Binomial Edge Ideals Research Articles

On the v-number of binomial edge ideals of some classes of graphs

On the v-number of binomial edge ideals of some classes of graphs

Generalized binomial edge ideals of bipartite graphs

Connected bipartite graphs whose binomial edge ideals are Cohen–Macaulay have been classified by Bolognini et al. In this paper, we compute the depth, Castelnuovo–Mumford regularity, and dimension of the generalized binomial edge ideals of these graphs.

Cohen-Macaulay property of binomial edge ideals with girth of graphs

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.

Regularity of Initial Ideal of Binomial Edge Ideals in Degree 2 and Their Powers

In this paper, we study the Castelnuovo-Mumford regularity of the initial ideal of binomial edge ideals in degree 2 ([in<(JG)]2), and their powers for some classes of graphs. Also, we obtain a lower bound for the regularity of powers of binomial edge ideal associated to complete bipartite graphs.

Cohen–Macaulay binomial edge ideals in terms of blocks with whiskers

For a graph [Formula: see text] and the binomial edge ideal [Formula: see text] of [Formula: see text], Bolognini et al. have proved the following: [Formula: see text] is strongly unmixed [Formula: see text][Formula: see text] is Cohen–Macaulay [Formula: see text][Formula: see text] 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 [Formula: see text] with Cohen–Macaulay [Formula: see text]. We show that accessible and strongly unmixed properties of [Formula: see text] 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 [Formula: see text]-regular [Formula: see text]-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 [Formula: see text] has a cut vertex [Formula: see text] such that [Formula: see text] is accessible. We show that any non-complete accessible graph [Formula: see text] having at most three cut vertices has a cut vertex [Formula: see text] for which [Formula: see text] is accessible.

Algebraic properties of binomial edge ideals of Levi graphs associated with curve arrangements

In this article, we study algebraic properties of binomial edge ideals of Levi graphs associated with certain plane curve arrangements. Using combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of binomial edge ideals of Levi graphs associated to some curve arrangements in the complex projective plane, like the d-arrangement of curves and the conic-line arrangements. We also discuss the existence of certain induced cycles in the Levi graphs of these arrangements and obtain lower bounds for the regularity of powers of the corresponding binomial edge ideals.

Cohen-Macaulay binomial edge ideals of small graphs

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal JG to special disconnecting sets of vertices of its underlying graph G, called cut sets. More precisely, the conjecture states that JG is Cohen-Macaulay if and only if JG is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results.

Arithmetical rank and cohomological dimension of generalized binomial edge ideals

Let [Formula: see text] be a connected and simple graph on the vertex set [Formula: see text]. To the graph [Formula: see text] one can associate the generalized binomial edge ideal [Formula: see text] in the polynomial ring [Formula: see text]. We provide a lower bound for the cohomological dimension of [Formula: see text]. We also study when [Formula: see text] is a cohomologically complete intersection. Finally, we show that the arithmetical rank of [Formula: see text] equals the projective dimension of [Formula: see text] in several cases.

Connected domination in graphs and v-numbers of binomial edge ideals

AbstractThe v-number of a graded ideal is an algebraic invariant introduced by Cooper et al., and originally motivated by problems in algebraic coding theory. In this paper we study the case of binomial edge ideals and we establish a significant connection between their v-numbers and the concept of connected domination in graphs. More specifically, we prove that the localization of the v-number at one of the minimal primes of the binomial edge ideal $$J_G$$ J G of a graph G coincides with the connected domination number of the defining graph, providing a first algebraic description of the connected domination number. As an immediate corollary, we obtain a sharp combinatorial upper bound for the v-number of binomial edge ideals of graphs. Lastly, building on some known results on edge ideals, we analyse how the v-number of $$J_G$$ J G behaves under Gröbner degeneration when G is a closed graph.

Level and pseudo-Gorenstein binomial edge ideals

We prove that level binomial edge ideals with regularity 2 and pseudo-Gorenstein binomial edge ideals with regularity 3 are cones, and we describe them completely. Also, we characterize level and pseudo-Gorenstein binomial edge ideals of bipartite graphs.

D-Sequence edge binomials, and regularity of powers of binomial edge ideals of trees

In this paper, we provide the necessary and sufficient conditions for the edge binomials of the tree forming a [Formula: see text]-sequence in terms of the degree sequence notion of a graph. We study the regularity of powers of the binomial edge ideals of trees generated by [Formula: see text]-sequence edge binomials.

Binomial Edge Ideals of Weakly Closed Graphs

Abstract Closed graphs have been characterized by Herzog et al. as the graphs whose binomial edge ideals have a quadratic Gröbner basis with respect to a diagonal term order. In this paper, we focus on a generalization of closed graphs, namely weakly closed graphs (or co-comparability graphs). Building on some results about Knutson ideals of generic matrices, we characterize weakly closed graphs as the only graphs whose binomial edge ideals are Knutson ideals for a certain polynomial $f$. In doing so, we re-prove Matsuda’s theorem about the F-purity of binomial edge ideals of weakly closed graphs in positive characteristic and we extend it to generalized binomial edge ideals. Furthermore, we give a characterization of weakly closed graphs in terms of the minimal primes of their binomial edge ideals and we characterize all minimal primes of Knutson ideals for this choice of $f$.

On the Equality of Symbolic and Ordinary Powers of Binomial Edge Ideals

On the Equality of Symbolic and Ordinary Powers of Binomial Edge Ideals

Regularity of powers of (parity) binomial edge ideals

In this paper, we provide exact formulas for the regularity of powers of an almost complete intersection ideal I which is generated by a homogeneous d-sequence. As applications, when I is an almost complete intersection, taking the form of a (parity) binomial edge ideal of a connected graph, we can describe explicitly $${\text {reg}}(I^t)$$ for $$t\ge 2$$ . The only exception is when I is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.

Binomial edge ideals of cographs

We determine the Castelnuovo-Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.

Hamiltonian paths, unit-interval complexes, and determinantal facet ideals

We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Pósa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and (tight and weak) Hamiltonian paths in simplicial complexes. Some important consequences of our work are:(1)Every unit-interval strongly-connected d-dimensional simplicial complex is traceable.(This extends the well-known result “unit-interval connected graphs are traceable”.)(2)Every unit-interval d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian.(This extends the fact that “unit-interval 2-connected graphs are Hamiltonian”.)(3)Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Gröbner basis for a diagonal term order which is compatible with the traceability of the complex.(This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa–Vandebogert.)(4)Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution.(This extends the result by Kiani and Saeedi-Madani that “only the complete graph has a binomial edge ideal with linear resolution”.)(5)The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure.

Sequentially Cohen–Macaulay binomial edge ideals of closed graphs

In this paper, we provide a full combinatorial characterization of sequentially Cohen–Macaulay binomial edge ideals of closed graphs. In addition, we show that a binomial edge ideal of a closed graph is approximately Cohen–Macaulay if and only if it is almost Cohen–Macaulay.

Rees algebra and special fiber ring of binomial edge ideals of closed graphs

In this paper, I compute the regularity of the Rees algebra of binomial edge ideals of closed graphs. I obtain a lower bound for the regularity of the Rees algebra of binomial edge ideals. I also present some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. I obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.

On the depth of binomial edge ideals of graphs

Let G be a graph on the vertex set [n] and $$J_G$$ the associated binomial edge ideal in the polynomial ring $$S=\mathbb {K}[x_1,\ldots ,x_n,y_1,\ldots ,y_n]$$ . In this paper, we investigate the depth of binomial edge ideals. More precisely, we first establish a combinatorial lower bound for the depth of $$S/J_G$$ based on some graphical invariants of G. Next, we combinatorially characterize all binomial edge ideals $$J_G$$ with $$\mathrm {depth} S/J_G=5$$ . To achieve this goal, we associate a new poset $$\mathscr {M}_G$$ with the binomial edge ideal of G and then elaborate some topological properties of certain subposets of $$\mathscr {M}_G$$ in order to compute some local cohomology modules of $$S/J_G$$ .

Cohen–Macaulay binomial edge ideals and accessible graphs

The cut sets of a graph are special sets of vertices whose removal disconnects the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen–Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen–Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.