Related papers: Koszulness of binomial edge ideals

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…

Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a study of…

The independence polynomial $i(G,x)$ of a graph $G$ is the generating function of the numbers of independent sets of each size. A graph of order $n$ is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu…

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I(R)^*$ be the set of all non-trivial ideals of $R$. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$, is a graph with the vertex set $I(R)^*$, and two…

We study powers of binomial edge ideals associated with closed and block graphs.

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…

Let $G$ be a graph which belongs to either of the following classes: (i) bipartite graphs, (ii) unmixed graphs, or (iii) claw--free graphs. Assume that $J(G)$ is the cover ideal $G$ and $J(G)^{(k)}$ is its $k$-th symbolic power. We prove…

We give an especially simple proof of a theorem in graph theory that forms the key part of the solution to a problem in commutative algebra, on how to characterize the integral closure of a polynomial ring generated by quadratic monomials.

An equigenerated monomial ideal $I$ in the polynomial ring $R=k[z_1,\ldots, z_n]$ is a Freiman ideal if $\mu(I^2)=\ell(I)\mu(I)-{\ell(I)\choose 2}$ where $\ell(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal…

Let $S$ be the power series ring or the polynomial ring over a field $K$ in the variables $x_1,\ldots,x_n$, and let $R=S/I$, where $I$ is proper ideal which we assume to be graded if $S$ is the polynomial ring. We give an explicit…

We associate to every graph a linear program for packings of vertex disjoint paths. We show that the optimal primal and dual values of the corresponding integer program are the binomial grade and height of the binomial edge ideal of the…

A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra is a coherent algebra. In this study we will find a necessary and sufficient condition for a graph to be a pattern polynomial graph. Some of the properties of the…

We give a complete characterization of graphs whose binomial edge ideal is licci. An important tool is a new general upper bound for the regularity of binomial edge ideals.

Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $I(G)$ and $J(G)$ denote the edge ideal and the cover ideal of $G$, respectively. We…

For any two integers $d,r \geq 1$, we show that there exists an edge ideal $I(G)$ such that the ${\rm reg}\left(R/I(G)\right)$, the Castelnuovo-Mumford regularity of $R/I(G)$, is $r$, and ${\rm deg} (h_{R/I(G)}(t))$, the degree of the…

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs,…

Let $K[G]$ denote the edge ring of a finite connected simple graph $G$ on $[d]$ and $\mat(G)$ the matching number of $G$. It is shown that $\reg(K[G]) \leq \mat(G)$ if $G$ is non-bipartite and $K[G]$ is normal, and that $\reg(K[G]) \leq…

We will study monomial ideals $I$ in the exterior algebra as well as in the polynomial ring whose generic initial ideal is constant for all term orders up to permutations of variables. First, in the exterior algebra, we determine all graphs…

Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \subseteq S$ denote the edge ideal of a graph $G$. We show that the $\ell$th symbolic power $I(G)^{(\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\ell)}$ is…

Let (P,<) be a finite poset and let G be its comparability graph. If cl(G) is the clutter of maximal cliques of G, we prove that cl(G) satisfies the max-flow min-cut property and that its edge ideal is normally torsion free. We prove that…