arXiv:1802.01884v2 [] 13 Feb 2018

Asymptotic invariants of ideals with Noetherian
symbolic Rees algebra and applications to cover ideals
Benjamin Drabkin

∗

Lorenzo Guerrieri

†

February 14, 2018

Abstract
Let I be an ideal whose symbolic Rees algebra is Noetherian. For m ≥ 1, the m-th
symbolic defect, sdefect(I, m), of I is defined to be the minimal number of generators
(m)
of the module II m . We prove that sdefect(I, m) is eventually quasi-polynomial as
a function in m. We compute the symbolic defect explicitly for certain monomial
ideals arising from graphs, termed cover ideals. We go on to give a formula for the
Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of
generators of symbolic powers, for ideals whose symbolic Rees algebra is Noetherian.
MSC: 13F20; 05C25.
Keywords: Symbolic powers, Symbolic defect, Waldschimdt constant, Monomial ideals,
Cover ideals of graphs.

1

Introduction

Let I be a homogeneous ideal in a commutative Noetherian graded ring R. Given m ∈ N,
the m-th symbolic power of I is defined to be
\
I (m) =
(I m Rp ∩ R).
p∈Ass(I)
From the definition is clear that I m ⊆ I (m) . The opposite containment, however, does not
hold in general. Much effort has been invested into determining for which values of r the
containment I (r) ⊆ I m holds. An overview of this topic, often called the “containment
problem”, can be found in papers like [13], [7], in the survey paper [20], and in [4].
While the containment problem is the most-explored line of inquiry into the relationship between symbolic and ordinary powers, there are other avenues to investigate this
relationship. One such method is to study the module I (m) /I m . This module is relatively
unexplored: Herzog in [10] studies the module using homological methods when I is a prime
∗

University of Nebraska – Lincoln, Lincoln Nebraska Email address: benjamin.drabkin@huskers.unl.edu
Università di Catania, Dipartimento di Matematica e Informatica, Viale A. Doria, 6, 95125 Catania,
Italy Email address: guerrieri@dmi.unict.it
†

1

ideal of height two in a three-dimensional local Noetherian ring. Arsie and Vatne study the
Hilbert function of I (m) /I m in [1], giving examples for ideals of coordinate planes and sets of
points in Pn . More recently, in [9] Galetto, Geramita, Shin, and Van Tuyl study this module
by focusing on the m-th symbolic defect, defined by
 (m) 
I
sdefect(I, m) := µ
,
Im
that is, the number of minimal generators of I (m) /I m . The symbolic defect measures how
different I (m) is from I m in that it counts the number of generators which must be added to
I m to make I (m) . For instance, the equality I (m) = I m is equivalent to sdefect(I) = 0. In
studying symbolic defect, there are many interesting questions which arise.
1. For which ideals is sdefect(I, m) bounded as a function of m?
2. For which ideals does sdefect(I, m) attain the value t, for given m, t ∈ N?
3. How does sdefect(I, m) grow as m grows?
In [9], the first two questions are analyzed for the defining ideal of a general set of points
in P2 or a set of points forming a star configuration. Interesting results about the symbolic
defect of edge ideals of graphs can also be found in the recent work [15]. In this paper, we
concern ourselves primarily with the asymptotic behavior of the symbolic defect, i.e., we
focus on the question of how the symbolic defect sdefect(I, m) grows as a function of m for
a fixed ideal I.
Another asymptotic invariant important for the study of the symbolic powers of a homogeneous ideal I is the Waldschimdt constant, defined as
α(I (m) )
,
m→∞
m

α̂(I) = lim

where α(I) = min {deg f | f ∈ I, f 6= 0} . This constant was introduced in [4] and is related
to work done by Waldschmidt in [23]. The Waldschimdt constant of squarefree monomial
ideals can be computed as the optimum value of a linear optimization problem as shown in
[3]. The Waldschimdt constant is connected to the containment problem by results of Bocci
and Harbourne [4]. They use it to find a lower bound for the resurgence of an ideal I, which
is defined as
nm
o
ρ(I) = sup
| I (m) 6⊆ I r .
r
In this paper, we study asymptotic invariants pertaining to the symbolic powers of I in
the case where I is an ideal with Noetherian symbolic Rees algebra. A notable class of ideals
having this property are the monomial ideals, as proven by Herzog, Hibi and Trung [12]. In
particular, we focus on characterizing the growth of sdefect(I, m) for any ideal satisfying the
assumption that the symbolic Rees algebra is Noetherian. We then specialize to the case in
which I is the cover ideal of a graph to obtain more refined results. In addition, we give a
formula for the Waldschimdt constant α̂(I) in the former setup.
We now describe our main results and the structure of this paper. In Section 2, we prove
in Theorem 2.4 that, when I is an ideal with Noetherian symbolic Rees algebra, sdefect(I, m)
is eventually quasi-polynomial as a function of m.
2

In Section 3, we prove that the Waldschmidt constant
of an ideal can be computed by
 (m) 
α(I
)
considering only the first few terms of the sequence
as long as I has Notherian
m
m∈N

symbolic Rees algebra. In particular, we prove in Theorem 3.6 that the Waldschimdt constant
is equal to
α(I (m) )
α̂(I) = minm≤n
,
m
where n is the highest degree of a generator of the symbolic Rees algebra of I.
In Section 4, we consider the cover ideal J(G) of a graph G. Cover ideals, which are
generated by monomials corresponding to vertex covers of graphs, are an interesting family of
squarefree monomial ideals. Their structure has been studied in the recent years due to the
relationships between their algebraic and combinatorial properties. After some preliminary
results, we state, in Theorem 4.15, a recursive formula which computes the symbolic defect
of cover ideals for certain graphs. Moreover, in Theorem 4.8 we describe a family of graphs
achieving arbitrarily large symbolic defects.
In Section 5, we compute the degree as a quasi-polynomial of the symbolic defect for
several families of graphs including complete, circulant and cyclic graphs.
Throughout this paper, for all the standard notations we refer to classical commutative
algebra textbooks such as [2] and [8] and for the theory of monomial ideals in polynomial
rings we refer to the book [11]. For an introductory text about cover and edge ideals we
refer to [22].

2

Asymptotic behavior of the symbolic defect

We start by considering the growth of sdefect(I, m) when I is an ideal (or homogeneous
ideal) in a Noetherian local (or graded-local) ring R.
Definition 2.1. A quasi-polynomial in Q is a function f : N → Q such that for some
n ∈ N there exist rational polynomials f0 , . . . fn−1 having the property that for all m ∈ N,
f (m) = fr (m) whenever m ≡ r mod n. The value n is called a quasi-period of f .
Definition 2.2. The Rees algebra of an ideal I is defined to be the graded ring
R(I) =

∞
M

I i ti ⊆ R[t]

i=0

and the symbolic Rees algebra of I is defined to be
Rs (I) =

∞
M

I (i) ti ⊆ R[t]

i=0

where t is an indeterminate tracking the grading in the graded families of the powers and
symbolic powers of I, respectively.
We recall the useful fact that the property of a function being eventually quasi-polynomial
can be read off its generating function.
3

Lemma 2.3. [18, 4.4.1] If a numerical function φ : N → Z has generating function
∞
X
n=0

q(z)
di
i=1 (1 − z )

φ(n)z n = Qs

for some d1 , d2 , . . . , ds ∈ N, and some polynomial q(z) ∈ Z[z], then φ(n) = Q(n) for n sufficiently large, where Q(n) is a quasi-polynomial with quasi-period given by the least common
multiple of d1 , . . . , dn .
One of our main results is that sdefect(I, m) grows quasi-polynomially in m if the symbolic
Rees algebra of I, Rs (I) is a Noetherian ring.
Theorem 2.4. Let R be a Noetherian local or graded-local ring with maximal (or homogeneous maximal) ideal m and residue field R/m = k. Let I be an ideal (or homogeneous ideal)
of R such that Rs (I) is a Noetherian ring, and let d1 , . . . , ds be the degrees of the generators
of Rs (I) as an R-algebra. Then sdefect(I, m) is eventually a quasi-polynomial in m with
quasi-period d = lcm(d1 , . . . , ds ).
Proof. We first note that for each m ∈ N,
0 −→ I m −→ I (m) −→

I (m)
−→ 0
Im

is an exact sequence of R-modules. Taking direct sums, this gives a short exact sequence of
R(I)-modules
0 −→ R(I) −→ Rs (I) −→ C(I) −→ 0
where
C(I) =

∞  (m) 
M
I

m=0

Im

tm .

Tensoring with R(I) = R(I)/mR(I) gives the exact sequence
0 −→ K −→ R(I) −→ Rs (I) −→ C(I) −→ 0
of R(I)-modules where


∞ 
M
I (m)
tm
Rs (I) =
(m)
mI
m=0

and
C(I) =

∞ 
M

m=0

I (m)
(I m + mI (m) )



tm .

In particular, each strand of the above sequence is of the form
0 −→ Km −→

I (m)
I (m)
Im
−→
−→
−→ 0
mI m
mI (m)
(I m + mI (m) )

where Km = (I m ∩ mI (m) )/mI m .
4

Let hC(I) (z) be the Hilbert series for C(I). We note that hC(I) (z) =
since, by Nakayama’s Lemma,
dimk



I (m)
I m + mI (m)



=µ



I (m)
Im



∞
P

sdefect(I, m)z m

m=0

.

Furthermore,
hC(I) (z) = hRs (I) (z) + hK (z) − hR(I) (z)
where hRs (I) (z), hK (z), and hR(I) (z) are the Hilbert series of Rs (I), K, and R(I) respectively.
Since R(I) is a Noetherian k-algebra generated in degree 1 and K is a homogeneous ideal
of this ring, hR(I) and hK are rational functions of z with denominators which are powers of
(1 − z). Since Rs (I) is a Noetherian ring, it follows that hRs (I) is a rational function with
Q
denominator given by si=1 (1 − z di ), where d1 , d2 , . . . , ds are the degrees of the generators of
Rs (I) as a Noetherian k-algebra. Thus hC(I) (z) is a rational function on z, and Lemma 2.3
show both that sdefect(I, m) is quasi-polynomial in m and that the quasi-period d is given
by lcm(d1 , . . . , ds ).
We note that, in particular, Theorem 2.4 holds for monomial ideals. Sections 4 and 5 of
this paper are dedicated to sharpening the results in Theorem 2.4 in the case of monomial
ideals arising from vertex covers of graphs.

3

Computing the Waldschmidt constant

In this section we prove a decomposition for the symbolic powers of an ideal I having
Noetherian symbolic Rees algebra and we give an useful application. In order to do this, we
use information about the relationships between different symbolic powers of I captured in
the symbolic Rees algebra. In particular, when Rs (I) is Noetherian, its maximum generating
degree can be used to understand the structure of I (m) , as shown in the following lemma.
Lemma 3.1. Let R be a positive-integer-graded Noetherian ring, and let I ⊆ R be a homogeneous ideal such that Rs (I) is generated in degree at most n. Then
I (m) = I m + I (2) I (m−2) + · · · + I (n) I (m−n)
for all m > n.
Proof. Since the symbolic powers of I form a graded family of ideals, we have that
I (m) ⊇ I m + I (2) I (m−2) + · · · + I (n) I (m−n) .
As the symbolic Rees algebra is generated in degree at most n, Rs (I) = R[I, I (2) t2 , . . . , I (n) tn ].
Therefore we have that
X
I (m) =
I i1 (I (2) )i2 · · · (I (n) )in .
i1 +2i2 +...+nin =m

Thus
I (m) ⊆ I m + I (2) I (m−2) + · · · + I (n) I (m−n) .

5

Decompositions of symbolic powers of ideals along the lines of Lemma 3.1 offer a method
to computing the Waldschmidt constant of ideals with finitely generated symbolic Rees
algebra in terms of the initial degrees of finitely many symbolic powers.
Definition 3.2. Given an ideal I ⊆ k[x1 , . . . , xn ], we define
α(I) = min{deg f | f ∈ I, f 6= 0}.
Definition 3.3. Given an ideal I, the Waldschmidt constant, α̂(I), is defined by
α(I (m) )
.
m→∞
m

α̂(I) = lim

For many details about Waldschmidt constant of squarefree monomial ideals see the paper
[3]. The resurgence of an ideal is another constant related to the containment problem.
Definition 3.4. Given an ideal I, its the resurgence, ρ(I), is defined by
o
nm
|I (m) 6⊆ I r .
ρ(I) = sup
r
The resurgence ρ(I) can be bounded below using the Waldschmidt constant. In particular
ρ(I) ≥

α(I)
α̂(I)

(see [4] Theorem 1.2).
Lemma 3.5. Let I be a homogeneous ideal such that Rs (I) is generated in degree at most
n. For each i ∈ {1, . . . , n}, let αi := α(I (i) ). Then
α1 m + (α2 − 2α1 )y2 + · · · + (αn − nα1 )yn
m→∞
m

α̂(I) = lim

where y2 , . . . , yn are positive integers minimizing α1 m + (α2 − 2α1 )y2 + · · · + (αn − nα1 )yn
with respect to the constraint 2y2 + 3y3 + · · · + nyn ≤ m.
Proof. Since Rs (I) = R[I, I (2) t2 , . . . , I (n) tn ], we have that
X
I (m) =
(I (2) )y2 (I (3) )y3 · · · (I (n) )yn I m−2y2 −...−nyn .
2y2 +3y3 +···+nyn ≤m

Thus


α(I (m) ) = min α (I (2) )y2 (I (3) )y3 . . . (I (n) )yn I m−2y2 −...−nyn |2y2 + 3y3 + . . . nyn ≤ m ,

and setting αi = α(I (i) )) this gives

α(I (m) ) = min {α1 m + (α2 − 2α1 )y2 + · · · + (αn − nα1 )yn |2y2 + 3y3 + . . . nyn ≤ m} .

6

We proceed to give a formula for the Waldschmidt constant in terms of finitely many
symbolic powers.
Theorem 3.6. Let I be a homogeneous ideal such that Rs (I) is generated in degree at most
n. Then
α(I (m) )
.
α̂(I) = minm≤n
m
Proof. By Lemma 3.5, α(I (m) ) is the minimum value of
α1 m + (α2 − 2α1 )y2 + · · · + (αn − nα1 )yn
subject to the condition 2y2 + 3y3 + · · · + nyn ≤ m, where αi = α(I (i) )). Equivalently,
assuming that n!|m and setting zi := iyi , we see that α(I (m) ) is the minimum value of
α1 m +

(α2 − 2α1 )
(αn − nα1 )
z2 + · · · +
zn
2
n

(3.1)

subject to z2 + z3 + · · · + zn ≤ m, and for each i, zi is a multiple of i. Let c ∈ {2, . . . , n}
1
such that αc −cα
is minimal. Then
c
n

α1 m +

X αc − cα1
(αn − nα1 )
(α2 − 2α1 )
z2 + · · · +
zn ≥ mα1 +
zi .
2
n
c
i=2

(3.2)

Since I i ⊆ I (i) , we have αi − iα1 ≤ 0 for all i. Thus Equation 3.2 is minimized when
z2 + · · · + zn = m. Hence
mα1 +

n
X
αc − cα1
i=2

c

zi ≥ mα1 + m

αc − cα1
.
c

Thus, when n!|m, (3.1) is minimized at zc = m and zi = 0 for i 6= c with a value of
Therefore
α(I (m) )
αc
lim
= .
m→∞
m
c

mαc
.
c

We note that, in particular, Theorem 3.6 holds for monomial ideals.

4

Symbolic defect of cover ideals of graphs

Definition 4.1. Let G be a graph with vertex set {x1 , . . . , xn } and edge set E. The cover
ideal of G is defined to be
\
J(G) :=
(xi , xj ) ⊆ R.
{xi ,xj }∈E

For m ≥ 1, we say that a monomial g ∈ R is an m-cover of G if for every edge {xi , xj } ∈ E,
there exists one monomial of the form hij = xai xbj with a + b ≥ m such that hij divides g.
7

The generators of J(G) are the monomials which correspond to the vertex 1-covers of G.
For any m ∈ N, we see that
\
J(G)(m) =
(xi , xj )m
{xi ,xj }∈E

is generated by the monomials which correspond to vertex m-covers of G and J(G)m is
generated by the monomials which correspond to vertex m-covers which decompose into the
product of m vertex 1-covers. We say that a vertex m-cover is minimal if it is not divisible
by any other different vertex m-cover. Thus, in the context of cover ideals, sdefect(J(G), m)
counts the number of minimal vertex m-covers of G which cannot be decomposed as a
product of m vertex 1-covers. We call such m-covers indecomposable.
Herzog, Hibi and Trung proved the next important result:
Theorem 4.2 ([12], Theorem 5.1). Let G be a graph and let I = J(G) be its cover ideal.
Then, the symbolic Rees algebra Rs (I) is generated in degree at most 2.
As a corollary to this result and Theorem 2.4, we have the following:
Corollary 4.3. Let G be a graph and let I = J(G) be its cover ideal. Then sdefect(I, m) is
eventually quasi-polynomial with quasi-period at most 2.
As another consequence of Theorem 4.2 and of Theorem 3.6, we have the following
information about the Waldschmidt constant of cover ideals.
Corollary 4.4. Let I = J(G) be the cover ideal of a graph G with n vertices. Then
α̂(I) =

α(I (2) )
2

Proof. By Theorem 4.2, Rs (I) is generated in degree 2. Hence, Theorem 3.6 implies that
(2)
(m)
= α(I).
α̂(I) = minm≤2 α(Im ) . Now the first result follows, since α(I2 ) ≤ 2α(I)
2

Definition 4.5. A graph G = (V, E) is bipartite if there is a partition V = V1 ∪ V2 on the
vertex set, such that V1 ∩ V2 = ∅ and for any edge {xi , xj } of G, xi ∈ V1 and xj ∈ V2 .
The following theorem by Dupont and Villareal [5] characterizes the minimal indecomposable vertex covers of bipartite graphs and the minimal indecomposable vertex 0,1 and
2-covers of non-bipartite graphs. In the latter case, since Rs (I) is generated in degree 2,
these vertex covers generate all the vertex m-covers. Given a graph G = (V, E), the set of
neighbors of a vertex xi is the set of vertices xj adjacent to xi , which means {xi , xj } is an
edge of G. Given a set of vertices S ⊆ V in G, the induced subgraph on S is the graph with
vertex set S, and edge set {{x, y} ∈ E|x, y ∈ S}.
Theorem 4.6 ([5], Theorem 1.7). Let f = xa11 . . . xann ∈ k[x1 , . . . , xn ].

8

(i) If G is bipartite, then f is an indecomposable minimal vertex m-cover of G if and only
if m = 0 and f = xi for some 1 ≤ i ≤ n or m = 1 and f = xj1 . . . xjl is such that
f x−1
ji is not a vertex 1-cover of G for every i = j1 , . . . , jl . For m ≥ 2, all the vertex
m-covers of G are divisible by m vertex 1-covers.
(ii) If G is non-bipartite, then the minimal indecomposable vertex 0,1 and 2-covers are of
the form:
(a) (0-covers) m = 0 and f = xi for some i,
(b) (1-covers) m = 1 and f = xj1 . . . xjl is such that f x−1
ji is not a vertex 1-cover of
G for every i = j1 , . . . , jl ,
(c) (2-covers) m = 2 and f = x1 x2 · · · xn is the product of all the variables,
(d) (2-covers) m = 2 and
f = x0i1 · · · x0is x2is+1 · · · x2is+t xis+t+1 · · · xis+t+u
is such that:
(1)
(2)
(3)
(4)
(5)

each xij is distinct,
s + t + u = n,
{xis+1 , . . . , xis+t } is the set of neighbors of {xi1 , . . . , xis } in G,
g = xis+1 · · · xis+t is not a vertex cover of G, and u 6= 0,
the induced subgraph on {xis+t+1 , . . . , xis+t+u } has no isolated vertices and is
not bipartite.

An important consequence for the theory of cover ideals of graph is the following result
which is also a corollary of work by Sullivant in [19]:
Corollary 4.7. Let G be a bipartite graph, and let I = J(G). Then sdefect(I, m) = 0 for
all m ∈ N.
Proof. As G is bipartite, by Theorem 4.6(i), the graph G does not have indecomposable
m-covers for m > 1. Thus sdefect(I, m) = 0.
In contrast to this fact, we show that in general sdefect(J(G), 2) can be arbitrarily large.
Let for n ∈ N, let Tn be the graph on vertices {x1 , x2 , x3 , y1, . . . , yn } such that the induced
subgraph on {x1 , x2 , x3 } is C3 , the induced subgraph on {y1 , . . . , yn } is a path of length n−1,
and x3 is adjacent to y1 . In the case of n = 3, this graph is pictured below:
x1

x2

x3

y1

y2

y3

Theorem 4.8. For all n ≥ 5, sdefect(J(Tn ), 2) = sdefect(J(Tn−1 ), 2) + µ(J(Pn−4 )2 ) where
Pi is the path of length i.
9

Proof. Let C = xa11 xa22 xa33 y1b1 y2b2 · · · ynbn be a minimal indecomposable vertex 2-cover of Tn .
Since every non-bipartite induced subgraph of Tn contains the vertices x1 , x2 , x3 , by Theorem
4.6(ii.5), we have that a1 = a2 = a3 = 1. Thus, b1 is nonzero. If b1 = 1, then b2 > 0. Consider
bn
.
C ′ = xa11 xa22 xa33 y1b2 y2b3 · · · yn−1

We claim that C ′ is an indecomposable vertex 2-cover of Tn−1 . Suppose that bi = 2 for
some i. Then, as C is a minimal indecomposable vertex 2-cover for Tn , we know that either
bi−1 = 0 or bi+1 = 0. Thus, either yi−2 or yi has exponent equal to zero in C ′ , Thus C ′ is a
vertex 2-cover of Tn−1 .
On the other hand, if b1 = 2, then by Theorem 4.6(ii) we know that b2 = 0 and b3 = 2.
The vertices y4 , . . . , yn form the path of length n − 4, that is Pn−4 , and y4b4 · · · ynbn is a vertex
2-cover of this path. Thus, we see that
sdefect(J(Tn ), 2) ≥ sdefect(J(Tn−1 ), 2) + µ(J(Pn−4 )2 ).
a

n−1
To show the other inequality, let G = x1 x2 x3 y1a1 · · · yn−1
be an indecomposable 2-cover
a1
an−1
for Tn−1 . Then C = x1 x2 x3 y1 y2 · · · yn
is a vertex 2-cover for Tn . Moreover, we note that
the set {y | y 2 divides C} is exactly the set of neighbors of {y | y does not divide C}. Thus,
again by Theorem 4.6, we see that C is an indecomposable 2-cover of Tn .
Let D = y4d4 · · · yndn be a minimal 2-cover of Pn−4 . We claim that H = x1 x2 x3 y12y20 y32D is
an indecomposable 2-cover for Tn . Certainly H is a 2-cover of Tn . As {y | y 2 divides C} is
exactly the set of neighbors of {y | y does not divide C} and H is divisible by x1 x2 x3 , we see
that H is indeed indecomposable by Theorem 4.6.

It is interesting to characterize which graphs have cover ideal I with sdefect(I, 2) = 1. A
similar characterization of such graphs using the language of symbolic Rees algebras can be
found in [12] (Proposition 5.3). Our proof is different from the proof presented in [12], so we
include it below.
Recall that a graph is not bipartite if and only if it contains an odd cycle, which means
that there is an odd integer l ≥ 3 and l vertices xi1 , xi2 , . . . , xil such that for 1 ≤ j < l,
{xij , xij+1 } are edges and {xil , xi1 } is an edge.
Theorem 4.9. Let G be a graph with n vertices and let I = J(G). Then sdefect(I, 2) = 1 if
and only if G is non-bipartite and every vertex in G is adjacent to every odd cycle in G.
Proof. First assume sdefect(I, 2) = 1. Hence G is not bipartite by Corollary 4.7 and hence
there is at least an odd cycle contained in G. Assume by way of contradiction that one vertex
xi of G is not adjacent to the odd cycle and let xj1 , . . . , xjc be the vertices of G adjacent
to xi . Clearly F = x1 x2 · · · xn is an indecomposable vertex 2-cover by Theorem 4.6(ii), but
in this case, by the same characterization of vertex 2-covers of graph, we also have that
F x−1
i xj1 · · · xjc is an indecomposable vertex 2-cover and hence sdefect(I, 2) ≥ 2.
Conversely, assume G is non-bipartite and every vertex in G is adjacent to every odd
cycle in G and let f = xa11 · · · xann ∈ I (2) \ I 2 be an indecomposable minimal vertex 2-cover.
The set {xi : ai = 1} is not bipartite again by Theorem 4.6(ii), hence the induced graph on
this set contains an odd cycle. Since any vertex of the graph is adjacent to this odd cycle,
then ai 6= 0 for every i = 1, . . . , n. But this also implies ai 6= 2 for every i since f is a
minimal vertex 2-cover. Hence f = F and sdefect(I, 2) = 1.
10

Remark 4.10. From the proof of Theorem 4.9 it is clear that when sdefect(I, 2) = 1 where
(2)
I = J(G), the unique generator of II 2 is F = x1 x2 . . . xn . Hence, for m ≥ 3,, using the
decomposition of I (m) given in Lemma 3.1 together with the fact that the symbolic Rees
algebra Rs (I) is generated in degree 2 (see Theorem 4.2), we obtain
I (m) = (F )I (m−2) + I m .
A consequence of Remark 4.10 is a lower bound on the resurgence of the cover ideal of
this kind of graphs. Such a bound is interesting when the degree of F is strictly smaller than
2α(I), where α(I) is the minimal degree of an element of I.
Corollary 4.11. Let I = J(G) be the cover ideal of a graph G with n vertices such that
sdefect(I, 2) = 1. Then
 2α(I)
if n2 < α(I)
n
ρ(I) ≥
1
if n2 ≥ α(I).
Proof. By Remark 4.10, since the degree of F = x1 x2 . . . xn is n and F is the unique generator
(2)
of II 2 , we get α(I (2) ) = min{n, 2α(I)}. Hence the result follows from Definition 3.4 and
Corollary 4.4.
For some of the graphs having sdefect(J(G), 2) = 1 we can give an explicit recursive
formula to compute the symbolic defect of any symbolic power of the cover ideal.
Using the same notation as before, for a graph G on n vertices, we set F = x1 · · · xn
and I = J(G) = (g1 , . . . , gt ). Since by Remark 4.10, I (m) = (F )I (m−2) + I m , the possible
elements of I (m) \ I m are of the form F k gi1 · · · gis for some integers k, s such that m = 2k + s.
It is possible to find an exact formula for the symbolic defect for the graphs in which all the
elements of this form are not in the ordinary power I m . For this reason we give the following
definition:
Definition 4.12. Let G be a graph of n vertices and I = J(G) be its cover ideal. Assume
I = (g1 , . . . , gt ) and sdefect(I, 2) = 1 and let F = x1 · · · xn . We say that G satisfies the
Indecomposability Property if for any integers k ≥ 1 and s ≥ 0, the monomial
F k gi1 · · · gis 6∈ I 2k+s .
The following lemma describes some graphs satisfying Indecomposability Property.
Lemma 4.13. Let I = J(G) ⊆ R = k[x1 . . . xn ] be the cover ideal of a graph G. Assume
sdefect(I, 2) = 1 and let F = x1 · · · xn . Then G satisfies the Indecomposability Property if at
least one of the following conditions is satisfied:
1. I = (g1 , . . . , gt ), deg F < 2α(I) and deg gi = α(I) for every i = 1, . . . , c.
2. I = (g1 , . . . , gt , h1 , . . . , hs ) is generated in two different degrees deg gi = α1 < α2 =
deg hl for all i, l and deg F < α1 + α2 . Moreover there exists j ∈ {1, . . . , n} such that
the variable xj divides gi for every i but it does not divide hl for every l.

11

3. I = (g1 , . . . , gt , h1 , . . . , hs ) is generated in two different degrees deg gi = α1 < α2 =
deg hl for all i, l. Moreover deg F < α1 + α2 , there are c indices j1 , . . . , jc , such that
the variables xj1 , . . . , xjc divide hl for every l and do not divide gi for every i and
α2 − α1 ≤ c.
Proof. 1) The thesis follows since deg F k gi1 · · · gis < (2k + s)α(I).
2) Take an element of the form
q = F k gi1 · · · gis1 hi1 · · · his2 .
1
It follows that xk+s
divides q but there is not bigger power of xj dividing q. By way of
j
contradiction suppose q ∈ I 2k+s . Hence q is divisible by at most k + s1 generators of the
form gi and by at least k + s2 generators of the form hl . Hence there are two integers a, b
such that a ≤ k + s1 , b ≥ k + s2 and a + b = 2k + s and q is divisible by a generators of the
form gi and by b generators of the form hl . Then,

deg q = k deg F + s1 α1 + s2 α2 < (k + s1 )α1 + (k + s2 )α2 ,
but there exists an integer c ≥ 0 such that (k + s1 )α1 + (k + s2 )α2 = (a + c)α1 + (b − c)α2
and hence, since α1 < α2 ,
deg q < (k + s1 )α1 + (k + s2 )α2 ≤ aα1 + bα2 ≤ deg q
and this is a contradiction for what assumed on a and b.
3) As in the proof of (2), take
q = F k gi1 · · · gis1 hi1 · · · his2 .
Let X := xk1 · · · xkc . Thus X k+s2 divides q and no bigger power of X divides it. Now,
assuming by way of contradiction q ∈ I 2k+s , we get that q is divisible by at least k + s1
generators of the form gi and by at most k + s2 generators of the form hl . As before there
are two integers a, b such that a ≤ k + s1 , b ≥ k + s2 , a + b = 2k + s and m is divisible by a
generators of the form gi and by b generators of the form hl .
Hence there exists an integer d ≥ 0 such that k + s1 = a − d and k + s2 = b + d and we
can write deg q = aα1 + bα2 + w with w ≥ dc since we need to have X k+s2 dividing q. It
follows that
deg q = k deg F + s1 α1 + s2 α2 < (k + s1 )α1 + (k + s2 )α2 = aα1 + bα2 + d(α2 − α1 ) ≤ deg q
since α2 − α1 ≤ c and this is a contradiction.
We can give examples of graphs satisfying each one of the conditions of Lemma 4.13.
Recall that a graph is complete is for every two distinct vertices xi , xj , the pair {xi , xj } is
an edge. We denote the complete graph with n vertices by Kn . A graph is a cycle with n
vertices x1 , x2 , . . . , xn if its edges are of the form {xi , xi+1 } modulo n. (i.e also {xn , x1 } is an
edge). We denote the cycle with n vertices by Cn . Every cyclic graph with an even number
of vertices is bipartite.
In the next section, complete graphs and the cycles C3 , C5 , C7 will be shown to satisfy
condition 1 of Lemma 4.13. The following graph satisfies condition 2 of Lemma 4.13.
12

a
e

b
x
c

f
d

Similarly, any graph consisting of 3-cycles all joined at a single vertex satisfies Lemma
4.13. The following graph satisfies condition 3 of Lemma 4.13:
x2

x3

x1

x4

We now concern ourselves with the computation of the symbolic defect for graphs satisfying the Indecomposability Property.
Lemma 4.14. Let I = J(G) ⊆ R = k[x1 , . . . , xn ] be a cover ideal of a graph, G, on n
vertices and let f ∈ k[x1 , . . . , xn ] be such that
Y
xi ∈ I (m)
f
i6=a,b

for all a, b ∈ {1, . . . , n}. Then f ∈ I (m) .
a2
an
(m)
Proof. Suppose f = xa11 x
exists an edge {xs , xt } such that
Q2 · · · xn 6∈ I .m Then there Q
as + at < m. But then f i6=s,t xi 6∈ (xs , xt ) , and thus f i6=s,t xi 6∈ I (m) .

Theorem 4.15. Let I = J(G) ⊆ R = k[x1 . . . xn ] where G is a graph such that sdefect(I, 2) =
1 and let F = x1 · · · xn . Assume that the graph G satisfies the Indecomposability Property.
Let ν(I, m) be the number of minimal generators of I m which are not divisible by F .
Then for m ≥ 3,
sdefect(I, m) = sdefect(I, m − 2) + ν(I, m − 2).
Proof. By Remark 4.10 and by assuming sdefect(I, 2) = 1, we have I (m) = (F )I (m−2) + I m .
Then the minimal elements of I (m) \ I m are of the form F k gi1 · · · gis where gi are minimal
generators of I and m = 2k+s. Conversely, since the graph G satisfies the Indecomposability
Property, any such monomial is in I (m) \ I m . Hence, for g such that g is a minimal generator
(m−2)
of II m−2 , we have that F g is in I (m) \ I m .
The module I (m) /I m is generated by the images of the minimal generators of (F )I (m−2) ,
which are either generators of (F )(I (m−2) /I m−2 ) or images of generators of (F )I m−2 . It
13

follows that sdefect(I, m) is equal to sdefect(I, m − 2) plus the number of minimal generator
of (F )I m−2 which are not already multiples of F g for some g ∈ I (m−2) \ I m−2 .
By dividing F on both sides, we have that the last number coincides with the number of
minimal generator of I m−2 which are not multiple of any element of I (m−2) \ I m−2 and we
will show that this number is ν(I, m − 2). Let f be a minimal generator of I m−2 . If f 6∈ (F ),
then f is not a multiple of an element of I (m−2) \ I m−2 since we showed, in Remark 4.10,
that I (m−2) \ I m−2 ⊆ (F ).
On the other hand, assume f ∈ (F ). Then f = F h for some h ∈ R. Since F h ∈ I m−2 , we
know F h ∈ (xs , xt )m−2 for all xs , xt where {xs , xt } is an edge of G. Thus xsFxt h ∈ (xs , xt )m−4
and so h ∈ I (m−4) by Lemma 4.14.
Then f ∈ F I (m−4) , and so by Remark 4.10, we get
m

f ∈ (F )I (m−4) = (F )I m−4 + (F 2 )I m−6 + . . . + (F )⌊ 2 ⌋ J ∗
where J ∗ = R for m even and J ∗ = I for m odd. Notice now that the minimal generators
of all the summands of the previous equation are contained in I (m−2) \ I m−2 (since F 6∈ I 2 ).
Therefore f is divisible by some element in I (m−2) \ I m−2 . This proves the formula and the
theorem.
Remark 4.16. The formula for the symbolic defect presented in Theorem 4.15 does not
apply to every graph. For instance consider the graph G = (V, E), pictured below.
y1

x1
x2

x3

y2

y3

For this graph, the cover ideal is
I = (g1 , g2 , g3 , g4) = (x1 x2 x3 , x1 x2 y3 , x1 x3 y2 , x2 x3 y1 )
and we observe that, since F = x1 x2 x3 y1 y2 y3 has degree 6, the ideal I does not satisfy any
of the conditions of Lemma 4.13. Moreover we see that F g1 = g2 g3 g4 ∈ I 3 and therefore
sdefect(I, m) is less than what predicted by the formula of Theorem 4.15 for m ≥ 3.

5

Computing symbolic defect for certain cover ideals

Recall that by Theorem 2.4, the symbolic defect of the cover ideal, I, of a graph grows quasipolynomially in m, since Rs (I) is Noetherian (see Theorem 4.2). After some preliminary
results, we discuss some examples of cover ideals of graphs for which we are able to compute
the symbolic defect or its degree as quasi-polynomial in m. In the simple case of complete
graphs it is possible to find the explicit value of the symbolic defect using Theorem 4.15.
14

Theorem 5.1. Let Kn be the complete graph with n vertices and let I be its cover ideal.
Call F = x1 x2 · · · xn . Then sdefect(I, m) grows as a linear polynomial in m.
Proof. Observe that I = (g1 , . . . , gn ) where gi = F x−1
i . Since I satisfies condition 1 of
Lemma 4.13, it satisfies the Indecomposability Property.
Since F divides gi gj for i 6= j, the minimal generators of I m which are not divisible by F
are g1m, g2m , . . . , gnm.
It follows that ν(I, m) = n for every m ≥ 1 and by Theorem 4.15, sdefect(I, m) = nk + 1
for m = 2k + 2 and sdefect(I, m) = nk for m = 2k + 1. Thus sdefect(I, m) is eventually a
linear polynomial in m.
For the computation in other cases of graphs with sdefect(J(G), 2) = 1, we need some
further result about the degree in m of ν(I, m). For an ideal I ⊆ R = k[x1 . . . xn ], the
number of generators of the power µ(I m ) grows as a polynomial in m for m >> 0. This is
clear since the fiber cone of I is a standard-graded Noetherian k-algebra and thus its Hilbert
function is eventually given by a polynomial (see [16] theorem 13.2). We call the degree of
this polynomial the degree of µ(I m ).
In Proposition 5.2 we relate the degree in m of ν(I, m) is related to the degree of µ(I m ).
In the case in which the generators of I are algebraically independent over the base field k
we recall how to compute µ(I m ) in Proposition 5.3.
Proposition 5.2. Let I = J(G) ⊆ R = k[x1 . . . xn ] where G is a graph such that sdefect(I, 2) =
1 and assume that G satisfies Indecomposability Property.
Then for every i = 1, . . . , n we have
m 

m 

n
X
I + xj R
I + xi R
µ
≤ ν(I, m) ≤
µ
.
(5.1)
xi R
x
R
j
j=1
I+x R

iR
iR m
Moreover, take i such that µ( I+x
) ≥ µ( xj Rj ) for all j and let d be the degree of µ(( I+x
) )
xi R
xi R
seen as a polynomial in m for m >> 0. Then for m >> 0, sdefect(I, m) is a quasi-polynomial
in m of degree d + 1.

Proof. Recall that from Remark 4.10, it follows that when the sdefect(I, 2) = 1, the unique
(2)
generator of II 2 is F = x1 x2 · · · xn . Hence ν(I, m) is the number of minimal generators of I m
iR m
) ) is the number of minimal generators
not divisible by at least one variable. Since µ(( I+x
xi R
m
of I not divisible by xi , we can see easily the inequalities in (5.1).
P
I+x R
I+x R
iR
iR m
Now if µ( I+x
) ≥ µ( xj Rj ) for all j, then µ(( I+x
) ) and nj=1 µ(( xj Rj )m ) have both
xi R
xi R
degree d as polynomials in m.
The sequence {ν(I, m)}m≥1 is non-decreasing since F is not in the set of minimal generators of I, and it is bounded by a polynomial. Hence it admits a (possibly infinite) limit as
m grows to infinity. Therefore, dividing both sides of (5.1) by md and passing to the limit
for m going to infinity, we can see that ν(I, m) is quasi-polynomial in m of degree d.
]. By Theorem 4.15 we have
For a given m, let k = [ m−2
2
sdefect(I, m) =

k
X
i=0

15

ν(I, 2i + c)

where c = 0 if m is even and c = 1 if m is odd (recall also that ν(I, 0) = 1 and ν(I, 1) = µ(I)).
It follows that sdefect(I, m) is quasi-polynomial in m of degree one more than the degree of
ν(I, m).
Proposition 5.3. Let I = (g1 , . . . , gs ) be a squarefree monomial ideal in a polynomial ring
over a field k. Then:
∂gi
1. The generators of I are algebraically independent over k if the matrix Mij = { ∂x
} has
j
maximal rank.

2. If the generators of I are algebraically independent over k, then


m+s−1
m
µ(I ) =
s−1
is a polynomial in m of degree s − 1.
Proof. We note that 1 follows immediately from Jacobi’s criterion [14].
To prove 2, we first note that, since g1 , . . . , gs are algebraically independent, g1a1 · · · gsas =
g1b1 · · · gsbs if and only if ai = bi for 1 ≤ i ≤ s. Thus the number of minimal generators of I m
is equal to the number of ways to distribute m objects between s sets. Therefore


(m + s − 1) · · · (m + 1)
m+s−1
m
=
µ(I ) =
(s − 1)!
s−1
which is a polynomial in m of degree s − 1 .
We can use Propositions 5.2 and 5.3 in order to explicitly compute the degree in m of
sdefect(I, m) when I is the cover ideal of a cycle in Theorem 5.6.
Let Cn be the n-cycle and let I be its cover ideal. Since an even cycle is a bipartite
graph, it is enough to consider n = 2k + 1 to be an odd number. For every i = 1, . . . , n the
monomials
gi = xi xi+2 xi+4 · · · xi+n−1 mod n
are 1-vertex covers of Cn and their degree is minimal among the degrees of the minimal
generators of I. Indeed, any other generator of I, if exists, has degree greater than α(I). We
prove the following lemmas:
Lemma 5.4. Let n = 2k + 1 and let Cn be the n-cycle. Let I be the cover ideal of Cn and
call F = x1 x2 · · · xn . Assume G is a minimal generators of I such that deg G > α(I). Then:
1. There exists a minimal generator H of I and a positive even integer s < n such that
deg H = deg G − 1 and
−1
−1
G = Hxj x−1
j+1 xj+2 xj+3 · · · xj+s−1 xj+s .

2. F G ∈ I 3 . In particular F G is equal to the product of three minimal generators of I.

16

Proof. 1) All the generators of I of degree α(I) are of the form gi = xi xi+2 xi+4 · · · xi+n−1 mod n,
as mentioned in the paragraph above. Hence, if we assume deg G > α(I), we have that there
are two different monomials xi xi+1 and xj xj+1 dividing G, with i + 1 < j. We can also
assume without loss of generality that, for every i + 1 < h < j, xh divides G if and only if
xh−1 and xh+1 do not divide G. Clearly, since G is a minimal 1-cover of Cn , xi+2 and xj−1
do not divide G. Thus, we have a sequence of indices i + 1, i + 2, i + 3, . . . , j − 2, j − 1, j such
that the variables xi+1 , xi+3 , . . . , xj−3 , xj divide G and all the others in the sequence do not
divide G. It follows that j − i is an odd number, otherwise we would have two consecutive
variables xh xh+1 not dividing G and this is impossible since {h, h + 1} is an edge of Cn .
Hence the monomial
−1
−1
H = Gx−1
i+1 xi+2 xi+3 · · · xj−1 xj
is a well defined 1-cover of Cn and deg H = deg G − 1. This proves item 1 taking s = j − i.
2) We claim that
−1
−1
F G = F Hxj x−1
j+1 xj+2 xj+3 · · · xj+s−1 xj+s = Hgj gj+s+1

and hence F G ∈ I 3 . Indeed, this is equivalent to prove
−1
−1
gj gj+s+1 = F xj x−1
j+1 xj+2 xj+3 · · · xj+s−1 xj+s .

But this follows since, in the case j is odd, j + s + 1 is even and by definition
gj = xj xj+2 xj+4 · · · xn x2 x4 · · · xj−3 xj−1
and
gj+s+1 = xj+s+1 xj+s+3 · · · xn−1 x1 x3 · · · xj+s−2xj+s
and hence in the products, all the variables with odd index between j and j + s appear twice
and all the variables with even index in the same interval do not appear. If j is even, the
situation is reversed and the result follows in the same way.
Lemma 5.5. Let n = 2k + 1 and let Cn be the n-cycle. Let I be the cover ideal of Cn and
call F = x1 x2 · · · xn . For m ≥ 2, take q = F h f1 · · · fs where fi are minimal generators of I
and m = 2h + s.
Then, q ∈ I (m) \ I m if and only if deg fi = α(I) for every i. Moreover, all the monomials
generating I (m) /I m are of this form.
Proof. A cyclic graph of odd order has sdefect(I, 2) = 1 by Theorem 4.9. Hence, by Remark
4.10, we have I (m) = (F )I (m−2) + I m and the possible minimal generators of I (m) /I m are all
of the form q = F h f1 · · · fs described in the statement above. It is clear that q ∈ I (m) . By
item 2 of Lemma 5.4, if some fi has degree greater than α(I) , we note that F fi is a product
of three minimal generators of I, and thus we can rewrite q as a product of F h−1 and s + 2
minimal generators of I. Iterating this process as far as we can, we find two possibilities for
q:
i) Either q = F a f1 · · · fb with 2a + b = m and deg fi = α(I) for every i = 1, . . . , b, or
ii) q = F f1 · · · fb with b + 2 = m and there exists at least one fi such that deg fi > α(I).
17

In the first case, since deg F = n < n + 1 = 2α(I), we have
deg q = a deg F + bα(I) < (2a + b)α(I) = α(I m ).
and therefore q 6∈ I m .
In the second case, assuming deg f1 > α(I), we have again by item 2 of Lemma 5.4,
q = h1 h2 h3 f2 . . . fb with hi ∈ I, and hence q ∈ I b+2 = I m . This proves the lemma.
Theorem 5.6. Let n = 2k + 1 and let Cn be the n-cycle. Let I be the cover ideal of Cn and
call F = x1 x2 · · · xn . Define I ⋆ = (g1 , . . . , gn ) where gi = xi xi+2 xi+4 · · · xi+n−1 mod n and let
ν(I ⋆ , m) be the number of minimal generators of (I ⋆ )m which are not divisible by F . Then:
1. For m ≥ 3, sdefect(I, m) = sdefect(I, m − 2) + ν(I ⋆ , m − 2).
2. The degree of sdefect(I, m) as a quasipolynomial in m is k.
Proof. 1) In the case n ≤ 7, it is possible to observe that all the minimal generators of I have
the same degree, equal to k + 1. Hence the graph satisfies the Indecomposability Property,
because it satisfies condition 1 of Lemma 4.13. In this case, by definition I = I ⋆ and thus
the result follows applying Theorem 4.15.
Now suppose that n ≥ 9. We proceed along the method use to prove Theorem 4.15. Let
m ≥ 2 and let q ∈ I (m) \ I m a minimal generator of I (m) /I m . By Lemma 5.5, q = F h f1 · · · fs
where fi are minimal generators and of I, m = 2h + s and deg fi = α(I) for every i. Hence
for such a monomial q ∈ I (m−2) \ I m−2 , we have F q ∈ I (m) \ I m , again by Lemma 5.5.
Thus we have that, as in the proof of Theorem 4.15, the symbolic defect sdefect(I, m) is
equal to sdefect(I, m − 2) plus the number of minimal generators of (F )(I ⋆ )m−2 which are
not multiples of F g for some g ∈ I (m−2) \ I m−2 .
Again as in the proof of Theorem 4.15, we have that the last number coincides with
the number of minimal generators of (I ⋆ )m−2 which are not multiples of any element of
I (m−2) \ I m−2 and we can prove that this number is ν(I ⋆ , m − 2) and this proves the formula
in this case, using the same argument.
2) Applying the proof of Proposition 5.2 to the ideal I ⋆ , we see that this fact is equivalent
⋆
iR m
to showing that µ(( I x+x
) ) is a quasipolynomial of degree k − 1 for some variable xi which
iR
⋆
iR m
maximizes such degree. The symmetry of the cyclic graphs tell us that µ(( I x+x
) ) =
iR
I ⋆ +x1 R
I ⋆ +x1 R m
µ(( x1 R ) ) for every i. Hence we just consider the ideal x1 R = (f3 , f5 , . . . , fn ).
Notice that µ(I ⋆ /x1 ) = k and we want to conclude the proof applying item 2 of Proposition 5.3. In order to do this, by item (i) of Proposition 5.3, we need to show that the matrix
∂gi
} has maximal rank.
M = Mij = { ∂x
j
The matrix Mij has k rows and n − 3 columns. Consider the maximal square submatrix
H obtained taking the columns 3, 5, . . . , n − 2, n − 1 of M and let 3 ≤ j ≤ n − 2 an odd
number. Then, by definition of gi for i 6= 1 and odd, we have
∂gi
= 0 ⇐⇒ xj does not divide gi ⇐⇒ j < i.
∂xj
n
i
= 0 for i 6= 1 odd and ∂x∂gn−1
6= 0. Hence the matrix H is upper
It is also easy to see that ∂x∂gn−1
triangular with non zero elements on the diagonal and hence the rank of M is maximal.

18

A natural generalization of complete graphs and cyclic graphs is given by the class of
circulant graphs. Results about ideals related to circulant graphs are given for instance in
[17], [21], [6]. Here we state a result about their cover ideals.
Definition 5.7. Let n be a positive integer and let 1 ≤ s ≤ ⌊n/2⌋. Denote by Zn the cyclic
group with n elements let S = {1, 2, . . . s} ⊆ Zn .
The circulant graph Cn (1, 2, . . . , s) is defined as the graph with vertex set {x1 , . . . , xn }
and with the edge set such that, for i > j, {xi , xj } is an edge if and only if i − j ∈ S. Note
that the cycle Cn is equal to Cn (1) and the complete graph Kn is Cn (1, 2, . . . , ⌊n/2⌋).
The graph in the next picture is the circulant graph C6 (1, 2).
x1
x5

x2

x6

x3
x4

Now characterize which circulant graphs have cover ideals with sdefect(I, 2) = 1 and compute
sdefect(I, m) in such cases. In next proposition we will consider circulant graphs which are
neither cyclic nor complete since we have already dealt with those cases.
Theorem 5.8. Let n ≥ 6 and 2 ≤ s < k := ⌊ n2 ⌋ two integers. Let G = Cn (1, 2, . . . , s) and
let I = J(G). Then:
1. sdefect(I, 2) = 1 if and only if s = k − 1.
2. If s = k − 1, then for m >> 0, sdefect(I, m) grows as a quasi-polynomial in m of
degree 1 if m is even and of degree 2 if m is odd.
Proof. 1) We will use the characterization of graphs such that sdefect(J(G), 2) = 1 given
in Theorem 4.9. Since s ≥ 2, every three consecutive vertices of G form an odd cycle. By
definition of circulant graph for every i = 1, . . . n, {xi , xi+j } and {xi , xi−j } are edges of G for
all positive j ≤ s (notice that i + j and i − j are considered modulo n).
For simplicity, using the symmetry of G we can assume without loss of generality that
i = 1. Then, the vertices xs+2 , xs+3 , xs+4 form an odd cycle and hence, if sdefect(I, 2) = 1,
it follows that x1 has to be adjacent to such odd cycle. This means that xs+4 has to be
adjacent to x1 and thus is of the form xn+1−j for 1 ≤ j ≤ s. It follows that s + 4 ≥ n − s + 1
which implies n ≤ 2s + 3.
Conversely, assume n ≤ 2s + 3. Hence, for every vertex v of G there are at most two
vertices not adjacent to v and this implies that v is adjacent to every odd cycle of G.
It is easy to see that if n ≤ 2s + 3, then n−3
≤ s < k and hence s = k − 1.
2
19

2) In this case, we first show that the graph Cn (1, 2, . . . , k − 1) satisfies the Indecomposability Property and hence sdefect(I, m) is given by the formula of Theorem 4.15. As usual
let F = x1 x2 · · · xn .
−1
−1 −1
The generators of I are of the form F x−1
i xk+i if n is even and of the form F xi xk+i and
−1
F x−1
i xk+i+1 if n is odd, with i ∈ {1, 2, . . . , n} and where the sums k + i and k + i + 1 are
considered modulo n. Hence every generators of I has degree n − 2 and the graph satisfies
condition 1 of Lemma 4.13 and therefore the Indecomposability Property.
Like we did in Example 5.1 for cyclic graph we use the symmetry of the graph and
Proposition 5.2 to see that the degree as a quasipolynomial in m of sdefect(I, m) is one more
1R m
then the degree as a quasipolynomial in m of µ(( I+x
) ).
x1 R
When n is even, since there is only one vertex vk+1 not adjacent to v1 , we have that
−1
I+x1 R
I+x1 R m
is principal generated by F x−1
1 xk+1 and then µ(( x1 R ) ) is constant. If instead n
x1 R
1R
is
is odd there are exactly two vertices vk+1 and vk+2 not adjacent to v1 and hence I+x
x1 R
−1 −1
−1 −1
generated by F x1 xk+1 and F x1 xk+2 which are monomials algebraically independents over
1R m
K and hence the degree in m of µ(( I+x
) ) is 1 by item 2 of Proposition 5.3.
x1 R
We conclude this section by showing how the symbolic defect of a cover ideal can behaves
under a graph operation.
Let G = (V = {x1 , . . . , xn }, E) be a graph. Consider the graph
′

G = (V ∪ {y}, E ∪

n
[

{xi , y})

i=1

obtained by adding one vertex y to G and connecting all the vertices of G to the new
vertex y. We can compute the symbolic defect of G′ assuming that both G and G′ satisfy
the Indecomposability Property. For instance, it happens if the cover ideal I of G fulfills
condition 1 or 2 of Lemma 4.13, thus it is generated in one degree, and therefore the cover
ideal I ′ of G′ satisfies condition 2 of Lemma 4.13.
Theorem 5.9. Let G = (V = {x1 , . . . , xn }, E) be a graph and let
G′ = (V ∪ {y}, E ∪

n
[

{xi , y}).

i=1

Assume both G and G′ satisfy the Indecomposability Property.
Then, for m ≥ 3, sdefect(I ′ , m) = sdefect(I, m) + ⌊ m−1
⌋.
2
Proof. Clearly by Theorem 4.9, if sdefect(I, 2) = 1, then also sdefect(I ′ , 2) = 1.
Following the formula given in Theorem 4.15, it suffices to compute ν(I ′ , m) in terms of
ν(I, m). First we observe that I ′ = yI + (F ) where F = x1 · · · xn . Hence
(I ′ )m = y m I m + F y m−1 I m−1 + · · · + F m−1 yI + (F m ).
All the terms except the first and the last are divisible by F y and therefore by definition they
are not counted in ν(I ′ , m). Moreover, F m is not divisible by F y and a generator y m g of y m I m
is divisible by F y if and only if g is divisible by F . It follows that ν(I ′ , m) = ν(I, m) + 1.
20

Acknowledgements
Most of this work was carried out at the PRAGMATIC workshop at the Universitá di Catania
in the summer of 2017. We would like to thank the workshop organizers, Alfio Ragusa, Elena
Guardo, Francesco Russo, and Giuseppe Zappalá, as well as the lecturers and collaborators,
Brian Harbourne, Adam Van Tuyl, Enrico Carlini, and Tài Huy Hà. Special thanks go
to Adam Van Tuyl, Alexandra Seceleanu, Tài Huy Hà, Jonathan Montaño, and Giancarlo
Rinaldo for helpful conversations and guidance.

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