Related papers: Strongly Koszul edge rings
A cut ideal of a graph was introduced by Sturmfels and Sullivant. In this paper, we give a necessary and sufficient condition for toric rings associated to the cut ideal to be Strongly Koszul.
We give necessary and sufficient conditions for strong Koszulness of toric rings associated with the stable set polytope of graphs.
Let $G$ be a simple graph on the vertex set $V(G) = [n] = \{1,...,n\}$ and edge ideal $E(G)$. We consider the class of closed graphs. A closed graph is a simple graph satisfying the following property: for all edges $\{i, j\}$ and $\{k,…
As the binomial edge ideal of a graph is always generated by homogeneous quadratic polynomials corresponding to the edges of the graph, the question of when a binomial edge ideal defines a Koszul algebra has been studied by many authors…
We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Gr\"obner basis with respect to the reverse lexicographic…
Koszul algebras with quadratic Groebner bases, called strong Koszul algebras, are studied. We introduce affine algebraic varieties whose points are in one-to-one correspondence with certain strong Koszul algebras and we investigate the…
We study the Koszul property of a standard graded $K$-algebra $R$ defined by the binomial edge ideal of a pair of graphs $(G_1,G_2)$. We show that the following statements are equivalent: (i) $R$ is Koszul; (ii) the defining ideal…
It is shown that if the binomial edge ideal of a graph $G$ defines a Koszul algebra, then $G$ must be chordal and claw free. A converse of this statement is proved for a class of chordal and claw free graphs.
In this note, it is proved that a graphs is $(2K_2,P_4)$-free if and only if its edge ring is universally Koszul. Using properties of this family of graphs, we show that Universally Koszul algebras defined by graphs have linear minimal free…
This work concerns the study of properties of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain…
Restuccia and Rinaldo proved that a standard graded $K$-algebra $K[x_1, ... x_n]/I$ is strongly Koszul if the reduced Gr\"obner basis of $I$ with respect to any reverse lexicographic order is quadratic. In this paper, we give a sufficient…
In this article, we investigate the strongly robust property of toric ideals associated with weighted oriented graphs. We establish that the toric ideals of a broad class of monomial ideals are strongly robust; this class encompasses the…
A toric ideal is called robust if its universal Gr\"obner basis is a minimal set of generators, and is called generalized robust if its universal Gr\"obner basis equals its universal Markov basis (the union of all its minimal sets of…
Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${\cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that…
Closed graphs have been characterized by Herzog et al. as the graphs whose binomial edge ideals have a quadratic Gr\"obner basis with respect to a diagonal term order. In this paper, we focus on a generalization of closed graphs, namely…
Let $G$ be a finite connected simple graph and $I_{G}$ the toric ideal of the edge ring $K[G]$ of $G$. In the present paper, we study finite graphs $G$ with the property that $I_{G}$ is generated by quadratic binomials and $I_{G}$ possesses…
Let I be a toric ideal. We say I is robust if its universal Groebner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely…
Strongly Koszul algebras were introduced by Herzog, Hibi and Restuccia in 2000. The goal of the present paper is to provide an in-depth study of such algebras and to investigate how strong Koszulness interacts with the existence of a…
Toric face rings is a generalization of the concepts of affine monoid rings and Stanley-Reisner rings. We consider several properties which imply Koszulness for toric face rings over a field $k$. Generalizing works of Laudal, Sletsj\o{}e…
We define the notion of a twisted topological graph algebra associated to a topological graph and a $1$-cocycle on its edge set. We prove a stronger version of a Vasselli's result. We expand Katsura's results to study twisted topological…