arXiv subjects

Kamalesh Saha

Publications and source records attributed to Kamalesh Saha.

At least 19 recordsLinked to original sources

arXiv2026

Lattice points arising from regularity and $\mathrm{v}$-number of Graphs: Whisker and Cameron-Walker

Let $G$ be a simple graph on $n$ vertices and $I(G)\subseteq R$ be its edge ideal. In this paper, we initiate the study of determining lattice points in $\mathbb{N}^2$ that appear as a pair $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$, where $G$ ranges over all connected graphs on $n$ vertices, and we denote this set by $\mathcal{RV}(n)$. Here `$\mathrm{reg}$' denotes the (Castelnuovo-Mumford) regularity and `$\mathrm{v}$' denotes the $\mathrm{v}$-number. We establish general bounds for $\mathcal{RV}(n)$ by identifying two sets $A(n)$ and $B(n)$ satisfying $A(n)\subseteq \mathcal{RV}(n)\subseteq B(n)$. Furthermore, we explicitly determine the subsets of $\mathcal{RV}(n)$ consisting of all possible pairs $(\mathrm{reg}(R/I(G)), \mathrm{v}(I(G)))$ arising from whisker graphs and Cameron-Walker graphs on $n$ vertices. Finally, we propose a conjecture on the subset of $\mathcal{RV}(n)$ arising from connected chordal graphs.

math.AC↗

arXiv2026

Generalized binomial edge ideals of whisker graphs via an extension of generalized corona products

In this paper, we initiate a systematic study of generalized binomial edge ideals of whisker graphs by working within a substantially broader class of graphs. We extend the notion of generalized corona products, and through this enlarged framework, investigate fundamental algebraic invariants such as depth, (Castelnuovo-Mumford) regularity, and the Cohen-Macaulay property. In particular, we establish a sharp lower bound on the depth of generalized binomial edge ideals for our extended class, and further obtain explicit depth formula for a broad subclass of this family, which in turn recovers the depth formula for whisker graphs. We also establish sharp upper bounds for the regularity, and in the case of binomial edge ideals of whisker graphs over gap-free graphs, determine the exact value of the regularity. Finally, for our extended class, we provide a combinatorial classification of all Cohen-Macaulay binomial edge ideals, which in turn yields a new construction of Cohen-Macaulay binomial edge ideals.

math.AC↗

arXiv2025

Stanley-Reisner ideals of higher independence complexes of chordal graphs

For $t\geq 2$, the $t$-independence complex $\mathrm{Ind}_t(G)$ of a graph $G$ is the collection of all $A\subseteq V(G)$ such that each connected component of the induced subgraph $G[A]$ has at most $t-1$ vertices. The topology of $\mathrm{Ind}_t(G)$ is intimately related to the combinatorial property of $G$. In this article, we consider the Stanley-Reisner ideal $J_{t}(G)$ of $\mathrm{Ind}_t(G)$ and focus on its algebraic properties. We prove that for a chordal graph $G$ and for all $t$ \[ \mathrm{reg}(R/J_{t}(G))=(t-1)ν_{t}(G) \text{ and } \mathrm{pd}(R/J_{t}(G))=\mathrm{bight}(J_{t}(G)), \] where $ν_{t}(G)$ denotes the induced matching number of the corresponding hypergraph of $J_{t}(G)$, and $\mathrm{reg}$, $\mathrm{pd}$ and $\mathrm{bight}$ stand for the regularity, projective dimension, and big height, respectively. As a consequence of the above results, we combinatorially characterize when the Stanley-Reisner ideal of the $t$-independence complex of a chordal graph has a linear resolution as well as when it satisfies the Cohen-Macaulay property. The above formulas and their consequences can be seen as a nice generalization of the classical results corresponding to the edge ideals of chordal graphs.

math.CO↗

arXiv2025

On a regularity-conjecture of generalized binomial edge ideals

In this paper, we prove the upper bound conjecture proposed by Saeedi Madani \& Kiani on the Castelnuovo-Mumford regularity of generalized binomial edge ideals. We give a combinatorial upper bound of regularity for generalized binomial edge ideals, which is better than the bound claimed in that conjecture. Also, we show that the bound is tight by providing an infinite class of graphs.

math.AC↗

arXiv2025

Equality of ordinary and symbolic powers and the Conforti-Cornuéjols conjecture for $(n-2)$-uniform clutters

Let $I$ be an equigenerated squarefree monomial ideal in the polynomial ring $\mathbb{K}[x_1,\ldots,x_n]$, and let $\mathcal{H}$ be a uniform clutter on the vertex set $\{x_1,\ldots,x_n\}$ such that $I=I(\mathcal{H})$ is its edge ideal. A central and challenging problem in combinatorial commutative algebra is to classify all clutters $\mathcal{H}$ for which $I(\mathcal{H})^{(k)} = I(\mathcal{H})^{k}$ for a fixed positive integer $k$, where $I(\mathcal{H})^{(k)}$ denotes the $k^{\text{th}}$ symbolic power of $I(\mathcal{H})$. In this article, we give a complete solution to this problem for $(n-2)$-uniform clutters. Moreover, we provide a simple combinatorial classification of all $(n-2)$-uniform clutters having the packing property. As a consequence, we confirm the celebrated Conforti-Cornuéjols conjecture for $(n-2)$-uniform clutters. We also compare our results with the known families of clutters for which the conjecture is known to be true. Finally, we present an application of our results to the theory of Linear Programming duality problems.

math.AC↗

arXiv2025

Admissible set and squarefree-power-like function with applications to squarefree symbolic powers

We introduce the abstract notion of squarefree-power-like functions, which unify the sequences of squarefree ordinary and symbolic powers of squarefree monomial ideals. By employing the Tor-vanishing criteria for mixed sums of ideals, we establish sharp lower bounds for their Castelnuovo-Mumford regularity in terms of what we call the admissible set of the associated hypergraph. As an application, we derive the first general combinatorial lower bound for the regularity of squarefree symbolic powers of monomial ideals. In the setting of edge ideals, by exploiting the special combinatorial structures of block graphs and Cohen-Macaulay chordal graphs, we show that this bound turns into an exact formula for all squarefree symbolic powers of block graphs, as well as for the second squarefree symbolic powers of edge ideals of Cohen-Macaulay chordal graphs.

math.AC↗

arXiv2025

Square-free powers of Cohen-Macaulay simplicial forests

Let $I(Δ)^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $Δ$ in a polynomial ring $R$. Square-free powers are intimately related to the `Matching Theory' and `Ordinary Powers'. In this article, we show that if $Δ$ is a Cohen-Macaulay simplicial forest, then $R/I(Δ)^{[k]}$ is Cohen-Macaulay for all $k\ge 1$. This result is quite interesting since all ordinary powers of a graded radical ideal can never be Cohen-Macaulay unless it is a complete intersection. To prove the result, we introduce a new combinatorial notion called special leaf, and using this, we provide an explicit combinatorial formula of $\mathrm{depth}(R/I(Δ)^{[k]})$ for all $k\ge 1$, where $Δ$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing.

math.AC↗

arXiv2025

Comparing the $\mathrm{v}$-number and $h$-polynomials of edge ideals

In this paper, we compare the $\mathrm{v}$-numbers and the degree of the $h$-polynomials associated with edge ideals of connected graphs. We prove that the $\mathrm{v}$-number can be arbitrarily larger or smaller than the degree of the $h$-polynomial for the edge ideal of a connected graph. We also establish that for any pair of positive integers $(v,d)$ with $v \leq d$, there exists a connected graph $H(v,d)$ with the $\mathrm{v}$-number equal to $v$ and the degree of $h$-polynomial equal to $d$. Additionally, we show that the sum of the $\mathrm{v}$-number and the degree of the $h$-polynomial is bounded above by $n$, the number of vertices of $G$, and we classify all graphs for which this sum is exactly $n$. Finally, we show that all thirteen possible inequalities among the three invariants, the $\mathrm{v}$-number, the degree of the $h$-polynomial, and the Castelnuovo-Mumford regularity, can occur in the case of edge ideals of connected graphs. Many of these examples rely on a minimal example of a graph whose $\mathrm{v}$-number is more than the degree of its $h$-polynomial. Using a computer search, we show that there are exactly two such graphs on 11 vertices and 25 edges, and no smaller example on fewer vertices, or 11 vertices and less than 25 edges.

math.AC↗

arXiv2025

On the homological shifts of cover ideals of Cohen-Macaulay graphs

For a non-negative integer $k$, let $\mathrm{HS}_{k}(J(G))$ denote the $k^{\text{th}}$ homological shift ideal of the vertex cover ideal $J(G)$ of a graph $G$. For each $k\geq 2$, we construct a Cohen-Macaulay very well-covered graph $G_k$ which is both Cohen-Macaulay bipartite and a whiskered graph so that $\mathrm{HS}_{k}(J(G))$ does not have a linear resolution. This contradicts several results as well as disproves a conjecture in [J. Algebra, $\mathbf{629}$, (2023), 76-108] and [Mediterr. J. Math., $\mathbf{21}$, 135 (2024)]. The graphs $G_k$ are also examples of clique-whiskered graphs introduced by Cook and Nagel, which include Cohen-Macaulay chordal graphs, Cohen-Macaulay Cameron-Walker graphs, and clique corona graphs. Surprisingly, for Cohen-Macaulay chordal graphs, we can use a special ordering on the minimal generators to show that $\mathrm{HS}_{k}(J(G))$ has linear quotients for all $k$. Moreover, for all Cohen-Macaulay Cameron-Walker graphs and certain clique corona graphs, we show that $\mathrm{HS}_{k}(J(G))$ is weakly polymatroidal, and thus, has linear quotients for all $k$.

math.AC↗

arXiv2025

Admissible matchings and the Castelnuovo-Mumford regularity of square-free powers

Let $I$ be any square-free monomial ideal, and $\mathcal{H}_I$ denote the hypergraph associated with $I$. Refining the concept of $k$-admissible matching of a graph defined by Erey and Hibi, we introduce the notion of generalized $k$-admissible matching for any hypergraph. Using this, we give a sharp lower bound on the (Castelnuovo-Mumford) regularity of $I^{[k]}$, where $I^{[k]}$ denotes the $k^{\text{th}}$ square-free power of $I$. In the special case when $I$ is equigenerated in degree $d$, this lower bound can be described using a combinatorial invariant $\mathrm{aim}(\mathcal{H}_I,k)$, called the $k$-admissible matching number of $\mathcal{H}_I$. Specifically, we prove that $\mathrm{reg}(I^{[k]})\ge (d-1)\mathrm{aim}(\mathcal{H}_I,k)+k$, whenever $I^{[k]}$ is non-zero. Even for the edge ideal $I(G)$ of a graph $G$, it turns out that $\mathrm{aim}(G,k)+k$ is the first general lower bound for the regularity of $I(G)^{[k]}$. In fact, when $G$ is a forest, $\mathrm{aim}(G,k)$ coincides with the $k$-admissible matching number introduced by Erey and Hibi. Next, we show that if $G$ is a block graph, then $\mathrm{reg}(I(G)^{[k]})= \mathrm{aim}(G,k)+k$, and this result can be seen as a generalization of the corresponding regularity formula for forests. Additionally, for a Cohen-Macaulay chordal graph $G$, we prove that $\mathrm{reg}(I(G)^{[2]})= \mathrm{aim}(G,2)+2$. Finally, we propose a conjecture on the regularity of square-free powers of edge ideals of chordal graphs.

math.AC↗

arXiv2024

Square-free powers of Cohen-Macaulay forests, cycles, and whiskered cycles

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-Macaulay 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. Next, we give an exact formula for the regularity and tight bounds on the depth of square-free powers of edge ideals of cycles. In the case of whiskered cycles, we obtain tight bounds on the regularity and depth of square-free powers, which aids in identifying when such ideals have linear resolutions. Additionally, we compute depth of $I(G)^{[2]}$ when $G$ is a cycle or whiskered cycle, and regularity of $I(G)^{[2]}$ when $G$ is a whiskered cycle.

math.AC↗

arXiv2024

Cohen-Macaulay Property of Binomial Edge Ideals with Girth of Graphs

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.

math.AC↗

arXiv2024

Regularity of two classes of Cohen-Macaulay binomial edge ideals

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 $r$.

math.AC↗

arXiv2024

Binomial expansion and the $\mathrm{v}$-number

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 $(I+J)^k$.

math.AC↗

arXiv2024

On the $\mathrm{v}$-number of binomial edge ideals of some classes of graphs

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.

math.AC↗

arXiv2024

On the path ideals of chordal graphs

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).

math.CO↗

arXiv2024

The slope of v-function and Waldschmidt constant

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 $\displaystyle \lim_{k \rightarrow \infty}\frac{α(I_{[k]})}{k}$, where $α(I)$ 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 $\displaystyle \lim_{k \rightarrow \infty}\frac{\mathrm{v}(I^{(k)})}{k}=\hatα(I)$, 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 $\mathrm{v}(J(G)^{(k)}) = \mathrm{reg}(R/J(G)^{(k)})=α(J(G)^{(k)})-1$ 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 $G$.

math.AC↗

arXiv2024

Asymptotic behaviour and stability index of v-numbers of graded ideals

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 $\mathrm{v}(I^k)=α(I)k-1$ for all $k\geq 1$, where $α(I)$ 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.

math.AC↗