arXiv subjects
Publications and source records attributed to Takayuki Hibi.
Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.
Let $G$ be a finite simple non-complete connected graph on $\{1, \ldots, n\}$ and $κ(G) \geq 1$ its vertex connectivity. Let $f(G)$ denote the number of free vertices of $G$ and $\mathrm{diam}(G)$ the diameter of $G$. Being motivated by the computation of the depth of the binomial edge ideal of $G$, the possible sequences $(n, q, f, d)$ of integers for which there is a finite simple non-complete connected graph $G$ on $\{1, \ldots, n\}$ with $q = κ(G), f = f(G), d = \mathrm{diam}(G)$ satisfying $f + d = n + 2 - q$ will be determined. Furthermore, finite simple non-complete connected graphs $G$ on $\{1, \ldots, n\}$ satisfying $f(G) + \mathrm{diam}(G) = n + 2 - κ(G)$ will be classified.
Inspired by the notion of König graphs we introduce graded ideals of König type with respect to a monomial order $<$. It is shown that if $I$ is of König type, then the Cohen--Macaulay property of $\ini_<(I)$ does not depend on the characteristic of the base field. This happens to be the case also for $I$ itself when $I$ is a binomial edge ideal. Attached to an ideal of König type is a sequence of linear forms, whose elements are variables or differences of variables. This sequence is a system of parameters for $\ini_<(I)$, and is a potential system of parameters for $I$ itself. We study in detail the ideals of König type among the edge ideals, binomial edge ideals and the toric ideal of a Hibi ring and use the König property to determine explicitly their canonical module.
Let $\operatorname{pd}(I(G))$ and $\operatorname{reg}(I(G))$ respectively denote the projective dimension and the regularity of the edge ideal $I(G)$ of a graph $G$. For any positive integer $n$, we determine all pairs $(\operatorname{pd}(I(G)),\, \operatorname{reg}(I(G)))$ as $G$ ranges over all connected bipartite graphs on $n$ vertices.
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gröbner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $R(I)$ of a graded ideal or the symmetric algebra $Sym(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.
Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, \ldots, n \}$ and $K[X, Y] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\mathrm{deg} x_i = \mathrm{deg} y_j = 1$. The binomial edge ideal of $G$ is the binomial ideal $J_G \subset K[X, Y]$ which is generated by those binomials $x_iy_j - x_jy_i$ for which $\{i, j\}$ is an edge of $G$. The Hilbert series $H_{K[X, Y]/J_G}(λ)$ of $K[X, Y]/J_G$ is of the form $H_{K[X, Y]/J_G}(λ) = h_{K[X, Y]/J_G}(λ)/(1 - λ)^d$, where $d = \mathrm{dim} K[X, Y]/J_G$ and where $h_{K[X, Y]/J_G}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$ with each $h_i \in \mathbb{Z}$ and with $h_s \neq 0$ is the $h$-polynomial of $K[X, Y]/J_G$. It is known that, when $K[X, Y]/J_G$ is Cohen-Macaulay, one has $\mathrm{reg}(K[X, Y]/J_G) = \mathrm{deg} h_{K[X, Y]/J_G}(λ)$, where $ \mathrm{reg}(K[X, Y]/J_G)$ is the (Castelnuovo-Mumford) regularity of $K[X, Y]/J_G$. In the present paper, given arbitrary integers $r$ and $s$ with $2 \leq r \leq s$, a finite simple graph $G$ for which $\mathrm{reg}(K[X, Y]/J_G) = r$ and $\mathrm{deg} h_{K[X, Y]/J_G}(λ) = s$ will be constructed.
Let $G$ be a finite simple connected graph on $[n]$ and $R = K[x_1, \ldots, x_n]$ the polynomial ring in $n$ variables over a field $K$. The edge ideal of $G$ is the ideal $I(G)$ of $R$ which is generated by those monomials $x_ix_j$ for which $\{i, j\}$ is an edge of $G$. In the present paper, the possible tuples $(n, {\rm depth} (R/I(G)), {\rm reg} (R/I(G)), \dim R/I(G), {\rm deg} \ h(R/I(G)))$, where ${\rm deg} \ h(R/I(G))$ is the degree of the $h$-polynomial of $R/I(G)$, arising from Cameron--Walker graphs on $[n]$ will be completely determined.
For a numerical semigroup ring $K[H]$ we study the trace of its canonical ideal. The colength of this ideal is called the residue of $H$. This invariant measures how far is $H$ from being symmetric, i.e. $K[H]$ from being a Gorenstein ring. We remark that the canonical trace ideal contains the conductor ideal, and we study bounds for the residue. For $3$-generated numerical semigroups we give explicit formulas for the canonical trace ideal and the residue of $H$. Thus, in this setting we can classify those whose residue is at most one (the nearly-Gorenstein ones), and we show the eventual periodic behaviour of the residue in a shifted family.
The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen--Macaulay ring. We call a local Cohen--Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen--Macaulay $K$-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We study connections between the class of nearly Gorenstein rings and that of almost Gorenstein rings. We show that in dimension one, the former class includes the latter.
Let $Δ$ be a simplicial complex on $[n]$. The $\mathcal{N}\mathcal{F}$-complex of $Δ$ is the simplicial complex $δ_{\mathcal{N}\mathcal{F}}(Δ)$ on $[n]$ for which the facet ideal of $Δ$ is equal to the Stanley--Reisner ideal of $δ_{\mathcal{N}\mathcal{F}}(Δ)$. Furthermore, for each $k = 2,3,\ldots$\,, we introduce {\em $k^{th}$ $\mathcal{N}\mathcal{F}$-complex} $δ^{(k)}_{\mathcal{N}\mathcal{F}}(Δ)$ which is inductively defined by $δ^{(k)}_{\mathcal{N}\mathcal{F}}(Δ) = δ_{\mathcal{N}\mathcal{F}}(δ^{(k-1)}_{\mathcal{N}\mathcal{F}}(Δ))$ with setting $δ^{(1)}_{\mathcal{N}\mathcal{F}}(Δ) = δ_{\mathcal{N}\mathcal{F}}(Δ)$. One can set $δ^{(0)}_{\mathcal{N}\mathcal{F}}(Δ) = Δ$. The $\mathcal{N}\mathcal{F}$-number of $Δ$ is the smallest integer $k > 0$ for which $δ^{(k)}_{\mathcal{N}\mathcal{F}}(Δ) \simeq Δ$. In the present paper we are especially interested in the $\mathcal{N}\mathcal{F}$-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the $\mathcal{N}\mathcal{F}$-number of the finite graph $K_n\coprod K_m$ on $[n + m]$, which is the disjoint union of the complete graphs $K_n$ on $[n]$ and $K_m$ on $[m]$, where $n \geq 2$ and $m \geq 2$ with $(n,m) \neq (2,2)$, is equal to $n + m + 2$. Its corollary says that the $\mathcal{N}\mathcal{F}$-number of the complete bipartite graph $K_{n,m}$ on $[n+m]$ is also equal to $n + m + 2$.
In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gröbner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gröbner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange.
Let $G$ be a finite simple graph on $n$ vertices, that contains no isolated vertices, and let $I(G) \subseteq S = K[x_1, \dots, x_n]$ be its edge ideal. In this paper, we study the pair of integers that measure the projective dimension and the regularity of $S/I(G)$. We show that if the projective dimension of $S/I(G)$ attains its minimum value $2\sqrt{n}-2$ then, with only one exception, the its regularity must be 1. We also provide a full description for the spectrum of the projective dimension of $S/I(G)$ when the regularity attains its minimum value 1.
Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a study of the set $\mathcal{RD}(n) \subseteq \mathbb{N}^2$ consisting of all the pairs $(r,d)$ where $r = {\rm reg}(R/I(G))$, the Castelnuovo-Mumford regularity, and $d = {\rm deg} h_{R/I(G)}(t)$, the degree of the $h$-polynomial, as we vary over all the connected graphs on $n$ vertices. In particular, we identify sets $A(n)$ and $B(n)$ such that $A(n) \subseteq \mathcal{RD}(n) \subseteq B(n)$. When we restrict to the family of Cameron-Walker graphs on $n$ vertices, we can completely characterize all the possible $(r,d)$.
Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that, for each $1 \leq s \leq 5$, an $O$-sequence $(h_0, h_1, \ldots, h_{2s-1}) \in \mathbb{Z}_{\geq 0}^{2s}$ satisfying the properties that (i) $h_0 \leq h_1 \leq \cdots \leq h_{s-1}$, (ii) $h_{2s-1} = h_0$, $h_{2s-2} = h_1$ and (iii) $h_{2s - 1 - i} = h_i + (-1)^{i}$, $2 \leq i \leq s - 1$, can be the $h$-vector of a Cohen--Macaulay standard $G$-domain.
Let $K[G]$ denote the edge ring of a finite connected simple graph $G$ on $[d]$ and $\mat(G)$ the matching number of $G$. It is shown that $\reg(K[G]) \leq \mat(G)$ if $G$ is non-bipartite and $K[G]$ is normal, and that $\reg(K[G]) \leq \mat(G) - 1$ if $G$ is bipartite.
Squarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly related or have linear resolution. We also consider the so-called squarefree Ratliff property.
Given an arbitrary integer $d>0$, we construct a homogeneous ideal $I$ of the polynomial ring $S = K[x_1, \ldots, x_{3d}]$ in $3d$ variables over a filed $K$ for which $S/I$ is a Cohen--Macaulay ring of dimension $d$ with the property that, for each of the integers $0 \leq r \leq d$, there exists a monomial order $<_r$ on $S$ with ${\rm depth} (S/{\rm in}_{<_r}(I)) = r$, where ${\rm in}_{<_r}(I)$ is the initial ideal of $I$ with respect to $<_r$.
To classify the lattice polytopes with a given $δ$-polynomial is an important open problem in Ehrhart theory. A complete classification of the Gorenstein simplices whose normalized volumes are prime integers is known. In particular, their $δ$-polynomials are of the form $1+t^k+\cdots+t^{(v-1)k}$, where $k$ and $v$ are positive integers. In the present paper, a complete classification of the Gorenstein simplices with the above $δ$-polynomials will be performed, when $v$ is either $p^2$ or $pq$, where $p$ and $q$ are prime integers with $p \neq q$. Moreover, we consider the number of Gorenstein simplices, up to unimodular equivalence, with the expected $δ$-polynomial.