Related papers: Cohen-Macaulay Weighted Oriented Chordal and Simpl…
Let $G$ be a finite simple graph on a vertex set $V(G)=\{x_{11}, \ldots, x_{n1}\}$. Also let $m_1, \ldots,m_n \geq 2$ be integers and $G_1, \ldots, G_n$ be connected simple graphs on the vertex sets $V(G_i)=\{x_{i1}, \ldots, x_{im_i}\}$. In…
We associate a {\it skew tableau ideal} to each filling of a skew Ferrers diagram with positive integers. We classify all unmixed and sequentially Cohen-Macaulay skew tableau ideals. Consequently, we classify all Cohen-Macaulay, Buchsbaum,…
Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which…
We study the licci property for several classes of squarefree monomial ideals arising from graphs and related combinatorial structures. We characterize licci bi-Cohen-Macaulay squarefree monomial ideals, complementary edge ideals, $t$-path…
We describe combinatorially the Cohen-Macaulay type of edge-weighted r-path suspensions of edge-weighted graphs for an arbitrary positive integer r. The computation of the Cohen-Macaulay type of edge-weighted suspensions of edge-weighted…
Let $G_\omega$ be an edge-weighted graph whose underlying graph is $G$. In this paper, we enlarge the class of Cohen-Macaulay edge-weighted graphs $G_\omega$ by classifying completely them when the graph $G$ has girth $5$ or greater.
In this paper we extend one direction of Fr\"oberg's theorem on a combinatorial classification of quadratic monomial ideals with linear resolutions. We do this by generalizing the notion of a chordal graph to higher dimensions with the…
We introduce and study the weighted $r$-path ideal of a weighted graph $G_\omega$, which is a common generalization of Conca and De Negri's $r$-path ideal for unweighted graphs and Paulsen and Sather-Wagstaff's edge ideal of the weighted…
We graph-theoretically characterize triangle-free Gorenstein graphs $G$. As an application, we classify when $I(G)^2$ is Cohen-Macaulay.
In this short note we prove that the projective dimension of a sequentially Cohen-Macaulay square-free monomial ideal is equal to the maximal height of its minimal primes (also known as the big height), or equivalently, the maximal…
Let $D=(G,\mathcal{O},w)$ be a weighted oriented graph whose edge ideal is $I(D)$. In this paper, we characterize the unmixed property of $I(D)$ for each one of the following cases: $G$ is an $SCQ$ graph; $G$ is a chordal graph; $G$ is a…
We introduce a new family of graphs, namely, hybrid graphs. There are infinitely many hybrid graphs associated to a single graph. We show that every hybrid graph associated to a given graph is Cohen Macaulay. Furthermore, we show that every…
In this paper, our main focus is to explore different classes of nearly normally torsion-free ideals. We first characterize all finite simple connected graphs with nearly normally torsion-free cover ideals. Next, we characterize all…
We prove that wheels and block graphs have sequentially Cohen-Macaulay binomial edge ideals. Moreover, we provide a construction of new families of sequentially Cohen-Macaulay graphs by cones.
Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and…
In this paper we discuss the problem of characterizing the Cohen-Macaulay property of certain families of monomial ideals with fixed radical. More precisely, we consider generically complete intersection monomial ideals whose radical…
We prove the multiplicity bounds conjectured by Herzog-Huneke-Srinivasan and Herzog-Srinivasan in the following cases: the strong conjecture for edge ideals of bipartite graphs, and the weaker Taylor bound conjecture for all quadratic…
Scattered over the past few years have been several occurrences of simplicial complexes whose topological behavior characterize the Cohen-Macaulay property for quotients of polynomial rings by arbitrary (not necessarily squarefree) monomial…
We identify several classes of monomial ideals that possess minimal generalized Barile-Macchia resolutions. These classes of ideals include generic monomial ideals, monomial ideals with linear quotients, and edge ideals of hypertrees. We…
Let $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will give sufficient and necessary combinatorial conditions of $G$ in which the second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp. Buchsbaum,…