arXiv:1609.03710v4 [] 18 Sep 2017

ARITHMETICAL RANK OF BINOMIAL IDEALS
ANARGYROS KATSABEKIS
Abstract. In this paper, we investigate the arithmetical rank of a binomial
ideal J. We provide lower bounds for the binomial arithmetical rank and the
J-complete arithmetical rank of J. Special attention is paid to the case where
J is the binomial edge ideal of a graph. We compute the arithmetical rank of
such an ideal in various cases.

1. Introduction
Consider the polynomial ring K[x1 , . . . , xm ] in the variables x1 , . . . , xm over a
field K. For the sake of simplicity we will denote by xu the monomial xu1 1 · · · xumm
of K[x1 , . . . , xm ], with u = (u1 , . . . , um ) ∈ Nm , where N stands for the set of nonnegative integers. A binomial in the sense of [12, Chapter 8] is a difference of two
monomials, i.e. it is of the form xu − xv . A binomial ideal is an ideal generated by
binomials.
Toric ideals serve as important examples of binomial ideals. Let A = {a1 , . . . , am }
be a subset of Zn . The toric ideal IA is the kernel of the K-algebra homomorphism
±1
φ : K[x1 , . . . , xm ] → K[t±1
1 , . . . , tn ] given by
a

φ(xi ) = tai = t1 i,1 · · · tani,n for all i = 1, . . . , m,
where ai = (ai,1 , . . . , ai,n ).
We grade K[x1 , . . . , xm ] by the semigroup NA := {l1 a1 + · · · + lm am |li ∈ N}
setting degA (xi ) = ai for i = 1, . . . , m. The A-degree of a monomial xu is defined
by
degA (xu ) = u1 a1 + · · · + um am ∈ NA.
A polynomial F ∈ K[x1 , . . . , xm ] is A−homogeneous if the A−degrees of all the
monomials that occur in F are the same. An ideal is A-homogeneous if it is generated by A-homogeneous polynomials. The ideal IA is generated by all the binomials xu − xv such that degA (xu ) = degA (xv ) (see [11, Lemma 4.1]), thus IA is
A-homogeneous.
Let J ⊂ K[x1 , . . . , xm ] be a binomial ideal. There exist a positive integer n and
a vector configuration A = {a1 , . . . , am } ⊂ Zn such that J ⊂ IA , see for instance
[7, Theorem 1.1]. We say that a polynomial F = c1 M1 + . . . + cs Ms ∈ J, where
ci ∈ K and M1 , . . . , Ms are monomials, is J-complete if Mi − Ml ∈ J for every
1 ≤ i < l ≤ s. Clearly every J-complete polynomial F is also A-homogeneous.
Computing the least number of polynomial equations defining an algebraic set is
a classical problem in Algebraic Geometry which goes back to Kronecker [9]. This
problem is equivalent, over an algebraically closed field, with the corresponding
problem in Commutative Algebra of the determination of the smallest integer s for
2010 Mathematics Subject Classification. 13F20, 14M12, 05C25.
Key words and phrases. Arithmetical rank; binomial ideals; graphs; indispensable monomials.
1

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A. KATSABEKIS

which there exist polynomials F1 , . . . , Fs in J such that rad(J) = rad(F1 , . . . , Fs ).
The number s is commonly known as the arithmetical rank of J and will be denoted
by ara(J). Since J is generated by binomials, it is natural to define the binomial
arithmetical rank of J, denoted by bar(J), as the smallest integer s for which there
exist binomials B1 , . . . , Bs in J such that rad(J) = rad(B1 , . . . , Bs ). Furthermore
we can define the J-complete arithmetical rank of J, denoted by arac (J), as the
smallest integer s for which there exist J-complete polynomials F1 , . . . , Fs in J such
that rad(J) = rad(F1 , . . . , Fs ). Finally we define the A-homogeneous arithmetical
rank of J, denoted by araA (J), as the smallest integer s for which there exist
A-homogeneous polynomials F1 , . . . , Fs in J such that rad(J) = rad(F1 , . . . , Fs ).
From the definitions and [2, Corollary 3.3.3] we deduce the following inequalities:
cd(J) ≤ ara(J) ≤ araA (J) ≤ arac (J) ≤ bar(J)
where cd(J) is the cohomological dimension of J.
In section 2 we introduce the simplicial complex ∆J and use combinatorial invariants of the aforementioned complex to provide lower bounds for the binomial
arithmetical rank and the J-complete arithmetical rank of J. In particular we prove
that bar(J) ≥ δ(∆J ){0,1} and arac (J) ≥ δ(∆J )Ω , see Theorem 2.6.
In section 3 we study the arithmetical rank of the binomial edge ideal JG of a
graph G. This class of ideals generalize naturally the determinantal ideal generated
by the 2-minors of the matrix


x1
x2
. . . xn
.
xn+1 xn+2 . . . x2n
We prove (see Theorem 3.3) that, for a binomial edge ideal JG , both the binomial
arithmetical rank and the JG -complete arithmetical rank coincide with the number
of edges of G. If G is the complete graph on the vertex set {1, . . . , n}, then, from [3,
Theorem 2], the arithmetical rank of JG equals 2n − 3. It is still an open problem
to compute ara(JG ) when G is not the complete graph. We show that ara(JG ) ≥
n + l − 2, where n is the number of vertices of G and l is the vertex connectivity of
G. Furthermore we prove that in several cases ara(JG ) = cd(JG ) = n + l − 2, see
Theorem 3.7, Theorem 3.9 and Theorem 3.13.
2. Lower bounds
First we will use the notion of indispensability to introduce the simplicial complex
∆J . Let J ⊂ K[x1 , . . . , xm ] be a binomial ideal containing no binomials of the form
xu − 1, where u 6= 0. A binomial B = M − N ∈ J is called indispensable of J if
every system of binomial generators of J contains B or −B, while a monomial M
is called indispensable of J if every system of binomial generators of J contains a
binomial B such that M is a monomial of B. Let MJ be the ideal generated by all
monomials M for which there exists a nonzero M − N ∈ J. By [7, Proposition 1.5]
the set G(MJ ) of indispensable monomials of J is the unique minimal generating
set of MJ .
The support of a monomial xu of K[x1 , . . . , xm ] is supp(xu ) := {i|xi divides xu }.
Let T be the set of all E ⊂ {1, . . . , m} for which there exists an indispensable
monomial M of J such that E = supp(M ). Let Tmin denote the set of minimal
elements of T .

ARITHMETICAL RANK OF BINOMIAL IDEALS

3

Definition 2.1. We associate to J a simplicial complex ∆J with vertices the elements of Tmin . Let T = {E1 , . . . , Ek } be a subset of Tmin , then T ∈ ∆J if there exist
Mi , 1 ≤ i ≤ k, such that supp(Mi ) = Ei and Mi − Ml ∈ J for every 1 ≤ i < l ≤ k.
Next we will study the connection between the radical of J and ∆J . The induced
subcomplex ∆′ of ∆J by certain vertices V ⊂ Tmin is the subcomplex of ∆J with
vertices V and T ⊂ V is a simplex of the subcomplex ∆′ if T is a simplex of ∆J . A
subcomplex H of ∆J is called a spanning subcomplex if both have exactly the same
set of vertices.
Let F be a polynomial in K[x1 , . . . , xm ]. We associate to F the induced subcomplex ∆J (F ) of ∆J consisting of those vertices Ei ∈ Tmin with the property:
there exists a monomial Mi in F such that Ei = supp(Mi ). The next theorem
provides a necessary condition under which a set of polynomials in the binomial
ideal J generates the radical of J up to radical.
Proposition 2.2. Let K be any field. If rad(J) = rad(F1 , . . . , Fs ) for some polynomials F1 , . . . , Fs in J, then ∪si=1 ∆J (Fi ) is a spanning subcomplex of ∆J .
Proof. Let E = supp(xu ) ∈ Tmin , where B = xu − xv ∈ J and xu is an indispensable monomial of J. We will show that there exists a monomial M in some
Fl , 1 ≤ l ≤ s, such that E = supp(M ). Since rad(J) = rad(F1 , . . . , Fs ), there
is a power B r , r ≥ 1, which belongs to the ideal generated by F1 , . . . , Fs . Thus
there is a monomial M in some Fl dividing the monomial (xu )r and therefore
supp(M ) ⊆ supp(xu ). But Fl ∈ J and J is generated by binomials, so there exists
xz − xw ∈ J such that xz divides M . Since xz ∈ MJ and G(MJ ) generates MJ ,
there is an indispensable monomial N dividing xz , thus
supp(N ) ⊆ supp(xz ) ⊆ supp(M ) ⊆ E.
Since E ∈ Tmin , we deduce that E = supp(N ), and therefore E = supp(M ).



Remark 2.3. (1) If F is a J-complete polynomial of J, then ∆J (F ) is a simplex.
To see that ∆J (F ) is a simplex suppose that ∆J (F ) 6= ∅ and let T = {E1 , . . . , Ek }
be the set of vertices of ∆J (F ). For every 1 ≤ i ≤ k there exists a monomial Mi ,
1 ≤ i ≤ k, in F such that Ei = supp(Mi ). Since F is J-complete, we have that
Mi − Ml ∈ J for every 1 ≤ i < l ≤ k. Thus ∆J (F ) is a simplex.
(2) If B is a binomial of J, then ∆J (B) is either a vertex, an edge or the empty
set.
Remark 2.4. If the equality rad(J) = rad(F1 , . . . , Fs ) holds for some J-complete
polynomials F1 , . . . , Fs in J, then ∪si=1 ∆J (Fi ) is a spanning subcomplex of ∆J and
each ∆J (Fi ) is a simplex.
For a simplicial complex ∆ we denote by r∆ the smallest number s of simplices Ti
of ∆, such that the subcomplex ∪si=1 Ti is spanning and by b∆ the smallest number
s of simplices Ti of ∆, such that the subcomplex ∪si=1 Ti is spanning and each Ti is
either an edge, a vertex or the empty set.
Theorem 2.5. Let K be any field, then b∆J ≤ bar(J) and r∆J ≤ arac (J).
It turns out that both b∆J and r∆J have a combinatorial interpretation in terms
of matchings in ∆J .
Let ∆ be a simplicial complex on the vertex set Tmin and Q be a subset of
Ω := {0, 1, . . . , dim(∆)}. A set N = {T1 , . . . , Ts } of simplices of ∆ is called a Qmatching in ∆ if Tk ∩ Tl = ∅ for every 1 ≤ k, l ≤ s and dim(Tk ) ∈ Q for every

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A. KATSABEKIS

1 ≤ k ≤ s; see also Definition 2.1 in [8]. Let supp(N ) = ∪si=1 Ti , which is a subset
of the vertices Tmin . We denote by card(N ) the cardinality s of the set N . A
Q-matching N in ∆ is called a maximal Q-matching if supp(N ) has the maximum
possible cardinality among all Q-matchings. By δ(∆)Q , we denote the minimum of
the set
{card(N )|N is a maximal Q − matching in ∆}.
Theorem 2.6. Let K be any field, then bar(J) ≥ δ(∆J ){0,1} and arac (J) ≥
δ(∆J )Ω .
Proof. By Proposition 3.3 in [8], b∆J = δ(∆J ){0,1} and r∆J = δ(∆J )Ω . Now the
result follows from Theorem 2.5.

Proposition 2.7. Let J be a binomial ideal. Suppose that there exists a minimal
generating set S of J such that every element of S is a difference of two squarefree
monomials. Assume that J is generated by the indispensable binomials, namely
S consists precisely of the indispensable binomials (up to sign). Then bar(J) =
card(S).
Proof. Let card(S) = t. Since S is a generating set of J, we have that bar(J) ≤ t.
It is enough to prove that t ≤ bar(J). Let |Tmin | = g. By [4, Corollary 3.6] it holds
that card(G(MJ )) = 2t, so g = 2t. For every maximal
{0, 1}-matching M in ∆J
 
we have that supp(M) = Tmin , so δ(∆J ){0,1} ≥ g2 and therefore δ(∆J ){0,1} ≥ t.
Thus, from Theorem 2.6, bar(J) ≥ t.

Example 2.8. Let J be the binomial ideal generated by f1 = x1 x6 − x2 x5 , f2 =
x2 x7 − x3 x6 , f3 = x1 x8 − x4 x5 , f4 = x3 x8 − x4 x7 and f5 = x1 x7 − x3 x5 . Actually
J is the binomial edge ideal of the graph G with edges {1, 2}, {2, 3}, {1, 4}, {3, 4}
and {1, 3}, see section 3 for the definition of such an ideal. Note that J is Ahomogeneous where A = {a1 , . . . , a8 } is the set of columns of the matrix


1 0 0 0 1 0 0 0
0 1 0 0 0 1 0 0

D=
0 0 1 0 0 0 1 0 .
0 0 0 1 0 0 0 1
By [4, Theorem 3.3] every binomial fi is indispensable of J. Thus

Tmin = {E1 = {1, 6}, E2 = {2, 5}, E3 = {2, 7}, E4 = {3, 6}, E5 = {1, 8},
E6 = {4, 5}, E7 = {3, 8}, E8 = {4, 7}, E9 = {1, 7}, E10 = {3, 5}}.
By Proposition 2.7 the binomial arithmetical rank of J equals 5. The simplicial complex ∆J has 5 connected components and all of them are 1-simplices,
namely ∆1 = {E1 , E2 }, ∆2 = {E3 , E4 }, ∆3 = {E5 , E6 }, ∆4 = {E7 , E8 } and
∆5 = {E9 , E10 }. Consequently
δ(∆J )Ω =

5
X

δ(∆i )Ω = 1 + 1 + 1 + 1 + 1 = 5

i=1

and therefore 5 ≤ arac (J). Since arac (J) ≤ bar(J), we get that arac (J) = 5. We
will show that araA (J) = 5. Suppose that araA (J) = s < 5 and let F1 , . . . , Fs be
A-homogeneous polynomials in J such that rad(J) = rad(F1 , . . . , Fs ). For every
vertex Ei ∈ Tmin there exists, from Proposition 2.2, a monomial Mi in Fk such that
Ei = supp(Mi ). But s < 5, so there exist Ei ∈ Tmin and Ej ∈ Tmin such that

ARITHMETICAL RANK OF BINOMIAL IDEALS

5

(1) {Ei , Ej } is not a 1-simplex of ∆J ,
(2) Ei = supp(Mi ), Ej = supp(Mj ) and
(3) Mi and Mj are monomials of some Fk .
Since Fk is A-homogeneous, it holds that degA (Mi ) = degA (Mj ). Considering
all possible combinations of Ei and Ej we finally arrive at a contradiction. Thus
araA (J) = 5. Note that J is B-homogeneous where B is the set of columns of the
matrix


1 1 1 1 0 0 0 0
1 0 0 0 1 0 0 0



N =
0 1 0 0 0 1 0 0 .
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
Since every row of D is a row of N , we deduce that every B-homogeneous polynomial
in J is also A-homogeneous. So araB (J) is an upper bound for araA (J), therefore
araB (J) = 5. We have that rad(J) = rad(f1 , f2 + f3 , f4 , f5 ), since the second power
of both binomials f2 and f3 belongs to the ideal generated by the polynomials
f1 , f2 + f3 , f4 , f5 . Remark that the polynomials f1 , f2 + f3 , f4 and f5 are Chomogeneous, where C is the set of columns of the matrix


1 2 3 4 1 2 3 4
.
5 6 7 8 5 6 7 8
Thus araC (J) ≤ 4, so ara(J) ≤ 4. A primary decomposition of J is
J = (f1 , f2 , f3 , f4 , f5 , x2 x8 − x4 x6 ) ∩ (x1 , x3 , x5 , x7 ).
Hence, by [2, Proposition 19.2.7], it follows that ara(J) ≥ 4. Thus
ara(J) = araC (J) = 4 < 5 = araA (J) = araB (J) = arac (J) = bar(J).
3. Binomial edge ideals of graphs
In this section we consider a special class of binomial ideals, namely binomial
edge ideals of graphs. This ideal was introduced in [6] and independently at the
same time in [10].
Let G be an undirected connected simple graph on the vertex set [n] := {1, . . . , n}
and with edge set E(G). Consider the polynomial ring
R := K[x1 , . . . , xn , xn+1 , . . . , x2n ]
in 2n variables, x1 , . . . , xn , xn+1 , . . . , x2n , over K.
Definition 3.1. The binomial edge ideal JG ⊂ R associated to the graph G is the
ideal generated by the binomials fij = xi xn+j − xj xn+i , with i < j, such that {i, j}
is an edge of G.
Remark 3.2. From [7, Corollary 1.13] every binomial fij , where {i, j} is an edge
of G, is indispensable of JG . Thus
1
2
Tmin = {Eij
= {i, n + j}, Eij
= {j, n + i}|{i, j} ∈ E(G)}.

We recall some fundamental material from [6]. Let G be a connected graph on
[n] and let S ⊂ [n]. By G \ S, we denote the graph that results from deleting all
vertices in S and their incident edges from G. Let c(S) be the number of connected

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A. KATSABEKIS

components of G \ S and let G1 , . . . , Gc(S) denote the connected components of
∼

G \ S. Also let Gi denote the complete graph on the vertices of Gi . We set
PS (G) = (∪i∈S {xi , xn+i }, J ∼ , . . . , J ∼
G1

Gc(S)

)R.

Then PS (G) is a prime ideal for every S ⊂ [n]. The ring R/P∅ (G) has Krull
dimension n+1. For S 6= ∅ the ring R/PS (G) has Krull dimension n−card(S)+c(S).
The ideal PS (G) is a minimal prime of JG if and only if S = ∅ or S 6= ∅ and for
each i ∈ S one has c(S \ {i}) < c(S). Moreover JG is a radical ideal and it admits
the minimal primary decomposition JG = ∩S∈M(G) PS (G), where M(G) = {S ⊂
[n] : PS (G) is a minimal prime of JG }.
Theorem 3.3. Let G be a connected graph on the vertex set [n] with m edges.
Then bar(JG ) = arac (JG ) = m.
Proof. Every binomial fij , where {i, j} is an edge of G, is indispensable of JG , thus,
1
2
from Proposition 2.7, bar(JG ) = m. Note that, for every edge {i, j} of G, {Eij
, Eij
}
is a 1-simplex of ∆JG . Furthermore ∆JG has exactly m connected components and
all of them are 1-simplices. Thus δ(∆JG )Ω = m and therefore, from Theorem 2.6,
arac (JG ) ≥ m. Consequently arac (JG ) = m.

Theorem 3.4. Let G be a connected graph on the vertex set [n] with m edges. Consider the canonical basis {e1 , . . . , en } of Zn and the canonical basis {w1 , . . . , wn+1 }
of Zn+1 . Let A = {a1 , . . . , a2n } ⊂ Nn be the set of vectors where ai = ei , 1 ≤ i ≤ n,
and an+i = ei for 1 ≤ i ≤ n. Let B = {b1 , . . . , b2n } ⊂ Nn+1 be the set of vectors where bi = w1 + wi+1 , 1 ≤ i ≤ n, and bn+i = wi+1 for 1 ≤ i ≤ n. Then
araA (JG ) = araB (JG ) = m.
Proof. Suppose that araA (JG ) = t < m and let F1 , . . . , Ft be A-homogeneous
polynomials in JG such that JG = rad(F1 , . . . , Ft ). For every edge {i, j} of G with
i < j there exist, from Proposition 2.2, monomials Mijk and Nijl in Fk and Fl ,
2
1
= supp(Nijl ). But t < m, so there
= supp(Mijk ) and Eij
respectively, such that Eij
1
exists Ers
∈ Tmin , where {r, s} is an edge of G with r < s, such that
1
1
} is not a 1-simplex of ∆JG ,
(1) {Eij
, Ers
1
1
k
(2) Eij = supp(Mijk ), Ers
= supp(Mrs
) and
k
k
(3) Mij and Mrs are monomials of some Fk .
g

j
s
k
and Mrs
= xgrr xgn+s
Let Mijk = xgi i xn+j
. Since Fk is A-homogeneous, we deduce
k
k
that degA (Mij ) = degA (Mrs ), and therefore gi ei +gj ej = gr er +gs es . Consequently
k
, a contradiction. Let D and Q be the matrices with
i = r, j = s and also Mijk = Mrs
columns A and B, respectively. Since every row of D is a row of Q, we deduce that
every B-homogeneous polynomial in JG is also A-homogeneous. Thus araB (JG ) is
an upper bound for araA (JG ), so m ≤ araB (JG ) and therefore araB (JG ) = m. 
The graph G is called l-vertex-connected if l < n and G \ S is connected for every
subset S of [n] with card(S) < l. The vertex connectivity of G is defined as the
maximum integer l such that G is l-vertex-connected.
In [1] the authors study the relationship between algebraic properties of a binomial edge ideal JG , such as the dimension and the depth of R/JG , and the vertex
connectivity of the graph. It turns out that this notion is also useful for the computation of the arithmetical rank of a binomial edge ideal.

ARITHMETICAL RANK OF BINOMIAL IDEALS

7

Theorem 3.5. Let K be a field of any characteristic and G be a connected graph
on the vertex set [n]. Suppose that the vertex connectivity of G is l. Then ara(JG ) ≥
n + l − 2.
Proof. If G is the complete graph on the vertex set [n], its vertex connectivity is
n − 1, then ara(JG ) = 2n − 3 = n + l − 2 by [3, Theorem 2]. Assume now that
G is not the complete graph. Let P∅ (G), W1 , . . . , Wt be the minimal primes of
JG . It holds that JG = P∅ (G) ∩ L where L = ∩ti=1 Wi . First we will prove that
dim (R/(P∅ (G) + L)) ≤ n−l +1. For every prime ideal Q such that P∅ (G)+L ⊆ Q,
we have that L ⊆ Q, so there is 1 ≤ i ≤ t such that Wi ⊆ Q. Thus P∅ (G) + Wi ⊆ Q
and therefore dim (R/(P∅ (G) + L)) ≤ dim (R/(P∅ (G) + Wi )). It is enough to show
that dim (R/(P∅ (G) + Wi )) ≤ n − l + 1. Let Wi = PS (G) for ∅ 6= S ⊂ [n]. We have
that P∅ (G) + PS (G) is generated by
{xi xn+j − xj xn+i : i, j ∈ [n] \ S} ∪ {xi , xn+i : i ∈ S}.
Then dim (R/(P∅ (G) + PS (G))) = n − card(S) + 1. If l = 1, then card(S) ≥ 1
since S 6= ∅, and therefore dim (R/(P∅ (G) + Wi )) ≤ n. Suppose that l ≥ 2 and
also that card(S) < l. Since PS (G) is a minimal prime, for every i ∈ S we have
that c(S \ {i}) < c(S). But G is l-vertex-connected, namely G \ S is connected, so
P∅ (G) ⊂ PS (G) a contradiction to the fact that PS (G) is a minimal prime. Thus
dim (R/(P∅ (G) + Wi )) ≤ n − l + 1 and therefore dim (R/(P∅ (G) + L)) ≤ n − l + 1.
Next we will show that min{dim (R/P∅ (G)) , dim (R/L)} > dim (R/(P∅ (G) + L)).
Recall that dim (R/P∅ (G)) = n + 1, so dim (R/(P∅ (G) + L)) < dim (R/P∅ (G)).
Since L ⊂ P∅ (G) + L, we deduce that dim (R/(P∅ (G) + L)) ≤ dim (R/L). Suppose
that dim (R/(P∅ (G) + L)) = dim (R/L), say equal to s, and let Q1 $ Q2 $ · · · $ Qs
be a chain of prime ideals containing P∅ (G) + L. Then there is 1 ≤ j ≤ t such that
Q1 = Wj . So P∅ (G) ⊂ Wj , a contradiction. By [2, Proposition 19.2.7] it holds that
cd(JG ) ≥ dim(R) − dim (R/(P∅ (G) + L)) − 1 = 2n − dim (R/(P∅ (G) + L)) − 1 ≥
2n − (n − l + 1) − 1 = n + l − 2.
Consequently ara(JG ) ≥ n + l − 2.



Example 3.6. Let G be the graph on the vertex set [5] with edges {1, 2}, {2, 3},
{1, 3}, {2, 4}, {4, 5} and {3, 5}. Here the vertex connectivity is l = 2. By Theorem
3.5, ara(JG ) ≥ 5. The ideal JG is generated up to radical by the polynomials
2
2
f12 , f23 , f13 + f24 , f35 and f45 , since both f13
and f24
belong to the ideal generated
by f12 , f23 , f13 + f24 , f35 and f45 . Thus ara(JG ) = 5 < 6 = bar(JG ).
Theorem 3.7. If G is a cycle of length n ≥ 3, then ara(JG ) = bar(JG ) = n.
Proof. The vertex connectivity of G is 2, so, from Theorem 3.5, the inequality
n ≤ ara(JG ) holds. Since G has n edges, we have that ara(JG ) ≤ bar(JG ) = n and
therefore ara(JG ) = n.

Proposition 3.8. Let G be a connected graph on [n], with m edges and n ≥ 4. If
G contains an odd cycle of length 3, then ara(JG ) ≤ m − 1.
Proof. Let C be an odd cycle of G of length 3, with edge set {{1, 2}, {2, 3}, {1, 3}}.
Since G is connected, without loss of generality there is a vertex 4 ≤ i ≤ n such
that {1, i} is an edge of G. We will show that (x1 xn+i − xi xn+1 )2 belongs to the
ideal L generated by the polynomials f12 , f13 , f1i + f23 . We have that
x21 x2n+i ≡ x1 xn+i xi xn+1 − x1 x2 xn+i xn+3 + x1 x3 xn+i xn+2 ≡

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A. KATSABEKIS

x1 xi xn+i xn+1 − x2 xn+i x3 xn+1 + x2 x3 xn+1 xn+i ≡ x1 xi xn+i xn+1 mod L.
Similarly we have that x2i x2n+1 ≡ x1 xi xn+i xn+1 mod L. Thus x21 x2n+i + x2i x2n+1 ≡
2x1 xi xn+i xn+1 mod L, so (x1 xn+i − xi xn+1 )2 belongs to L. Next we prove that
(x2 xn+3 − x3 xn+2 )2 belongs to L. We have that
x22 x2n+3 ≡ x2 xn+3 x3 xn+2 − x2 xn+3 x1 xn+i + x2 xn+3 xi xn+1 ≡
x2 xn+3 x3 xn+2 − x2 xn+i x3 xn+1 + xn+3 xi x1 xn+2 ≡
x2 xn+3 x3 xn+2 − x1 xn+2 xn+i x3 + xi xn+2 x3 xn+1 mod L.
Furthermore
x23 x2n+2 ≡ x2 xn+3 x3 xn+2 − x3 xn+2 xi xn+1 + x3 xn+2 x1 xn+i mod L.
Thus x22 x2n+3 + x23 x2n+2 ≡ 2x2 xn+3 x3 xn+2 mod L, so (x2 xn+3 − x3 xn+2 )2 ∈ L. Let
H be the subgraph of G consisting of the cycle C and the edge {1, i}. Then JG is
generated up to radical by the following set of m − 1 binomials:
{fkl |{k, l} ∈ E(G) \ E(H)} ∪ {f12 , f13 , f1i + f23 }.
Therefore ara(JG ) ≤ m − 1.



Let G1 = (V (G1 ), E(G1 )), G2 = (V (G2 ), E(G
L2 )) be graphs such that G1 ∩ G2
is a complete graph. The new graph G = G1 G2 with the vertex set V (G) =
V (G1 ) ∪ V (G2 ) and edge set E(G) = E(G1 ) ∪ E(G2 ) is called the clique sum of G1
and G2 in G1 ∩ G2 . If the cardinality of V (G1 ) ∩ V (G2 ) is k + 1, then this L
operation
is called a k-clique sum of the graphs G1 and G2 . We write G = G1 vb G2 to
indicate that G is the clique sum of G1 and G2 and that V (G1 ) ∩ V (G2 ) = vb.

Theorem 3.9. Let G be a connected graph on the vertex set [n]. Suppose that G
has exactly one cycle C. If n ≥ 4 and C is odd of length 3, then ara(JG ) = n − 1.

Proof. The graph G can be written as the 0-clique sum of the cycle C and some
trees. More precisely,
M M
M
G=C
T1
···
Ts
v1

v2

vs

for some vertices v1 , . . . , vs of C. The vertex connectivity of G is 1. By Theorem 3.5, the inequality n − 1 ≤ ara(JG ) holds. Since G has exactly one cycle, we
have that card(E(G)) = n. From Proposition 3.8, ara(JG ) ≤ n − 1, and therefore
ara(JG ) = n − 1.

Let ht(JG ) be the height of JG , then, we have, from the generalized Krull’s
principal ideal theorem, that ht(JG ) ≤ ara(JG ). We say that JG is a set-theoretic
complete intersection if ara(JG ) = ht(JG ).
Corollary 3.10. Let G be a connected graph on the vertex set [n] with n ≥ 4.
Suppose that G has exactly one cycle C and its length is 3. Then the following
properties are equivalent:
(a) JG is unmixed,
(b) JG is Cohen-Macaulay,
(c) JG is a L
set-theoretic
intersection,
L
L complete
(d) G = C v1 T1 v2 · · · vs Ts , where {v1 , . . . , vs } ⊂ V (C), s ≥ 1, vh are
pairwise distinct and Th are paths.
In particular, if one of the above conditions is true, then ara(JG ) = ht(JG ) = n − 1.

ARITHMETICAL RANK OF BINOMIAL IDEALS

9

Proof. The implication (b)⇒(a) is well known. If JG is a set-theoretic complete
intersection, then, from Theorem 3.9, ht(JG ) = n − 1 and dim(R/JG ) = n + 1.
Also depth(R/JG ) = n + 1 by [5, Theorem 1.1], so JG is Cohen-Macaulay, whence
(c)⇒(b). Recall that M(G) = {S ⊂ [n] : PS (G) is a minimal prime of JG }. If
JG is unmixed, then every vertex v of Th , v 6= vh , has degree at most 2. In fact,
{v} ∈ M(G) and, if degG (v) ≥ 3, then by [6, Lemma 3.1], one has ht(P{v} (G)) =
n+card({v})−c({v}) = n+1−degG (v) ≤ n−2 < n−1 = ht(P∅ (G)), a contradiction.
Moreover, vh has degree at most 3 for every h. In fact, {vh } ∈ M(G) and, if
degG (vh ) ≥ 4, then by [6, Lemma 3.1], one has ht(P{vh } (G)) = n + card({vh }) −
c({vh }) = n + 1 − (degG (vh ) − 1) ≤ n − 2 < n − 1 = ht(P∅ (G)), a contradiction.
Thus, (d) follows. Finally, assuming (d), JG is unmixed by [5, Theorem 1.1] and
ht(JG ) = n − 1. By Theorem 3.9, it follows that
ara(JG ) = n − 1 = ht(JG ).



If C1 and C2 are cycles of G having no common vertex, then a bridge between
C1 and C2 is an edge {i, j} of G with i ∈ V (C1 ) and j ∈ V (C2 ).
Proposition 3.11. Let G be a connected graph on the vertex set [n] with m edges.
Suppose that G contains a subgraph H consisting of two vertex disjoint odd cycles
of length 3, namely C1 and C2 , and also two bridges between the cycles C1 and C2 .
Then ara(JG ) ≤ m − 2.
Proof. Let E(C1 ) = {{1, 2}, {2, 3}, {3, 1}} and E(C2 ) = {{4, 5}, {5, 6}, {4, 6}}.
Suppose first that the bridges have no common vertex. Let e1 = {1, 4} and e2 =
2
{3, 6} be the bridges of the two cycles. Then f14
belongs to the ideal generated by
2
belongs to the ideal generated by
the polynomials f12 , f13 , f14 +f23 . Furthermore f36
the polynomials f46 , f56 , f36 + f45 . Thus JG is generated up to radical by the union
of {f12 , f13 , f14 + f23 , f46 , f56 , f36 + f45 } and {fij |{i, j} ∈ E(G) and {i, j} ∈
/ E(H)}.
If the bridges have a common vertex, then without loss of generality we can assume
that e1 = {1, 4} and e2 = {3, 4} are the bridges of the two cycles. Applying similar
arguments as before, we deduce that ara(JG ) ≤ m − 2.

Example 3.12. Suppose that G is a graph with 6 vertices and 8 edges consisting
of two vertex disjoint odd cycles of length 3, namely C1 and C2 , and also two vertex
disjoint bridges between the cycles C1 and C2 . Here the vertex connectivity is l = 2.
Thus ara(JG ) ≥ 6. By Proposition 3.11, ara(JG ) ≤ 6 and therefore ara(JG ) = 6.
Theorem 3.13. Let Gk be a graph containing k odd cycles C1 , . . . , Ck of length 3
such that the cycles Ci and Cj have disjoint vertex sets, for every 1 ≤ i < j ≤ k.
Suppose that there exists exactly one path Pi,i+1 of length ri ≥ 2 connecting a vertex
of Ci with a vertex of Ci+1 , 1 ≤ i ≤ k − 1. If Gk has no more vertices or edges, then
Pk−1
ara(JGk ) = ht(JGk ) = 2k + i=1 ri . In particular, JGk is a set-theoretic complete
intersection.
Pk−1
Proof. The graph Gk has 3k + i=1 (ri − 1) vertices. Here the vertex connectivity
is l = 1, so
k−1
k−1
X
X
(ri − 1) + 1 − 2 ≤ ara(JGk ).
ri = 3k +
2k +
i=1

i=1

Pk−1
We will prove that ara(JGk ) ≤ 2k + i=1 ri by induction on k ≥ 2. Suppose that
k = 2 and let E(C1 ) = {{1, 2}, {2, 3}, {1, 3}}, P1,2 = {{3, 4}, {4, 5}, . . . , {r + 2, r +

10

A. KATSABEKIS

3}} and C2 = {{r + 3, r + 4}, {r + 4, r + 5}, {r + 3, r + 5}}. Then JG2 is generated
up to radical by the union of
{f12 + f34 , xr+2 xn+r+3 − xr+3 xn+r+2 + xr+4 xn+r+5 − xr+5 xn+r+4 }
and
{fij |{i, j} ∈ E(G2 ) \ {{1, 2}, {3, 4}, {r + 2, r + 3}, {r + 4, r + 5}}}.
Pk−1
Thus ara(JG2 ) ≤ 4 + r. Assume that the inequality ara(JGk ) ≤ 2k + i=1 ri
Pk
holds for k and we will prove that ara(JGk+1 ) ≤ 2(k + 1) + i=1 ri . We have that
JGk+1 = JGk + JH where H is the graph consisting of the path Pk,k+1 and the cycle
Ck+1 . By Theorem 3.9, ara(JH ) = rk + 2. Then, from the induction hypothesis,
ara(JGk+1 ) ≤ ara(JGk ) + ara(JH ) ≤ 2k +

k−1
X

ri + rk + 2 = 2(k + 1) +

k
X

ri .

i=1

i=1

Since JGk is unmixed by [5, Theorem 1.1], we have that
ht(JGk ) = card(V (Gk )) − 1 = 2k +

k−1
X

ri . 

i=1

Remark 3.14. All the results presented are independent of the field K.
Acknowledgments
The author is grateful to an anonymous referee for useful suggestions and comments that helped improve an earlier version of the manuscript. This work was supported by the Scientific and Technological Research Council of Turkey (TÜBITAK)
through BIDEB 2221 grant.
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Department of Mathematics, Bilkent University, 06800 Ankara, Turkey
E-mail address: katsampekis@bilkent.edu.tr

