In this paper, we prove the upper bound conjecture proposed by Saeedi Madani and 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.

In this paper, we study the asymptotic behavior of the v-number of a Noetherian graded filtration I={I[k]}k≥0 of a Noetherian N-graded domain R. Recently, it was shown that v(I[k]) is periodically linear in k for k≫0. We show that all these linear functions have the same slope, i.e. [Formula presented] exists, which is equal to [Formula presented], where α(I) denotes the minimum degree of a non-zero element in I. In particular, for any Noetherian symbolic filtration I={I(k)}k≥0 of R, it follows that [Formula presented], the Waldschmidt constant of I. Next, for a non-equigenerated square-free monomial ideal I, we prove that v(I(k))≤reg(R/I(k)) for k≫0. Also, for an ideal I having the symbolic strong persistence property, we give a linear upper bound on v(I(k)). As an application, we derive some criteria for a square-free monomial ideal I to satisfy v(I(k))≤reg(R/I(k)) for all k≥1, and provide several examples in support. In addition, for any simple graph G, we establish that v(J(G)(k))≤reg(R/J(G)(k)) for all k≥1, and v(J(G)(k))=reg(R/J(G)(k))=α(J(G)(k))−1 for all k≥1 if and only if G is a Cohen-Macaulay very-well covered graph, where J(G) is the cover ideal of G.

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. We give combinatorial characterizations of componentwise linearity of I(D) if the vertices in V+ are sinks or |V+|≤1. Furthermore, if G belongs to certain chordal graphs or G is bipartite or the vertices in V+ are sinks or |V+|≤1, then we show the following equivalence for I(D): Vertex splittable ⟺ Linear quotient ⟺ Componentwise linear.

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

Let G be a simple graph and It(G) denote the t-path ideal of G. It is well known that the Castelnuovo–Mumford regularity reg(R/It(G)) and the projective dimension pd(R/It(G)) are bounded below by the quantities (t-1)νt(G) and the big height bight(It(G)), respectively, where νt(G) denotes induced matching number of the hypergraph corresponding to It(G). We show that if t≥4, then the difference between reg(R/It(G)) and (t-1)νt(G), and the difference between pd(R/It(G)) and bight(It(G)) can be arbitrarily large even if we take G to be a tree. This, in particular, disproves a conjecture in Hang and Vu (Graphs Combin 41(1):18, 2025). However, when t=3 and G is chordal, we show that reg(R/I3(G))=2ν3(G) and pd(R/I3(G))=bight(I3(G)), extending the well-known formulas for the edge ideals of chordal graphs. 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 I3(G) is vertex splittable when G is a tree, thereby resolving the t=3 case of a recent conjecture in Abdelmalek et al. (Int J Algebra Comput 33(3):481–498, 2023). Also, for each t≥3, we give examples of chordal graphs G such that the duals of the corresponding t-path ideals are not vertex splittable. Furthermore, we extend the formula of the regularity of 3-path ideals of chordal graphs to all t-path ideals of caterpillar graphs.

For a graph G and the binomial edge ideal JG of G, Bolognini et al. have proved the following: JG is strongly unmixed ⇒JG is Cohen–Macaulay ⇒G is accessible. Moreover, they have conjectured that the converse of these implications is true. 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 JG. We show that accessible and strongly unmixed properties of G depend only on the corresponding properties of its blocks with whiskers and vice versa. We give a new family of graphs whose binomial edge ideals are Cohen–Macaulay, and from that family, we classify all r-regular r-connected graphs, with the property that, after attaching some special whiskers to it, the binomial edge ideals become Cohen–Macaulay. To prove the Cohen–Macaulay conjecture, it is enough to show that every non-complete accessible graph G has a cut vertex v such that G / {v} is accessible. We show that any non-complete accessible graph G having at most three cut vertices has a cut vertex v for which G / {v} is accessible.

Let CCM denote the class of closed graphs with Cohen-Macaulay binomial edge ideals and PIG denote the class of proper interval graphs. Then CCM ⊆ PIG. The PIG-completion problem is a classical problem in graph theory as well as in molecular biology, and this problem is known to be NP-hard. In this paper, we study the CCM-completion problem. We give a method to construct all possible CCM-completions of a graph. We find the CCM-completion number and the set of all minimal CCM-completions for a large class of graphs. Moreover, for this class, we give a polynomial-time algorithm to compute the CCM-completion number and a minimum CCM-completion of a given graph. The unmixedness and Cohen-Macaulay properties of binomial edge ideals of induced subgraphs are investigated. Also, we discuss the accessible graph completion and the Cohen-Macaulay property of binomial edge ideals of whisker graphs.

The study of the edge ideal I(DG) of a weighted oriented graph DG with underlying graph G started in the context of Reed–Muller type codes. We generalize some Cohen–Macaulay constructions for I(DG), which Villarreal gave for edge ideals of simple graphs. Our constructions can be used to produce large classes of Cohen–Macaulay weighted oriented edge ideals. We use these constructions to classify all the Cohen–Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We also show that I(DCn) is Cohen–Macaulay if and only if I(DCn) is unmixed and I(Cn) is Cohen–Macaulay, where Cn 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(DG) and its conditions to be Cohen–Macaulay.

This research focuses on analyzing the depth of generalized binomial edge ideals. We extend the notion of d-compatible map and use it to 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 cycles and graphs with Cohen-Macaulay binomial edge ideals.

The v-number of a graded ideal I ⊆ R, denoted by 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 v-number of edge ideals. In this paper, we study the v number of the cover ideal J(G) of a graph G. The main result shows that v(J(G)) ≤ 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 v(J(G)) with the Cohen-Macaulay property of R/I(G), where I(G) denotes the edge ideal of G. We provide an infinite class of connected graphs, which satisfy v(J(G)) = reg(R/J(G)). Also, we show that for every positive integer k, there exists a connected graph Gk such that reg(R/J(Gk)) − v(J(Gk)) = k. Also, we explicitly compute the v-number of cover ideals of cycles.

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.

In this paper, we show the equality of the (local) 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) v-number serves as an upper bound for the regularity. As an application, we get the equality between the v-number and regularity for Stanley-Reisner rings of matroid complexes. Furthermore, this paper investigates the v-number of Frobenius powers of graded ideals in prime characteristic setup. In this direction, we demonstrate that the 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 v-number without prior knowledge of the associated primes.

Conca and Varbaro (2020) [7] 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 is 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.

The invariant 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 v-number of edge ideals. Inspired by their work, we take the initiation to study the v-number of binomial edge ideals in this paper. We discuss some properties and bounds of the v-number of binomial edge ideals. We explicitly find the v-number of binomial edge ideals locally at the associated prime corresponding to the cutset ∅. We show that the v-number of Knutson binomial edge ideals is less than or equal to the v-number of their initial ideals. Also, we classify all binomial edge ideals whose v-number is 1. Moreover, we try to relate the v-number with the Castelnuvo-Mumford regularity of binomial edge ideals and give a conjecture in this direction.

We show that the v -number of an arbitrary monomial ideal is bounded below by the v -number of its polarization and also find a criteria for the equality. By showing the additivity of associated primes of monomial ideals, we obtain the additivity of the v-numbers for arbitrary monomial ideals. We prove that the v -number v (I(G)) of the edge ideal I(G), the induced matching number im (G) and the regularity reg (R/ I(G)) of a graph G, satisfy v (I(G)) ≤ im (G) ≤ reg (R/ I(G)) , where G is either a bipartite graph, or a (C4, C5) -free vertex decomposable graph, or a whisker graph. There is an open problem in Jaramillo and Villarreal (J Combin Theory Ser A 177:105310, 2021), whether v (I) ≤ reg (R/ I) + 1 , for any square-free monomial ideal I. We show that v (I(G)) > reg (R/ I(G)) + 1 , for a disconnected graph G. We derive some inequalities of v -numbers which may be helpful to answer the above problem for the case of connected graphs. We connect v (I(G)) with an invariant of the line graph L(G) of G. For a simple connected graph G, we show that reg (R/ I(G)) can be arbitrarily larger than v (I(G)). Also, we try to see how the v -number is related to the Cohen–Macaulay property of square-free monomial ideals.