Gorenstein binomial edge ideals associated with scrolls

Abstract

Let I G be the binomial edge ideal on the generic 2 × n -Hankel matrix associated with a closed graph G on the vertex set [n]. We characterize the graphs G for which I G has maximal regularity and is Gorenstein.2010 Mathematics Subject Classification. 13H10,13P10,13D02.

Hankel edge ideals of trees and (semi-)Hamiltonian graphs

“…For some properties of the ideal I X , see for example [2,5,8]. Note that in [3] and [7], the ideal I G was called the binomial edge ideal of X and also the scroll binomial edge ideal of G. But, in this paper, we chose to call I G the Hankel edge ideal of G.…”

“…They, indeed, gave a combinatorial upper bound for reg(S/I G ) where G is a closed graph. Later, in [7], the authors characterized all closed graphs for which the aforementioned upper bound is attained. In the same paper, the graded Betti numbers of I G and in < (I G ) were also considered.…”

“…In the same paper, the graded Betti numbers of I G and in < (I G ) were also considered. In [7], a combinatorial characterization for Gorenstein Hankel edge ideals was also given. A generalization of Hankel edge ideals was also introduced and studied in [4].…”

A Theorem on Indifference Graphs

“…Since each vertex is within distance one of itself, we have a loop at each vertex. Indifference graphs are an important class of graphs with many applications to order theory and algebra among other areas, and are the subject of considerable study (see, e.g., [3,4,5,6,9,10,12,14]). The case k = 2 in Theorem 1.1 was proved by Balof and appears in [8] in contrapositive form.…”