Content uploaded by Hong Wang
Author content
All content in this area was uploaded by Hong Wang on Dec 25, 2023
Content may be subject to copyright.
Algebra
Colloquium
c
2023 AMSS CAS
& SUZHOU UNIV
Algebra Colloquium 30 : 4 (2023) 649–666
DOI: 10.1142/S1005386723000494
Algebraic Properties of Powers of Edge Ideals
of Vertex-Weighted Oriented Cycles
Hong Wang Guangjun Zhu†Li Xu
School of Mathematical Sciences, Soochow University
Suzhou, Jiangsu 215006, China
E-mail: 651634806@qq.com zhuguangjun@suda.edu.cn
Received 18 May 2021
Revised 3 November 2022
Communicated by Zhongming Tang
Abstract. Let tbe any positive integer and I(Cn) the edge ideal of a vertex-weighted
oriented n-cycle graph Cn. We provide explicit formulas for the regularity and depth of
I(Cn)t. In particular, we find that the regularity of I(Cn)tis a linear function; Ass(I(Cn)t),
the set of associated prime ideals of I(Cn)t, equals Ass(I(Cn)); and the depth of I(Cn)t
is a constant for all t. We also give some examples to show that these results are related
to the direction selection of edges and the weight of vertices.
2020 Mathematics Subject Classification: primary 13F20; secondary 05C20, 05C22,
05E40
Keywords: depth, regularity, powers of an ideal, vertex-weighted oriented cycles
1 Introduction
An oriented graph D= (V(D), E(D)) consists of an underlying simple graph G
on which each edge is given an orientation (i.e., a directed graph without multiple
edges and without loops). If (u, v)∈E(D), then we write uv for (u, v) to indicate
that it is an oriented edge where the orientation is from uto v. Now for any
x∈V(D), we set N+
D(x) = {y|xy ∈E(D)},N−
D(x) = {y|yx ∈E(D)}and
ND(x) = N+
D(x)∪N−
D(x). A vertex-weighted (or simply, weighted) oriented graph
is a triplet D= (V(D), E(D), w), wis a weight function w:V(D)→N+, where
N+={1,2, . . .}. Sometimes we denote the vertex set V(D) and edge set E(D) by
Vand E, respectively, for short. For any xi∈V, its weight w(xi) is denoted by wi
or wxi. An oriented path or oriented cycle of length nis an orientation of a path
or cycle of length nin which each vertex dominates its successor in the sequence.
Given a weighted oriented graph D= (V(D), E(D), w) and the polynomial ring
S=k[x1, . . . , xn] on the vertex set V(D) over a field k, the edge ideal of Dis
†Corresponding author.
650 H. Wang, G.J. Zhu, L. Xu
defined to be
I(D)=(xixwj
j|xixj∈E(D)) ⊆S.
If wj= 1 for all j, then I(D) is the edge ideal of an unweighted unoriented graph
which was introduced by Villarreal in [28] and has been studied extensively (see
[1–4, 11, 16, 22, 31, 32]).
The edge ideal of a weighted oriented graph, which arose in the theory of Reed-
Muller codes (see [23]), was first introduced by Gimenez et al. [12]. Such codes
arise as the image of a degree-devaluation map of a given set of projective points
over a finite field. The regularity of the vanishing ideal provides a threshold for the
degree of the map indicating when a Reed-Muller-type code has sufficiently large
minimal distance, but it is hard to calculate. For a set of projective points, the
initial ideal of its vanishing ideal is exactly the edge ideal of a weighted oriented
graph with respect to a proper monomial order. Moreover, the regularity of edge
ideal of a weighted oriented graph is easier to calculate than that of vanishing ideal
and provides some valuable information on Reed-Muller-type codes.
For a homogeneous ideal I⊂S, its regularity and depth are two central in-
variants associated to I. It is well known that reg(It) is asymptotically a linear
function for t0, i.e., there exist constants a,band a positive integer t0such that
for all t≥t0, reg(It) = at +b(see [10, 20, 27]). The coefficient ais well-understood,
but the constants band t0are quite mysterious. In this regard, there has been
an interest in finding the exact form of this linear function and determining the
stabilization index t0for which reg(It) becomes linear (cf. [1]–[4]). It turns out that
even in the case of monomial ideals it is challenging to find the linear function and
t0(cf. [9]). In [6], Brodmann showed that depth(S/It) is a constant for t0, and
this constant is bounded above by n−`(I), where `(I) is the analytic spread of I.
It was shown in [16, Theorem 1.2] that depth(S/It) is a nonincreasing function of t
when all powers of Ihave linear resolutions, and in that paper conditions were given
under which all powers of Ihave linear quotients. In this regard, there has been
an interest in determining the smallest value t0such that depth(S/It) is a constant
for all t≥t0(see [11, 16, 24]). Let I⊂Sbe an ideal. The sets of associated prime
ideals of powers of Iare the sets
Ass(It) = {p⊂S|pis prime and p= (It:u) for some u∈S}for t≥1.
In [6], Brodmann showed that Ass(It) stabilize for large t. That is, there exists
a positive integer n0such that Ass(It) = Ass(In0) for all t≥n0. The minimum
of such n0is called the index of stability of I. Finding the stable set Ass(In0) is
complicated by the fact that a prime ideal pthat is associated to a low power of an
ideal Ineed not be associated to higher powers. If p∈Ass(It) implies p∈Ass(It+1)
for all t≥1, then one says that the sets Ass(It) form an ascending chain. In the
case where Ass(It)⊆Ass(I) for all t≥1, the ideal Iis said to be normally torsion-
free. Although these properties are highly desirable, few classes of ideals are known
to possess them. When Iis the edge ideal of a simple graph G, Mart´ınez-Bernal et
al. [22] proved that the sets Ass(It) form an ascending chain, and Simis et al. [26]
proved that Iis normally torsion-free if and only if Gis a bipartite graph.
Algebraic Properties of Powers of Edge Ideals 651
The study of edge ideals of weighted oriented graphs is much more recent and
consequently there are many fewer results in this direction. In [29, 33, 34] we
provided some formulas for the projective dimension and regularity of edge ideals
of weighted oriented cyclic graphs with a common vertex or edge, rooted forests,
oriented cycles, and three types of m-partite graphs. In [30, 35] we gave some
exact formulas for the regularity and projective dimension of powers of edge ide-
als of weighted oriented gap-free bipartite graphs and rooted forests. The Cohen-
Macaulayness of edge ideals of weighted oriented graphs has been studied in [12,
14, 25]. To the best of our knowledge, there is no result about Ass(It) if Iis the
edge ideal of a weighted oriented graph.
Beyarslan et al. showed in [4] that if Cnis an n-cycle with w(x) = 1 for any
x∈V(Cn), then reg(I(Cn)t)=2t−2 + reg(I(Cn)). In this paper, we are interested
in the regularity, depth and the sets of associated prime ideals of I(Cn)tfor a
weighted oriented n-cycle Cnwith w(x)≥2 for any x∈V(Cn) and with the natural
orientation of all edges pointing in the same direction. If there exist x, y ∈V(Cn)
such that w(x) = 1 and w(y)≥2, by the method of this paper, we also provide
some formulas for an upper bound of reg(I(Cn)t), a lower bound of depth(I(Cn)t)
and obtain Ass(I(Cn)) ⊆Ass(I(Cn)t) for all t≥1, but the statements are very
complicated. Therefore, we always assume that w(x)≥2 for any x∈V(Cn)
throughout the paper. By using the approach of polarization, we derive an exact
formula for reg(I(Cn)t). Moreover, we show that Ass(I(Cn)t) stabilize from the
beginning. The results are as follows:
Theorem 1.1. Let tbe a positive integer and Cn= (V, E , w)a weighted oriented
n-cycle satisfying w(x)≥2for any x∈V. Then the following hold:
(i) reg(I(Cn)t) = (t−1)(w+ 1) + reg(I(Cn)), where w= max{w(x)|x∈V}.
(ii) depth(I(Cn)t) = 1.
(iii) Ass(I(Cn)t) = Ass(I(Cn)); in particular, I(Cn)is normally torsion-free.
Our paper is organized as follows. In Section 2, we recall some definitions and
basic facts used in the paper. In Section 3, we provide a special order on the set of
minimal monomial generators of powers of edge ideals of weighted oriented cycles.
Using this order, we give formulas for the regularity of powers of the edge ideals
of weighted oriented cycles in Section 4. In Section 5, first we prove that for any
t≥1, Ass(I(Cn)t) = Ass(I(Cn)), where Cnis a weighted oriented n-cycle, and then
using this result, we obtain formulas for depth(I(Cn)t). By some examples, we show
that these formulas are related to the direction of edges and the assumption that
w(x)≥2 for any x∈Vcannot be dropped.
For all unexplained terminology and additional information, refer to [19] (for the
theory of digraphs), [5] (for graph theory), and [7, 17] (for the theory of edge ideals
of graphs and monomial ideals). We gratefully acknowledge the use of computer
algebra systems CoCoA [8] and Macaulay 2 [13] for our experiments.
2 Preliminaries
In this section, we gather together some of the needed definitions and basic facts,
which will be used throughout this paper. For more details, we refer the reader to
652 H. Wang, G.J. Zhu, L. Xu
[3, 5, 17, 19, 23, 32, 34].
For any homogeneous ideal Iof the polynomial ring S=k[x1, . . . , xn], there
exists a finite minimal graded free resolution
0→M
j
S(−j)βp,j (I)→M
j
S(−j)βp−1,j (I)→ · · · → M
j
S(−j)β0,j (I)→I→0,
where p≤nand S(−j) is the S-module obtained by shifting the degrees of Sby j.
The number βi,j(I), the (i, j )th graded Betti number of I, is an invariant of Ithat
equals the number of minimal generators of degree jin the ith syzygy module of I.
Of particular interest is the regularity of I, which measures the size of the minimal
graded free resolution of I. The regularity of Iis defined as
reg(I) := max{j−i|βi,j (I)6= 0}.
Calculating or even estimating regularity for a general ideal is a difficult problem.
Thus, we restrict our focus to powers of the edge ideal of a weighted oriented n-cycle
Cn. We can exploit the combinatorial structure of Cnto characterize the regularity
of I(Cn)t.
Let Ibe a monomial ideal, and G(I) denote the unique minimal set of monomial
generators of I. To compute reg(I), we often use the approach of polarization. Here
we recall its definition.
Definition 2.1. Let I⊂Sbe a monomial ideal with G(I) = {u1, . . . , um}, where
ui=Qn
j=1 xaij
jfor i= 1, . . . , m. We define the polarization of I,IP, to be a square-
free monomial ideal in the polynomial ring SP=k[xj1, . . . , xjaj|j= 1, . . . , n] :
IP= (P(u1),...,P(um)),
where P(ui) = Qn
j=1 Qaij
k=1 xjk is a squarefree monomial in the polynomial ring SP
and aj= max{aij |i= 1, . . . , m}for 1 ≤j≤n.
The following result is a very useful property of polarization.
Lemma 2.2. [17, Corollary 1.6.3] Let I⊂Sbe a monomial ideal and IP⊂SPits
polarization. Then
(i) βij (I) = βij (IP)for all iand j;
(ii) reg(I) = reg(IP).
The following lemmas can be used for computing the regularity of an ideal.
Lemma 2.3. [15, Lemma 3.2] Let S1=k[x1, . . . , xm],S2=k[xm+1 , . . . , xn]be two
polynomial rings, and I⊂S1,J⊂S2be two nonzero homogeneous ideals. Then
reg(I+J) = reg(I) + reg(J)−1.
Lemma 2.4. [18, Lemma 3.1] Let 0→A→B→C→0be a short exact sequence
of finitely generated graded S-modules. Then reg(C)≤max{reg(A)−1,reg(B)}.
The equality holds if reg(A)6= reg(B).
Algebraic Properties of Powers of Edge Ideals 653
Let G= (V, E ) be a simple graph. A subset C⊂Vis called a vertex cover of
Gif xy ∈Eimplies x∈Cor y∈C. A vertex cover Cof Gis minimal if no proper
subset of Cis a vertex cover of G.
Definition 2.5. [14, Definition 2.9] Let D= (V , E, w) be a weighted oriented
graph and Gits underlying graph. For a vertex cover Cof G, define
L1(C) = {x∈C|there exists some ysuch that xy ∈Eand y /∈C},
L3(C) = {x∈C|NG(x)⊂C},
L2(C) = C\(L1(C)∪L3(C)).
A vertex cover Cof Gis called a strong vertex cover of Dif Cis a minimal
vertex cover of Gor for each x∈L3(C) there is an edge yx ∈E(D) such that
y∈L2(C)∪L3(C) with w(y)≥2.
Lemma 2.6. [14, Theorem 2.11] Let D= (V , E, w)be a weighted oriented graph.
Let pbe a prime ideal of S. Then p∈Ass(I(D)) if and only if p= (C)for some
strong vertex cover Cof D.
From the above lemma, we have
Proposition 2.7. Let tbe a positive integer and D= (V, E, w)be a weighted
oriented graph. If p∈Ass(I(D)t), then p= (C)for some vertex cover Cof D.
Proof. Since pI(D)t=pI(D), we get Min(I(D)t) = Min(I(D)), where Min(I(D))
is the set of minimal prime ideals of I(D). If p∈Ass(I(D)t), then there exists a
minimal prime ideal p0of I(D)tsuch that p0⊂p. It follows that p0∈Min(I(D)).
Let p0= (C0), p= (C) with C0, C ⊆V. Then C0⊆C. It follows that Cis a vertex
cover of Dsince C0is a strong vertex cover of Dby Lemma 2.6.
3 Ordering the Minimal Generators of Powers of the Edge Ideal of a
Weighted Oriented n-Cycle
In this section, we provide a special order on the unique minimal set of monomial
generators of powers of the edge ideal of a weighted oriented n-cycle. Using this
order, we obtain two main results (Theorems 3.5 and 3.6), which are the basis for
computing the regularity of powers of the edge ideal of a weighted oriented n-cycle
in the next section.
Let Cn= (V, E , w) be a weighted oriented cycle with nvertices (an n-cycle
for short) such that w(x)≥2 for any x∈Vand V={x1, . . . , xn}. Then we
obtain I(Cn)=(xnxw1
1, x1xw2
2, . . . , xn−1xwn
n). Throughout this paper, we write
xj=xiif j≡imod nand Li=xi−1xwi
ifor i= 1, . . . , n. We stipulate an order
L1>· · · > Lnon the set G(I(Cn)). For any integer t≥1, we define an order
on the set G(I(Cn)t) as follows: Let M, N ∈ G(I(Cn)t), and we write M, N as
M=La1
1· · · Lan
n,N=Lb1
1· · · Lbn
nsuch that Pn
i=1 ai=Pn
i=1 bi=t. We say
M > N if and only if (a1, . . . , an)>lex (b1, . . . , bn). We denote by L(t)the totally
ordered set of G(I(Cn)t) ordered in the way above and by L(t)
kthe kth element of
the set L(t). It follows from Theorem 3.1 that this order is well-defined.
654 H. Wang, G.J. Zhu, L. Xu
We have the following fundamental fact.
Theorem 3.1. Let tbe a positive integer and M∈ G(I(Cn)t). Then Mcan
be shown as M=Lai1
i1· · · Lai`
i`with P`
p=1 aip=t,aip>0for 1≤p≤`and
1≤i1<· · · < i`≤n. Moreover, the expression of this form is unique.
Proof. Since M∈ G(I(Cn)t), we may write Mas M=Lk1Lk2· · · Lkt, where
Lks ∈ G(I(Cn)) for 1 ≤s≤t. Therefore, by reordering Lk1, Lk2, . . . , Lkt and
merging the same elements, we obtain M=Lai1
i1· · · Lai`
i`with P`
p=1 aip=t,aip>0
for 1 ≤p≤`and 1 ≤i1<· · · < i`≤n.
Let M=Lbj1
j1· · · Lbjm
jmbe another expression of Mwith Pm
q=1 biq=t, where
bjq>0 for any 1 ≤q≤mand 1 ≤j1<· · · < jm≤n. We will show that `=m,
ip=jpand aip=bjpfor 1 ≤p≤`.
We use induction on t. The case t= 1 is clear. Now we assume that t≥2. We
claim: {i1, . . . , i`}∩{j1, . . . , jm} 6=∅. Thus, there exist 1 ≤p≤`and 1 ≤q≤m
such that ip=jq. It follows that
Lai1
i1· · · Laip−1
ip−1Laip−1
ipLaip+1
ip+1 · · · Lai`
i`=Lbj1
j1· · · Lbjq−1
jq−1Lbjq−1
jqLbiq+1
jq+1 · · · Lbjm
jm.
Therefore, by the induction hypothesis, we obtain `=m,is=jswith s= 1, . . . , `.
Since ip=jq, we have ip=jp=jq=iq. This implies p=q. Hence, by using the
induction hypothesis again and the above formula, we get ais=bjs,s= 1, .. . , `, as
desired.
In fact, if {i1, . . . , i`}∩{j1, . . . , jm}=∅, from M=Lai1
i1· · · Lai`
i`=Lbj1
j1· · · Lbjm
jm
it follows that Laip
ip= (xip−1xwip
ip)aipis a factor of the monomial Lbj1
j1· · · Lbjm
jmfor
any 1 ≤p≤`. Hence, there exists some 1 ≤s≤msuch that Ljs=xipxwip+1
ip+1
and Laipwip
jsis a factor of the monomial Lbj1
j1· · · Lbjm
jm. By the arbitrariness of p, we
obtain
Lbj1
j1· · · Lbjm
jm= (xi1xwi1+1
i1+1 )ai1wi1· · · (xi`xwi`+1
i`+1 )ai`wi`M0,
where M0is a monomial. By comparing the degrees of monomials Lai1
i1· · · Lai`
i`and
Lbj1
j1· · · Lbjm
jm, we get
`
X
p=1
aip(1 + wip)≥
`
X
p=1
aipwip(1 + wip+1).
This implies P`
p=1 aip(1 −wipwip+1)≥0, which yields wipwip+1 ≤1, a contradic-
tion.
Definition 3.2. Let k, t be two integers with 1 ≤k < t, and let M1∈ G(I(Cn)k),
M2∈ G(I(Cn)t). We write M1|edgeM2if there is an element M3∈ G(I(Cn)t−k)
such that M2=M1M3. Otherwise, we write M1-edgeM2.
The next two theorems are the most important technical results of this section.
Let u∈Sbe a monomial. We set supp(u) = {xi|xi|u}. If G(I) = {u1, . . . , um},
then we set supp(I) = Sm
i=1 supp(ui).
Algebraic Properties of Powers of Edge Ideals 655
Theorem 3.3. Let tbe a positive integer, `= min{t, bn/2c} − 1, where bn/2c
denotes the largest integer ≤n/2, and let L(t)
i, L(t)
j∈L(t)with L(t)
i> L(t)
j. Then
there exists some L(t)
k∈L(t)with L(t)
k< L(t)
isuch that (L(t)
j:L(t)
i)⊆(L(t)
k:L(t)
i)
and (L(t)
k:L(t)
i)has one of the following two forms:
(i) (L(t)
k:L(t)
i)=(L`2:L`1), where L`1> L`2,L`2|edgeL(t)
kand L`1|edgeL(t)
i;
(ii) (L(t)
k:L(t)
i) = Qq
s=0 Ln−2s:Qq
s=0 Ln+1−2sfor some integer q≤`, where
Ln−2s|edgeL(t)
k,Ln+1−2s|edgeL(t)
ifor any 0≤s≤q.
Proof. Use induction on t. The case t=1 holds because we may choose k=`2=jand
`1=i. Now suppose that t≥2. Set L(t)
j=Lj1· · · Ljtwith 1 ≤j1≤ · · · ≤ jt≤n.
We consider the following two cases:
Case I. If there exists some 1 ≤a≤tsuch that Lja|edgeL(t)
i, then
(L(t)
j:L(t)
i)=(L(t−1)
j0:L(t−1)
i0),
where L(t−1)
j0=L(t)
j/Ljaand L(t−1)
i0=L(t)
i/Lja. By induction hypothesis, there
exists some k0such that L(t−1)
k0∈L(t−1) with L(t−1)
k0< L(t−1)
i0and (L(t−1)
k0:L(t−1)
i0)
has one of the following two forms:
(a) (L(t−1)
k0:L(t−1)
i0)=(L`2:L`1) with L`1> L`2,L`2|edgeL(t−1)
k0,L`1|edgeL(t−1)
i0.
(b) (L(t−1)
k0:L(t−1)
i0) = Qq0
s=0 Ln−2s:Qq0
s=0 Ln+1−2sfor some q0≤`, where
Ln−2s|edgeL(t−1)
k0,Ln+1−2s|edgeL(t−1)
i0for any 0 ≤s≤q0.
We choose L(t)
k=LjaL(t−1)
k0. Then L(t)
i>L(t)
kand (L(t)
k:L(t)
i)=(L(t−1)
k0:L(t−1)
i0),
as desired.
Case II. If Lja-edgeL(t)
ifor any 1 ≤a≤t, then j1≥2 because L(t)
i> L(t)
j. We
consider the following two situations:
(a) If (L(t)
j:L(t)
i)⊆(xwjr
jr) for some 1 ≤r≤t, then set L(t)
i=Li1· · · Litwith
1≤i1≤ · · · ≤ it≤n. Choose L(t)
k=L(t)
i
Lib
Ljr, where ib= max{c|c<jr, Lc|edgeL(t)
i}.
Thus, L(t)
k< L(t)
iand (L(t)
k:L(t)
i) = (Ljr:Lib).
Next we will prove that (L(t)
j:L(t)
i)⊆(Ljr:Lib). This implies the containment
(L(t)
j:L(t)
i)⊆(L(t)
k:L(t)
i). Notice a fact that
(L(t)
j:L(t)
i)= L(t)
j
gcd(L(t)
j, L(t)
i),(Ljr:Lib)= (xwjr
jr) if Ljr=Lib+1,
(xjr−1xwjr
jr) if Ljr< Lib+1.
If Ljr=Lib+1, then the result is true. Otherwise, it is enough to show that xjr−1
is not a factor of gcd(L(t)
j, L(t)
i), i.e., we only need to show that xjr−1is not a factor
of L(t)
i. In fact, if xjr−1is a factor of L(t)
i, then Ljr−1|edgeL(t)
iby the expression
of L(t)
jand the hypothesis that Lja-edgeL(t)
ifor any 1 ≤a≤t. It follows by the
definition of bthat Ljr−1=Lib, contradicting the hypothesis Lib+1 > Ljr.
(b) If (L(t)
j:L(t)
i)*(xwjr
jr) for any 1 ≤r≤t, then xjris a factor of L(t)
ibecause
(L(t)
j:L(t)
i)=(L(t)
j/gcd(L(t)
j, L(t)
i)). This implies Ljr+1|edgeL(t)
ifor 1 ≤r≤tby
656 H. Wang, G.J. Zhu, L. Xu
the hypothesis that Lja-edgeL(t)
ifor any 1 ≤a≤t. Thus, L(t)
ihas the form
L(t)
i=Lj1+1 · · · Ljt+1.
This implies Ljt+1 =L1by the expression of L(t)
jand L(t)
i> L(t)
j. It follows that
Ljt=Ln, i.e., jt=n.
Choose q= max{c|Ljt−s=Ln−2sfor any 0 ≤s≤c}. Thus, Ln−2s|edgeL(t)
j
and Ln+1−2s|edgeL(t)
ifor any 0 ≤s≤q. We consider the following two possibilities:
If q=t−1, then `=t−1. In this case, one has L(t)
j=Qt−1
s=0 Ln−2sand
L(t)
i=Qt−1
s=0 Ln−2s+1. Choose k=j, as desired.
If q≤t−2, then jt−q−1≤n−2qand jt−q−16=n−2q−2 by the definition of q. We
claim that jt−q−16=n−2q, n−2q−1. In fact, if jt−q−1=n−2q, then L2
n−2q|edgeL(t)
j
in this case. This implies (L(t)
j:L(t)
i)⊆(xwn−2q
n−2q) because wn−2q≥2, contradicting
the hypothesis (L(t)
j:L(t)
i)*(xwjr
jr) for any 1 ≤r≤t. If jt−q−1=n−2q−1,
then we obtain jt−q−1+ 1 = n−2q, which implies Ln−2q|edgeL(t)
i, contradicting the
hypothesis Lja-edgeL(t)
ifor any 1 ≤a≤t. Thus, it follows that jt−q−1< n −2q−2.
Since we have Ln−2s|edgeL(t)
jand Ln+1−2s|edgeL(t)
ifor any 0 ≤s≤q, we may
write L(t)
jand L(t)
ias L(t)
j=Q1Qq
s=0 Ln−2sand L(t)
i=Q2Qq
s=0 Ln−2s+1, where
Q1=Lj1· · · Ljt−q−1and Q2=Lj1+1 · · · Ljt−q−1+1. Choose L(t)
k=Q2Qq
s=0 Ln−2s.
Then L(t)
k< L(t)
i, (L(t)
k:L(t)
i) = Qq
s=0 Ln−2s:Qq
s=0 Ln+1−2sand Ln−2s|edgeL(t)
k,
Ln+1−2s|edgeL(t)
ifor any 0 ≤s≤q.
Now we prove (L(t)
j:L(t)
i)⊆(L(t)
k:L(t)
i). Since jt−q−1< n −2q−2, we have
xn−2q−1/∈supp(Q2). This implies supp Qq
s=0 Ln−2s∩supp(Q2)= ∅. Accordingly,
(L(t)
j:L(t)
i) = L(t)
jgcd(L(t)
j, L(t)
i)
=Q1
q
Y
s=0
Ln−2s.gcdQ1
q
Y
s=0
Ln−2s, Q2
q
Y
s=0
Ln−2s+1
⊆(L(t)
k:L(t)
i).
The proof is complete.
Theorem 3.4. Let tbe a positive integer and L(t)={L(t)
1, . . . , L(t)
r}a totally
ordered set of all elements of G(I(Cn)t)such that L(t)
1>· · · >L(t)
r. For any 1≤i≤r,
we write L(t)
ias L(t)
i=Lai1
i1· · · Laiki
ikiwith 1≤i1<· · · < iki≤n,Pki
j=1 aij=t
and aij>0for j= 1, . . . , ki. For 1≤i≤r−1, let Ji= (L(t)
i+1, . . . , L(t)
r)and
Ki= ((Li1+1, . . . , Ln) : Li1) + Ppi
j=1(Lij+1 :Lij), where pi=ki−1if iki=n,
kiotherwise.
(i) If i1= 1, then (Ji:L(t)
i) = Ki+Qiand
Qi=
qi
X
j=0
j
Y
s=0
Ln−2s:
j
Y
s=0
Ln+1−2s,
Algebraic Properties of Powers of Edge Ideals 657
where qi= max{bi|Ln+1−2s|edgeL(t)
ifor any 0≤s≤bi}.
(ii) If i1≥2, then (Ji:L(t)
i) = Ki.
Proof. We only prove (i); (ii) can be shown by arguments similar to those of (i).
The case t= 1 is obvious. Now assume t≥2. By the hypothesis and choice
of piand qi, we can set Mj=L(t)
i
L1Ljfor any 2 ≤j≤n,Nj=L(t)
i
Lij
Lij+1 for any
1≤j≤piand Tj=L(t)
i
Qj
s=0 Ln+1−2sQj
s=0 Ln−2sfor any 0 ≤j≤qi. It is clear that
(M2, . . . , Mn, N1, . . . , Npi, T0, . . . , Tqi)∈Ji.It follows that
Ki+Qi= ((L2, . . . , Ln) : L1) +
pi
X
j=1
(Lij+1 :Lij) +
qi
X
j=0
j
Y
s=0
Ln−2s:
j
Y
s=0
Ln+1−2s
= ((M2, . . . , Mn) : L(t)
i) + ((N1, . . . , Npi) : L(t)
i) +
qi
X
j=0
(Tj:L(t)
i)
= ((M2, . . . , Mn, N1, . . . , Npi, T0, . . . , Tqi) : L(t)
i)⊆(Ji:L(t)
i).
Note that (Ji:L(t)
i) = Pr
j=i+1(L(t)
j:L(t)
i). In order to prove Ki+Qi= (Ji:L(t)
i),
we only need to prove (L(t)
j:L(t)
i)⊆Ki+Qifor any i+ 1 ≤j≤r. By Theorem
3.3, (L(t)
j:L(t)
i) falls into one of the following two cases:
(a) (L(t)
j:L(t)
i)⊆(L`2:L`1) for some `1, `2with L`1> L`2,L`2|edgeL(t)
jand
L`1|edgeL(t)
i. In this case we have `1≤ipi. If supp(L`1)∩supp(L`2)6=∅, then
`1=`2−1 or `1= 1, `2=n. Therefore, (L(t)
j:L(t)
i)⊆Kiby
(L`2:L`1) = (L`2:L`1) if `1=`2−1,
(Ln:L1) if `1= 1, `2=n.
Otherwise, (L`2:L`1)=(L`2). Hence, (L(t)
j:L(t)
i)⊆Ki.
(b) (L(t)
j:L(t)
i)⊆Qq
s=0 Ln−2s:Qq
s=0 Ln+1−2sfor some integer q≤`with
Ln−2s|edgeL(t)
j,Ln+1−2s|edgeL(t)
ifor any 0 ≤s≤qand `= min{t, bn/2c} − 1. By
the choice of qi, we get qi≥q, which implies (L(t)
j:L(t)
i)⊆Qi.
4 Regularity of Powers of Edge Ideals of Weighted Oriented n-Cycles
In this section, we provide some formulas for the regularity of powers of the edge
ideal of a weighted oriented n-cycle. We need the following lemma.
Lemma 4.1. [21, Proposition 4.1] Let I⊂Sbe a squarefree monomial ideal
satisfying that every element of G(I)contains at least one variable not dividing any
other element of G(I). Then reg(I) = |X| − |G(I)|+ 1, where X= supp(I).
Remark 4.2. By polarizing a monomial ideal I, another expression of the above
lemma can be obtained as follows: Let I⊂Sbe a monomial ideal, and define
658 H. Wang, G.J. Zhu, L. Xu
ai= max{p|xp
i|ufor some u∈ G(I)}for any 1 ≤i≤n. If for each 1 ≤i≤nthere
exists a unique element u∈ G(I) such that xai
i|u, then reg(I) = Pn
i=1 ai−|G(I)|+ 1.
For convenience, all notations used in the next two propositions and in Theorem
4.5 are as those in Theorem 3.4.
Proposition 4.3. Let L(t),L(t)
i,Ji,Kiand Qibe as in Theorem 3.4. Then for
any 1≤i≤r−1, the following hold:
(i) reg((Ji:L(t)
i)) = Pn
j=2 wj−n+ 1 if i1= 1 and qi= 0;
(ii) reg((Ji:L(t)
i)) = Pn
j=i1+1 wj−(n−i1)+1if i1≥2.
Proof. We only prove (i); (ii) can be shown by similar arguments to those in (i).
Since qi= 0, we obtain Ln−1-edgeL(t)
iand Qi= (Ln:L1). Moreover, ipi< n−1
by the choice of pi. Since i1= 1, we have
Ki= ((L2, . . . , Ln) : L1) +
pi
X
j=1
(Lij+1 :Lij)
= (xw2
2, x2xw3
3, . . . , xn−1xwn−1
n) +
pi
X
j=1
(xwij+1
ij+1 ),
(Ji:L(t)
i) = Ki+Qi=Ki.
Since 1 ≤ipi< n −1 and wj≥2 for 2 ≤j≤n−1, each xwij+1
ij+1 is a factor of some
element in {xw2
2, x2xw3
3, . . . , xn−1xwn−1
n}, which implies |G(Ki)|=n−1. By Remark
4.2, reg((Ji:L(t)
i)) = reg(Ki) = Pn
j=2 wj−1− |G(Ki)|+ 1 = Pn
j=2 wj−n+ 1.
Proposition 4.4. Let L(t),L(t)
i,Ji,Kiand Qibe as in Theorem 3.4. Then for
any 1≤i≤r−1, we have reg((Ji:L(t)
i)) ≤Pn
s=2 ws−n+ 1 if i1= 1 and qi≥1.
Proof. Since i1= 1 and qi≥1, we have Ls|edgeL(t)
ifor s= 1 and n−1. It follows
that ipi=n−1. Thus,
Ki= (xw2
2, x2xw3
3, . . . , xn−1xwn−1
n) +
pi
X
s=1
(xwis+1
is+1 ),(1)
Qi=
qi
X
j=0
j
Y
s=0
Ln−2s:
j
Y
s=0
Ln+1−2s= (u0, u1, . . . , uqi),(2)
where
uj=
j
Q
s=0
Ln−2s
gcd
j
Q
s=0
Ln−2s,
j
Q
s=0
Ln+1−2s
=xn−2j−1
j
Y
s=0
xwn−2s−1
n−2s,0≤j≤qi.
Let Tj=Ki+(u0, u1, . . . , uj) for any 0 ≤j≤qi. Then (Ji:L(t)
i) = Ki+Qi=Tqi
Algebraic Properties of Powers of Edge Ideals 659
by Theorem 3.4(i). For 0 ≤j≤qi, we will prove by induction on jthat
reg(Tj)≤
n
X
s=2
ws−n+ 1,(3)
and thus the result follows.
In fact, if j= 0, then we observe that T0=Ki+ (u0) = Ki=K0
i+ (xwn
n),
where the second equality holds because (u0)=(xn−1xwn−1
n)⊆Ki, and we have
K0
i= (xw2
2, x2xw3
3, . . . , xn−1xwn−1
n) + Ppi−1
s=1 (xwis+1
is+1 ) because xwipi+1
ipi+1 =xwn
n. By
similar arguments to those in the proof of Proposition 4.3 and Remark 4.2, we get
reg(K0
i) = n
X
s=2
ws−1− |G(K0
i)|+ 1
=n
X
s=2
ws−1−(n−1) + 1 =
n
X
s=2
ws−n+ 1.
(4)
Notice that (K0
i:xwn
n) = Pi+ (xn−1), where Pi= (xw2
2, x2xw3
3, . . . , xn−3xwn−2
n−2) +
Pp0
i
s=1(xwis+1
is+1 ) with p0
i=pi−2 if ipi−1=n−2 and p0
i=pi−1 otherwise. Thus, we
get |G(Pi)|=n−3 by similar arguments to the proof of Proposition 4.3. It follows
from Lemma 2.3 and Remark 4.2 that
reg((K0
i:xwn
n)(−wn)) = reg(Pi+ (xn−1)) + wn
= reg(Pi) + wn
=
n−2
X
s=2
ws− |G(Pi)|+1+wn
=
n−2
X
s=2
ws−(n−3) + 1 + wn≤
n
X
s=2
ws−n+ 2,
(5)
where the inequality holds because wn−1≥2.
Using the formulas (4) and (5), and Lemma 2.4 on the short exact sequence
0−→ S
(K0
i:xwn
n)(−wn)·xwn
n
−−−→ S
K0
i
−→ S
T0
−→ 0,
we have reg(T0)≤Pn
s=2 ws−n+ 1.
Now suppose that the formula (3) is true for any 1 ≤j≤qi−1. We consider
the condition that j=qi. First we compute (Tqi−1:uqi).
Note that uj=xn−2j−1Qj
s=0 xwn−2s−1
n−2sfor 0 ≤j≤qi. Thus,
(u0, u1, . . . , uqi−1) : uqi= (xn−2qi+1, xn−2qi+3 , . . . , xn−1).
It is also easy to see that {n−2qi+ 2, n −2qi+ 4, . . . , n} ⊆ {is+ 1 |1≤s≤pi}
by Theorem 3.4, which implies (xn−2qi+2, xn−2qi+4, . . . , xn)⊆Ppi
s=1(xwis+1
is+1 ) : uqi.
Moreover, (xn−2qi−1xwn−2qi
n−2qi) : uqi= (xn−2qi). Thus, Pn
k=n−2qi(xk)⊆(Tqi−1:uqi).
660 H. Wang, G.J. Zhu, L. Xu
Hence, from the formulas (1) and (2) it follows that
(Tqi−1:uqi)
= (Ki+ (u0, u1, . . . , uqi−1)) : uqi
= ((xw2
2, x2xw3
3, . . . , xn−1xwn−1
n) : uqi) +
pi
X
s=1
(xwis+1
is+1 )+(u0, u1, . . . , uqi−1):uqi
=(xw2
2, x2xw3
3, . . . , xn−2qi−2xwn−2qi−1
n−2qi−1) : uqi
+
p00
i
X
s=1
(xwis+1
is+1 ) : uqi+
n
X
k=n−2qi
(xk)
=
n
P
k=2
(xk) if n= 2m, qi=m−1,
(xw2−1
2) +
n
P
k=3
(xk) if n= 2m+ 1, qi=m−1,
B+
p00
i
P
k=1
(xwik+1
ik+1 ) +
n
P
k=n−2qi
(xk) otherwise,
where B= (xw2
2, xn−2qi−2xwn−2qi−1−1
n−2qi−1) + Pn−2qi−3
k=2 (xkxwk+1
k+1 ),
p00
i= max{a|1≤a≤pi, ia≤n−2qi−2},
and, by Pn
k=n−2qi(xk)⊆(Tqi−1:uqi), the third equality holds.
Next we compute reg((Tqi−1:uqi)). Let dbe the degree of monomial uqi. Then
d=Pm
s=1 w2s−mif n= 2m,qi=m−1, and d=Pqi
s=0 wn−2s−qiotherwise. We
distinguish the following three case.
(i) If n= 2mand qi=m−1, then by Lemma 2.3 we have
reg((Tqi−1:uqi)(−d))
= reg((Tqi−1:uqi)) + d
= reg n
X
k=2
(xk)+d= 1 + m
X
s=1
w2s−m
=n
X
s=2
ws−2m+ 1+m−
m
X
s=2
w2s−1
=n
X
s=2
ws−n+ 1+m−
m
X
s=2
w2s−1≤
n
X
s=2
ws−n+ 1.
(ii) If n= 2m+ 1 and qi=m−1, then by Lemma 2.3 we have
reg((Tqi−1:uqi)(−d))
= reg((Tqi−1:uqi)) + d= reg (xw2−1
2) +
n
X
k=3
(xk)+d
= (w2−1) + m−1
X
s=0
wn−2s−(m−1)
Algebraic Properties of Powers of Edge Ideals 661
=n
X
s=2
ws−(2m+ 1) + 1+m−
m−1
X
s=1
wn−2s+1
=n
X
s=2
ws−n+ 1+m−
m−1
X
s=1
wn−2s+1≤
n
X
s=2
ws−n+ 1.
(iii) In the other situations, using Lemma 2.3, we have
reg((Tqi−1:uqi)(−d))
= reg((Tqi−1:uqi)) + d
= reg B+
p00
i
X
k=1
(xwik+1
ik+1 ) +
n
X
k=n−2qi
(xk)+d= reg B+
p00
i
X
k=1
(xwik+1
ik+1 )+d
≤
n−2qi−1
X
k=2
wk−(n−2qi−1) + 1+
qi
X
s=0
wn−2s−qi
=n
X
s=2
ws−n+ 1+qi+ 1 −
qi
X
s=1
wn−2s+1≤
n
X
s=2
ws−n+ 1,
where the first inequality holds, by similar arguments to the calculation of reg(T0),
because regB+Pp00
i
k=1(xwik+1
ik+1 )≤Pn−2qi−1
k=2 wk−(n−2qi−1) + 1.
Using the above formulas of reg((Tqi−1:uqi)(−d)), Lemma 2.4 and the induction
hypothesis on the short exact sequence
0−→ S
(Tqi−1:uqi)(−d)·uqi
−−−→ S
Tqi−1
−→ S
Tqi
−→ 0,
we have reg((Ji:L(t)
i)) = reg(Tqi)≤Pn
s=2 ws−n+ 1. The proof is completed.
The following theorem is the main result in this section.
Theorem 4.5. Let tbe a positive integer and Cn= (V, E , w)a weighted oriented
n-cycle with w(x)≥2for any x∈V. Then
reg(I(Cn)t) = X
x∈V
w(x)− |E|+ 1 + (t−1)(w+ 1),
where w= max{w(x)|x∈V}.
Proof. The case t= 1 follows from [34, Theorem 4.1]. Now assume that t≥2.
Assume w=w1without loss of generality and let L(t)={L(t)
1, . . . , L(t)
r}be a totally
ordered set of all elements of G(I(Cn)t) such that L(t)
1>· · · > L(t)
r. For 1 ≤i≤r,
we write L(t)
ias L(t)
i=Lai1
i1· · · Laiki
ikiwith 1 ≤i1<· · · < iki≤n,Pki
j=1 aij=t
and aij>0 for j= 1, . . . , ki. Let dibe the degree of monomial L(t)
i. Then we get
d1= (w1+1)+(t−1)(w+ 1) and di≤(wi1+1)+(t−1)(w+ 1) for 2 ≤i≤r−1
by the choice of w.
662 H. Wang, G.J. Zhu, L. Xu
Set Ji= (L(t)
i+1, . . . , L(t)
r) for 1 ≤i≤r−1. Consider the exact sequence
0−→ S
(J1:L(t)
1)(−d1)·L(t)
1
−−−−→ S
J1
−→ S
I(Cn)t−→ 0.(6)
By Proposition 4.3, we obtain
reg((J1:L(t)
1)(−d1)) = reg((J1:L(t)
1)) + d1
=
n
X
j=1
wj−n+ 1 + (t−1)(w+ 1) + 1
=X
x∈V
w(x)− |E|+ 1 + (t−1)(w+ 1) + 1.
Thus, by Lemma 2.4 and the exact sequence (6), it is enough to prove
reg(J1)≤
n
X
j=1
wj−n+ 1 + (t−1)(w+ 1).
Since Jr−1= (L(t)
r)=(xt
n−1xtwn
n), we have
reg(Jr−1) = t(wn+ 1) = n
X
j=1
wj−n+ 1 + (t−1)(wn+ 1)+n−
n−1
X
j=1
wj
≤
n
X
j=1
wj−n+ 1 + (t−1)(w+ 1),(7)
where the inequality holds because of the definitions of wand wj≥2 for 1 ≤j≤
n−1.
For 2 ≤i≤r−1, by Propositions 4.3 and 4.4, we obtain
reg((Ji:L(t)
i))
≤
n
P
j=2
wj−n+ 1 if i1= 1,
n
P
j=i1+1
wj−(n−i1) + 1 if i1≥2
≤
n
P
j=1
wj−n+ 1 −w1if i1= 1,
n
P
j=1
wj−n+ 1 −wi1if i1≥2,
where the last inequality holds because wj≥2 for each j. It follows that
reg((Ji:L(t)
i)(−di)) = reg((Ji:L(t)
i)) + di
≤n
X
j=1
wj−n+ 1 −wi1+ ((wi1+ 1) + (t−1)(w+ 1))
=
n
X
j=1
wj−n+ 1 + (t−1)(w+ 1) + 1.(8)
Algebraic Properties of Powers of Edge Ideals 663
Using the formulas (7) and (8), Lemma 2.4 and the exact sequences
0−→ S
(J2:L(t)
2)(−d2)·L(t)
2
−−−−→ S
J2
−→ S
J1
−→ 0,
0−→ S
(J3:L(t)
3)(−d3)·L(t)
3
−−−−→ S
J3
−→ S
J2
−→ 0,
.
.
..
.
..
.
.
0−→ S
(Jr−1:L(t)
r−1)(−dr)·L(t)
r−1
−−−−→ S
Jr−1
−→ S
Jr−2
−→ 0,
we obtain reg(J1)≤Pn
j=1 wj−n+ 1 + (t−1)(w+ 1).
As a consequence of Theorem 4.5, we have
Corollary 4.6. Let tbe a positive integer and Cn= (V, E , w)be a weighted
oriented n-cycle as in Theorem 4.5. Then reg(I(Cn)t) = reg(I(Cn))+ (t−1)(w+ 1),
where w= max{w(x)|x∈V}.
5 Depth of Powers of Edge Ideals of Weighted Oriented n-Cycles
In this section we will prove Ass(I(Cn)t) = Ass(I(Cn)) for any t≥1. By using this
property, we derive an explicit formula for the depth of I(Cn)t. We also give some
examples to show that the formulas of Theorems 4.5, 5.1 and 5.2 are related to the
direction of edges and the weight of vertices.
Theorem 5.1. Let tbe a positive integer and Cn= (V, E , w)a weighted oriented
n-cycle with w(x)≥2for x∈V. Then
Ass(I(Cn)t) = Ass(I(Cn)).
In particular, I(Cn)is normally torsion-free.
Proof. The case t= 1 is trivial. Now assume that t≥2. Let p∈Ass(I(Cn)t) and
p= (xi1, . . . , xir). Choose
u=Y
xp∈V\C
xwp
p Y
xp∈W
xwp−1
p,
where C={xi1, . . . , xir}and W={xp∈C|xp+1 ∈C}. Then p= (I(Cn) : u),
which means p∈Ass(I(Cn)). In fact, for any p∈ {1, . . . , r}, we consider the
following two cases: (i) If xip∈W, then xip−1|uand xwip−1
ip
u. It follows that
xip−1xwip
ip
xipu, i.e., the edge Lip|xipu. (ii) If xip/∈W, then xip+1 /∈C. Thus,
xwip+1
ip+1
u, which imples xipxwip+1
ip+1
xipu, i.e., the edge Lip+1|xipu. Therefore, we see
that p⊆(I(Cn) : u).
Conversely, notice that (I(Cn) : u)=(xp−1xwp
p/gcd(xp−1xwp
p, u)|p= 1, . . . , n).
Let Mp=xp−1xwp
p/gcd(xp−1xwp
p, u) for any p∈ {1, . . . , n}. Then xp−1|Mpor
664 H. Wang, G.J. Zhu, L. Xu
xp|Mp. We consider the following two cases: (i) If xp−1|Mp, then xp−1-u. This
implies xp−1∈C\W. (ii) If xp−1-Mp, then xp|Mpand xp−1|u. This implies
xp−1∈(V\C)∪W. If xp−1∈W, then xp∈C. Otherwise, we have xp∈Cby
Proposition 2.7. Therefore, (I(Cn) : u)⊆p.
On the other hand, let p∈Ass(I(Cn)) and p= (xj1, . . . , xjm). We set C0=
{xj1, . . . , xjm},W0={xp∈C0|xp+1 ∈C0}and u0=Qxp∈V\C0xwp
pQxp∈W0xwp−1
p.
Similarly to the above arguments, we get p= (I(Cn) : u0).
Note that C0is a strong vertex cover of Cn, and we get x1∈C0or x2∈C0.
Without loss of generality, we set x2∈C0. Then either x2-u0, or xw2−1
2|u0and
xw2
2-u0. Thus, (x2w2
2, xn−1xwn−1
nxw2
2) : u0= (x2)⊆pbecause x2∈C0. Choose
v= (xnxw1
1)t−1u0. Then
(I(Cn)t:v)=(I(Cn)t: (xnxw1
1)t−1u0) = ((I(Cn)t: (xnxw1
1)t−1) : u0)
= ((I(Cn)+(x2w2
2, xn−1xwn−1
nxw2
2)) : u0)
= (I(Cn) : u0) + ((x2w2
2, xn−1xwn−1
nxw2
2) : u0)
=p.
Hence, p∈Ass(I(Cn)t).
Theorem 5.2. Let tbe a positive integer and Cn= (V, E , w)a weighted oriented
n-cycle with w(x)≥2for x∈V. Then
(i) depth(I(Cn)t) = 1;
(ii) pd(I(Cn)t) = |E| − 1.
Proof. (i) Let V={x1, . . . , xn}. Choose C=Vin Definition 2.5, and we observe
that Cis a strong vertex cover of Cn. It follows that (x1, . . . , xn)∈Ass(I(Cn)t)
from Lemma 2.6 and Theorem 5.1. Hence, depth(S/I(Cn)t) = 0. This implies
depth(I(Cn)t) = 1.
(ii) follows from the Auslander-Buchsbaum formula.
The following example shows that the assumption w(x)≥2 for any x∈Vin
Theorems 4.5, 5.1 and 5.2 cannot be dropped.
Example 5.3. Let I(C5)=(x5x1, x1x3
2, x2x3
3, x3x4, x4x3
5) be the edge ideal of a
weighted oriented 5-cycle C5, with its weight function defined by w2=w3=w5= 3,
w1=w4= 1. By using CoCoA, we obtain reg(I(C5)2) = 10, depth(I(C5)2) = 2
and pd(I(C5)2) = 3, which do not satisfy the formulas of Theorems 4.5 and 5.2.
Let p1= (x1, x2, x3, x5) and p2= (x1, x3, x4, x5). By using Macaulay 2, we obtain
p1,p2∈Ass(I(C5)2)\Ass(I(C5)). Hence, Ass(I(C5)2)6= Ass(I(C5)).
The following example shows that the conclusions in Theorems 4.5, 5.1 and 5.2
are related to the direction of edges.
Example 5.4. Let I(C5)=(x1x3
5, x1x3
2, x2x3
3, x3x3
4, x4x3
5) be the edge ideal of a
weighted oriented 5-cycle C5with w1= 1, w2=w3=w4=w5= 3. By using
CoCoA, we obtain reg(I(C5)2) = 14, depth(I(C5)2) = 2 and pd(I(C5)2) = 3, which
do not satisfy the formulas of Theorems 4.5 and 5.2. Let p= (x1, x2, x3, x4, x5).
Algebraic Properties of Powers of Edge Ideals 665
By using Macaulay 2, we observe that p∈Ass(I(C5)2)\Ass(I(C5)). Accordingly,
Ass(I(C5)2)6= Ass(I(C5)).
Acknowledgements. This research is supported by the Natural Science Foundation of
Jiangsu Province (BK20221353) and the foundation of the Priority Academic Program
Development of Jiangsu Higher Education Institutions. Finally, the authors would like
to thank the referees who read the manuscript carefully and gave very helpful comments,
which improved the paper both in mathematics and presentation.
References
[1] A. Alilooee, A. Banerjee, S. Selvaraja, Regularity of powers of edge ideal of unicyclic
graphs, Rocky Mountain J. Math. 49 (3) (2019) 699–728.
[2] A. Banerjee, The regularity of powers of edge ideals, J. Algebraic Combin. 41 (2015)
303–321.
[3] A. Banerjee, S. Beyarslan, H.T. H`a, Regularity of edge ideals and their powers, in:
Advances in Algebra, Springer Proc. Math. Stat., 277, Springer, Cham, 2019, pp. 17–
52.
[4] S. Beyarslan, H.T. H`a, T.N. Trung, Regularity of powers of forests and cycles, J.
Algebraic Comb. 42 (2015) 1077–1095.
[5] J.A. Bondy, U.S.R. Murty, Graph Theory, Springer, New York, 2008.
[6] M. Brodmann, The asymptotic nature of the analytic spread, Math. Proc. Cambridge
Philos. Soc. 86 (1979) 35–39.
[7] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Rev. ed., Cambridge University Press,
Cambridge, 1998.
[8] CoCoATeam, CoCoA: a system for doing computations in commutative algebra (avail-
able at http://cocoa.dima.unige.it).
[9] A. Conca, Regularity jumps for powers of ideals, in: Commutative Algebra, Lect.
Notes Pure Appl. Math., 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 21–
32.
[10] S.D. Cutkosky, J. Herzog, N.V. Trung, Asymptotic behaviour of the Castelnuovo-
Mumford regularity, Compos. Math. 118 (3) (1999) 243–261.
[11] L. Fouli, S. Morey, A lower bound for depths of powers of edge ideals, J. Algebraic
Comb. 42 (2015) 829–848.
[12] P. Gimenez, J.M. Bernal, A. Simis, R.H. Villarreal, C.E. Vivares, Symbolic powers
of monomial ideals and Cohen-Macaulay vertex-weighted digraphs, in: Singularities,
Algebraic Geometry, Commutative Algebra, and Related Topics, Springer, Cham,
2018, pp. 491–510.
[13] D. Grayson, M. Stillman, Macaulay 2: a software system for research in algebraic
geometry (available at www.math.uiuc.edu/Macaulay2/).
[14] H.T. H`a, K.N. Lin, S. Morey, E. Reyes, R.H. Villarreal, Edge ideals of oriented graphs,
Internat. J. Algebra Comput. 29 (3) (2019) 535–559.
[15] H.T. H`a, A. Van Tuyl, Splittable ideals and the resolutions of monomial ideals, J.
Algebra 309 (1) (2007) 405–425.
[16] J. Herzog, T. Hibi, The depth of powers of an ideal, J. Algebra 291 (2) (2005) 534–
550.
[17] J. Herzog, T. Hibi, Monomial Ideals, Springer-Verlag, New York, 2011.
[18] L.T. Hoa, N.D. Tam, On some invariants of a mixed product of ideals, Arch. Math.
94 (4) (2010) 327–337.
666 H. Wang, G.J. Zhu, L. Xu
[19] J.B. Jensen, G. Gutin, Digraphs. Theory, Algorithms and Applications, Springer
Monographs in Mathematics, Springer, London, 2006.
[20] V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer.
Math. Soc. 128 (1999) 407–411.
[21] Kuei-Nuan Lin, J. McCullough, Hypergraphs and regularity of square-free monomial
ideals, Internat. J. Algebra Comput. 23 (7) (2013) 1573–1590.
[22] J. Mart´ınez-Bernal, S. Morey, R. H. Villarreal, Associated primes of powers of edge
ideals, Collect. Math. 63 (2012) 361–374.
[23] J. Mart´ınez-Bernal, Y. Pitones, R.H. Villarreal, Minimum distance functions of grad-
ed ideals and Reed-Muller-type codes, J. Pure Appl. Algebra 221 (2017) 251–275.
[24] S. Morey, Depths of powers of the edge ideal of a tree, Comm. Algebra 38 (2010)
4042–4055.
[25] Y. Pitones, E. Reyes, J. Toledo, Monomial ideals of weighted oriented graphs, Elec-
tron. J. Combin. 26 (3) (2019) Paper No. 3.44, 18 pp.
[26] A. Simis, W.V. Vasconcelos, R.H. Villarreal, On the ideal theory of graphs, J. Algebra
167 (1994) 389–416.
[27] N.V. Trung, H. Wang, On the asymptotic behavior of Castelnuovo-Mumford regular-
ity, J. Pure Appl. Algebra 201 (2005) 42–48.
[28] R.H. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (3) (1990) 277–293.
[29] H. Wang, G.J. Zhu, L. Xu, J.Q. Zhang, Algebraic properties of edge ideals of some
vertex-weighted oriented cyclic graphs, Algebra Colloq. 28 (2) (2021) 253–268.
[30] L. Xu, G.J. Zhu, H. Wang, J.Q. Zhang, Projective dimension and regularity of powers
of edge ideals of vertex-weighted rooted forests, Bull. Malays. Math. Sci. Soc. 44
(4) (2021) 2215–2233.
[31] G.J. Zhu, Projective dimension and regularity of the path ideal of the line graph, J.
Algebra Appl. 17 (4) (2018) 1850068, 15 pp.
[32] G.J. Zhu, Projective dimension and regularity of path ideals of cycles, J. Algebra
Appl. 17 (10) (2018) 1850188, 22 pp.
[33] G.J. Zhu, H. Wang, L. Xu, J.Q. Zhang, Algebraic properties of edge ideals of some
vertex-weighted oriented m-partite graphs, Bull. Braz. Math. Soc. (N.S.) 52 (4)
(2021) 1005–1023.
[34] G.J. Zhu, L. Xu, H. Wang, Z.M. Tang, Projective dimension and regularity of edge
ideals of some weighted oriented graphs, Rocky Mountain J. Math. 49 (4) (2019)
1391–1406.
[35] G.J. Zhu, L. Xu, H. Wang, J.Q. Zhang, Regularity and projective dimension of powers
of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs,
J. Algebra Appl. 19 (12) (2020) 2050233, 23 pp.
