Related papers: On a regularity-conjecture of generalized binomial…

In this article, we survey the recent results on the Castelnuovo-Mumford regularity of binomial edge ideals and generalized binomial edge ideals. We also generalize some of the known upper bounds for binomial edge ideals to the case of…

Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in $S = K[x_1, \ldots, x_n, y_1,\ldots, y_n].$ We prove that the Castelnuovo-Mumford regularity of $J_G$ is bounded above by $c(G)+1$ when $G$…

We prove a conjectured upper bound for the Castelnuovo-Mumford regularity of binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that $\mathrm{reg}(J_G)\leq n-1$ for any graph $G$ with $n$ vertices, which is not a…

We prove two recent conjectures on some upper bounds for the Castelnuovo-Mumford regularity of the binomial edge ideals of some different classes of graphs. We prove the conjecture of Matsuda and Murai for graphs which has a cut edge or a…

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo-Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by $m(G)+1$, where…

We determine the Castelnuovo-Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph theoretic…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of $I(G)^q$ in terms of certain combinatorial invariants associated with $G$. We…

In this paper, we mainly study the Castelnuovo-Mumford regularity of the generalized binomial edge ideals of graphs. We show that this number can be any integer number from $2$ to $n-1$ where $n$ is the number of vertices in the underlying…

In this paper we prove the conjectured upper bound for Castelnuovo-Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the…

We show that the Castelnuovo-Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices.

A famous theorem of Kalai and Meshulam is that $\mathrm{reg}(I + J) \leq \mathrm{reg}(I) + \mathrm{reg}(J) -1$ for any squarefree monomial ideals $I$ and $J$. This result was subsequently extended by Herzog to the case where $I$ and $J$ are…

Connected bipartite graphs whose binomial edge ideals are Cohen--Macaulay have been classified by Bolognini et al. In this paper, we compute the depth, Castelnuovo--Mumford regularity, and dimension of the generalized binomial edge ideals…

We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of symbolic powers of certain…

Let $I$ and $J$ be edge ideals in a polynomial ring $R = \mathbb{K}[x_1,\ldots,x_n]$ with $I \subseteq J$. In this paper, we obtain a general upper and lower bound for the Castelnuovo-Mumford regularity of $IJ$ in terms of certain…

In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its…

We study the regularity of binomial edge ideals. For a closed graph $G$ we show that the regularity of the binomial edge ideal $J_G$ coincides with the regularity of $\ini_{\lex}(J_G)$ and can be expressed in terms of the combinatorial data…

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs…

In this article, we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique. As a consequence, we obtain an upper bound for the regularity of binomial edge ideal of a…

This research focuses on analyzing the depth of generalized binomial edge ideals. We extend the notion of $d$-compatible map for the pairs of a complete graph and an arbitrary graph, and using it, we give a combinatorial lower bound for the…