1. Introduction
Let
be an integral convex polytope, i.e., a convex polytope whose vertices have integer coordinates, and let
. Let
be the Laurent polynomial ring in
variables over a field
K. Given an integer vector
, we set
. Then, the
toric ring of
is the subalgebra
of
generated by
over
K. Here, we need the variable
t in order to regard
as a homogeneous algebra by setting each
. The
toric ideal of
is the kernel of a surjective homomorphism
defined by
for
. In general,
is generated by homogeneous binomials and any reduced Gröbner basis of
consists of homogeneous binomials; see [
1]. A simplex
is called a
subsimplex of
if the set of vertices of
is contained in
. A set
of subsimplices of
is called a
covering of
if
. A covering
of
is called a
triangulation of
if
is a simplicial complex. A covering (triangulation)
of
is called
unimodular if the normalized volume of each maximal simplex in
is equal to 1. The following properties of an integral convex polytope
have been investigated in many papers on commutative algebra and combinatorics:
- (i)
is unimodular (any triangulation of
is unimodular), the initial ideal of
is generated by squarefree monomials with respect to any monomial order ([
2] Section 4.3);
- (ii)
is compressed (any “pulling” triangulation is unimodular), the initial ideal of
is generated by squarefree monomials with respect to any reverse lexicographic order [
3,
4];
- (iii)
has a regular unimodular triangulation, there exists a monomial order such that the initial ideal of
is generated by squarefree monomials ([
2] Theorem 4.17);
- (iv)
has a unimodular triangulation ([
2] Section 4.2.4);
- (v)
has a unimodular covering ([
2] Section 4.2.4);
- (vi)
is normal,
is a normal semigroup ring ([
2] Section 4.2.3).
Details for these conditions are explained in ([
1] Chapter 8) and ([
2] Chapter 4). The hierarchy (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v) is trivial by their definition. The implication (v) ⇒ (vi) is explained in ([
2] Theorem 4.11). Note that the converse of each of the above implications is false. On the other hand, the following properties of
are studied by many authors:
- (a)
has a Gröbner bases consisting of quadratic binomials;
- (b)
is Koszul ([
2] Definition 2.20);
- (c)
is generated by quadratic binomials.
The hierarchy (a) ⇒ (b) ⇒ (c) is known, and the converse of each of the two implications is false; see ([
5] Examples 2.1 and 2.2) and [
6].
The purpose of this paper is to study such properties of toric rings and ideals of stable set polytopes of simple graphs. Let G be a finite simple graph on the vertex set , and let denote the set of edges of G. Given a subset , we associate the vector , where is the ith unit vector of . In particular, . A subset is said to be stable (or independent) if for all with . In particular, Ø and each with are stable. Let be the set of all stable sets of G. The stable set polytope (independent set polytope) of a simple graph G is the convex hull of .
Example 1. If G is a complete graph, then is a unit simplex, and hence and .
Stable set polytopes are very important in many areas, such as optimization theory as well as combinatorics and commutative algebra. Below, we present a list of results on the toric ring and the toric ideals of the stable set polytope of a simple graph G.
The stable set polytope
is compressed if and only if
G is perfect ([
3,
4,
7]).
Let
G be a perfect graph. Then, the toric ring
is Gorenstein if and only if all maximal cliques of
G have the same cardinality ([
8]).
The toric ring
is strongly Koszul ([
2] p. 53) if and only if
G is trivially perfect ([
9] Theorem 5.1).
Let
be a comparability graph of a poset
P. Then,
is called a
chain polytope of
P. It is known that the toric ideals of a chain polytope have a squarefree quadratic initial ideal (see [
10] Corollary 3.1). For example, if a graph
G is bipartite, then there exists a poset
P such that
.
Suppose that a graph
G on the vertex set
is an almost bipartite graph, i.e., there exists a vertex
v such that the induced subgraph of
G on the vertex set
is bipartite. Then,
has a squarefree quadratic initial ideal ([
11] Theorem 8.1). For example, any cycle is almost bipartite.
Let
G be the complement of an even cycle of length
. Then, the maximum degree of a minimal set of binomial generators of
is equal to
k ([
11] Theorem 7.4).
In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite (nonsimple) graphs and the results on their toric ideals. Here, the
edge polytope of a graph
G allowing loops and having no multiple edges is the convex hull of
This paper is organized as follows. In
Section 2, fundamental properties of
and
are studied. In particular, the relationship between the stable set polytopes and the edge polytopes are given (Lemma 1). In addition, it is shown that
is unimodular if and only if the complement of
G is bipartite (Proposition 5). We also point out that, by the results in [
11], it is easy to see that
has a squarefree quadratic initial ideal if
G is either a chordal graph or a ring graph (Proposition 2). In
Section 3, we discuss the normality of the stable set polytopes. We prove that, for a simple graph
G of stability number two,
is normal if and only if the complement of
G satisfies the “odd cycle condition” (Theorem 1). Using this criterion, we construct an infinite family of normal stable set polytopes without regular unimodular triangulations (Theorem 2). For general simple graphs, some necessary conditions for
to be normal are also given. In
Section 4, we study the set of generators and Gröbner bases of toric ideals of stable set polytopes. It is shown that for a simple graph
G of stability number two, the set of binomial generators of
are described in terms of even closed walks of a graph (Theorem 3). If
is bipartite and if
is generated by quadratic binomials, then
has a quadratic Gröbner basis (Corollary 1). Finally, using the results on normality, generators, and Gröbner bases, we present an infinite family of non-normal stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases (Theorem 4).
2. Fundamental Properties of the Stable Set Polytopes
In this section, we give some fundamental properties of and . In particular, a relation between the stable set polytopes and the edge polytopes is discussed. The stability number of a graph G is the cardinality of the largest stable set. If a simple graph G satisfies , then G is a complete graph in Example 1. Given lattice polytopes and , the product of and is defined by . Then the toric ring is called the Segre product of and .
Example 2. Suppose that a simple graph G is not connected. Let be the connected components of G. Then, it is easy to see that , and hence is the Segre product of .
Thus, it is enough to study stable set polytopes of connected simple graphs
G such that
. We use the notion of toric fiber products to study toric rings of stable set polytopes. This notion is first introduced in [
12] as a generalization of the Segre product. Since the definition of toric fiber products is complicated, we give an example.
Example 3. Let and be cycles of length 4, where and . Then and . We define the multigrading by . Let . Then, the toric fiber product is generated by the monomials such that and have the same multidegree . It is easy to see that , and A corresponds to the lattice point in , which is a simplex.
The application of toric fiber products to toric rings of stable set polytopes was studied in [
11]. For
, let
be a simple graph on the vertex set
and the edge set
. If
is a clique of both
and
, then we construct a new graph
on the vertex set
and the edge set
, which is called the
clique sum of
and
along
.
Proposition 1. Let be the clique sum of simple graphs and . Then, is a toric fiber product of and . We can construct a set of binomial generators (or a Gröbner basis) of from that of ’s and some quadratic binomials. Moreover, is normal if and only if both and are normal.
Proof. Note that
. Hence, this is a special case of ([
11] Proposition 5.1) by ([
11] Proposition 9.6). □
A simple graph
G is called
chordal if any induced cycle of
G is of length 3. A graph
G is called a
ring graph if each block of
G that is not a bridge or a vertex can be constructed from a cycle by successively adding cycles of length
using the clique sum construction. Ring graphs are introduced in [
13,
14].
Proposition 2. Suppose that a simple graph G is either a chordal graph or a ring graph. Then, has a squarefree quadratic initial ideal.
Proof. It is known by ([
15] Proposition 5.5.1) that a graph
G is chordal if and only if
G is a clique sum of complete graphs. By the statement in Example 1 and Proposition 1,
has a squarefree quadratic initial ideal if
G is chordal.
Suppose that
G is a ring graph. Then,
G is a clique sum of trees and cycles since any graph is a clique sum (along a vertex) of trees and its blocks. Since trees and cycles are almost bipartite, by ([
11] Theorem 8.1), the toric ideal
has a squarefree quadratic initial ideal if
H is either a tree or a cycle. Thus, by Proposition 1,
has a squarefree quadratic initial ideal if
G is a ring graph. □
A graph
G is called
perfect if the chromatic number of every induced subgraph of
G is equal to the size of the largest clique of that subgraph; see [
15]. We recall the following result on perfect graphs and their stable set polytopes; see [
3,
4,
7].
Proposition 3. Let G be a simple graph. Then, is compressed if and only if G is perfect. In particular, if G is perfect, then is normal.
For a graph G on the vertex set , let denote the complement of a graph G. An induced cycle of G of length is called a hole of G and an induced cycle of of length is called an antihole of G. Below we combine two important characterizations of perfect graphs, where the first part is the strong perfect graph theorem and the second part considers just the perfect graphs with stability number 2.
Proposition 4. Let G be a simple graph. Then G is a perfect graph if and only if G has no odd holes and no odd antiholes. In particular, G is a perfect graph with if and only if is bipartite and not empty.
For a graph G, let be the nonsimple graph on the vertex set whose edge (and loop) set is The following lemma plays an important role when we study the stable set polytope of G.
Lemma 1. Let G be a simple graph with . Then we have . Moreover, if is bipartite, then there exists a bipartite graph H such that .
Proof. Let
be the injective ring homomorphism defined by
Then
,
for
, and
for each stable set
of
G. Note that
is a stable set of
G if and only if
is an edge of
. Hence, the image of
is
.
Suppose that
is bipartite. Then
has no odd cycles. Hence, any odd cycle of
have the vertex
. (Note that
is an odd cycle of length 1.) Thus, in particular, any two odd cycles of
has a common vertex. We now show that there exists a bipartite graph
H such that
(by a similar argument in ([
16] Proof of Proposition 5.5)). Let
be a partition of the vertex set of the bipartite graph
. Let
H be a bipartite graph on the vertex set
and the edge set
Let
be the ring homomorphism defined by
Since
is obtained from
H by identifying the vertices
and
, it follows that the image of
is
. Hence, it is enough to show that
is injective. Suppose that
,
satisfies
. Then
and
for
. Since
is a partition for
, we have
Thus,
and
, as desired. □
The first application of Lemma 1 is as follows:
Proposition 5. Let G be a simple graph. Then the following conditions are equivalent:
- (i)
is unimodular;
- (ii)
is bipartite.
Moreover, if , then the conditions
- (iii)
is compressed;
- (iv)
G is perfect
are also equivalent to conditions (i) and (ii).
Proof. We may assume that
G is not complete (i.e.,
is not empty and
). Let
A be the matrix whose columns are vertices of
, and let
Then
B is a submatrix of
. Since
, the rank of
is
. It is known by ([
1] p. 70) that
is unimodular if and only if the absolute value of any nonzero
-minor of the matrix
is 1.
Suppose that
is not bipartite. Then
has an odd cycle
. Then the absolute value of the
-minor of
that corresponds to
equals 2. Hence,
is not unimodular. Thus, we have (i) ⇒ (ii).
Suppose that
is bipartite. By Lemma 1, there exists a bipartite graph
H such that
. It is well known that the edge polytope of a bipartite graph is unimodular; see ([
2] Theorem 5.24). Thus,
is unimodular, and hence we have (ii) ⇒ (i).
Suppose that . By Proposition 3, conditions (iii) and (iv) are equivalent. In addition, by Proposition 4, conditions (ii) and (iv) are equivalent. □
We close this section with the following fundamental fact on stable set and edge polytopes.
Proposition 6. Let be an induced subgraph of a graph G. Then
- (i)
the edge polytope is a face of ;
- (ii)
if G is a simple graph, then is a face of .
It can be seen from Proposition 6 that several properties of
(resp.
) are inherited to
(resp.
), for example, normality of the toric ring, the existence of a squarefree initial ideal, existence of a quadratic Gröbner basis, and the existence of the set of quadratic binomial generators of the toric ideal; see [
17].
3. Normality of Stable Set Polytopes
In this section, we study the normality of stable set polytopes. Normality of edge polytopes is studied in [
18,
19], and we make use of the normality conditions of edge polytopes while working with stable set polytopes, due to the relation discovered in Lemma 1. If
and
are cycles in a graph
G, then
is called a
bridge of
and
if
i is a vertex of
and
j is a vertex of
. We say that a graph
G satisfies the
odd cycle condition if any induced odd cycles
and
in
G have either a common vertex or a bridge. For the sake of simplicity, assume that a graph
H has at most one loop. Then, it is known by [
18,
19] that
is normal if and only if each connected component of
H satisfies the odd cycle condition. By ([
18] Corollary 2.3) and Lemma 1, we have the following. (Note that
below is not necessarily connected.)
Theorem 1. Let G be a simple graph with . Then the following conditions are equivalent.
- (i)
is normal;
- (ii)
has a unimodular covering;
- (iii)
satisfies the odd cycle condition, i.e., if two odd holes and in have no common vertices, then there exists a bridge of and in .
In particular, if is normal, then is normal.
Proof. By Lemma 1, we have
. Hence, by ([
18] Corollary 2.3), conditions (i) and (ii) are equivalent, and they hold if and only if
satisfies the odd cycle condition. (Note that
is connected.) Since the vertex
is incident to any vertex of
, it is easy to see that
satisfies the odd cycle condition if and only if
satisfies the odd cycle condition. □
It is shown in [
20] that there exists a graph
G such that
is normal and that
has no squarefree initial ideals. Examples on infinite families of such edge polytopes are given in [
21]. We can construct the stable set polytopes with the same properties. Let
be the graph defined in ([
21] Theorem 3.10).
Theorem 2. Let G be a graph such that with for . Then is normal, and has no squarefree initial ideals.
Proof. Since has no triangles, we have . Since satisfies the odd cycle condition, is normal by Theorem 1. On the other hand, has no squarefree initial ideals. Since is an induced subgraph of , has no squarefree initial ideals by Lemma 1 and Proposition 6. □
It seems to be a challenging problem to characterize the normal stable set polytopes with large stability numbers. We give several necessary conditions. The following is a consequence of Proposition 6 and Theorem 1.
Proposition 7. Let G be a simple graph. Suppose that is normal. Then any two odd holes of without a common vertex have a bridge in .
Proof. Suppose that two odd holes , of without common vertices have no bridges in . Let H be an induced subgraph of G whose vertex set is that of . Then , and hence is not normal by Theorem 1. Thus, is not normal by Proposition 6. □
Similar conditions are required for antiholes of .
Proposition 8. Let G be a simple graph. Suppose that is normal. Then G satisfies all of the following conditions:
- (i)
Any two odd antiholes of having no common vertices have a bridge in .
- (ii)
Any two odd antiholes of of length having exactly one common vertex have a bridge in .
- (iii)
Any odd hole and odd antihole of having no common vertices have a bridge in .
Proof. Let
G be a graph on the vertex set
. Let
. It is known by ([
1] Proposition 13.5) that
is normal if and only if we have
.
(i) Let
and
be odd antiholes in
having no common vertices and no bridges in
. By Proposition 6, we may assume that
. Then,
Since
, we have
. Hence,
belongs to
. Suppose that
belongs to
. Since the
-th coordinate of
is 5, there exist
such that
where each
belongs to either
or
. It then follows that
and
. Since
,
, we have
This is a contradiction. Thus,
is not in
.
(ii) Let
and
be odd antiholes in
of length
having exactly one common vertex
and no bridges in
. By Proposition 6, we may assume that
. Let
Then,
Since
, we have
. Hence,
belongs to
. Since the
-th coordinate of
is 5, there exist
such that
. Then, each
belongs to either
or
. Since
,
, we have
Thus,
is either
or
. Changing indices if necessary, we may assume that
and
. It then follows that
, and hence
. This implies that
. Thus,
, a contradiction. Therefore, we have
.
(iii) Let
be an odd hole and
an odd antihole in
having no common vertices. By Proposition 6, we may assume that
. Then,
Hence,
belongs to
. However, this vector is not in
since
,
, and
. □
Unfortunately, the above conditions are not sufficient to be normal in general. For example, if the length of the two odd antiholes of without common vertices are long, then a lot of bridges in seem to be needed.
4. Generators and Gröbner Bases of
For a toric ideal I, let be the maximum degree of binomials in a minimal set of binomial generators of I. If , then we set . In this section, we study by using results on the toric ideals of edge polytopes.
Let
G be a graph on the vertex set
allowing loops and having no multiple edges. Let
be a set of all edges and loops of
G. The toric ideal
is the kernel of a homomorphism
defined by
where
. A finite sequence of the form
with each
is called a
walk of length
q of
G connecting
and
. A walk
of the form (
1) is called
even (resp.
odd) if
q is even (resp. odd). A walk
of the form (
1) is called
closed if
. Given an even closed walk
of
G, we write
for the binomial
We regard a loop as an odd cycle of length 1. We recall the following result from [
1,
5,
22].
Proposition 9. Let G be a graph having at most one loop. Then is generated by all the binomials , where Γ is an even closed walk of G. In particular, if and only if each connected component of G has at most one cycle and the cycle is odd.
The following theorem implies that the set of binomial generators of can also be characterized by the graph-theoretical terminology if .
Theorem 3. Let G be a simple graph with . Then, where J is an ideal generated by quadratic binomials where Γ is an even closed walk of that satisfies one of the following:
- (i)
is a cycle where , ;
- (ii)
where .
In particular, .
Proof. Since , we have by Lemma 1. Since is a subgraph of , it follows that . Thus, it is enough to show that .
Let
be an even closed walk of
. It is enough to show that
belongs to
. Suppose that
does not belong to
. Then, the vertex
belongs to
. We may assume that the degree of
is minimum among binomials in
that do not belong to
. Then,
is irreducible. Let
where
is an odd subwalk of
from the vertex
r to the vertex
. Since
is irreducible, it follows that
are distinct vertices and that
. Then,
where
,
, and
. Since
, the binomial
belongs to
by the assumption on
. Moreover,
satisfies one of the conditions (i) or (ii), and hence
. Thus, we have
, a contradiction. □
Remark 1. A graph-theoretical characterization of a simple
graph G such that is given in [5]. Then, by making use of Theorem 3, one can provide a similar characterization of a simple graph G where and . It is known by ([
11] Theorem 7.4) that if the complement of a graph
G is an even cycle of length
, then we have
. By Theorem 3, we can generalize this result for a graph whose complement is an arbitrary bipartite graph.
Corollary 1. Let G be a simple graph such that is bipartite. Then we havewhere is the maximum length of induced cycles of . Moreover, the following conditions are equivalent: - (i)
, i.e., is generated by quadratic binomials;
- (ii)
is Koszul;
- (iii)
has a quadratic Gröbner basis;
- (iv)
the length of any induced cycle of is 4.
Proof. Let
G be a simple graph such that
is bipartite. Then
. Since
is bipartite, it is known (see [
23] Lemma 2.4) that
is generated by
where
is an induced even cycle of
. Note that
if the length of
is
. Hence, by Theorem 3, we obtain the desired formula for
.
It follows from the formula of
that (i) and (iv) are equivalent. Moreover, (iii) ⇒ (ii) ⇒ (i) holds in general. By Lemma 1, there exists a bipartite graph
H such that
. By ([
24] Theorem),
has a quadratic Gröbner basis if and only if
. Thus, we have (i) ⇒ (iii). □
If
is not bipartite, then condition (i) and (iii) in Corollary 1 are not equivalent. In order to construct an infinite family of counterexamples, the following proposition is important. (Proof is essentially given in ([
5] Proof of Proposition 1.6)).
Proposition 10. Let be a -polytope. If has a quadratic Gröbner basis, then the initial ideal is generated by squarefree monomials, and hence is normal.
Theorem 4. Let G be a simple graph such that , where and are odd holes without common vertices. Then , and
- (a)
is generated by quadratic binomials;
- (b)
has no quadratic Gröbner bases;
- (c)
is not normal.
Proof. Since has no triangles, we have . Since each connected component of is an odd cycle, by Proposition 9. It follows from Theorem 3 that is generated by quadratic binomials. By Theorem 1, is not normal since and have no bridges. Thus, by Proposition 10, has no quadratic Gröbner bases. □
The graphs in Theorem 4 are not strongly Koszul by ([
9] Theorem 1.3). However, we do not know whether they are Koszul or not in general.
Remark 2. It is known by ([5] Theorem 1.2) that if G is a simple connected graph and is generated by quadratic binomials, then G satisfies the odd cycle condition, and hence is normal. It seems to be a challenging problem to characterize the graphs
G such that
and
. The following is a consequence of Proposition 6, Theorem 3 and ([
5] Theorem 1.2).
Proposition 11. Let G be a simple graph. If is generated by quadratic binomials, then satisfies the following conditions:
- (i)
Any even cycle of of length has a chord;
- (ii)
Any two odd holes of having exactly one common vertex have a bridge;
- (iii)
Any two odd holes of having no common vertex have at least two bridges.
Proof. Suppose that
does not satisfy one of the conditions above. If
does not satisfy condition (i), then let
H be an induced subgraph of
G whose vertex set is that of the even cycle. If
does not satisfy either condition (ii) or (iii), then let
H be an induced subgraph of
G whose vertex set is that of two odd holes. Then
, and hence
is not generated by quadratic binomials by Theorem 3 and ([
5] Theorem 1.2). Thus, it follows from Proposition 6 that
is not generated by quadratic binomials. □