Related papers: Bounds on the regularity of toric ideals of graphs

For all integers $4 \leq r \leq d$, we show that there exists a finite simple graph $G= G_{r,d}$ with toric ideal $I_G \subset R$ such that $R/I_G$ has (Castelnuovo-Mumford) regularity $r$ and $h$-polynomial of degree $d$. To achieve this…

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 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…

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…

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…

Let $G$ be a finite simple graph and let $I_G$ denote its associated toric ideal in the polynomial ring $R$. For each integer $n\geq 2$, we completely determine all the possible values for the tuple $({\rm reg}(R/I_G), {\rm…

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…

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 $G$…

Let G be a finite simple graph and let indm(G) and ordm(G) denote the induced matching number and the ordered matching number of G, respectively. We characterize all bipartite graphs G with indm(G) = ordm(G). We establish the…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is the graph…

We study the Castelnuovo-Mumford regularity of powers of edge ideals. We prove that if G is a bipartite graph, then reg(I(G)^s) \leq 2s + reg I(G) - 2 for all s \geq 2, which is the best possible upper bound for any s. Suspension plays a…

Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this article, we study the Castelnuovo-Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh's conjecture for wheel graphs,…

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…

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…

Let G be a graph obtained by taking r>=2 paths and identifying all first vertices and identifying all the last vertices. We compute the Castelnuovo--Mumford regularity of the quotient S/I(X), where S is the polynomial ring on the edges of G…

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…

Let $G$ be a finite simple graph on $n$ vertices. Let $J_G \subset K[x_1, \ldots, x_n]$ be the cover ideal of $G$. In this article, we obtain syzygies, Betti numbers and Castelnuovo-Mumford regularity of $J_G^s$ for all $s \geq 1$ for…

Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in…

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 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.