# Graded ideals of König type

Jürgen Herzog, Takayuki Hibi and Somayeh Moradi  Jürgen Herzog, Fachbereich
Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany
<EMAIL_ADDRESS>Takayuki Hibi, Department of Pure and Applied
Mathematics, Graduate School of Information Science and Technology, Osaka
University, Suita, Osaka 565-0871, Japan<EMAIL_ADDRESS>Somayeh
Moradi, Department of Mathematics, School of Science, Ilam University, P.O.Box
69315-516, Ilam, Iran<EMAIL_ADDRESS>
###### Abstract.

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 $\operatorname{in}_{<}(I)$ 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
$\operatorname{in}_{<}(I)$, 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.

###### Key words and phrases:

binomial ideals, Cohen-Macaulay rings, ideals of König type, systems of
parameters

###### 2010 Mathematics Subject Classification:

Primary 13H10, 13C15; Secondary 05C25

The second author was supported by JSPS KAKENHI 19H00637.

## Introduction

For a graded ideal $I$ of a polynomial ring $S=K[x_{1},\ldots,x_{n}]$ over a
field $K$, the Cohen-Macaulay property of the ring $S/I$ may depend on the
characteristic of the base field $K$ in general. Even when we restrict to the
case that $I$ is the edge ideal of a graph $G$, this dependency may occur.
Such an example is given in [12]. Nevertheless, for some families of graphs it
is shown that the Cohen-Macaulayness of $S/I(G)$ is independent of $K$. When
$G$ is a bipartite graph, the first and second authors of this paper in [6]
presented a combinatorial condition on the graph $G$ equivalent to $S/I(G)$ be
Cohen-Macaulay. In [3] characterizing Cohen-Macaulay graphs was extended to
very well-covered graphs which implies that the Cohen-Macaulay property of
$S/I(G)$ is independent of $K$, when $G$ is a very well-covered graph (see
also [2]). If $G$ is a graph with no isolated vertex, $G$ is very well-covered
if and only if it is an unmixed König graph. Recall that a graph $G$ is called
König, if its matching number coincides with its vertex cover number.
Motivated by this, we look in a more general frame for conditions on a graded
ideal $I\subset S$ for which the Cohen-Macaulay property of the ring $S/I$ is
independent of $K$ and to this aim we introduce graded ideals of König type
which generalize edge ideals of König graphs.

In [10], the first and third authors studied monomial ideals of König type and
characterized them in terms of the existence of systems of parameters for
$S/I$ which consist of elements of the form $x_{i}-x_{j}$. Bringing to mind
the characteristic independency of König graphs together with this result,
this natural question arises: for a graded ideal $I$, does the existence of
systems of parameters in a particular form for $S/I$ imply the characteristic
independence of Cohen-Macaulayness? We investigate this question when $S/I$
admits a special system of parameters. By a special system of parameters we
mean a system of parameters $s_{1},\ldots,s_{d}$, where each $s_{j}$ is either
of the form $x_{i}$ or of the form $x_{i}-x_{j}$. Indeed we study binomial
ideals whose quotient rings admit special systems of parameters. Here we call
an ideal $I$ a binomial ideal if the generators of $I$ are monomials or
binomials of the form $u-v$ with $u$ and $v$ monomials. This includes the case
that all generators are monomials or all generators are binomials. It turns
out that for a binomial ideal $I$, the existence of a special system of
parameters for $S/I$ gives a positive answer to the above question. For
binomial edge ideals of König type as well as monomial ideals of König type,
such special systems of parameters always exist. We study further monomial
ideals and binomial edge ideals of König type and investigate when the
defining ideal of a Hibi ring is of König type.

The paper proceeds as follows. In the first section we define a graded ideal
$I$ to be of König type with respect to a homogeneous sequence
$f_{1},\ldots,f_{h}$ coming from a minimal generating set of $I$ and an order
$<$ if $h=\operatorname{height}I$ and
$\operatorname{in}_{<}(f_{1}),\ldots,\operatorname{in}_{<}(f_{h})$ is a
regular sequence. Attached to this sequence and this order we consider a
natural sequence of linear forms of length $d=\dim S/I$ which has the
potential to be a special system of parameters for $S/I$. In Theorem 1.5, we
show that for a binomial ideal $I$ if $S/I$ admits a special system of
parameters, the Cohen-Macaulay property of $S/I$ does not depend on the base
field. Using this theorem and the characterization of monomial ideals of König
type, we deduce that for any graded ideal of König type with respect to $<$,
the Cohen–Macaulayness of $\operatorname{in}_{<}(I)$ does not depend on the
base field (see Corollary 1.11).

In Section 2, we concentrate on monomial ideals of König type. In Theorem 2.1,
a combinatorial condition is given which is equivalent to the Cohen-
Macaulayness of $S/I$, when $I$ is a monomial ideal of König type. This
generalizes [3, Proposition 3.2]. In Theorem 2.2, for a Cohen-Macaulay König
graph, combinatorial descriptions for the canonical module and the Cohen-
Macaulay type of $S/I(G)$ are presented. Next in Theorem 2.3, for a König
graph $G$ we show that $S/I(G)$ is Cohen–Macaulay if and only if
$\alpha(G)=|\Delta_{G_{0}}|$, where $\alpha(G)$ is the number of vertex covers
of $G$ of cardinality $\tau(G)$, $G_{0}$ is some graph attached to $G$ and
$\Delta_{G_{0}}$ is the independence complex of $G_{0}$. When $I$ is the edge
ideal of an unmixed graph $G$ without isolated vertices, it is known that $I$
is of König type if and only if $G$ is a very well-covered graph (see [2,
Lemma 17]). In Theorem 2.5, we consider an unmixed monomial ideal of König
type $I$ and show that the polarization $I^{\wp}$ of $I$ is very well-covered.
An example follows which shows that the converse may not hold in general.

In Section 3, we study binomial edge ideals of König type. We call a subgraph
$P$ of $G$ a semi-path if each component of $P$ is a path graph. In Theorem
3.5 it is shown that for a graph $G$ on $[n]$ the binomial edge ideal $J_{G}$
is of König type if and only if there exists a semi-path in $G$ of length
$2n-d$, where $d=\dim S/J_{G}$. There is a close relation between binomial
edge ideals of König type and traceable graphs , i.e., graphs which contain a
path meeting all vertices of the graph. Indeed if $G$ is traceable, then
$J_{G}$ is of König type. Also if $J_{G}$ is of König type and unmixed, then
$G$ is traceable (see Proposition 3.6). Giving a special system of parameters
for a binomial edge ideal of König type in Theorem 3.7, we recover a result in
[1] which indicates that for a traceable graph, the Cohen-Macaulay property of
$S/J_{G}$ does not depend on the base field. We close this section by a
proposition which gives a description for the canonical module of $S/J_{G}$,
when $G$ is traceable and $J_{G}$ is Cohen-Macaulay.

Finally in Section 4, we study when the toric ideal of a Hibi ring
${\mathcal{R}}_{K}[L]$ is of König type with respect to $f_{i,j}$ and $<_{\rm
rev}$, where $f_{i,j}=x_{i}x_{j}-(x_{i}\wedge x_{j})(x_{i}\vee x_{j})$ for
which $x_{i}$ and $x_{j}$ are incomparable in $L$. We show in Theorem 4.5 that
$I_{L}$ is of König type with respect to $f_{i,j}$ and $<_{\rm rev}$ if and
only if $L$ is thin and this is the case when the incomparability graph ${\rm
incom(L)}$ of $L$ is bipartite.

##  1\. Graded ideals of König type and parameter systems

We recall some concepts about graphs. Let $G$ be a finite simple graph with
the vertex set $V(G)$ and the edge set $E(G)$. A matching of $G$ is a subset
$\\{e_{1},\ldots,e_{m}\\}$ of $E(G)$ such that $e_{i}$’s are pairwise
disjoint. The maximum cardinality of matchings of $G$ is called the matching
number of $G$ and is denoted by $m(G)$. A vertex cover of $G$ is a subset $C$
of $V(G)$ which has at least one element from each edge of $G$. A vertex cover
$C$ is called a minimal vertex cover if no proper subset of $C$ is a vertex
cover. The minimum cardinality of vertex covers of $G$ is called the vertex
cover number of $G$ and is denoted by $\tau(G)$.

Let $G$ be a graph on $[n]$. The edge ideal $I(G)$ of $G$ is the monomial
ideal whose generators are the monomials $x_{i}x_{j}$ for which $\\{i,j\\}\in
E(G)$. It is easy to see that the height of $I(G)$ is equal to $\tau(G)$ and
$m(G)$ is the maximal length of a regular sequence among the generators of
$I(G)$. It follows that $m(G)\leq\tau(G)$ since
$m(G)\leq\operatorname{grade}(I(G))=\operatorname{height}I(G)=\tau(G)$. In
1931, König showed that $m(G)=\tau(G)$, if $G$ is bipartite. Now, any graph
$G$, satisfying $m(G)=\tau(G)$ is called a König graph.

This concept, as defined for graphs, leads us to the following

######  Definition 1.1.

Let $K$ be field, $S=K[x_{1},\ldots,x_{n}]$ be the polynomial ring in $n$
variables over $K$, and let $I\subset S$ be a graded ideal of height $h$. We
say that $I$ is of König type, if there exists a sequence
${\mathbf{f}}=f_{1},\ldots,f_{h}$ of homogeneous polynomials which forms part
of a minimal system of generators of $I$ and a monomial order $<$ on $S$ such
that $\operatorname{in}_{<}(f_{1}),\ldots,\operatorname{in}_{<}(f_{h})$ is a
regular sequence.

If we want to be more specific we say that $I$ is of König type with respect
to the sequence ${\mathbf{f}}$ and the monomial order $<$.

######  Remark 1.2.

If $K$ is infinite, one can always find a homogeneous regular sequence
$f_{1},\ldots,f_{h}\in I$. The condition that
$\operatorname{in}_{<}(f_{1}),\ldots,\operatorname{in}_{<}(f_{h})$ be a
regular sequence for some monomial order and part of a minimal system of
generators is much stronger. This is why not all graded ideals are of König
type.

######  Example 1.3.

Consider the ideal $I=(f_{1},f_{2})$ with $f_{1}=x_{1}x_{2}-x_{2}x_{3}$ and
$f_{2}=x_{1}x_{3}-x_{3}^{2}$. Then $I$ is of height 2. The only two monomials
in the support of $f_{1}$ and $f_{2}$ which form a regular sequence are the
monomials $x_{1}x_{2}$ and $x_{3}^{2}$. But there is no monomial order $<$
with
$\\{\operatorname{in}_{<}(f_{1}),\operatorname{in}_{<}(f_{2})\\}=\\{x_{1}x_{2},x_{3}^{2}\\}$.
Thus $I$ is not of König type.

Suppose now that $I$ is of König type with respect to
${\mathbf{f}}=f_{1},\ldots,f_{h}$ and $<$, and let
$u_{i}=\operatorname{in}_{<}(f_{i})$ for all $i$. Then we define a sequence
$C$ of linear forms attached to ${\mathbf{f}}$ and $<$ as follows: let

$A=\\{x_{i}\colon\;x_{i}\nmid u_{j}\text{ for all $j$}\\},$

and for each $j$ let

$B_{j}=\\{x_{i}\colon\;x_{i}\mid u_{j}\\}.$

Since $u_{1},\ldots,u_{h}$ is a regular sequence, the sets $B_{j}$ are
pairwise disjoint and of course $A\cap B_{j}=\emptyset$ for all $j$.
Therefore, $n=|A|+\sum_{j=1}^{h}|B_{j}|=n$.

Finally for each $j$ let $i_{j}$ be the smallest integer with $x_{i_{j}}\in
B_{j}$ and let

$C_{j}=\\{x_{k}-x_{i_{j}}\colon\;x_{k}\in B_{j},k\neq i_{j}\\}.$

Then we set

(1) $\displaystyle C=A\cup C_{1}\cup\cdots\cup C_{h}.$

Note that $|C|=n-h=\dim S/I$, because $|C_{j}|=|B_{j}|-1$ for $j=1,\ldots,h$.
Hence $C$ is a potential system of parameters of $S/I$ attached to
$f_{1},\ldots,f_{h}$ and $<$.

######  Example 1.4.

Let $R=K[x_{1},x_{2}]/I$, where $I=(x_{1}x_{2}-x_{2}^{2})$. Then $I$ is of
König type with respect to $f=x_{1}x_{2}-x_{2}^{2}$ for any monomial order. If
$x_{1}>x_{2}$, then the attached sequence is $x_{2}-x_{1}$ and
$R/(x_{2}-x_{1})R\cong K[x_{1}]$. On the other hand, if $x_{2}>x_{1}$, then
the attached sequence is $x_{1}$, and $R/(x_{1})R\cong K[x_{2}]/(x_{2}^{2})$.
Hence in the second case, the attached sequence (1) is a system of parameters
of $R$, while in the first case it is not.

Let $R$ be a standard graded $K$-algebra of dimension $d$. Having in mind
attached sequences of linear forms attached to ideals of König type, we call a
system of parameters $s_{1},\ldots,s_{d}$ of $R$ special with respect to a
given $K$-basis $x_{1},\ldots,x_{n}$ of $R_{1}$, if each $s_{j}$ is either of
the form $x_{i}$ or of the form $x_{k}-x_{i}$.

For binomial ideals, special systems of parameters have nice properties. Here
we call an ideal $I$ a binomial ideal if the generators of $I$ are monomials
and binomials of the form $u-v$ with $u$ and $v$ monomials. This includes the
case that all generators are monomials or all generators are binomials.

######  Theorem 1.5.

Let $I\subset S$ be a binomial ideal and suppose that $R=S/I$ admits a special
system of parameters. Then the Cohen-Macaulay property of $R$ does not depend
on the base field.

###### Proof.

Let $L\subset S$ be the ideal generated by the special system of parameters.
Then $S/L\cong T$ where $T$ is a polynomial ring over $K$ in $h$ variables,
and $R/LR$ can be written as $T/J$, where $J$ is again a binomial ideal
because the generators of $J$ are binomials which are obtained from the
binomials generating $I$ by replacing some variables by $0$ or identifying
different variables.

Denote by $e(A)$ the multiplicity of a standard graded $K$-algebra $A$. Then
it is known (see for example [10, Proposition 1.2]) that $R$ is Cohen-Macaulay
if and only if $e(R)=e(R/LR)$ (in which case the system of parameters forms a
regular sequence). The multiplicities $e(R)$ and $e(R/LR)$ are determined by
their corresponding Hilbert series. Thus Lemma 1.6 completes the proof.

######  Lemma 1.6.

Let $I\subset S$ be a binomial ideal. Then the Hilbert series of $S/I$ does
not depend on the base field.

###### Proof.

First of all, it is known [8, Corollary 6.1.5] that, for an arbitrary monomial
order $<$ on $S$, the Hilbert series of $S/I$ coincides with that of
$S/\operatorname{in}_{<}(I)$.

Now, recall from [8, Theorem 2.2.1] what the division algorithm is. Let
$f=u_{0}-v_{0}$ and $g=u-v$ be binomials of $S$, where each of
$u_{0},v_{0},u,v$ are monomials. It may happen that either $v_{0}=0$ or $v=0$.
Let $\operatorname{in}_{<}(g)=u$. Suppose that either $u_{0}$ or $v_{0}$ is
divisible by $u$ and, say, $u_{0}=wu$ for some monomial $w$. Then dividing $f$
by $g$ is the process

$f=u_{0}-v_{0}=wu-v_{0}=w(g+v)-v_{0}=wg+(wv-v_{0})$

and the binomial $f_{1}=wv-v_{0}=u_{1}-v_{1}$, where $u_{1}=wv$ and
$v_{1}=v_{0}$, is called a remainder. Let $f\rightarrow_{g}f_{1}$ denote the
procedure of division. If, furthermore, $f_{1}\neq 0$ and if either $u_{1}$ or
$v_{1}$ is divisible by $u$, then dividing $f_{1}$ by $g$ yields $f_{2}$. Thus

$f\rightarrow_{g}f_{1}\rightarrow_{g}f_{2}.$

A division algorithm guarantees that, after $N$ steps of divisions, a
remainder $f_{N}=u_{N}-v_{N}$ possesses the property that either $f_{N}=0$ or,
neither $u_{N}$ nor $v_{N}$ is divisible by $u$. Clearly the process of
divisions

$f\rightarrow_{g}f_{1}\rightarrow_{g}f_{2}\rightarrow_{g}\cdots\rightarrow_{g}f_{N}$

is independent of the base field. More generally, if $g_{1},\ldots,g_{s}$ are
binomials of $S$, one can divide $f$ by $g_{1},\ldots,g_{s}$ and $N$ steps of
divisions

$f\rightarrow_{g_{i_{1}}}f_{1}\rightarrow_{g_{i_{2}}}f_{2}\rightarrow_{g_{i_{3}}}\cdots\rightarrow_{g_{i_{N}}}f_{N}$

yield a binomial $f_{N}=u_{N}-v_{N}$ for which either $f_{N}=0$ or, neither
$u_{N}$ nor $v_{N}$ can be divisible by each of
$\operatorname{in}_{<}(g_{1}),\ldots,\operatorname{in}_{<}(g_{s})$. We say
that $f$ reduces to $f_{N}$ with respect to $g_{1},\ldots,g_{s}$. It may
happen that another process of divisions

$f\rightarrow_{g_{i^{\prime}_{1}}}f_{1}\rightarrow_{g_{i^{\prime}_{2}}}f_{2}\rightarrow_{g_{i^{\prime}_{3}}}\cdots\rightarrow_{g_{i^{\prime}_{N^{\prime}}}}f_{N^{\prime}}$

yields a binomial $f_{N^{\prime}}$ with $f_{N}\neq f_{N^{\prime}}$ to which
$f$ reduces with respect to $g_{1},\ldots,g_{s}$. Especially, one says that
$f$ reduces to $0$ with respect to $g_{1},\ldots,g_{s}$ if one can find a
process of divisions of the form

$f\rightarrow_{g_{j_{1}}}f_{1}\rightarrow_{g_{j_{2}}}f_{2}\rightarrow_{g_{j_{3}}}\cdots\rightarrow_{g_{j_{N^{\prime\prime}}}}0.$

Since the process of divisions is independent of the base field, it follows
that the property that $f$ reduces $0$ with respect to $g_{1},\ldots,g_{s}$ is
independent of the base field.

We turn to the discussion of Buchberger criterion [8, Theorem 2.3.2] together
with Buchberger algorithm [8, pp. 37–38]. Let $I=(f_{1},\ldots,f_{s})$, where
each $f_{i}=u_{i}-v_{i}$ is a binomial with
$\operatorname{in}_{<}(f_{i})=u_{i}$. The $S$-polynomial $S(f_{i},f_{j})$ is
the binomial

$S(f_{i},f_{j})=uf_{i}-vf_{j}=vv_{j}-uv_{i},$

where $u$ and $v$ are monomials with $uu_{i}=vu_{j}$ for which $uu_{i}=vu_{j}$
is the least common multiple of $u_{i}$ and $u_{j}$. Buchberger criterion
guarantees that $\\{f_{1},\ldots,f_{s}\\}$ is a Gröbner basis of $I$ with
respect to $<$ if and only if each of the $S$-polynomials $S(f_{i},f_{j})$
with $1\leq i<j\leq s$ reduces to $0$ with respect to $f_{1},\ldots,f_{s}$.
Thus, if $\\{f_{1},\ldots,f_{s}\\}$ is not a Gröbner basis of $I$, then there
is $S(f_{i},f_{j})$ which reduces to a binomial $f_{s+1}=u_{s+1}-v_{s+1}\neq
0$ with $\operatorname{in}_{<}(f_{s+1})=u_{s+1}$ with respect to
$f_{1},\ldots,f_{s}$, where neither $u_{s+1}$ nor $v_{s+1}$ can be divisible
by each of $u_{1},\ldots,u_{s}$. Since $f_{s+1}$ belongs to $I$, one may
consider $I=(f_{1},\ldots,f_{s},f_{s+1})$. If each of the $S$-polynomials
$S(f_{i},f_{j})$ with $1\leq i<j\leq s+1$ reduces to $0$ with respect to
$f_{1},\ldots,f_{s},f_{s+1}$, then $\\{f_{1},\ldots,f_{s},f_{s+1}\\}$ is a
Gröbner basis of $I$ with respect to $<$. Buchberger algorithm guarantees
that, after a finite number of repeating the procedure of adding a remainder
to a system of binomial generators of $I$, one has
$I=(f_{1},\ldots,f_{s},f_{s+1},\ldots,f_{s+q})$ for which each of the
$S$-polynomials $S(f_{i},f_{j})$ with $1\leq i<j\leq s+q$ reduces to $0$ with
respect to $f_{1},\ldots,f_{s},f_{s+1},\ldots,f_{s+q}$ and
$\\{f_{1},\ldots,f_{s},f_{s+1},\ldots,f_{s+q}\\}$ is a Gröbner basis of $I$
with respect to $<$. The procedure of Buchberger algorithm, which consists of
the computation of $S$-polynomials and their remainders, is independent of the
base field. Thus in particular, the property that a system of generators of
$I$ is a Gröbner basis of $I$ with respect to $<$ is independent of the base
field.

It then follows that $S/\operatorname{in}_{<}(I)$ is independent of the base
field. Since the Hilbert function of a monomial ideal is independent of the
base field, the Hilbert function of $S/I$ is independent of the base field, as
desired.

For a monomial ideal $I$ we denote by $G(I)$ the unique minimal set of
monomial generators of $I$.

######  Lemma 1.7.

Let $I$ be a monomial ideal of height $h$. Then $I$ is of König type for any
monomial order, if and only if there exists a regular sequence of monomials
$u_{1},\ldots,u_{h}$ in $G(I)$.

###### Proof.

Suppose $I$ is of König type, and let $f_{1},\ldots,f_{h}$ be homogeneous
polynomials such that
$u_{1}=\operatorname{in}_{<}(f_{1}),\ldots,u_{h}=\operatorname{in}_{<}(f_{h})$
is a regular sequence for some monomial order $<$. Since $I$ is a monomial
ideal, it follows that $u_{i}\in I$ for all $i$. Hence there exist $v_{i}\in
G(I)$ with $v_{i}\mid u_{i}$ for all $i$. The sequence $v_{1},\ldots,v_{h}$ is
again a regular sequence.

The converse direction is trivial.

######  Lemma 1.8.

Let $I\subset S$ be a graded ideal and $<$ a monomial order on $S$. If $I$ is
of König type with respect to $<$, then $\operatorname{in}_{<}(I)$ is of König
type.

###### Proof.

Since $\dim S/I=\dim S/\operatorname{in}_{<}(I)$ (see [8, Theorem 3.3.4]), it
follows that
$h=\operatorname{height}I=\operatorname{height}\operatorname{in}_{<}(I)$.
Suppose that $I$ is of König type. Then there exist homogeneous polynomials
$f_{1},\ldots,f_{h}\in I$ such that
$\operatorname{in}_{<}(f_{1}),\ldots,\operatorname{in}_{<}(f_{h})$ is a
regular sequence. Hence by Lemma 1.7, $\operatorname{in}_{<}(I)$ is of König
type.

The converse of Lemma 1.8 does not hold in general. Indeed if
$\operatorname{in}_{<}(I)$ is of König type, then always there exist
$f_{1},\ldots,f_{h}\in I$ of length $h=\operatorname{height}I$ such that
$\operatorname{in}_{<}(f_{1}),\ldots,\operatorname{in}_{<}(f_{h})$ is a
regular sequence, but there may not exist such a sequence among the elements
of a minimal generating set of $I$. See the following

######  Example 1.9.

Consider the ideal $I=(x_{1}x_{2}-x_{4}^{2},x_{2}x_{3})$. Then with the
reverse lex order
$\operatorname{in}_{<}(I)=(x_{2}x_{3},x_{1}x_{2},x_{3}x_{4}^{2})$ which is an
ideal of height $2$ and $x_{1}x_{2},x_{3}x_{4}^{2}$ is the only regular
sequence of length $2$ among its minimal generators. While $I$ is not of König
type. Indeed any minimal generator of $I$ is of degree $2$, so its initial
term is of degree $2$ as well. Thus there exists no sequence of length $2$
belonging to a minimal generating set of $I$ such that their initials form a
regular sequence.

In [10, Theorem 3.2] the first and third authors of this paper proved the
following result.

######  Theorem 1.10.

Let $I\subset S$ be a monomial ideal. The following conditions are equivalent:

  1. (a)

$I$ is of König type.

  2. (b)

$S/I$ admits a special system of parameters.

Moreover, if the equivalent conditions hold, then the sequence $C$ as given in
(1) is a special system of parameters.

Combining Theorem 1.5, Lemma 1.8 and Theorem 1.10 we obtain

######  Corollary 1.11.

Suppose $I$ is of König type with respect to $<$. Then the Cohen–Macaulayness
of $\operatorname{in}_{<}(I)$ does not depend on the base field.

The following lemma is useful, when we deal with ideals of König type. Recall
that the unmixed part of an ideal in a Noetherian ring is the intersection of
its primary components of minimal height.

######  Lemma 1.12.

Let $J\subset I\subset S$ be ideals with
$\operatorname{height}J=\operatorname{height}I=h$ such that $J$ is a complete
intersection and a radical ideal. Then

  1. (a)

$J:I=\bigcap_{P\in\min(J)\setminus\min(I)}P$. In particular $J:I$ is a radical
ideal and unmixed.

  2. (b)

$J:(J:I)$ is the unmixed part of $\sqrt{I}$.

###### Proof.

(a) It is easy to see that $\bigcap_{P\in\min(J)\setminus\min(I)}P\subseteq
J:I$. Now by contradiction suppose that there exists
$f\in(J:I)\setminus\bigcap_{P\in\min(J)\setminus\min(I)}P$. Then $f\notin P$
for some $P\in\min(J)\setminus\min(I)$. On the other hand
$f/1\in(J:I)_{P}=J_{P}:I_{P}=PS_{P}:S_{P}=PS_{P}$. Thus $f\in P$, which is a
contradiction.

(b) By (a),

$J:(J:I)=\bigcap_{P\in\min(J)\setminus(\min(J)\setminus\min(I))}P.$

Since all minimal prime ideals in $\min(J)$ have height $h$, it follows that

$\min(J)\setminus(\min(J)\setminus\min(I))=\min(J)\cap\min(I)=\\{P\in\min(I):\operatorname{height}P=h\\}.$

This proves (b).

######  Corollary 1.13.

Let $I$ be a graded ideal of König type with respect to $f_{1},\ldots,f_{h}$
and assume that $\operatorname{in}_{<}(f_{i})$ is a squarefree monomial for
all $i$. Then the unmixed part of $\sqrt{I}$ is given by $J:(J:I)$, where
$J=(f_{1},\ldots,f_{h})$.

###### Proof.

Since $\operatorname{in}_{<}(f_{1}),\ldots,\operatorname{in}_{<}(f_{h})$ is a
regular sequence, it follows that $f_{1},\ldots,f_{h}$ forms a Gröbner basis
of $J$. Since $\operatorname{in}_{<}(f_{i})$ is a squarefree monomial for all
$i$, it follows that $\operatorname{in}_{<}(J)$ is a squarefree monomial
ideal. This implies that $J$ is a radical ideal, see for example the proof of
[9, Corollary 2.2]. Now by Lemma 1.12 the result holds.

##  2\. Properties of monomial ideals of König type

In this section we give a combinatorial characterization for Cohen-Macaulay
monomial ideals of König type. When $I$ is the edge ideal of a graph $G$, a
description of Cohen-Macaulayness and the canonical module of $S/I(G)$, when
$G$ is Cohen-Macaulay are given in terms of data from $G$. We consider the
following setting.

Let $I$ be a monomial ideal of König type with respect to
$u_{1},\ldots,u_{h}\in G(I)$. Then $S/I$ has a special system of parameters
$f_{1},\ldots,f_{d}$ in the form of (1) attached to $u_{1},\ldots,u_{h}$ (see
the proof of [10, Theorem 3.2]). For any $f_{k}=x_{i}-x_{j}$ we may assume
that $i<j$. For any $B\subseteq\\{f_{1},\ldots,f_{d}\\}$, we set $I_{B}$ be
the ideal obtained from $I$ by replacing $x_{j}$ by $x_{i}$ for any
$x_{i}-x_{j}\in B$. We call a monomial $g$ a modification of $f$ by $B$, if we
get $g$ when we replace $x_{j}$ by $x_{i}$ in $f$ for any $x_{i}-x_{j}\in B$.

The following theorem gives a combinatorial characterization for Cohen-
Macaulay monomial ideals of König type and is a generalization of [3,
Proposition 3.2].

######  Theorem 2.1.

Let $I$ be a monomial ideal of König type with a special system of parameters
$f_{1},\ldots,f_{d}$ attached to $u_{1},\ldots,u_{h}\in G(I)$. The following
conditions are equivalent:

  1. (i)

$S/I$ is Cohen-Macaulay.

  2. (ii)

$S/I_{B}$ is Cohen-Macaulay for any $B\subseteq\\{f_{1},\ldots,f_{d}\\}$.

  3. (iii)

$I_{B}$ is unmixed for any $B\subseteq\\{f_{1},\ldots,f_{d}\\}$.

###### Proof.

Without loss of generality we may assume that any variable $x_{i}$ belongs to
the support of some $u_{j}$. Thus each $f_{k}$ is of the form $x_{i}-x_{j}$
with $i<j$.

(i)${}\Rightarrow{}$(ii): Let
$B=\\{g_{1},\ldots,g_{r}\\}\subseteq\\{f_{1},\ldots,f_{d}\\}$. Since $R=S/I$
is a Cohen-Macaulay ring, $g_{1},\ldots,g_{r}$ is a regular sequence, which
implies that $R/(g_{1},\ldots,g_{r})R$ is Cohen-Macaulay. Notice that
$R/(g_{1},\ldots,g_{r})R\cong T/I_{B}$, where $T=S/(x_{j}:\ j\in A)$ and
$A=\\{1\leq j\leq n:\ x_{i}-x_{j}\in B\textrm{ for some }i\\}$. Thus $T/I_{B}$
is Cohen-Macaulay, and hence $S/I_{B}$ is Cohen-Macaulay.

(ii)${}\Rightarrow{}$(iii): Any Cohen-Macaulay ideal is unmixed.

(iii)${}\Rightarrow{}$(i): By contradiction assume that $R=S/I$ is not Cohen-
Macaulay. Then $f_{1},\ldots,f_{d}$ is not a regular sequence. So there exists
$k$ with $f_{k}\in Z(R/(f_{1},\ldots,f_{k-1})R)$. Since
$R/(f_{1},\ldots,f_{k-1})R\cong T/I_{B}$ for $B=\\{f_{1},\ldots,f_{k-1}\\}$
and some polynomial ring $T$, we have $f_{k}\in P$ for some minimal prime
ideal $P$ of $I_{B}$. By assumption $u_{1},\ldots,u_{h}$ is a regular sequence
in $G(I)$ such that for each $i$,
$\operatorname{Supp}(f_{i})\subseteq\operatorname{Supp}(u_{j})$ for some
$1\leq j\leq h$, where $h=\operatorname{height}I$. If $f_{k}=x_{i}-x_{j}$,
then $x_{i},x_{j}\in P$, since $P$ is a monomial ideal. Let $u^{\prime}_{t}$
be the modification of $u_{t}$ by $B$ for any $1\leq t\leq h$. Then
$(u^{\prime}_{1},\ldots,u^{\prime}_{h})+(x_{i},x_{j})\subseteq P$. Let
$1\leq\ell\leq h$ be the unique integer with $x_{i}x_{j}|u_{\ell}$. Then
$u^{\prime}_{1},\ldots,u^{\prime}_{\ell-1},u^{\prime}_{\ell+1},\ldots,u^{\prime}_{h},x_{i},x_{j}$
is a regular sequence in $P$. So we obtain $\operatorname{height}P>h$. Notice
that $\operatorname{height}I_{B}=\operatorname{height}I=h$. Since $I_{B}$ is
unmixed, $h=\operatorname{height}I_{B}=\operatorname{height}P>h$, which is a
contradiction.

For a graph $G$ we show the set of all minimal vertex covers of $G$ by
$\min(G)$ and $|\min(G)|$ is denoted by $c_{G}$. For a minimal vertex cover
$C\subseteq V(G)=\\{x_{1},\ldots,x_{n}\\}$ of $G$, we set $L_{C}=(x_{i}:\
x_{i}\in C)$. A perfect matching of König type for a graph $G$, is a matching
$\\{e_{1},\ldots,e_{h}\\}$ with $h=\tau(G)$ and $V(G)=\cup_{i=1}^{h}e_{i}$. It
is known that any unmixed König graph without isolated vertices has a perfect
matching of König type, see for example [2, Proposition 15].

######  Theorem 2.2.

Let $G$ be an Cohen-Macaulay König graph and $\\{e_{1},\ldots,e_{n}\\}$ be a
perfect matching of König type for $G$. Let $R=S/I(G)$ and $H$ be a graph with
$V(H)=V(G)$ and

$E(H)=\\{\\{z,w\\}:\ z\in e_{i},\ w\in e_{j},\ i\neq j,\
(e_{i}\setminus\\{z\\})\cup(e_{j}\setminus\\{w\\})\in E(G)\\}.$

Then

  1. (a)

$\omega_{R}\cong\overline{I(H)^{\vee}}$, where $\overline{I}$ denotes the
image of the ideal $I$ in $R$ under the canonical epimorphism $S\rightarrow
R$.

  2. (b)

$\operatorname{type}(R)=c_{H}$.

###### Proof.

(a) We use a basic fact from linkage theory (first observed by [14]): let
$I\subset S$ be a graded Cohen–Macaulay ideal, $J\subset I$ a complete
intersection with $\operatorname{height}I=\operatorname{height}J$, and let
$R=S/I$. Then

$\omega_{R}\cong(J:I)/J.$

For any $1\leq i\leq n$, let $e_{i}=\\{x_{i},y_{i}\\}$. Set
$J=(x_{1}y_{1},\ldots,x_{n}y_{n})$ and let $T$ be the graph with the vertex
set $V(G)$ and the edge set $E(T)=\\{e_{1},\ldots,e_{n}\\}$. Then $J=I(T)$.
Note that any minimal vertex cover $C$ of $G$ has cardinality $\tau(G)=n$ and
for any $1\leq i\leq n$, $|e_{i}\cap C|=1$. So $\min(G)\subseteq\min(T)$. By
Lemma 1.12,

(2) $\displaystyle
J:I(G)=\bigcap_{C\in\min(T)}L_{C}:\bigcap_{C\in\min(G)}L_{C}=\bigcap_{C\in\min(T)\setminus\min(G)}L_{C}.$

We have $C\in\min(T)\setminus\min(G)$, if and only if $C\in\min(T)$ and there
exists an edge $e\in E(G)\setminus E(T)$ such that $C\cap e=\emptyset$. Let
$C\in\min(T)\setminus\min(G)$ and $e=\\{z,w\\}\in E(G)\setminus E(T)$ with
$C\cap e=\emptyset$. Since $V(G)=\cup_{i=1}^{n}e_{i}$, we have $z\in e_{i}$
and $w\in e_{j}$ for some $i$ and $j$. Clearly $i\neq j$ and
$(e_{i}\setminus\\{z\\})\cup(e_{j}\setminus\\{w\\})\subseteq C$. In the sequel
we denote the set $(e_{i}\setminus\\{z\\})\cup(e_{j}\setminus\\{w\\})$ by
$\\{\tilde{z},\tilde{w}\\}$. Then we conclude that
$\min(T)\setminus\min(G)=\bigcup A_{z,w}$, where the union is taken over
$\\{z,w\\}\in E(G)\setminus E(T)$ and

$A_{z,w}=\\{D\cup\\{\tilde{z},\tilde{w}\\}:D\in\min(T\setminus\\{e_{i},e_{j}\\}),z\in
e_{i},w\in e_{j}\\}.$

So

$\bigcap_{C\in\min(T)\setminus\min(G)}L_{C}=\bigcap_{\\{z,w\\}\in
E(G)\setminus E(T)}(\bigcap_{C\in A_{z,w}}L_{C}).$

Note that for $z\in e_{i}$ and $w\in e_{j}$, we have

$\bigcap_{C\in
A_{z,w}}L_{C}=(\tilde{z},\tilde{w})+I(T\setminus\\{e_{i},e_{j}\\}).$

Therefore

(3)
$\displaystyle\bigcap_{C\in\min(T)\setminus\min(G)}L_{C}=(\bigcap_{\\{z,w\\}\in
E(G)\setminus E(T)}(\tilde{z},\tilde{w}))+I^{\prime}=I(H)^{\vee}+I^{\prime},$

for some ideal $I^{\prime}\subseteq J\subseteq I(G)$. By second isomorphism
Theorem,

(4)
$\displaystyle(\bigcap_{C\in\min(T)\setminus\min(G)}L_{C})/J\cong(\bigcap_{C\in\min(T)\setminus\min(G)}L_{C}+I(G))/I(G).$

Now by (2), (3) and (4) we get

$\omega_{R}\cong(\bigcap_{C\in\min(T)\setminus\min(G)}L_{C}+I(G))/I(G)\cong\overline{I(H)^{\vee}}.$

(b) By (a) we get
$\omega_{R}\cong(\bigcap_{C\in\min(T)\setminus\min(G)}L_{C})/J=(I(H)^{\vee}+I^{\prime})/J$,
where $I^{\prime}\subseteq J$. So $\omega_{R}\cong(I(H)^{\vee}+J)/J$ and then
$\operatorname{type}(R)$ is equal to the number of minimal generators of
$(I(H)^{\vee}+J)/J$. Thus it is enough to show that this number is equal to
the number of minimal generators $I(H)^{\vee}$. Let $\\{u_{1},\ldots,u_{m}\\}$
be the minimal generating set of monomials of $I(H)^{\vee}$, where
$u_{i}={\mathbf{x}}_{C_{i}}$ for each $i$ and
$\\{C_{1},\ldots,C_{m}\\}=\min(H)$. Here by ${\mathbf{x}}_{C}$ we mean
$\prod_{x_{i}\in C}x_{i}$. Let $\overline{u}_{i}=u_{i}+J$ for all $i$. We show
that $\\{\overline{u}_{1},\ldots,\overline{u}_{m}\\}$ is a minimal generating
set of $(I(H)^{\vee}+J)/J$. Clearly it is a generating set. By contradiction
assume that it is not minimal. Then $u_{j}-\sum_{i\neq j}r_{i}u_{i}\in J$ for
some $1\leq j\leq m$ and $r_{i}\in S$. Since $J$ is a monomial ideal and none
of the $u_{i}$’s with $i\neq j$ divides $u_{j}$, we have $u_{j}\in J$. This
means that $x_{k}y_{k}|u_{j}$ for some $1\leq k\leq n$, or equivalently
$x_{k},y_{k}\in C_{j}$. Thus $N_{H}(x_{k})\nsubseteq C_{j}$, otherwise
$C_{j}\setminus\\{x_{k}\\}$ is a vertex cover of $H$ as well, which
contradicts to the minimality of $C_{j}$. Similarly $N_{H}(y_{k})\nsubseteq
C_{j}$. So there exist $z\in N_{H}(x_{k})\setminus C_{j}$ and $w\in
N_{H}(y_{k})\setminus C_{j}$. Then $\\{x_{k},z\\},\\{y_{k},w\\}\in E(H)$. Let
$z\in e_{r}=\\{z,z^{\prime}\\}$ and $w\in e_{s}=\\{w,w^{\prime}\\}$. Then by
the definition of $H$ we have
$\\{y_{k},z^{\prime}\\},\\{x_{k},w^{\prime}\\}\in E(G)$. Since $G$ is an
unmixed König graph, by [13, Corollary 2.11], we should have
$\\{z^{\prime},w^{\prime}\\}\in E(G)$. This means that $\\{z,w\\}\in E(H)$.
But since $z,w\notin C_{j}$, this contradicts to $C_{j}$ be a vertex cover of
$H$.

We use Theorem 1.10 to derive a combinational Cohen–Macaulay criterion for
König graphs. The reader may compare our criterion with [2, Proposition 28]
and [10, Theorem 2.7].

Let $G$ be König graph on $[n]$, and let $\\{e_{1},\ldots,e_{m}\\}$ be a
matching of $G$ with $m=\tau(G)$. Let $e_{k}=\\{i_{k},j_{k}\\}$ with
$i_{k}<j_{k}$ for $k=1,\ldots,m$. We define a new graph $G_{0}$ with
$V(G_{0})=\\{i_{1},\ldots,i_{m}\\}$ for which $e=\\{i_{k},i_{l}\\}\in
E(G_{0})$ if and only if $e_{k}$ and $e_{l}$ are adjacent in $G$, i.e., if and
only if there exists an edge $e^{\prime}\in G$ such that $e_{k}\cap
e^{\prime}\neq\emptyset$ and $e_{l}\cap e^{\prime}\neq\emptyset$.

Part (b) of the following result is due to [3, Corollary 4.4]. For the
convenience of the reader we include its short proof.

######  Theorem 2.3.

Let $G$ be a König graph. Let $\alpha(G)$ be the number of vertex covers $D$
of $G$ with $|D|=\tau(G)$, and let $\Delta_{G_{0}}$ be the independence
complex of $G_{0}$. With the assumptions and notation introduced we have:

  1. (a)

$S/I(G)$ is Cohen–Macaulay if and only if $\alpha(G)=|\Delta_{G_{0}}|$.

  2. (b)

If $S/I(G)$ is Cohen–Macaulay, then $\operatorname{type}(S/I(G))$ is the
number of facets of $\Delta_{G_{0}}$ and
$\operatorname{reg}I(G)=\dim\Delta_{G_{0}}+1$.

###### Proof.

(a) We notice that $\alpha(G)$ coincides with the number of minimal prime
ideals $P$ of $I(G)$ with $\operatorname{height}P=\operatorname{height}I(G)$.
This shows that $\alpha(G)=e(R)$, where $R=S/I(G)$. By Theorem 1.10, the
sequence $C$ in (1) is a system of parameters for $R$. Let $L$ be the ideal
generated by this sequence. Then $R/LR\cong
K[\Delta_{G_{0}}]/(x_{1}^{2},\ldots,x_{n}^{2})$. Thus the set of monomials
$\\{\prod_{i\in F}x_{i}\colon\;F\in\Delta_{G_{0}}\\}$ is a $K$-basis of
$R/LR$. It follows that $\operatorname{\ell}(R/LR)=|\Delta_{G_{0}}|$. Here
$\operatorname{\ell}(M)$ denotes the length of a module $M$. As observed in
Theorem 1.5, $R$ is Cohen–Macaulay if and only if $e(R)=e(R/LR)$. Since $\dim
R/LR=0$ we have $e(R)=\operatorname{\ell}(R/LR)$. Thus, the desired conclusion
follows.

(b) If $R$ is Cohen–Macaulay, then
$\operatorname{type}(R)=\operatorname{type}(R/LR)$. Since $\dim R/LR=0$ it
follows that $\operatorname{type}(R/LR)=\dim_{K}\operatorname{Soc}(R/LR)$.
Here $\operatorname{Soc}(R/LR)$ denotes the socle of $R/IR$. It follows from
the structure of $R/LR$, as described in part (a) of the proof, that the
monomials $\prod_{i\in F}x_{i}$ with $F$ a facet of $\Delta_{G_{0}}$ form a
$K$-basis of $\operatorname{Soc}(R/LR)$. This shows that
$\operatorname{type}(I(G))$ is the number of facets of $\Delta_{G_{0}}$.
Keeping in mind that $\operatorname{reg}(R)$ coincides with the maximal degree
of a socle element of $R/LR$, we see that
$\operatorname{reg}I(G)=\dim\Delta_{G_{0}}+1$.

We end this section by proving a property for an unmixed monomial ideal of
König type generated in one degree.

######  Remark 2.4.

Let $I$ be a monomial ideal and $I^{\wp}$ be its polarization. For a monomial
$u=\prod_{i=1}^{n}x_{i}^{a_{i}}$, let
$u^{\wp}=\prod_{i=1}^{n}\prod_{j=1}^{a_{i}}x_{ij}$. Then it is easy to see
that $u_{1},\ldots,u_{h}$ is a regular sequence in $I$ if and only if
$u_{1}^{\wp},\ldots,u_{h}^{\wp}$ is a regular sequence in $I^{\wp}$. So from
the equality $\operatorname{height}I=\operatorname{height}I^{\wp}$, we deduce
that $I$ is of König type if and only if $I^{\wp}$ is of König type.

A squarefree monomial ideal $I$ generated in degree $d$ is called very well-
covered, if $|\bigcup_{u\in
G(I)}\operatorname{Supp}(u)|=(\operatorname{height}I)d$.

######  Theorem 2.5.

Let $I$ be an unmixed monomial ideal generated in degree $d$. If $I$ is of
König type, then $I^{\wp}$ is very well-covered.

###### Proof.

Note that $I$ is unmixed if and only if $I^{\wp}$ is unmixed (see [5,
Corollary 2.6]). So in view of Remark 2.4 we may assume that $I$ is
squarefree. Let $I$ be of König type and set $h=\operatorname{height}I$. Then
there exists a regular sequence $u_{1},\ldots,u_{h}\in G(I)$. Set
$A=\bigcup_{u\in G(I)}\operatorname{Supp}(u)$ and by contradiction assume that
$hd\neq|A|$. Since $u_{1},\ldots,u_{h}$ is a regular sequence of squarefree
monomials, this implies that $hd<|A|$. Then there exists $x_{\ell}\in A$ which
does not divide $u_{1},\ldots,u_{h}$. Then $x_{\ell}$ divides $v$ for some
$v\in G(I)$. So there exists a minimal prime ideal $P$ of $I$ which contains
$x_{\ell}$. Therefore $(x_{\ell},u_{1},\ldots,u_{h})\subseteq P$, which
implies that $\operatorname{height}P\geq h+1$. This is a contradiction, since
$I$ is unmixed. So $|A|=hd$ as desired.

The converse of Theorem 2.5 is true in the case that $I$ is generated in
degree $2$. Indeed, since $I$ is unmixed, then $I^{\wp}$ is unmixed as well.
So by [3, Lemma 2.1], if $I^{\wp}$ is very well-covered, then $I^{\wp}$ is of
König type. Therefore, Remark 2.4 implies that $I$ is of König type. But this
is not the case in general. To see this, consider the ideal
$I=(x_{1}x_{2}x_{3},x_{1}x_{3}x_{4},x_{1}x_{4}x_{6},x_{3}x_{4}x_{5})$ in
$S=K[x_{1},\ldots,x_{6}]$. Then $I$ is unmixed with
$\operatorname{height}I=2$, $d=3$ and $n=6=(\operatorname{height}I)d$,. So $I$
is unmixed and very well-covered, while any regular sequence of monomials in
$I$ has length $1$. So $I$ is not of König type.

##  3\. Binomial edge ideals of König type

Let $K$ be a field and $S=K[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ the
polynomial ring over $K$ in the variables $x_{1},\ldots,y_{n}$. For $1\leq
i<j\leq n$ we set $f_{ij}=x_{i}y_{j}-x_{j}y_{i}$.

Let $G$ be a simple graph on the vertex set $[n]$ with set of edges $E(G)$.
The ideal $J_{G}=(f_{ij}\colon i<j,\\{i,j\\}\in E(G))$ is the binomial edge
ideal of $G$. Throughout this section $<$ denotes the lex order induced by
$x_{1}>\cdots>x_{n}>y_{1}>\cdots>y_{n}$.

######  Lemma 3.1.

Let $G$ be a graph on $[n]$. Then the set $\\{f_{ij}\colon i<j,\\{i,j\\}\in
E(G)\\}$ is a minimal set of generators of $J_{G}$.

###### Proof.

The initial monomials of the generators $f_{ij}$ are pairwise distinct. This
yields the desired conclusion.

The binomials which belong to a minimal set of generators of $J_{G}$ are
described in the following

######  Lemma 3.2.

Let $G$ be a graph on $[n]$ and $f\in J_{G}$ a binomial. If $f$ is part of a
minimal set of generators of $J_{G}$, then the exists and edge $\\{i,j\\}$ of
$G$ with $i<j$ such that $f=\pm f_{ij}$.

###### Proof.

Let $\varepsilon_{i}$ be the $i$th canonical unit vector of
${\mathbb{Z}}^{n}$. We set $\deg(x_{i})=\deg(y_{i})=\varepsilon_{i}$. Each
$f_{ij}$ is homogeneous of degree $\varepsilon_{i}+\varepsilon_{j}$, and hence
$J_{G}$ is a ${\mathbb{Z}}^{n}$-graded ideal.

Let $f=u-v\in J_{G}$, then the total degree of $f$ is $2$. Otherwise,
$f\in{\mathfrak{m}}J_{G}$, where ${\mathfrak{m}}$ is the graded maximal ideal
of $S$, which by Nakayama’s lemma is a contradiction. It follows that $f$ is a
$K$-linear combination of the binomials $f_{ij}$. Hence, the monomials $u$ and
$v$ are of the form $x_{k}y_{l}$.

Since $J_{G}$ is ${\mathbb{Z}}^{n}$-graded, all ${\mathbb{Z}}^{n}$-graded
components belong to $J_{G}$. Suppose $u$ and $v$ have different
${\mathbb{Z}}^{n}$-degree, then $u,v\in J_{G}$. If we apply the map $\varphi$
which replaces each $y_{i}$ by $x_{i}$, then $\varphi(J_{G})=0$ while
$\varphi(u)\neq 0$, a contradiction. It follow that $u$ and $v$ have the same
${\mathbb{Z}}^{n}$ degree, say
$\deg(u)=\deg(v)=\varepsilon_{i}+\varepsilon_{j}$. This implies that
$u,v\in\\{x_{i}y_{j},x_{j}y_{i}\\}$. We may assume that $u=x_{i}y_{j}$. Then
$v=x_{j}y_{i}$, since $f\neq 0$. Therefore, $f=\pm f_{ij}$. Now we show that
if $f_{ij}\in J_{G}$, then $\\{i,j\\}$ is an edge of $G$. The set
$\\{x_{k}y_{l}\colon\\{k,l\\}\in E(G)\\}$ is the set of monomials of degree
$2$ of $\operatorname{in}_{<}(J_{G})$. Since $f_{ij}\in J_{G}$, it follows
that $x_{i}y_{j}$ belongs to the ideal generated by this set of monomials.
This is only possible, if $x_{i}y_{j}$ belongs to this set, and this implies
that $\\{i,j\\}\in E(G)$.

A _path graph_ on the vertex set $\\{i_{1},\ldots,i_{n}\\}=[n]$ is a graph
with the edge set $\\{\\{i_{k},i_{k+1}\\}:\ 1\leq i\leq n-1\\}$ and is denoted
by $P_{n}$. We call the vertices $i_{1}$ and $i_{n}$ the endpoints and $n-1$
the length of the path. Since the edge set of a path graph can be seen from
its vertex set, we usually denote a path graph by $P_{n}:i_{1},\ldots,i_{n}$.

For a graph $G$ and $i,j\in V(G)$, a path from $i$ to $j$ (a path connecting
$i$ and $j$) in $G$ is a subgraph of $G$, which is a path graph with the
endpoints $i$ and $j$. We call a subgraph $P$ of $G$ a semi-path if each
component of $P$ is a path graph. By definition, the length of the semi-path
is its number of edges.

######  Lemma 3.3.

$G$ is a semi-path if and only if $J_{G}$ is a complete intersection.

###### Proof.

Suppose that $G$ is a semi-path graph. We want to show that $J_{G}$ is a
complete intersection. This is the case if the binomial edge ideal of each
component of $G$ is a complete intersection. We may therefore assume that $G$
is connected and that $\\{i,i+1\\}$ for $i=1,\ldots,n$ are the edges of $G$.
Then $J_{G}=(f_{1,2},\ldots,f_{n,n+1})$. We consider the lexicographic order
on $S$ induced by $x_{1}>x_{2}>\cdots>x_{n+1}>y_{1}>\cdots>y_{n+1}$. Then
$\operatorname{in}_{<}(f_{i,i+1})=x_{i}y_{i+1}$ for all $i$. Since
$x_{1}y_{2},\ldots,x_{n}y_{n+1}$ is a regular sequence, it follows
$f_{1,2},\ldots,f_{n,n+1}$ is a regular sequence.

Conversely, assume that $J_{G}$ is a complete intersection. Then
$J_{G}=(g_{1},\ldots,g_{n})$ where $g_{1},\ldots,g_{n}$ is a regular sequence.
Then $\operatorname{height}J_{G}=n$ and $J_{G}$ is minimally generated by $n$
elements. Since the set of binomials $\\{f_{ij}\colon\;\\{i,j\\}\in E(G),\
i<j\\}$ is a minimal set of generators of $J_{G}$ it follows that the elements
of this set form a regular sequence. Thus we may assume that
$g_{k}=f_{i_{k}j_{k}}$ with $\\{i_{k},j_{k}\\}\in E(G)$ and $i_{k}<j_{k}$ for
$k=1,\ldots,n$.

We proceed by induction on the number of edges of $G$. If $G$ has only one or
two edges, then $G$ is a semi-path and there is nothing to prove. So now
assume that $G$ have more than two edges. Then $n\geq 3$. Since
$g_{1},\ldots,g_{n-1}$ is also a complete intersection, our induction
hypothesis implies that the subgraph $G^{\prime}$ of $G$ with
$E(G^{\prime})=\\{\\{i_{k},j_{k}\\}\colon k=1,\ldots,n-1\\}$ is a semi-path.
If the edge $\\{i_{n},j_{n}\\}$ is a component of $G$ or if $i_{n}$ or $j_{n}$
is an endpoint of one of the components of $G^{\prime}$, then $G$ is a semi-
path. Suppose these cases do not occur. Then there exists and integer $k$ such
that $j_{k}=i_{k+1}$ and $i_{n}=j_{k}$ or $j_{n}=j_{k}$. Assume that
$i_{n}=j_{k}$. Then $g_{n}=x_{i_{n}}y_{j_{n}}-x_{j_{n}}y_{i_{n}}$,
$g_{k}=x_{i_{k}}y_{i_{n}}-x_{i_{n}}y_{i_{k}}$ and
$g_{k+1}=x_{i_{n}}y_{j_{k+1}}-x_{j_{k+1}}y_{i_{n}}$. This implies that
$g_{k},g_{k+1},g_{n}\in(x_{i_{n}},y_{i_{n}})$ which is a contradiction since
$g_{k},g_{k+1},g_{n}$ is a regular sequence which implies that
$\operatorname{height}(g_{k},g_{k+1},g_{n})=0$, while
$\operatorname{height}(x_{i_{n}},y_{i_{n}})=2$. The case $j_{n}=j_{k}$ is
treated similarly.

######  Corollary 3.4.

Let $G$ be a simple graph. Then the binomial edge ideal $J_{G}$ is of König
type, if and only if $J_{G}$ contains a regular sequence $g_{1},\ldots,g_{h}$
such that each $g_{k}$ is a binomial of the form $f_{ij}$, where $\\{i,j\\}\in
E(G)$ and $h=\operatorname{height}J_{G}$.

###### Proof.

By Lemma 3.2, the minimal binomial generators of $J_{G}$ are of the form
$f_{ij}$, where $\\{i,j\\}\in E(G)$ and by Lemma 3.3 such a sequence is a
regular sequence if and only if the initial monomials of this sequence form a
regular sequence. This yields the desired conclusion.

######  Theorem 3.5.

Let $G$ be a graph on $[n]$ and $d=\dim S/J_{G}$.

  1. (a)

Let $P$ be a semi-path in $G$ of length $r$. Then $r\leq 2n-d$.

  2. (b)

The following conditions are equivalent:

    1. (i)

$J_{G}$ is of König type.

    2. (ii)

There exists a semi-path in $G$ of length $2n-d$.

###### Proof.

(a) By Lemma 3.3, $J_{P}$ is generated by a regular sequence whose length is
$r$. It follows that
$r=\operatorname{height}J_{P}\leq\operatorname{height}J_{G}=2n-d$.

(b) First note that by Lemma 3.2, if $f_{ij}\in J_{G}$, then $\\{i,j\\}$ is an
edge of $G$. Let $h=\operatorname{height}J_{G}=2n-d$. By Corollary 3.4,
$J_{G}$ is of König type if and only if $J_{G}$ contains a regular sequence
$g_{1},\ldots,g_{h}$ where each $g_{k}$ is a of the form $f_{ij}$ for some
$\\{i,j\\}\in E(G)$. Now, by Lemma 3.3, $g_{1},\ldots g_{h}$ is a regular
sequence in $J_{G}$, if and only if the subgraph $P$ of $G$ whose edges
correspond to the leading terms of the $g_{i}$’s is a semi-path of length $h$.
This proves the equivalence of (i)${}\Longleftrightarrow{}$(ii).

Recall that a subgraph $H$ of $G$ is called a spanning subgraph, whenever
$V(H)=V(G)$. A connected graph $G$ is called traceable, if it has a spanning
path $P$ as a subgraph. More generally, $G$ is called traceable if each of its
connected components is traceable.

######  Proposition 3.6.

Let $G$ be a graph.

  1. (a)

If $G$ is traceable, then $J_{G}$ is of König type. In particular, closed
graphs are of König type.

  2. (b)

If $J_{G}$ unmixed and of König type, then $G$ is traceable.

###### Proof.

Let $V(G)=[n]$. Without loss of generality we may assume that $G$ is
connected.

(a) Since $G$ is traceable, there exists a path $P$ in $G$ with $V(P)=[n]$.
Then by Lemma 3.3, the ideal $J_{P}\subset J_{G}$ is generated by a regular
sequence of length $n-1$ of binomials of the form $f_{ij}$. In particular,
$\operatorname{height}J_{G}\geq n-1$. Since $G$ is connected, we have $\dim
S/J_{G}\geq n+1$ (see [9, Corollary 3.3]). Therefore,
$\operatorname{height}J_{G}\leq n-1$. Hence we conclude that
$\operatorname{height}J_{G}=n-1$. This shows that $J_{G}$ is of König type.
The second statement follows from the observation that any closed graph is
traceable.

(b) By Theorem 3.5, there exists a semi-path $P$ in $G$ of length $r=2n-d$,
where $d=\dim S/J_{G}$. Since $J_{G}$ is unmixed we have $d=n+1$, see proof of
[9, Corollary 3.4]. It follows that $r=n-1$. Any semi-path of $G$ of length
$n-1$ is indeed a path. This implies that $P$ is a path of length $n-1$ and
hence is a spanning path of $G$.

There are graphs which are neither closed, nor bipartite, nor chordal, nor
traceable, but whose binomial edge ideal is of König type. The graph $G$ in
Figure 1 is such a graph. Indeed, it can be seen that $\dim S/J_{G}=9$, so
that $2n-d=5$ which is the length of the path $P:1,2,3,4,5,6$. On the other
hand, it cannot be unmixed because it is not traceable.

$5$$3$$4$$1$$2$$6$$7$ Figure 1. $5$-cycle with whiskers

Consider the graph $G$ in Figure 2. The semi-path $P$ with the edges
$\\{1,2\\},\\{2,3\\},\\\ \\{3,4\\},\\{5,6\\}$ has length 4 and it is a semi-
path in $G$ of largest length in $G$. On the other hand, $\dim S/J_{G}=7$.
Since $2n-d=12-7=5>4$, $J_{G}$ is not of König type.

$1$$2$$4$$5$$3$$6$ Figure 2. Triangle with whiskers

The following theorem shows the existence of special system of parameters for
binomial edge ideals of König type.

######  Theorem 3.7.

Let $G$ be a graph such that $J_{G}$ is of König type. Then $S/J_{G}$ has a
special system of parameters of the form (1).

###### Proof.

Let $V(G)=[n]$. By Theorem 3.5, $G$ has a semi-path $P$ of length $h=2n-d$,
where $d=\dim S/J_{G}$. Let

$E(P)=\\{\\{i_{1},j_{1}\\},\ldots,\\{i_{h},j_{h}\\}:\ i_{\ell}<j_{\ell}\\}.$

Then $J_{G}$ is of König type with respect to
$f_{i_{1}j_{1}},\ldots,f_{i_{h}j_{h}}$. We show that the sequence $C$ attached
to $f_{i_{1}j_{1}},\ldots,f_{i_{h}j_{h}}$ as in (1) is a system of parameters
of $S/J_{G}$. Since $|C|=d$, it suffices to show that $\dim S/(J_{G}+L)=0$,
where $L=(C)$.

Since $J_{P}\subseteq J_{G}$ it suffices to show that $\dim S/(J_{P}+L)=0$.
Let $P_{1},\ldots,P_{k}$ be the components of $P$. Without loss of
generalities we may assume
$P_{1}:1,\ldots,a_{1},P_{2}:a_{1}+1,\ldots,a_{2},\ldots,P_{i}:a_{i-1}+1,\ldots,a_{i},\ldots,P_{k}:a_{k-1}+1,\ldots,n$,
where $a_{i-1}<a_{i}$ for $i=2,\ldots,k$ and $a_{k}=n$. Then
$J_{P}+L=\sum_{s=1}^{k}(J_{P_{s}}+L_{s})$, where

$L_{s}=(x_{i}-y_{j}:\ \\{i,j\\}\in E(P_{s}),i<j)+(x_{a_{s}},y_{a_{s-1}+1}).$

Note that $J_{P_{1}}+L_{1},\ldots,J_{P_{k}}+L_{k}$ are ideals in the
polynomial rings with pairwise disjoint sets of variables. This implies that
$S/(J_{P}+L)\cong\bigotimes_{i=1}^{k}S_{i}/(J_{P_{i}}+L_{i})$, where
$S_{i}=K[x_{j},y_{j}:j\in V(P_{i})]$. Hence it is enough to show that $\dim
S/(J_{P}+L)=0$, when $P$ is a path and $S=K[V(P)]$. Let $P:1,\ldots,n$ be a
path graph. Then $x_{i}y_{i+1}-x_{i+1}y_{i}\in J_{P}$ for $i=1,\ldots,n-1$ and
$L=(x_{i}-y_{i+1}:1\leq i\leq n-1)+(x_{n},y_{1})$. Then $S/(J_{P}+L)\cong
K[y_{1},\ldots,y_{n}]/H$, where $H=(y_{i+1}^{2}-y_{i}y_{i+2}:\ 2\leq i\leq
n-2)+(y_{2}^{2},y_{n}^{2})$. By induction assume that $y_{i}^{r}\in H$ for
some positive integer $r$. Then $y_{i+1}^{2r}-y_{i}^{r}y_{i+2}^{r}\in H$,
because it is divided by $y_{i+1}^{2}-y_{i}y_{i+2}$. Thus $y_{i+1}^{2r}\in H$.
Since $y_{2}^{2}\in H$, this argument shows that $y_{i}^{2^{i-1}}\in H$ for
all $i$. So $\dim K[y_{1},\ldots,y_{n}]/H=0$, which completes the proof.

As a consequence of Theorems 1.5 and 3.7 we recover with a different proof the
following result, due to [1]. In fact by Proposition 3.6 an unmixed binomial
edge ideal $J_{G}$ is of König type if and only if $J_{G}$ is traceable.

######  Corollary 3.8.

Let $G$ be graph such that $J_{G}$ is of König type. Then the Cohen–Macaulay
property of $S/J_{G}$ does not depend on the base field $K$.

We recall a few facts from [9]. Let $G$ be a graph on $[n]$, and let
$T\subset[n]$. We denote by $c(T)$ the number of connected components of
$G_{[n]\setminus T}$. The set $T$ is called a cut set of $G$ if
$c(T\setminus\\{i\\})<c(T)$ for all $i\in T$. We consider $T=\emptyset$ also
as a cut set of $G$, and denote by $\operatorname{cut}(G)$ the set of all cut
sets of $G$.

Let $T\subset[n]$, and let $G_{1},\ldots,G_{c(T)}$ be the connected components
of $G_{[n]\setminus T}$. We set

$P_{T}=(\bigcup_{i\in
T}\\{x_{i},y_{i}\\},J_{\tilde{G}_{1}},\ldots,J_{\tilde{G}_{c(T)}}),$

where $\tilde{G}_{i}$ denotes the complete graph on the vertex set of $G_{i}$.
Then $P_{T}$ is a minimal prime ideal of $J_{G}$ if and only if $T$ is a cut
set, and one has

(5) $\displaystyle J_{G}=\bigcap_{T\in\operatorname{cut}(G)}P_{T}.$

Suppose that $J_{G}$ is unmixed and $G$ is traceable. Let $P\subset G$ be path
with $V(P)=V(G)$. Since $J_{P}$ is a complete intersection, it is
Cohen–Macaulay and hence unmixed. If $h=\operatorname{height}J_{G}$, then all
minimal prime ideals of $J_{P}$ and $J_{G}$ are of height $h$. Hence if
$P_{T}$ is a minimal prime ideal of $J_{G}$, then it is also a minimal prime
ideal of $J_{P}$. This shows that

$\operatorname{cut}(G)\subseteq\operatorname{cut}(P).$

Suppose now that $J_{G}$ is Cohen–Macaulay. When $G$ is traceable, the
canonical module $\omega_{R}$ of $R=S/J_{G}$ can be determined as follows.

######  Proposition 3.9.

Let $G$ be a traceable graph and $P$ be a spanning path of $G$. Suppose that
$J_{G}$ is Cohen–Macaulay and let $R=S/J_{G}$. Then

$\omega_{R}\cong(\bigcap_{T\in\operatorname{cut}(P)\setminus\operatorname{cut}(G)}P_{T})/J_{P}.$

###### Proof.

Since $J_{P}$ is a complete intersection and a radical ideal with
$\operatorname{height}J_{P}=\operatorname{height}J_{G}$, we have

$\omega_{R}\cong(J_{P}:J_{G})/J_{P}.$

Thus, by using (5) and Lemma 1.12,

$\bigcap_{T\in\operatorname{cut}(P)}P_{T}:\bigcap_{T\in\operatorname{cut}(G)}P_{T}=\bigcap_{T\in\operatorname{cut}(P)\setminus\operatorname{cut}(G)}P_{T}.$

######  Example 3.10.

Let $G$ be the complete graph on the vertex set $[n]$ and $P\subset G$ the
path $1,2,\ldots,n$. Then $\operatorname{cut}(G)=\emptyset$ and
${i}\subset[n]$ is a cut set of $P$ if and only $2\leq i\leq n-2$. Hence
$\omega_{R}\cong\bigcap_{i=2}^{n-1}(\bar{x}_{i},\bar{y}_{i})$. Note that

$\bigcap_{i=2}^{n-1}(x_{i},y_{i})=\prod_{i=2}^{n-1}(x_{i},y_{i})=(\prod_{i\in
A}x_{i}\prod_{i\in[2,n-1]\setminus A}y_{i}\colon\;{A\subseteq[2,n-1]}).$

Since $\bar{x}_{i}\bar{y}_{j}=\bar{x}_{j}\bar{y}_{i}$ for all $j>i$, we
finally obtain

$\omega_{R}=(\bar{x}_{2}\cdots\bar{x}_{n-1},\ldots,\bar{x}_{2}\cdots\bar{x}_{n-2}\bar{y}_{n-1},\ldots,\bar{x}_{2}\cdots\bar{x}_{j}\bar{y}_{j+1}\cdots\bar{y}_{n-1},\ldots,\bar{y}_{2}\cdots\bar{y}_{n-1}).$

##  4\. Hibi ideals of König type

We refer the reader to [8, pp. 156–159] for fundamental materials on finite
partially ordered sets and finite lattices. A partially ordered set will be
called a poset.

Let $P=\\{x_{1},\ldots,x_{n}\\}$ be a finite poset with $|P|=n$. The rank of
each $x\in P$, written as $\operatorname{rank}(x)$, is the maximal length of
chains of $P$ of the form

$x=x_{i_{0}}>x_{i_{1}}>\cdots>x_{i_{a}}.$

Let $\operatorname{rank}(P)$ denote the rank of $P$, which is the maximal
length of chains of $P$. Let $d=\operatorname{rank}(P)$ and write $L_{i}$ for
the set of those $x\in P$ with $\operatorname{rank}(x)=i$, where $0\leq i\leq
d$.

The incomparability graph of $P$ is the finite simple graph ${\rm incom}(P)$
on $[n]=\\{1,\ldots,n\\}$ whose edges are those $\\{i,j\\}$ for which $p_{i}$
and $p_{j}$ are incomparable in $P$. For example,

$p_{2}$$p_{3}$$p_{4}$$p_{5}$$p_{6}$$p_{8}$$p_{7}$$p_{1}$$4$$1$$3$$6$$7$$8$$5$$2$$P$${\rm
incom}(P)$ Figure 3. A finite poset and its incomparability graph

######  Lemma 4.1.

Let $P$ be a finite poset of $\operatorname{rank}(P)=d$ and $|P|=n$. Then

$\tau({\rm incom}(P))=n-(d+1).$

###### Proof.

Since the induced subgraph of ${\rm incom}(P)$ on $L_{i}$ is the complete
graph on $L_{i}$, each vertex cover $C$ of ${\rm incom}(P)$ satisfies $|C\cap
L_{i}|\geq|L_{i}|-1$. It then follows that

$|C|\geq\sum_{i=0}^{d}(|L_{i}|-1)=n-(d+1).$

On the other hand, if

$x_{i_{0}}<x_{i_{1}}<\cdots<x_{i_{d}}$

is a maximal chain of $P$ of length $d$, then

$L\setminus\\{x_{i_{0}},x_{i_{1}},\ldots,x_{i_{d}}\\}$

is a vertex cover of ${\rm incom}(P)$. Hence $\tau({\rm incom}(P))=n-(d+1)$,
as desired.

A subset $A$ of a finite poset $P$ is called an antichain of $P$ if every two
distinct elements of $A$ are incomparable in $P$. A finite poset $P$ is called
thin if $P$ possesses no antichain $A$ with $|A|\geq 3$. We say that a finite
poset $P$ of $\operatorname{rank}(P)=d$ is pure if every maximal chain of $P$
is of length $d$.

######  Lemma 4.2.

Let $P$ be a finite pure poset. Then the following conditions are equivalent:

  * (i)

$P$ is thin.

  * (ii)

each $|L_{i}|$ satisfies $|L_{i}|\leq 2$.

  * (iii)

$P$ is the sum of at most two disjoint chains.

  * (iv)

${\rm incom}(P)$ is bipartite.

###### Proof.

The equivalence (i) $\Leftrightarrow$ (iii) is just a decomposition theorem of
Dilworth [4]. Furthermore, (iii) $\Rightarrow$ (ii) is clear. On the other
hand, in order to show (ii) $\Rightarrow$ (i), suppose that each $|L_{i}|\leq
2$ and that $\\{a,b,c\\}$ is an antichain of $P$. Since $P$ is pure, if
$\operatorname{rank}(a)=\operatorname{rank}(b)$ and, say,
$\operatorname{rank}(c)<\operatorname{rank}(a)$, then either $c<a$ or $c<b$, a
contradiction. If
$\operatorname{rank}(a)<\operatorname{rank}(b)<\operatorname{rank}(c)$, then
there is $b^{\prime}\in P$ with
$\operatorname{rank}(b^{\prime})=\operatorname{rank}(b)$ and $a<b^{\prime}<c$.
Hence $a<c$, a contradiction. As a result, no three-element subset of $P$ can
be an antichain of $P$. Finally, the equivalence (iii) $\Leftrightarrow$ (iv)
follows directly from the definition of ${\rm incom}(P)$.

The finite nonpure poset $P$ of $\operatorname{rank}(P)=d$ drawn below
satisfies each $|L_{i}|\leq 2$. However, there is an antichain $A$ of $P$ with
$|A|=d+1$.

$x_{1}$$x_{d-2}$$x_{d-1}$$y_{1}$$y_{2}$$y_{d-1}$$y_{d}$$x_{d}$$x_{0}$ Figure
4. A nonpure poset and an antichain

We now come to a characterization for ${\rm incom}(P)$ of a finite pure poset
to be König.

######  Theorem 4.3.

The incomparability graph ${\rm incom}(P)$ of a finite pure poset $P$ is König
if and only if ${\rm incom}(P)$ is bipartite.

###### Proof.

Every bipartite graph is König. Now, suppose that ${\rm incom}(P)$ is König.
Let $q$ denote the number of those $L_{i}$ with $|L_{i}|=1$. Since $P$ is
pure, if $|L_{i}|=1$ and $\operatorname{rank}(a)=i$, then $a$ is comparable to
every element of $P$. Thus

$n-q=\sum_{|L_{i}|>1}|L_{i}|\geq 2\cdot m({\rm incom}(P))=2\cdot\tau({\rm
incom}(L))=2(n-(d+1)).$

Hence

$\sum_{|L_{i}|>1}|L_{i}|\geq
2\big{(}\big{(}\sum_{|L_{i}|>1}|L_{i}|)+q\big{)}-(d+1)\big{)}.$

It then follows that

$\sum_{|L_{i}|>1}|L_{i}|\leq 2(d-q+1).$

Since the number of those $0\leq i\leq d$ with $|L_{i}|>1$ is $d-q+1$, it
follows that

$\sum_{|L_{i}|>1}|L_{i}|\geq 2(d-q+1).$

Hence

$\sum_{|L_{i}|>1}|L_{i}|=2(d-q+1).$

Thus $|L_{i}|=2$ if $|L_{i}|>1$. It then follows from Lemma 4.2 that ${\rm
incom}(P)$ is bipartite, as desired.

We now turn to the study of the Hibi ring [11] of a finite distributive
lattice. Let $P=\\{p_{1},\ldots,p_{d}\\}$ be a finite poset. A subset
$\alpha\subset P$ is called a poset ideal of $P$ if $\alpha$ enjoys the
property that if $p_{i}\in\alpha,p_{j}\in P$ together with $p_{j}\leq p_{i}$,
then $p_{j}\in\alpha$. Let ${\mathcal{J}}(P)$ denote the set of poset ideals
of $P$. It then follows that ${\mathcal{J}}(P)$ is a finite poset, in fact,
finite lattice, ordered by inclusion. Furthermore, the finite lattice
${\mathcal{J}}(P)$ is distributive. On the other hand, Birkhoff’s fundamental
structure theorem [8, Theorem 9.1.7] guarantees that, given a finite
distributive lattice $L$, there is a unique finite poset $P$ with
$L={\mathcal{J}}(P)$. Every finite distributive lattice is pure.

Let $T=K[z_{1},\ldots,z_{d},t]$ denote the polynomial ring in $d+1$ variables
over a field $K$. Given a poset ideal $\alpha$ of $P$, we associate the
squarefree monomial

$u_{\alpha}=(\prod_{p_{i}\in\alpha}z_{i})t\in T.$

In particular $u_{\emptyset}=t$ and $u_{P}=z_{1}\cdots z_{d}t$. In [11], the
toric ring ${\mathcal{R}}_{K}[{\mathcal{J}}(P)]$ which is generated by those
monomials $u_{\alpha}$ with $\alpha\in{\mathcal{J}}(P)$ is introduced:

${\mathcal{R}}_{K}[{\mathcal{J}}(P)]=K[\\{u_{\alpha}\\}_{\alpha\in{\mathcal{J}}(P)}].$

Let $S=K[\\{x_{\alpha}\\}_{\alpha\in{\mathcal{J}}(P)}]$ denote the polynomial
ring in $|{\mathcal{J}}(P)|$ variables over $K$. We introduce a surjective
ring homomorphism $\pi:S\rightarrow{\mathcal{R}}_{K}[{\mathcal{J}}(P)]$ by
setting

$\pi(x_{\alpha})=u_{\alpha}.$

It turns out that the toric ideal $I_{{\mathcal{J}}(P)}\subset S$ of
${\mathcal{R}}_{K}[{\mathcal{J}}(P)]$ which is the kernel of $\pi$ is

$I_{{\mathcal{J}}(P)}=(x_{\alpha_{i}}x_{\alpha_{j}}-x_{\alpha_{i}\cap\alpha_{j}}x_{\alpha_{i}\cup\alpha_{j}}\,:\,1\leq
i<j\leq n).$

Nowadays, the toric ring ${\mathcal{R}}_{K}[L]$ with $L={\mathcal{J}}(P)$ is
called the Hibi ring of the finite distributive lattice $L$. One can ask when
the toric ideal $I_{L}$ of ${\mathcal{R}}_{K}[L]$ is of König type.

Let $L={\mathcal{J}}(P)=\\{x_{1},\ldots,x_{n}\\}$ with $n=|{\mathcal{J}}(P)|$
and $S=K[x_{1},\ldots,x_{n}]$. We discuss a reverse lexicographic order
$<_{\rm rev}$ on $S$ with the property that $x_{i}<_{\rm rev}x_{j}$ if
$x_{i}<x_{j}$ in $L$. A basic fact on $<_{\rm rev}$ is

######  Lemma 4.4 ([11]).

The set of those binomials

$f_{i,j}=x_{i}x_{j}-(x_{i}\wedge x_{j})(x_{i}\vee x_{j})$

for which $x_{i}$ and $x_{j}$ are incomparable in $L$ is the reduced Gröbner
basis of $I_{L}$ with respect to $<_{\rm rev}$.

$a$$c$$b$$d$$\emptyset$$\\{b\\}$$\\{a\\}$$\\{a,b\\}$$\\{b,d\\}$$\\{a,b,d\\}$$\\{a,b,c\\}$$\\{a,b,c,d\\}$$P$$\mathcal{J}(P)$
Figure 5. A poset and its poset ideal lattice

We now come to a characterization for the toric ideal $I_{L}$ to be of König
type.

######  Theorem 4.5.

Let $L$ be a finite distributive lattice and $I_{L}$ the toric ideal of the
Hibi ring ${\mathcal{R}}_{K}[L]$ of $L$. Then the following conditions are
equivalent:

  * (i)

$I_{L}$ is of König type with respect to $f_{i,j}$ and $<_{\rm rev}$.

  * (ii)

$L$ is thin.

  * (iii)

${\rm incom(L)}$ is bipartite.

###### Proof.

Let $L={\mathcal{J}}(P)$ with $|P|=d$ and $|L|=n$. Since

$\dim{\mathcal{R}}_{K}[L]=\operatorname{rank}(L)+1=d+1$

and ${\mathcal{R}}_{K}[L]$ is Cohen–Macaulay, one has

$\operatorname{height}I_{L}=n-(d+1).$

Let, as before, $L_{i}$ denote the set of those $\alpha\in L={\mathcal{J}}(P)$
with $\operatorname{rank}(\alpha)=i$, where $0\leq i\leq d$. Let $q\geq 2$
denote the number of those $0\leq i\leq d$ with $|L_{i}|=1$.

(ii) $\Rightarrow$ (i): Suppose that $L$ is thin. Lemma 4.2 says that
$|L_{i}|\leq 2$ for each $0\leq i\leq d$. One has $n=q+2(d-q+1)=2d-q+2$. Hence
$\operatorname{height}I_{L}=n-(d+1)=d-q+1$. Furthermore, the number of those
$i$ with $|L_{i}|=2$ is $d-q+1$. If $|L_{i}|=2$ and
$L_{i}=\\{x_{r_{i}},x_{r^{\prime}_{i}}\\}$, then the binomial
$f_{r_{i},r^{\prime}_{i}}$ belongs to the reduced Gröbner basis of $I_{L}$
with respect to $<_{\rm rev}$ and $\operatorname{in}_{<_{\rm
rev}}(f_{r_{i},r^{\prime}_{i}})=x_{r_{i}}x_{r^{\prime}_{i}}$. Clearly, the
$d-q+1$ monomials $x_{r_{i}}x_{r^{\prime}_{i}}$ with $|L_{i}|=2$ form a
regular sequence. Hence $I_{L}$ is of König type, as desired.

(i) $\Rightarrow$ (iii): Suppose that $I_{L}$ is of König type. Then there are
$n-(d+1)$ binomials $f_{i,j}$ for which their initial monomials $x_{i}x_{j}$
form a regular sequence. Hence the matching number of ${\rm incom(L)}$ is at
least $n-(d+1)$. Since Lemma 4.1 says that the vertex cover number of ${\rm
incom(L)}$ is $n-(d+1)$, it follows that ${\rm incom(L)}$ is König. Now,
Theorem 4.3 guarantees that ${\rm incom(L)}$ is bipartite.

(iii) $\Leftrightarrow$ (ii): This follows from Lemma 4.2.

######  Corollary 4.6.

Let $S=K[x_{1},\ldots,x_{n}]$ and $T=K[y_{1},\ldots,y_{m}]$ denote polynomial
rings in $n\geq 2$ and in $m\geq 2$ variables over a field $K$. Then the
defining ideal of the Segre product $S\,\sharp\,T$ is of König type if and
only if either $n=2$ or $m=2$.

###### Proof.

Let $P$ be a finite poset which is the disjoint union of the chain of length
$n-2$ and the chain of length $m-2$. Then the Segre product $S\,\sharp\,T$
coincides with ${\mathcal{R}}_{K}[{\mathcal{J}}(P)]$. The finite distributive
lattice $L={\mathcal{J}}(P)$ is thin if and only if either $n=2$ or $m=2$.
Now, Theorem 4.5 guarantees that the defining ideal of $S\,\sharp\,T$ is of
König type if and only if either $n=2$ or $m=2$, as desired.

By using Lemma 1.12 can give an explicit description of the canonical module
of ${\mathcal{R}}_{K}[L]$ when $L$ is thin, in the same way as we did it for
edge ideals and binomial edge ideals. More generally, let $L$ be a
distributive lattice. We call two incomparable elements $\alpha,\beta\in L$ a
basic pair of $L$, if $\alpha$ and $\beta$ cover $\alpha\wedge\beta$ and
$\alpha\vee\beta$ covers $\alpha$ and $\beta$. If $L$ is thin, the binomials
$f_{i,j}$ in Theorem 4.5 are just the binomials
$f_{\alpha,\beta}=x_{\alpha}x_{\beta}-x_{\alpha\wedge\beta}x_{\alpha\vee\beta}$
where $\alpha$ and $\beta$ form a basic pair in $L$. The elements
$\alpha,\beta,\alpha\wedge\beta,\alpha\vee\beta$ are the corners of a square,
also called a cell, whose four edges are $\\{\alpha,\alpha\wedge\beta\\}$,
$\\{\alpha,\alpha\vee\beta\\}$, $\\{\beta,\alpha\wedge\beta\\}$ and
$\\{\beta,\alpha\vee\beta\\}$. For example, the lattice in Figure 5 has three
cells. One of the cells has the corners $\\{a,b\\}$, $\\{b\\}$, $\\{b,d\\}$
and $\\{a,b,d\\}$. The set of cells of $L$ is a configuration of cells
${\mathcal{C}}$, as defined in [7]. We denote by $I_{\mathcal{C}}$ the ideal
generated by the binomials $f_{\alpha,\beta}$ for which $\alpha$ and $\beta$
is a basic pair. By Theorem 4.5, the generators of $I_{\mathcal{C}}$ form a
regular sequence and
$\operatorname{height}I_{\mathcal{C}}=\operatorname{height}I_{L}$. Since
$I_{L}$ is Cohen–Macaulay, it follows that
$(I_{\mathcal{C}}:I_{L})/I_{\mathcal{C}}$ is the canonical module
${\mathcal{R}}_{K}[L]$.

In order compute $I_{\mathcal{C}}:I_{L}$ we use the fact that
$I_{\mathcal{C}}$ is a radical ideal because its initial ideal is squarefree,
and we use the presentation

$I_{\mathcal{C}}=\bigcap_{W}P_{W}({\mathcal{C}}),$

where intersection is taken over all admissible sets $W$ of ${\mathcal{C}}$,
see [7, Theorem 3.2]. Recall that $W$ is called admissible, if whenever $W$
intersected with the corners of a cell of ${\mathcal{C}}$ is nonempty, then
$W$ contains an edge of this cell. The prime ideal $P_{W}({\mathcal{C}})$ is
the ideal generated by the variables corresponding to $W$ and the inner minors
of ${\mathcal{C}}^{\prime}$, where ${\mathcal{C}}^{\prime}$ is the set of
cells of ${\mathcal{C}}$ which do not intersect $W$. Note that the union of
the cells of ${\mathcal{C}}^{\prime}$ is a disjoint union of distributive
lattices, say $L_{1},\ldots,L_{m}$. Then the ideal $J_{W}$ generated by the
inner minors of ${\mathcal{C}}^{\prime}$ is just $\sum_{i=1}^{m}I_{L_{i}}$.

By applying Lemma 1.12 we find that

$I_{\mathcal{C}}:I_{L}=\bigcap_{W\atop W\neq\emptyset}(W,J_{W}),\quad\text{$W$
is admissible and
$\operatorname{height}(W,J_{W})=\operatorname{height}I_{L}$}.$

We come back to the example given in Figure 5. In order to simplify notation
we relabel the vertices of $L$ as follows: $1=\emptyset$, $2=\\{a\\}$,
$3=\\{b\\}$, $4=\\{a,b\\}$, $5=\\{b,d\\}$, $6=\\{a,b,c\\}$, $7=\\{a,b,d\\}$
and $8=\\{a,b,c,d\\}$.

The admissible sets $W$ are $\\{1,2\\}$, $\\{6,8\\}$,
$\\{1,5,3\\}$,$\\{2,4,6\\}$, $\\{5,7,8\\}$, $\\{3,4,6\\}$ and $\\{3,4,7\\}$.
Then for example
$P_{W}({\mathcal{C}})=(x_{1},x_{2},x_{4}x_{5}-x_{3}x_{7},x_{6}x_{7}-x_{4}x_{8},x_{6}x_{5}-x_{3}x_{8})$
for $W=\\{1,2\\}$, while for $W=\\{3,4,7,\\}$ we have
$P_{W}({\mathcal{C}})=(x_{3},x_{4},x_{7})$. The only ideals of height $3$
(which is the height of $I_{\mathcal{C}}$) among the prime ideals
$P_{W}({\mathcal{C}})$ are the ideals $(x_{3},x_{4},x_{7})$,
$(x_{2},x_{4},x_{7})$ and $(x_{3},x_{4},x_{6})$. Therefore.

$I_{\mathcal{C}}:I_{L}=(x_{3},x_{4},x_{7})\cap(x_{2},x_{4},x_{7})\cap(x_{3},x_{4},x_{6})=(x_{4},x_{2}x_{3},x_{3}x_{7},x_{6}x_{7}).$

In [7, Theorem 3.3] the minimal prime ideals among the
${\mathcal{P}}_{W}({\mathcal{C}})$ are described, in general.

In Theorem 4.5 we characterized for which finite distributive lattices $L$ the
toric ideal $I_{L}$ of the Hibi ring ${\mathcal{R}}_{K}[L]$ is of König type
with respect to the reverse lexicographic monomial order. One may ask the same
question for any other monomial order. We do not have a complete answer to
this question. But it is quite obvious that $I_{L}$ being of König type is
rather restrictive, no matter which monomial order we choose. Indeed, suppose
that $I_{L}$ is of König type for the monomial order $<$. Then
$\operatorname{in}_{<}(I_{L})$ contains a regular sequence
$u_{1},\ldots,u_{h}$ of monomials of degree $2$, where
$h=\operatorname{height}I_{L}$. Let $T$ be the set of join irreducible
elements of $L$. Then $h=|L|-|T|-1$. It follows that if $I_{L}$ is of König
type, then $2(|L|-|T|-1)\leq|L|$, equivalently, $|L|\leq 2(|T|+1)$.

Consider for example the Boolean lattice $B_{n}$ of rank $n$. Then
$|B_{n}|=2^{n}$ and $|T|=n$ and the inequality $2^{n}\leq 2(n+1)$ is satisfied
only if $n\leq 3$. So for $n\geq 4$, $I_{B_{n}}$ is never of König type.

On the other hand, $I_{B_{2}}$ is a principle ideal, and so is of König type,
and $I_{B_{3}}$ is of König type with respect to the lexicographic monomial
order. Indeed, let us label $B_{3}$ as indicated in Figure 6.

$8$$5$$7$$6$$2$$4$$1$$3$ Figure 6. A Boolean lattice $B_{3}$

Then

$\displaystyle I_{B_{3}}$ $\displaystyle=$
$\displaystyle(x_{2}x_{3}-x_{1}x_{5},x_{3}x_{4}-x_{1}x_{7},x_{1}x_{6}-x_{2}x_{4},x_{5}x_{6}-x_{2}x_{8},x_{6}x_{7}-x_{4}x_{8},$
$\displaystyle
x_{2}x_{7}-x_{1}x_{8},x_{4}x_{5}-x_{1}x_{8},x_{5}x_{7}-x_{3}x_{8},x_{3}x_{6}-x_{1}x_{8}).$

With respect to the lexicographic monomial order induced by
$x_{1}>x_{2}>\cdots>x_{8}$ we find that

$\operatorname{in}_{<}(I_{B_{3}})=(x_{1}x_{8},x_{3}x_{8},x_{3}x_{6},x_{2}x_{7},x_{4}x_{8},x_{2}x_{8},x_{1}x_{6},x_{1}x_{7},x_{1}x_{5}).$

This initial ideal contains the regular sequence

$x_{3}x_{6},x_{2}x_{7},x_{4}x_{8},x_{1}x_{5}.$

The corresponding ideal whose initial monomials are this sequence of monomials
is the ideal

$J=(x_{3}x_{6}-x_{4}x_{5},x_{2}x_{7}-x_{3}x_{6},x_{4}x_{8}-x_{6}x_{7},x_{1}x_{5}-x_{2}x_{3}).$

Since $\operatorname{height}I_{B_{3}}=4$, this shows $I_{B_{3}}$ is of König
type with respect to the lexicographic monomial order.

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