
Citation: Zahid Iqbal, Muhammad Ishaq. Depth and Stanley depth of edge ideals associated to some line graphs[J]. AIMS Mathematics, 2019, 4(3): 686-698. doi: 10.3934/math.2019.3.686
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Let S:=K[x1,…,xn] be a polynomial ring over a field K, and A be a finitely generated Zn-graded S-module. Let a∈A be a homogeneous element and X⊂{x1,x2,…,xn}. We denote by aK[X] the K-subspace of A generated by all elements ab where b is a monomial in K[X]. The Zn-graded K-subspace aK[X] of A is called a Stanley space of dimension |X|, if aK[X] is a free K[X]−module. A Stanley decomposition of A is a presentation of K-vector space A as a finite direct sum of Stanley spaces
D:A=⨁ri=1aiK[Xi]. |
The number sdepth(D)=min{|Xi|:i=1,…,r} is called the Stanley depth of D. Let sdepth(A)=max{sdepth(D):DisaStanleydecompositionofA}, then sdepth(A) is called the Stanley depth of A. Stanley conjectured in [1] that sdepth(A)≥depth(A) for any Zn-graded S-module A. Let I⊂J⊂S are monomial ideals and A=J/I, Ichim et al. reduced this conjecture to the case when J and I are the squarefree monomial ideals; see [2]. The above conjecture was disproved by Duval et al. by providing a counterexample; see [3]. Let m be the unique graded maximal ideal of S. For an S-module A, the depth of A is an important algebraic invariant which is defined to be the maximal length of a regular sequence on A in m; see [4] for definition and results regarding depth. Herzog, Vladoiu and Zheng gave an algorithm for computing Stanley depth of modules of the type J/I by using some posets related to J/I; see [5]. However, it is too hard to compute Stanley depth by using their method, see for instance, [6,7,8,9,10]. Recently, Ichim et al. gave another algorithm in [11] for computing Stanley depth of any finitely generated Zn-graded S-module. But it is still hard to compute the Stanley depth even by using this new algorithm. Therefore, it's worth giving values and bounds for Stanley depth of some classes of modules. For some known results on Stanley depth, we refer the readers to [12,13,14,15,16].
The paper is organized as follows. In Section 2, we give definitions, notation, and discussion of some necessary results. In the third section, we find bounds for depth and Stanley depth of cyclic modules associated to line graphs of the ladder and circular ladder graphs; see Theorem 3.2 and 3.6. We also compute some bounds for Krull dimension of these cyclic modules; see Proposition 3.8 and 3.9.
Let G be a graph having vertex set V(G)={a1,a2,…,an} and edge set E(G), then the edge ideal I(G)=(xixj:{ai,aj}∈E(G)) associated with G is a squarefree monomial ideal of S. If G is a graph on vertices {a1,a2,…,an}, then G is said to be a path if E(G)={{ai,ai+1}:i∈[n−1]} and G is called a cycle if E(G)={{ai,ai+1}:i∈[n−1]}⋃{{a1,an}}. We use the notations Pn and Cn for path and cycle on n vertices respectively. A vertex cover of a graph G is a subset B of V(G) such that for every edge e∈E(G), e∩B≠∅ and B is minimal with respect to this property, that is for any proper subset B′ of B, then there exists an edge e∈E(G) with e∩B′=∅. A prime ideal Q is a minimal prime of an ideal I if I⊂Q and if Q′ is a prime ideal with I⊂Q′⊂Q, then Q′=Q. It is easy to verify that B is a minimal vertex cover of G if and only if the prime ideal Q generated by the variables corresponding to vertices of B is a minimal prime of I(G). Let α(G):={min|B|:BisaminimalvertexcoverofG}, then α(G)=height(I). For vertices a and b of a graph G, the length of a shortest path from a to b is called the distance between a and b and it denoted by dG(a,b). If no such path exists between a and b, then dG(a,b)=∞. The diameter of a connected graph G is diam(G):=max{dG(a,b):a,b∈V(G)}. For a detailed discussion on squarefree monomial ideals, see [17,18] and for definitions from graph theory, see [19,20].
Definition 2.1. [19] For a given graph G, the line graph L(G) of G is a graph whose vertex set is the edge set of G that is V(L(G))=E(G) and two vertices in L(G) are adjacent if and only if the corresponding edges in G share a vertex.
The order of a graph is the cardinality of its vertex set, and size of a graph is the number of edges in it. The degree of a vertex v is denoted by deg(v), and it is the number of edges that are incident with v. The following lemma, due to Euler (1736), tells that if several people shake hands, then the number of hands shaken is even.
Lemma 2.2. [20](Handshaking lemma) The sum of the degrees of the vertices of a graph G is twice the number of edges,
∑v∈V(G)deg(v)=2E(G). |
Definition 2.3. [19] The Cartesian product of two graphs H1 and H2, is a graph, represented by H1◻H2, which has vertex set V(H1)×V(H2)(the Cartesian product of sets), and for (v1,u1),(v2,u2)∈V(H1◻H2), (v1,u1)(v2,u2)∈E(H1◻H2), if either
● v1v2∈E(H1) and u1=u2 or
● v1=v2 and u1u2∈E(H2).
If n≥2, then the Cartesian product of P2 and Pn is called ladder graph. We denote this graph by Ln that is Ln:=P2◻Pn. For n≥3, the Cartesian product of P2 and Cn is said to be a circular ladder graph. We denote this graph by CLn that is CLn:=P2◻Cn. Clearly |V(P2)×V(Pn)|=|V(P2)×V(Cn)|=2n, thus we have |V(Ln)|=|V(CLn)|=2n. The graph Ln has four vertices of degree 2 and 2n−4 vertices of degree 3 so by using Lemma 2.2, we have |E(Ln)|=3n−2. By definition of line graph, it follows that |E(Ln)|=|V(L(Ln))|=3n−2. If n=2, then Ln≅L(Ln). Let n≥3, the graph L(Ln) has two vertices of degree 2, four vertices of degree 3 and 3n−8 vertices of degree 4, by Lemma 2.2 we have |E(L(Ln))|=6n−8. Similarly, one can show that |E(CLn)|=|V(L(CLn))|=3n, and |E(L(CLn))|=6n. For examples of the ladder, circular ladder graphs and their corresponding line graphs see Figures 1 and 2.
In the following, we recall several results that are used quite often in this paper.
Lemma 2.4. [21] Let
0→A→B→C→0 |
be a short exact sequence of Zn-graded S-modules. Then
sdepth(B)≥min{sdepth(A),sdepth(C)}. |
Lemma 2.5. (Depth Lemma) If 0→M→N→P→0 is a short exact sequence of modules over a local ring S, or a Noetherian graded ring with S0 local, then
1. depth(N)≥min{depth(P),depth(M)}.
2. depth(M)≥min{depth(N),depth(P)+1}.
3. depth(P)≥min{depth(M)−1,depth(N)}.
Lemma 2.6. ([5], Lemma 3.6]). Let J⊂S be a monomial ideal, and ˆS=S[xn+1,xn+2,…,xn+r] a polynomial ring of n+r variables then
depth(ˆS/JˆS)=depth(S/JS)+r and sdepth(ˆS/JˆS)=sdepth(S/JS)+r. |
Corollary 2.7. ([21], Corollary 1.3]). Let J⊂S be a monomial ideal. Then depth(S/(J:v))≥depth(S/J) for all monomials v∉J.
Proposition 2.8. ([22], Proposition 2.7]). Let J⊂S be a monomial ideal. Then sdepth(S/(J:v))≥sdepth(S/J) for all monomials v∉J.
Let ⌈t⌉, t∈Q, denotes the smallest integer which is greater than or equal to t. Using Depth Lemma, Morey showed the following result.
Lemma 2.9. ([23], Lemma 2.8]). Let n≥2, then depth(S/I(Pn))=⌈n3⌉.
Stefan proved a similar result for Stanley depth.
Lemma 2.10. ([24], Lemma 4]). Let n≥2, then sdepth(S/I(Pn))=⌈n3⌉.
Cimpoeas proved the following results for depth and Stanley depth of the edge ideals of the cyclic graph.
Proposition 2.11. ([25], Proposition 1.3]). Let n≥3, then depth(S/I(Cn))=⌈n−13⌉.
Theorem 2.12. ([25], Theorem 1.9]). Let n≥3, then
(1) sdepth(S/I(Cn))=⌈n−13⌉, if n≡0,2(mod3).
(2) sdepth(S/I(Cn))≤⌈n3⌉, if n≡1(mod3).
For the edge ideal of a graph G, Fouli and Morey gave the following lower bound for depth and Stanley depth in terms of the diameter of G.
Theorem 2.13. ([26], Theorems 3.1 and 4.18]). Let G be a connected graph and I=I(G) be the edge ideal of G. If d=diam(G), then
depth(S/I),sdepth(S/I)≥⌈d+13⌉. |
In this section, we find some bounds for depth and Stanley depth of the cyclic modules associated to the line graphs of Ln and CLn. We denote the edge ideals of the line graphs of Ln and CLn with In and Jn respectively. We label the vertices of the line graphs of Ln and CLn by using three sets of variables {a1,a2,…,an}, {b1,b2,…,bn} and {c1,c2,…,cn}, see Figures 1 and 2. Let Sn:=K[a1,a2,…,an−1,b1,b2,…,bn,c1,c2,…,cn−1] and ¯Sn=Sn[an,cn] be the rings of polynomials in these variables over the field K. Then In and Jn are squarefree monomial ideals of Sn and ¯Sn respectively. With the labeling as shown in Figures 1 and 2, we have:
G(In)=n−1⋃i=1{aibi,bici,aibi+1,bi+1ci}⋃n−2⋃i=1{aiai+1,cici+1}, |
G(Jn)=G(In)⋃{a1an,c1cn,an−1an,cn−1cn,b1an,b1cn,anbn,bncn}, |
where G(In) and G(Jn) stand for the minimal sets of monomial generators of monomial ideals In and Jn respectively.
Lemma 3.1. For 2≤n≤4 we have that depth(Sn/In)=sdepth(Sn/In)=n−1.
Proof. If n=2, then G(I2)={a1b1,b1c1,a1b2,b2c1}, which is a minimal generating set of the edge ideal of C4. Thus by Proposition 2.11 it follows that depth(S2/I2)=1.
If n=3, then G(I3)=G(I2)⋃{a2b2,b2c2,a2b3,b3c2,a1a2,c1c2}. Consider the following short exact sequence
0⟶S3/(I3:b2)⋅b2→S3/I3⟶S3/(I3,b2)⟶0. | (3.1) |
Here (I3:b2)=(a1,a2,c1,c2), so we have S3/(I3:b2)≅K[b1,b2,b3], thus depth(S3/(I3:b2))=3. Also (I3,b2)=(a1b1,b1c1,c1c2,c2b3,b3a2,a2a1,b2), so we have S3/(I3,b2)≅K[a1,a2,b1,b3,c1,c2]/(a1b1,b1c1,c1c2,c2b3,b3a2,a2a1)≅K[a1,a2,b1,b3,c1,c2]/I(C6), by Proposition 2.11, we have depth(S3/(I3,b2))=2. By using Depth lemma on the exact sequence (3.1), we obtain depth(S3/I3)≥2. For the upper bound, since b3∉I3, by Corollary 2.7, we get depth(S3/I3)≤depth(S3/(I3:b3)). As (I3:b3)=(a2,c2,I2), thus S3/(I3:b3)≅S2/I2[b3], by Lemma 2.6, it follows that depth(S3/(I3:b3))≤depth(S2/I2)+1=1+1=2. Hence depth(S3/I3)=2. If n=4, then G(I4)=G(I3)⋃{a3b3,b3c3,a3b4,b4c3,a2a3,c2c3}. Consider the following short exact sequence
0⟶S4/(I4:b3)⋅b3→S4/I4⟶S4/(I4,b3)⟶0. | (3.2) |
Here (I4:b3)=(I2,a2,c2,a3,c3), so we have S4/(I4:b3)≅S2/I2[b3,b4], thus Lemma 2.6 yields depth(S4/(I4:b3))=depth(S2/I2)+2=1+2=3. Let T:=(I4,b3)=(I2,a2b2,b2c2,a1a2,a2a3,c1c2,c2c3,a3b4,c3b4,b3). Again consider the following short exact sequence
0⟶S4/(T:b2)⋅b2→S4/T⟶S4/(T,b2)⟶0. | (3.3) |
Here (T:b2)=(a1,a2,c1,c2,b3,a3b4,c3b4), so we have S4/(T:b2)≅K[a3,b4,c3]/(a3b4,c3b4)[b1,b2], by Lemmas 2.6 and 2.9, depth(S4/(T:b2))=1+2=3. Also (T,b2)=(a1b1,b1c1,c1c2,c2c3,c3b4,b4a3,a3a2,a2a1,b2,b3), which implies that
S4/(T,b2)≅K[a1,a2,a3,b1,b4,c1,c2,c3]/(a1b1,b1c1,c1c2,c2c3,c3b4,b4a3,a3a2,a2a1)≅K[a1,a2,a3,b1,b4,c1,c2,c3]/I(C8) |
thus Proposition 2.11 gives that depth(S4/(T,b2))=3. By applying [15, Lemma 3.1] on the exact sequences (3.2), and (3.3), we have depth(S4/I4)=3.
For Stanley depth, if n=2, then by Theorem 2.12, we have sdepth(S2/I2)≤1. Also, we use [5] to show that there exist Stanley decompositions of desired Stanley depth.
S2/I2=K[a1]⊕b1K[b1,b2]⊕c1K[a1,c1]⊕b2K[b2]. |
Thus we have sdepth(S2/I2)=1. If n=3, then by applying Lemmas 2.4, 2.10, and Theorem 2.12 on the exact sequences (3.1), we have sdepth(S3/I3)≥2. For upper bound, since b3∉I3, by Proposition 2.8, we get sdepth(S3/I3)≤sdepth(S3/(I3:b3)). As (I3:b3)=(a2,c2,I2), thus S3/(I3:b3)≅S2/I2[b3], by Lemma 2.6, it follows that sdepth(S3/(I3:b3))≤sdepth(S2/I2)+1=1+1=2. Hence sdepth(S3/I3)=2. If n=4, by using Lemmas 2.4, 2.10, and Theorem 2.12 on the exact sequences (3.2) and (3.3), we have sdepth(S4/I4)≥3. For upper bound, since b4∉I4, by Proposition 2.8, we get sdepth(S4/I4)≤sdepth(S4/(I4:b4)). As (I4:b4)=(a3,c3,I3), thus S4/(I4:b4)≅S3/I3[b4], by Lemma 2.6, it follows that sdepth(S4/(I4:b4))≤sdepth(S3/I3)+1=2+1=3. Hence sdepth(S4/I4)=3. This completes the proof.
Let 1≤k≤n−1 and Ak:=K[an−1,an−2,…,an−k],Ck:=K[cn−1,cn−2,…,cn−k],Dk:=Ak⊗KCk and
¯Dk:=Dk⊗KK[b1] be the subrings of Sn. Let B0:=(0), Bj:=(bn,bn−1,…,bn−j+1), for 1≤j≤n, and for 3≤j≤n−2, Pj−1:=(an−j+1an−j+2,an−j+2an−j+3,…,an−2an−1) and ¯Pj−1=(cn−j+1cn−j+2,…,cn−2cn−1) are the squarefree monomial ideals of Sn. In the following theorem, we give some bounds for depth and Stanley depth of Sn/In.
Theorem 3.2. For n≥2 we have that ⌈n2⌉≤depth(Sn/In),sdepth(Sn/In)≤n−1.
Proof. If 2≤n≤4, then the result follows by Lemma 3.1. For n≥5, first we prove that ⌈n2⌉≤depth(Sn/In)≤n−1 using induction on n. For 0≤j≤n−2, consider the family of short exact sequences
0⟶Sn/((In,B0):bn)⋅bn→Sn/(In,B0)⟶Sn/(In,B1)⟶0 |
0⟶Sn/((In,B1):bn−1)⋅bn−1→Sn/(In,B1)⟶Sn/(In,B2)⟶0 |
0⟶Sn/((In,B2):bn−2)⋅bn−2→Sn/(In,B2)⟶Sn/(In,B3)⟶0 |
⋮ |
0⟶Sn/((In,Bj):bn−j)⋅bn−j→Sn/(In,Bj)⟶Sn/(In,Bj+1)⟶0 |
⋮ |
0⟶Sn/((In,Bn−2):b2)⋅b2→Sn/(In,Bn−2)⟶Sn/(In,Bn−1)⟶0 |
(1) If j=0, then (In:bn)=(In−1,an−1,cn−1), so we have Sn/(In:bn)≅Sn−1/In−1[bn], the induction hypothesis and Lemma 2.6 give that depth(Sn/(In:bn))≥⌈n−12⌉+1=⌈n+12⌉.
(2) If j=1, then ((In,B1):bn−1)=(In−2,an−1,an−2,cn−1,cn−2,B1), so we obtain Sn/((In,B1):bn−1)≅Sn−2/In−2[bn−1], by induction and Lemma 2.6, it follows that depth(Sn/((In,B1):bn−1))≥⌈n−22⌉+1=⌈n2⌉.
(3) If j=2, then ((In,B2):bn−2)=(In−3,an−2,an−3,cn−2,cn−3,B2), so we have Sn/((In,B2):bn−2)≅Sn−3/In−3[an−1,bn−2,cn−1], the induction hypothesis and Lemma 2.6 give that depth(Sn/((In,B2):bn−2))≥⌈n−32⌉+3=⌈n+12⌉+1.
(4) If 3≤j≤n−3, then ((In,Bj):bn−j)=(In−(j+1),(an−j+1an−j+2,an−j+2an−j+3,…,an−2an−1),(cn−j+1cn−j+2,cn−j+2cn−j+3,…,cn−2cn−1),an−j,an−(j+1),cn−j,cn−(j+1),Bj), that further implies
Sn/((In,Bj):bn−j)≅(Sn−(j+1)/In−(j+1))⊗K(Aj−1/Pj−1)⊗K(Cj−1/¯Pj−1)⊗KK[bn−j]. |
By [18], Theorem 2.2.21], we have
depth(Sn/((In,Bj):bn−j))=depth(Sn−(j+1)/In−(j+1))+depth(Aj−1/Pj−1)+depth(Cj−1/¯Pj−1)+1. |
By Lemma 2.9, we get depth(Aj−1/Pj−1)=⌈j−13⌉=depth(Cj−1/¯Pj−1) and by induction on n, depth(Sn−(j+1)/In−(j+1))≥⌈n−(j+1)2⌉. Thus we have
depth(Sn/((In,Bj):bn−j))≥⌈n−(j+1)2⌉+⌈j−13⌉+⌈j−13⌉+1. |
(5) If j=n−2, then
((In,Bn−2):b2)=((a3a4,a4a5,…,an−2an−1),an−j,an−(j+1),cn−j,cn−(j+1),(c3c4,…,cn−2cn−1),Bj), so we have Sn/((In,Bn−2):b2)≅(An−3/Pn−3)⊗K(Cn−3/¯Pn−3)⊗KK[b1,b2], by [18,Theorem 2.2.21], it follows that depth(Sn/((In,Bn−2):b2))=depth(An−3/Pn−3)+depth(Cn−3/¯Pn−3)+2. By Lemma 2.9, depth(An−3/Pn−3)=⌈n−33⌉=depth(Cn−3/¯Pn−3). Thus we have
depth(Sn/((In,Bn−2):b2))=⌈n−33⌉+⌈n−33⌉+2. |
Also (In,Bn−1)=(an−1an−2,…,a2a1,a1b1,b1c1,c1c2,c2c3,…,cn−2cn−1,Bn−1), so we have Sn/(In,Bn−1)≅¯Dn−1/I(P2n−1). Thus by Lemma 2.9, it follows that depth(Sn/(In,Bn−1))=⌈2n−13⌉. By applying Depth Lemma on the above family of short exact sequences, we obtain the required lower bound for depth. Now by using induction on n, we show that depth(Sn/In)≤n−1. For n≥5, as bn∉In, by Corollary 2.7, we have depth(Sn/In)≤depth(Sn/(In:bn)). Since Sn/(In:bn)≅Sn−1/In−1[bn], the induction hypothesis and Lemma 2.6 yield depth(Sn/(In:bn))≤n−1−1+1=n−1.
Now, it remains to show the result for Stanley depth. The required lower bound can be obtained by applying Lemmas 2.4, 2.10, and [21,Theorem 3.1] instead of Depth Lemma, Lemma 2.9, and [18,Theorem 2.2.21] respectively on above family of short exact sequences. Finally, we prove sdepth(Sn/In)≤n−1 by using induction on n. For n≥5, as bn∉In, from Proposition 2.8, we get sdepth(Sn/In)≤sdepth(Sn/(In:bn)). As Sn/(In:bn)≅Sn−1/In−1[bn], by induction and Lemma 2.6, it follows that sdepth(Sn/(In:bn))≤n−1−1+1=n−1. This finishes the proof.
Remark 3.3. Clearly diam(L(Ln))=n, then by Theorems 2.13 we have depth(Sn/In),sdepth(Sn/In)≥⌈n+13⌉. Our Theorem 3.2 shows depth(Sn/In),sdepth(Sn/In)≥⌈n2⌉. Thus we find a better lower bound for depth and Stanley depth of these classes of edge ideals.
In order to find bounds for depth and Stanley depth of the cyclic module ¯Sn/Jn, we consider two supergraphs Un and Vn of L(Ln). The vertex and edge sets of Un are V(Un)=V(L(Ln))∪{cn} and E(Un)=E(L(Ln))∪{cn−1cn,bncn} respectively. The vertex and edge sets of Vn are V(Vn)=V(Un)∪{cn+1} and E(Vn)=E(Un)∪{c1cn+1,b1cn+1} respectively. For examples of Un and Vn, see Figure 3. We denote the edge ideals of Un and Vn with I∗n and I∗∗n respectively. The minimal sets of monomial generators of I∗n and I∗∗n are G(I∗n)=G(In)⋃{cn−1cn,bncn} and G(I∗∗n)=G(I∗n)⋃{c1cn+1,b1cn+1}. First, we find bounds for depth and Stanley depth of the cyclic modules S∗n/I∗n and S∗∗n/I∗∗n, where S∗n=Sn[cn] and S∗∗n=Sn[cn,cn+1].
Proposition 3.4. Let n≥2. Then ⌈n2⌉≤depth(S∗n/I∗n),sdepth(S∗n/I∗n)≤n.
Proof. If n=2, then by using CoCoA, we obtain depth(S∗n/I∗n)=sdepth(S∗n/I∗n)=2. For n≥3, we first prove that ⌈n2⌉≤depth(S∗n/I∗n) by using induction on n. For this, we assume the following short exact sequence
0⟶S∗n/(I∗n:cn)⋅cn→S∗n/I∗n⟶S∗n/(I∗n,cn)⟶0. | (3.4) |
Here (I∗n:cn)=(n−2⋃i=1{aibi,bici,aibi+1,bi+1ci}⋃n−3⋃i=1{aiai+1,cici+1},an−1an−2,an−1bn−1,bn,cn−1), |
so we have S∗n/(I∗n:cn)≅S∗n−1/I∗n−1[cn]. By using induction and Lemma 2.6,
depth(S∗n/(I∗n:cn))≥⌈n−12⌉+1=⌈n+12⌉. |
As (I∗n,cn)=(n−1⋃i=1{aibi,bici,aibi+1,bi+1ci}⋃n−2⋃i=1{aiai+1,cici+1},cn)=(In,cn), |
so we obtain S∗n/(I∗n,cn)≅Sn/In. By Theorem 3.2, it follows that depth(S∗n/(I∗n,cn))≥⌈n2⌉. Therefore by applying Depth Lemma on the exact sequence (3.4), we get depth(S∗n/I∗n)≥⌈n2⌉. Now we prove depth(S∗n/I∗n)≤n by using induction on n. For n≥3, as cn∉I∗n, from Corollary 2.7, we have depth(S∗n/I∗n)≤depth(S∗n/(I∗n:cn)). Since S∗n/(I∗n:cn)≅S∗n−1/I∗n−1[cn], by induction and Lemma 2.6, depth(S∗n/I∗n)≤n−1+1=n. It remains to show the result for Stanley depth. For n≥3, by using induction on n, and by applying Lemma 2.4 on the exact sequence (3.4), we get sdepth(S∗n/I∗n)≥⌈n2⌉. For upper bound of Stanley depth, one can repeat the proof for depth by using Proposition 2.8 instead of Corollary 2.7.
Proposition 3.5. For n≥2, we have that ⌈n2⌉≤depth(S∗∗n/I∗∗n),sdepth(S∗∗n/I∗∗n)≤n+1.
Proof. If n=2, then by using CoCoA, we obtain depth(S∗∗n/I∗∗n)=sdepth(S∗∗n/I∗∗n)=2, and for n=3, depth(S∗∗n/I∗∗n)=sdepth(S∗∗n/I∗∗n)=3. For n≥4, we first prove that depth(S∗∗n/I∗∗n)≥⌈n2⌉ by using induction on n. Let us consider the following short exact sequence
0⟶S∗∗n/(I∗∗n:cn)⋅cn→S∗∗n/I∗∗n⟶S∗∗n/(I∗∗n,cn)⟶0. | (3.5) |
As (I∗∗n,cn)=(n−1⋃i=1{aibi,bici,aibi+1,bi+1ci}⋃n−2⋃i=1{aiai+1,cici+1},c1cn+1,b1cn+1,cn), |
so we have that S∗∗n/(I∗∗n,cn)≅S∗n/I∗n. Therefore by Proposition 3.4, it follows that
depth(S∗∗n/(I∗∗n,cn))≥⌈n2⌉. |
Let T=(I∗∗n:cn)=(n−2⋃i=1{aibi,bici,aibi+1,bi+1ci}⋃n−3⋃i=1{aiai+1,cici+1},an−1an−2,bn,an−1bn−1,c1cn+1,b1cn+1,cn−1)=(I∗n−1,an−1an−2,an−1bn−1,bn,cn−1). |
Now consider another short exact sequence
0⟶S∗∗n/(T:an−1)⋅an−1→S∗∗n/T⟶S∗∗n/(T,an−1)⟶0, | (3.6) |
(T:an−1)=(n−3⋃i=1{aibi,bici,aibi+1,bi+1ci}⋃n−4⋃i=1{aiai+1,cici+1},bn−2cn−2,cn−2cn−3,bn−1,c1cn+1,b1cn+1,bn,cn−1,an−2)=(I∗∗n−2,bn,cn−1,an−2,bn−1), |
so we have S∗∗n/(T:an−1)≅S∗∗n−2/I∗∗n−2[an−1,cn]. Thus induction on n and Lemma 2.6 give that depth(S∗∗n/(T:an−1))≥⌈n−22⌉+2=⌈n2⌉+1. As (T,an−1)=(I∗n−1,an−1,bn,cn−1), which implies S∗∗n/(T,an−1)≅S∗n−1/I∗n−1. By Proposition 3.4 and Lemma 2.6, we obtain depth(S∗∗n/(T,an−1))≥⌈n−12⌉+1=⌈n+12⌉. Therefore by applying Depth Lemma on the exact sequences (3.5) and (3.6), we get depth(S∗∗n/I∗∗n)≥⌈n2⌉. Now we prove depth(S∗∗n/I∗∗n)≤n+1. We show this by induction on n. For n≥4, as an−1cn∉I∗∗n, from Corollary 2.7, we have
depth(S∗∗n/I∗∗n)≤depth(S∗∗n/(I∗∗n:an−1cn)). |
Since S∗∗n/(I∗∗n:an−1cn)≅S∗∗n−2/I∗∗n−2[an−1,cn], by induction and Lemma 2.6, depth(S∗∗n/I∗∗n)≤n−2+1+2=n+1. It remains to prove the result for Stanley depth. For n≥4, by using induction on n, and by applying Lemma 2.4 on the exact sequences (2.10) and (2.11) we get sdepth(S∗∗n/I∗∗n)≥⌈n2⌉. Similarly, one can obtain the required upper bound for Stanley depth by using Proposition 2.8 instead of Corollary 2.7.
Theorem 3.6. Let n≥3. Then ⌈n2⌉≤depth(¯Sn/Jn)≤n−1, and ⌈n2⌉≤sdepth(¯Sn/Jn)≤n.
Proof. For 3≤n≤4, by using CoCoA, (for sdepth we use SdepthLib:coc [27]), depth(¯S3/J3)=sdepth(¯S3/J3)=2, depth(¯S4/J4)=sdepth(¯S4/J4)=3. Now for n≥5, we first show that depth(¯Sn/Jn)≥⌈n2⌉. Let us consider the following short exact sequence
0⟶¯Sn/(Jn:an)⋅an→¯Sn/Jn⟶¯Sn/(Jn,an)⟶¯Sn/(Jn,an)⟶0. | (3.7) |
Let U=(Jn,an)=(n−1⋃i=1{aibi,bici,aibi+1,bi+1ci}⋃n−2⋃i=1{aiai+1,cici+1},c1cn,cn−1cn,b1cn,bncn,an). |
Now assume another short exact sequence
0⟶¯Sn/(U:cn)⋅cn→¯Sn/U⟶¯Sn/(U,cn)⟶0. | (3.8) |
As (U,cn)=(n−1⋃i=1{aibi,bici,aibi+1,bi+1ci}⋃n−2⋃i=1{aiai+1,cici+1},cn,an), |
so we obtain ¯Sn/(U,cn)≅Sn/In. Thus Theorem 3.2 gives that depth(¯Sn/(U,cn))≥⌈n2⌉.
Also (U:cn)=(n−2⋃i=2{aibi,bici,aibi+1,bi+1ci}⋃n−3⋃i=2{aiai+1,cici+1},a1a2,a1b2,an−1an−2,an,an−1bn−1,b1,bn,c1,cn−1)=(I∗∗n−2,an,b1,bn,c1,cn−1), |
so we get ¯Sn/(U:cn)≅S∗∗n−2/I∗∗n−2[cn]. Thus by Proposition 3.5 and Lemma 2.6 we have
depth(¯Sn/(U:cn))≥⌈n−22⌉+1=⌈n2⌉. |
Let V=(Jn:an)=(n−2⋃i=2{aibi,bici,aibi+1,bi+1ci}⋃n−3⋃i=2{aiai+1,cici+1},c1b2,c1c2,cn−1cn−2,cn−1bn−1,c1cn,cncn−1,a1,an−1,b1,bn). |
Now consider the following short exact sequence
0⟶¯Sn/(V:cn)⋅cn→¯Sn/V⟶¯Sn/(V,cn)⟶0, | (3.9) |
(V:cn)=(n−2⋃i=2{aibi,bici,aibi+1,bi+1ci}⋃n−3⋃i=2{aiai+1,cici+1},c1,cn−1,a1,an−1,b1,bn)=(In−2,c1,cn−1,a1,an−1,b1,bn), |
so we have ¯Sn/(V:cn)≅Sn−2/In−2[an,cn]. Thus by Theorem 3.2 and Lemma 2.6, we have
depth(¯Sn/(V:cn))≥⌈n−22⌉+2=⌈n2⌉+1. |
As (V,cn)=(n−2⋃i=2{aibi,bici,aibi+1,bi+1ci}⋃n−3⋃i=2{aiai+1,cici+1},c1b2,c1c2,cn−1cn−2,cn−1bn−1,a1,an−1,b1,bn,cn), |
so we have that ¯Sn/(V,cn)≅S∗∗n−2/I∗∗n−2. By Proposition 3.5 and Lemma 2.6, we obtain depth(¯Sn/(V,cn))≥⌈n−22⌉+1=⌈n2⌉. Therefore by applying Depth Lemma on the exact sequences (3.7), (3.8) and (3.9), we get depth(¯Sn/Jn)≥⌈n2⌉. Now we prove depth(¯Sn/Jn)≤n−1. For n≥5, as ancn∉Jn, from Corollary 2.7, it follows that depth(¯Sn/Jn)≤depth(¯Sn/(Jn:ancn)). Since ¯Sn/(Jn:ancn)≅Sn−2/In−2[an,cn], by Theorem 3.2 and Lemma 2.6, we have depth(¯Sn/Jn)≤n−2−1+2=n−1.
It remains to show the result for Stanley depth. For n≥5, by applying Lemma 2.4 on the exact sequences (3.7), (3.8) and (3.9), we get that sdepth(¯Sn/Jn)≥⌈n2⌉. Similarly, one can obtain the required upper bound for Stanley depth by using Proposition 2.8 instead of Corollary 2.7.
Remark 3.7. It is easy to see that diam(L(CLn))=⌈n+12⌉, then by Theorems 2.13, we have depth(¯Sn/Jn),sdepth(¯Sn/Jn)≥⌈n+26⌉. Our Theorem 3.6 shows that depth(¯Sn/Jn),sdepth(¯Sn/Jn)≥⌈n2⌉. Thus we find a much better lower bound for depth and Stanley depth for these classes of edge ideals.
Proposition 3.8. For n≥2, we have that dim(Sn/In)≥n.
Proof. Let E={a1,a2,…,an−1,c1,c2,…,cn−1} be a subset of vertex set V(L(Ln)). The set E is a vertex cover because it covers all the edges. Now if we remove ai for some 1≤i≤n−1 from set E then the resulting set is not a vertex cover because the edges aibi and aibi+1 will not covered. Similarly by removing ci for some 1≤i≤n−1 from set E then the resulting set is not a vertex cover because the edges cibi and cibi+1 will not covered. This shows that the set E forms a minimal vertex cover of In. Thus we have height(In)≤2n−2. Since Sn is a polynomial ring of dimension 3n−2, which implies that dim(Sn/In)≥3n−2−(2n−2)=n.
Proposition 3.9. Let n≥3. Then dim(¯Sn/Jn)≥n.
Proof. As in the Proposition 3.8, one can shows in a similar way that the set F={a1,a2,…,an,c1,c2,…,cn} forms a minimal vertex cover of Jn, therefore height(Jn)≤2n. As ¯Sn is a polynomial ring of dimension 3n, thus dim(¯Sn/Jn)≥n.
Remark 3.10. By Theorem 3.2 and 3.6 we have that depth(Sn/In),depth(¯Sn/Jn)≤n−1, and by Proposition 3.8 and 3.9 we have dim(Sn/In),dim(¯Sn/Jn)≥n. Thus graphs L(Ln) and L(CLn) are not Cohen-Macaulay.
The authors declare that there is no conflicts of interest in this paper.
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