Search

In this paper we investigate the monomial ideals which satisfy the copersistence property or nearly copersistence property.

In this paper, we first show that any square-free monomial ideal in $K[x_1, x_2, x_3, x_4, x_5]$ has the strong persistence property. Next we will provide a criterion for a minimal counterexample to the Conforti-Cornuejols conjecture. Finally we give a necessary and sufficient condition to determine the normally torsion-freeness of a linear combination of two normally torsion-free square-free monomial ideals.

Let $I\subset R=K[x_1, \ldots, x_n]$ be a square-free monomial ideal, $\mathfrak{q}$ be a prime monomial ideal in $R$, $h$ be a square-free monomial in $R$ with $\mathrm{supp}(h) \cap (\mathrm{supp}(\mathfrak{q}) \cup \mathrm{supp}(I))=\emptyset$, and $L:=I\cap (\mathfrak{q}, h)$. In this paper, we first focus on the associated primes of powers of $L$ and explore the normally torsion-freeness of $L$. We also give an application on a comb inatorial result. Next, we study when a square-free monomial ideal is minimally not normally torsion-free. Particularly, we introduce a class of square-free monomial ideals, which are minimally not normally torsion-free., Comment: This paper will appear in "Journal of Algebra and its Applications" (accepted)

In this paper, we give a classification of all Mengerian $4$-uniform hypergraphs derived from graphs., Comment: to appear in Ars Combinatoria

In this paper, we unfold balanced and totally balanced neighborhood hypergraphs to discover new classes of normally torsion-free monomial ideals. As a consequence, we establish that the closed neighborhood ideals and the dominating ideals of strongly chordal graphs are normally torsion-free. We discuss the stable sets of associated primes of the dominating ideals of cycles and characterize all the cycles with normally torsion-free dominating ideals., Comment: Any comments are welcome

In this paper, we first give some sufficient criteria for normality of monomial ideals. As applications, we show that closed neighborhood ideals of complete bipartite graphs are normal, and hence satisfy the (strong) persistence property. We also prove that dominating ideals of complete bipartite graphs are nearly normally torsion-free. In addition, we show that dominating ideals of $h$-wheel graphs, under certain condition, are normal.

In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals., Comment: 15 pages, 2 figures; pre-published version

We study the closed neighborhood ideals and the dominating ideals of graphs, in particular, of trees and cycles. We prove that the closed neighborhood ideals and the dominating ideals of trees are normally torsion-free. The closed neighborhood ideals and the dominating ideals of cycles fail to be normally torsion-free. However, we prove that the closed neighborhood ideals of cycles admit the (strong) persistence property and the dominating ideals of cycles are nearly normally torsion-free., Comment: Accepted in the Mediterranean Journal of Mathematics Lemma 2.5, Corollary 2.6, Corollary 2.8, Lemma 2.18, Corollary 2.19, and Corollary 2.20. arXiv admin note: text overlap with arXiv:2106.10219

Let $I$ be a square-free monomial ideal in a polynomial ring $R=K[x_1,\ldots, x_n]$ over a field $K$, $\mathfrak{m}=(x_1, \ldots, x_n)$ be the graded maximal ideal of $R$, and $\{u_1, \ldots, u_{\beta_1(I)}\}$ be a maximal independent set of minimal generators of $I$ such that $\mathfrak{m}\setminus x_i \notin \mathrm{Ass}(R/(I\setminus x_i)^t)$ for all $x_i\mid \prod_{i=1}^{\beta_1(I)}u_i$ and some positive integer $t$, where $I\setminus x_i$ denotes the deletion of $I$ at $x_i$ and $\beta_1(I)$ denotes the maximum cardinality of an independent set in $I$. In this paper, we prove that if $\mathfrak{m}\in \mathrm{Ass}(R/I^t)$, then $t\geq \beta_1(I)+1$. As an application, we verify that under certain conditions, every unmixed K\"onig ideal is normally torsion-free, and so has the strong persistence property. In addition, we show that every square-free transversal polymatroidal ideal is normally torsion-free. Next, we state some results on the corner-elements of monomial ideals. In particular, we prove that if $I$ is a monomial ideal in a polynomial ring $R=K[x_1, \ldots, x_n]$ over a field $K$ and $z$ is an $I^t$-corner-element for some positive integer $t$ such that $\mathfrak{m}\setminus x_i \notin \mathrm{Ass}(I\setminus x_i)^t$ for some $1\leq i \leq n$, then $x_i$ divides $z$., Comment: To appear in Rocky Mountain Journal of Mathematics

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 normally torsion-free $t$-spread principal Borel ideals that can also be viewed as edge ideals of uniform multipartite hypergraphs.

We prove that the set of all paths of a fixed length in a complete multipartite graph is the bases of a matroid. Moreover, we discuss the Cohen-Macaulayness and depth of powers of $t$-path ideals of a complete multipartite graph., Comment: This paper has been published in Journal of Algebra and Its Applications