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Let $I(G)^{[k]}$ denote the $k^{th}$ square-free power of the edge ideal $I(G)$ of a graph $G$. In this article, we provide a precise formula for the depth of $I(G)^{[k]}$ when $G$ is a Cohen-Macaulay forest. Using this, we show that for a Cohen-Macaualy forest $G$, the $k^{th}$ square-free power of $I(G)$ is always Cohen-Macaulay, which is quite surprising since all ordinary powers of $I(G)$ can never be Cohen-Macaulay unless $G$ is a disjoint union of edges. Additionally, we provide tight bounds for the regularity and depth of $I(G)^{[k]}$ when $G$ is either a cycle or a whiskered cycle, which aids in identifying when such ideals have linear resolution. Furthermore, we provide combinatorial formulas for the depth of second square-free powers of edge ideals of cycles and whiskered cycles. We also obtained an explicit formula of the regularity of second square-free power for whiskered cycles., Comment: Comments are welcome

Some recent investigations indicate that for the classification of Cohen-Macaulay binomial edge ideals, it suffices to consider biconnected graphs with some whiskers attached (in short, `block with whiskers'). This paper provides explicit combinatorial formulae for the Castelnuovo-Mumford regularity of two specific classes of Cohen-Macaulay binomial edge ideals: (i) chain of cycles with whiskers and (ii) $r$-regular $r$-connected block with whiskers. For the first type, we introduce a new invariant of graphs in terms of the number of blocks in certain induced block graphs, and this invariant may help determine the regularity of other classes of binomial edge ideals. For the second type, we present the formula as a linear function of $r$., Comment: Comments are welcome

Let $I\subset A$ and $J\subset B$ be two monomial ideals, where $A$ and $B$ are two polynomial rings with disjoint variables. Considering a general set-up of monomial filtrations, we study the behaviour of the $\mathrm{v}$-function under binomial expansion. As an application, we get an explicit formula of $\mathrm{v}((I+J)^{(k)})$ in terms of $\mathrm{v}(I^{(i)})$ and $\mathrm{v}(J^{(j)})$, where $L^{(k)}$ denote the symbolic power of an ideal $L$. Furthermore, an analogous formula is extended for the $\mathrm{v}$-function of integral closure of $(I+J)^k$., Comment: comments are welcome

In this article, we give combinatorial formulas for the regularity and the projective dimension of $3$-path ideals of chordal graphs, extending the well-known formulas for the edge ideals of chordal graphs given in terms of the induced matching number and the big height, respectively. As a consequence, we get that the $3$-path ideal of a chordal graph is Cohen-Macaulay if and only if it is unmixed. Additionally, we show that the Alexander dual of the $3$-path ideal of a tree is vertex splittable, thereby resolving the $t=3$ case of a recent conjecture in [Internat. J. Algebra Comput., 33(3):481--498, 2023]. Also, we give examples of chordal graphs where the duals of their $t$-path ideals are not vertex splittable for $t\ge 3$. Furthermore, we extend the formula of the regularity of $3$-path ideals of chordal graphs to all $t$-path ideals of caterpillar graphs. We then provide some families of graphs to show that these formulas for the regularity and the projective dimension cannot be extended to higher $t$-path ideals of chordal graphs (even in the case of trees)., Comment: 22 pages. Comments are welcome

Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain the $\mathrm{v}$-number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for $\mathrm{v}(J_G)$ using the number of vertices of the graph. We characterize all connected graphs $G$ with $\mathrm{v}(J_G) = 2$. We show that for a given pair $(k,m), k\leq m$, there exists a graph $G$ with an associated monomial edge ideal $I$ having $\mathrm{v}$-number equal to $k$ and regularity $m$. If $2k \leq m$, then there exists a binomial edge ideal with $\mathrm{v}$-number $k$ and regularity $m$. Finally, we compute $\mathrm{v}$-number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the $\mathrm{v}$-number of powers of a graded ideal having linear powers, for the class of binomial edge ideals., Comment: 19 pages. Comments welcome

In this paper, we study the asymptotic behaviour of the v-number of a Noetherian graded filtration $\mathcal{I}= \{I_{[k]}\}_{k\geq 0}$ of a Noetherian $\mathbb{N}$-graded domain $R$. Recently, it is shown that $\mathrm{v}(I_{[k]})$ is periodically linear in $k$ for $k \gg 0$. We show that all these linear functions have the same slope, i.e. $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I_{[k]})}{k}$ exists, which is equal to $\displaystyle \lim_{k \rightarrow \infty}\frac{\alpha(I_{[k]})}{k}$, where $\alpha(I)$ denotes the minimum degree of a non-zero element in $I$. In particular, for any Noetherian symbolic filtration $\mathcal{I}= \{I^{(k)}\}_{k\geq 0}$ of $R$, it follows that $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I^{(k)})}{k}=\hat{\alpha}(I)$, the Waldschmidt constant of $I$. Next, for a non-equigenerated square-free monomial ideal $I$, we prove that $\mathrm{v}(I^{(k)}) \leq \mathrm{reg}(R/I^{(k)})$ for $k\gg 0$. Also, for an ideal $I$ having the symbolic strong persistence property, we give a linear upper bound on $\mathrm{v}(I^{(k)})$. As an application, we derive some criteria for a square-free monomial ideal $I$ to satisfy $\mathrm{v}(I^{(k)})\leq \mathrm{reg}(R/I^{(k)})$ for all $k\geq 1$, and provide several examples in support. In addition, for any simple graph $G$, we establish that $\mathrm{v}(J(G)^{(k)}) \leq \mathrm{reg}(R/J(G)^{(k)})$ for all $k \geq 1$, and $\mathrm{v}(J(G)^{(k)}) = \mathrm{reg}(R/J(G)^{(k)})=\alpha(J(G)^{(k)})-1$ for all $k\geq 1$ if and only if $G$ is a Cohen-Macaulay very-well covered graph, where $J(G)$ is the cover ideal of $G$., Comment: Some changes have been made

Recently, Ficarra and Sgroi initiated the study of v-numbers of powers of graded ideals. They proved that for a graded ideal $I$ in a polynomial ring $S$, $\mathrm{v}(I^k)$ is a linear function in $k$ for $k>>0$. Later, Ficarra conjectured that if $I$ is a monomial ideal with linear powers, then $\mathrm{v}(I^k)=\alpha(I)k-1$ for all $k\geq 1$, where $\alpha(I)$ denotes the initial degree of $I$. In this paper, we generalize this conjecture for graded ideals. We prove this conjecture for several classes of graded ideals: principal ideals, ideals $I$ with $\mathrm{depth}(S/I)=0$, cover ideals of graphs, $t$-path ideals, monomial ideals generated in degree $2$, edge ideals of weighted oriented graphs. We reduce the conjecture for several classes of graded ideals (including square-free monomial ideals) by showing it is enough to prove the conjecture for $k=1$ only. We define the stability index of the $\mathrm{v}$-number for graded ideals and investigate the stability index for edge ideals of graphs., Comment: comments are welcome

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 depth of generalized binomial edge ideals. Subsequently, we determine an upper bound for the depth of generalized binomial edge ideals in terms of the vertex-connectivity of graphs. We demonstrate that the difference between the upper and lower bounds can be arbitrarily large, even in cases when one of the bounds is sharp. In addition, we calculate the depth of generalized binomial edge ideals of certain classes of graphs, including cyclic graphs and graphs with Cohen-Macaulay binomial edge ideals., Comment: 15 pages, 5 figures. Comments are welcome

In this paper, we show the equality of the (local) $\mathrm{v}$-number and Castelnuovo-Mumford regularity of certain classes of Gorenstein algebras, including the class of Gorenstein monomial algebras. Also, for the same classes of algebras with the assumption of level, we show that the (local) $\mathrm{v}$-number serves as an upper bound for the regularity. Moreover, we investigate the $\mathrm{v}$-number of Frobenius powers of graded ideals in prime characteristic setup. In this study, we demonstrate that the $\mathrm{v}$-numbers of Frobenius powers of graded ideals have an asymptotically linear behaviour. In the case of unmixed monomial ideals, we provide a method for computing the $\mathrm{v}$-number without prior knowledge of the associated primes., Comment: 15 pages, comments are welcome

In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if $D$ is a weighted oriented graph whose edge ideal $I(D)$ is componentwise linear, then the underlying simple graph $G$ of $D$ is co-chordal. This is an analogue of Fr\"oberg's theorem for weighted oriented graphs. We give combinatorial characterizations of componentwise linearity of $I(D)$ if $V^+$ are sinks or $\vert V^+ \vert\leq 1$. Furthermore, if $G$ is chordal or bipartite or $V^+$ are sinks or $\vert V^+ \vert\leq 1$, then we show the following equivalence for $I(D)$: $$ \text{Vertex splittable}\,\, \Longleftrightarrow\,\, \text{Linear quotient}\,\, \Longleftrightarrow\,\, \text{Componentwise linear}.$$, Comment: 23 pages, 3 figures. Comments are welcome

The $\mathrm{v}$-number of a graded ideal $I\subseteq R$, denoted by $\mathrm{v}(I)$, is the minimum degree of a polynomial $f$ for which $I:f$ is a prime ideal. Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021) studied the $\mathrm{v}$-number of edge ideals. In this paper, we study the $\mathrm{v}$-number of the cover ideal $J(G)$ of a graph $G$. The main result shows that $\mathrm{v}(J(G))\leq \mathrm{reg}(R/J(G))$ for any simple graph $G$, which is quite surprising because, for the case of edge ideals, this inequality does not hold. Our main result relates $\mathrm{v}(J(G))$ with the Cohen-Macaulay property of $R/I(G)$. We provide an infinite class of connected graphs, which satisfy $\mathrm{v}(J(G))=\mathrm{reg}(R/J(G))$. Also, we show that for every positive integer $k$, there exists a connected graph $G_k$ such that $\mathrm{reg}(R/J(G_k))-\mathrm{v}(J(G_k))=k$. Also, we explicitly compute the $\mathrm{v}$-number of cover ideals of cycles., Comment: 12 pages, 1 figure. Comments are welcome

Herzog, Hibi, and Zheng classified the Cohen-Macaulay edge ideals of chordal graphs. In this paper, we classify Cohen-Macaulay edge ideals of (vertex) weighted oriented chordal and simplicial graphs, a more general class of monomial ideals. In particular, we show that the Cohen-Macaulay property of these ideals is equivalent to the unmixed one and hence, independent of the underlying field., Comment: Lemma 3.1 of the version 1 was wrong and the proof of Theorem 3.2 (v1) has been modified and became Theorem 3.1 in (v2)

The invariant $\mathrm{v}$-number was introduced very recently in the study of Reed-Muller-type codes. Jaramillo and Villarreal (J Combin. Theory Ser. A 177:105310, 2021) initiated the study of the $\mathrm{v}$-number of edge ideals. Inspired by their work, we take the initiation to study the $\mathrm{v}$-number of binomial edge ideals in this paper. We discuss some properties and bounds of the $\mathrm{v}$-number of binomial edge ideals. We explicitly find the $\mathrm{v}$-number of binomial edge ideals locally at the associated prime corresponding to the cutset $\emptyset$. We show that the $\mathrm{v}$-number of Knutson binomial edge ideals is less than or equal to the $\mathrm{v}$-number of their initial ideals. Also, we classify all binomial edge ideals whose $\mathrm{v}$-number is $1$. Moreover, we try to relate the $\mathrm{v}$-number with the Castelnuvo-Mumford regularity of binomial edge ideals and give a conjecture in this direction.

Conca and Varbaro (Invent. Math. 221 (2020), no. 3) showed the equality of depth of a graded ideal and its initial ideal in a polynomial ring when the initial ideal is square-free. In this paper, we give some beautiful applications of this fact in the study of Cohen-Macaulay binomial edge ideals. We prove that for the characterization of Cohen-Macaulay binomial edge ideals, it is enough to consider only "biconnected graphs with some whisker attached" and this done by investigating the initial ideals. We give several necessary conditions for a binomial edge ideal to be Cohen-Macaulay in terms of smaller graphs. Also, under a hypothesis, we give a sufficient condition for Cohen-Macaulayness of binomial edge ideals in terms of blocks of graphs. Moreover, we show that a graph with Cohen-Macaulay binomial edge ideal has girth less than $5$ or equal to infinity.

For a graph $G$, Bolognini et al. have shown $J_{G}$ is strongly unmixed $\Rightarrow$ $J_{G}$ is Cohen-Macaulay $\Rightarrow$ $G$ is accessible, where $J_{G}$ denotes the binomial edge ideals of $G$. Accessible and strongly unmixed properties are purely combinatorial. We give some motivations to focus only on blocks with whiskers for the characterization of all $G$ with Cohen-Macaulay $J_{G}$. We show that accessible and strongly unmixed properties of $G$ depend only on the corresponding properties of its blocks with whiskers and vice versa. Also, we give an infinite class of graphs whose binomial edge ideals are Cohen-Macaulay, and from that, we classify all $r$-regular $r$-connected graphs such that attaching some special whiskers to it, the binomial edge ideals become Cohen-Macaulay. Finally, we define a new class of graphs, called \textit{strongly $r$-cut-connected} and prove that the binomial edge ideal of any strongly $r$-cut-connected accessible graph having at most three cut vertices is Cohen-Macaulay.

The study of the edge ideal $I(D_{G})$ of a weighted oriented graph $D_{G}$ with underlying graph $G$ started in the context of Reed-Muller type codes. We generalize a Cohen-Macaulay construction for $I(D_{G})$, which Villarreal gave for edge ideals of simple graphs. We use this construction to classify all the Cohen-Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We show that the conjecture on Cohen-Macaulayness of $I(D_{G})$, proposed by Pitones et al. (2019), holds for $I(D_{C_{n}})$, where $C_{n}$ denotes the cycle of length $n$. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of $I(D_{G})$ and its conditions to be Cohen-Macaulay., Comment: 26 pages

We generalize some results of $\mathrm{v}$-number for arbitrary monomial ideals by showing that the $\mathrm{v}$-number of an arbitrary monomial ideal is the same as the $\mathrm{v}$-number of its polarization. We prove that the $\mathrm{v}$-number $\mathrm{v}(I(G))$ of the edge ideal $I(G)$, the induced matching number $\mathrm{im}(G)$ and the regularity $\mathrm{reg}(R/I(G))$ of a graph $G$, satisfy $\mathrm{v}(I(G))\leq \mathrm{im}(G)\leq \mathrm{reg}(R/I(G))$, where $G$ is either a bipartite graph, or a $(C_{4},C_{5})$-free vertex decomposable graph, or a whisker graph. There is an open problem in \cite{v}, whether $\mathrm{v}(I)\leq \mathrm{reg}(R/I)+1$ for any square-free monomial ideal $I$. We show that $\mathrm{v}(I(G))>\mathrm{reg}(R/I(G))+1$, for a disconnected graph $G$. We derive some inequalities of $\mathrm{v}$-numbers which may be helpful to answer the above problem for the case of connected graphs. We connect $\mathrm{v}(I(G))$ with an invariant of the line graph $L(G)$ of $G$. For a simple connected graph $G$, we show that $\mathrm{reg}(R/I(G))$ can be arbitrarily larger than $\mathrm{v}(I(G))$. Also, we try to see how the $\mathrm{v}$-number is related to the Cohen-Macaulay property of square-free monomial ideals., Comment: By adding Lemma 3.2, we have modified Proposition 3.2 (in v1) to Proposition 3.3 (in v2). Also, Theorem 3.3, Proposition 3.4, Proposition 3.5 have been modified to Theorem 3.4, Proposition 3.7, Proposition 3.9, respectively. We have added Lemma 3.8 to modify the proof of Proposition 3.9 (v2). Also, Corollary 3.5 and Example 3.6 have been added

Let $\mathbf{CCM}$ denote the class of closed graphs with Cohen-Macaulay binomial edge ideals and $\mathbf{PIG}$ denote the class of proper interval graphs. Then $\mathbf{CCM}\subseteq \mathbf{PIG}$. The $\mathbf{PIG}$-completion problem is a classical problem in molecular biology as well as in graph theory and this problem is known to be NP-hard. In this paper, we study the $\mathbf{CCM}$-completion problem. We give a method to construct all possible $\mathbf{CCM}$-completion of a graph. We find the $\mathbf{CCM}$-completion number and the set of all minimal $\mathbf{CCM}$-completions for a large class of graphs. Moreover, for that class, we give a polynomial-time algorithm to compute the $\mathbf{CCM}$-completion number and a minimum $\mathbf{CCM}$-completion of a given graph. We investigate unmixed and Cohen-Macaulay properties of binomial edge ideals of induced subgraphs. Also, we discuss the accessible graphs completion and the Cohen-Macaulay property of binomial edge ideals of whisker graphs., Comment: Substantially extended and revised version

In this paper, we introduce the notion of binomial edge ideals of a clutter and obtain results similar to those obtained for graphs by Rauf \& Rinaldo in \cite{raufrin}. We also answer a question posed in their paper.