…”

Krull dimension and regularity of binomial edge ideals of block graphs

“…Proof (1). If G is indecomposable, by[9, Theorem 2.4], the result follows. Otherwise, suppose G is decomposable into indecomposable graphs G 1 , .…”

Depth and extremal Betti number of binomial edge ideals

“…Extremal Betti numbers of binomial edge ideals of closed graphs were studied by de Alba and Hoang in [2]. In [7], Herzog and Rinaldo studied extremal Betti number of binomial edge ideal of block graphs. We compute all the Betti numbers of cone of a graph, (Theorem 3.10).…”