Research article

Depth and Stanley depth of edge ideals associated to some line graphs

  • Received: 21 January 2019 Accepted: 20 May 2019 Published: 19 June 2019
  • MSC : 13C15, 13P10, 13F20, 05E99

  • In this paper, we compute some upper and lower bounds for depth and Stanley depth of edge ideals associated to line graphs of the ladder and circular ladder graphs. Furthermore, we determine some bounds for the dimension of the quotient rings of the edge ideals associated to these graphs.

    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

    Related Papers:

    [1] Naeem Ud Din, Muhammad Ishaq, Zunaira Sajid . Values and bounds for depth and Stanley depth of some classes of edge ideals. AIMS Mathematics, 2021, 6(8): 8544-8566. doi: 10.3934/math.2021496
    [2] Ibad Ur Rehman, Mujahid Ullah Khan Afridi, Muhammad Ishaq, Asim Asiri, Aftab Hussain . Algebraic invariants of edge ideals of some bristled circulant graphs. AIMS Mathematics, 2025, 10(5): 11330-11348. doi: 10.3934/math.2025515
    [3] Tazeen Ayesha, Muhammad Ishaq . Some algebraic invariants of the edge ideals of perfect $ [h, d] $-ary trees and some unicyclic graphs. AIMS Mathematics, 2023, 8(5): 10947-10977. doi: 10.3934/math.2023555
    [4] Bakhtawar Shaukat, Muhammad Ishaq, Ahtsham Ul Haq . Algebraic invariants of edge ideals of some circulant graphs. AIMS Mathematics, 2024, 9(1): 868-895. doi: 10.3934/math.2024044
    [5] Malik Muhammad Suleman Shahid, Muhammad Ishaq, Anuwat Jirawattanapanit, Khanyaluck Subkrajang . Depth and Stanley depth of the edge ideals of multi triangular snake and multi triangular ouroboros snake graphs. AIMS Mathematics, 2022, 7(9): 16449-16463. doi: 10.3934/math.2022900
    [6] Samer Nofal . On finding a satisfactory partition in an undirected graph: algorithm design and results. AIMS Mathematics, 2024, 9(10): 27308-27329. doi: 10.3934/math.20241327
    [7] Jovanny Ibarguen, Daniel S. Moran, Carlos E. Valencia, Rafael H. Villarreal . The signature of a monomial ideal. AIMS Mathematics, 2024, 9(10): 27955-27978. doi: 10.3934/math.20241357
    [8] Tariq Alraqad, Hicham Saber, Rashid Abu-Dawwas . Intersection graphs of graded ideals of graded rings. AIMS Mathematics, 2021, 6(10): 10355-10368. doi: 10.3934/math.2021600
    [9] Meiqin Wei, Jun Yue, Xiaoyu zhu . On the edge metric dimension of graphs. AIMS Mathematics, 2020, 5(5): 4459-4465. doi: 10.3934/math.2020286
    [10] Yubin Zhong, Sakander Hayat, Suliman Khan, Vito Napolitano, Mohammed J. F. Alenazi . Combinatorial analysis of line graphs: domination, chromaticity, and Hamiltoniancity. AIMS Mathematics, 2025, 10(6): 13343-13364. doi: 10.3934/math.2025599
  • In this paper, we compute some upper and lower bounds for depth and Stanley depth of edge ideals associated to line graphs of the ladder and circular ladder graphs. Furthermore, we determine some bounds for the dimension of the quotient rings of the edge ideals associated to these graphs.


    Let S:=K[x1,,xn] be a polynomial ring over a field K, and A be a finitely generated Zn-graded S-module. Let aA 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 IJS 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[n1]} and G is called a cycle if E(G)={{ai,ai+1}:i[n1]}{{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 eE(G), eB and B is minimal with respect to this property, that is for any proper subset B of B, then there exists an edge eE(G) with eB=. A prime ideal Q is a minimal prime of an ideal I if IQ and if Q is a prime ideal with IQQ, 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,bV(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,

    vV(G)deg(v)=2E(G).

    Definition 2.3. [19] The Cartesian product of two graphs H1 and H2, is a graph, represented by H1H2, which has vertex set V(H1)×V(H2)(the Cartesian product of sets), and for (v1,u1),(v2,u2)V(H1H2), (v1,u1)(v2,u2)E(H1H2), if either

    v1v2E(H1) and u1=u2 or

    v1=v2 and u1u2E(H2).

    If n2, then the Cartesian product of P2 and Pn is called ladder graph. We denote this graph by Ln that is Ln:=P2Pn. For n3, the Cartesian product of P2 and Cn is said to be a circular ladder graph. We denote this graph by CLn that is CLn:=P2Cn. 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 2n4 vertices of degree 3 so by using Lemma 2.2, we have |E(Ln)|=3n2. By definition of line graph, it follows that |E(Ln)|=|V(L(Ln))|=3n2. If n=2, then LnL(Ln). Let n3, the graph L(Ln) has two vertices of degree 2, four vertices of degree 3 and 3n8 vertices of degree 4, by Lemma 2.2 we have |E(L(Ln))|=6n8. 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.

    Figure 1.  L2, L4, L6 and their line graphs L(L2), L(L4), L(L6).
    Figure 2.  From left to right, CL6 and L(CL6).

    In the following, we recall several results that are used quite often in this paper.

    Lemma 2.4. [21] Let

    0ABC0

    be a short exact sequence of Zn-graded S-modules. Then

    sdepth(B)min{sdepth(A),sdepth(C)}.

    Lemma 2.5. (Depth Lemma) If 0MNP0 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 JS 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 JS be a monomial ideal. Then depth(S/(J:v))depth(S/J) for all monomials vJ.

    Proposition 2.8. ([22], Proposition 2.7]). Let JS be a monomial ideal. Then sdepth(S/(J:v))sdepth(S/J) for all monomials vJ.

    Let t, tQ, 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 n2, then depth(S/I(Pn))=n3.

    Stefan proved a similar result for Stanley depth.

    Lemma 2.10. ([24], Lemma 4]). Let n2, 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 n3, then depth(S/I(Cn))=n13.

    Theorem 2.12. ([25], Theorem 1.9]). Let n3, then

    (1) sdepth(S/I(Cn))=n13, if n0,2(mod3).

    (2) sdepth(S/I(Cn))n3, if n1(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,,an1,b1,b2,,bn,c1,c2,,cn1] 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)=n1i=1{aibi,bici,aibi+1,bi+1ci}n2i=1{aiai+1,cici+1},
    G(Jn)=G(In){a1an,c1cn,an1an,cn1cn,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 2n4 we have that depth(Sn/In)=sdepth(Sn/In)=n1.

    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

    0S3/(I3:b2)b2S3/I3S3/(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 b3I3, 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

    0S4/(I4:b3)b3S4/I4S4/(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

    0S4/(T:b2)b2S4/TS4/(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 b3I3, 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 b4I4, 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 1kn1 and Ak:=K[an1,an2,,ank],Ck:=K[cn1,cn2,,cnk],Dk:=AkKCk and

    ¯Dk:=DkKK[b1] be the subrings of Sn. Let B0:=(0), Bj:=(bn,bn1,,bnj+1), for 1jn, and for 3jn2, Pj1:=(anj+1anj+2,anj+2anj+3,,an2an1) and ¯Pj1=(cnj+1cnj+2,,cn2cn1) 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 n2 we have that n2depth(Sn/In),sdepth(Sn/In)n1.

    Proof. If 2n4, then the result follows by Lemma 3.1. For n5, first we prove that n2depth(Sn/In)n1 using induction on n. For 0jn2, consider the family of short exact sequences

    0Sn/((In,B0):bn)bnSn/(In,B0)Sn/(In,B1)0
    0Sn/((In,B1):bn1)bn1Sn/(In,B1)Sn/(In,B2)0
    0Sn/((In,B2):bn2)bn2Sn/(In,B2)Sn/(In,B3)0
    0Sn/((In,Bj):bnj)bnjSn/(In,Bj)Sn/(In,Bj+1)0
    0Sn/((In,Bn2):b2)b2Sn/(In,Bn2)Sn/(In,Bn1)0

    (1) If j=0, then (In:bn)=(In1,an1,cn1), so we have Sn/(In:bn)Sn1/In1[bn], the induction hypothesis and Lemma 2.6 give that depth(Sn/(In:bn))n12+1=n+12.

    (2) If j=1, then ((In,B1):bn1)=(In2,an1,an2,cn1,cn2,B1), so we obtain Sn/((In,B1):bn1)Sn2/In2[bn1], by induction and Lemma 2.6, it follows that depth(Sn/((In,B1):bn1))n22+1=n2.

    (3) If j=2, then ((In,B2):bn2)=(In3,an2,an3,cn2,cn3,B2), so we have Sn/((In,B2):bn2)Sn3/In3[an1,bn2,cn1], the induction hypothesis and Lemma 2.6 give that depth(Sn/((In,B2):bn2))n32+3=n+12+1.

    (4) If 3jn3, then ((In,Bj):bnj)=(In(j+1),(anj+1anj+2,anj+2anj+3,,an2an1),(cnj+1cnj+2,cnj+2cnj+3,,cn2cn1),anj,an(j+1),cnj,cn(j+1),Bj), that further implies

    Sn/((In,Bj):bnj)(Sn(j+1)/In(j+1))K(Aj1/Pj1)K(Cj1/¯Pj1)KK[bnj].

    By [18], Theorem 2.2.21], we have

    depth(Sn/((In,Bj):bnj))=depth(Sn(j+1)/In(j+1))+depth(Aj1/Pj1)+depth(Cj1/¯Pj1)+1.

    By Lemma 2.9, we get depth(Aj1/Pj1)=j13=depth(Cj1/¯Pj1) and by induction on n, depth(Sn(j+1)/In(j+1))n(j+1)2. Thus we have

    depth(Sn/((In,Bj):bnj))n(j+1)2+j13+j13+1.

    (5) If j=n2, then

    ((In,Bn2):b2)=((a3a4,a4a5,,an2an1),anj,an(j+1),cnj,cn(j+1),(c3c4,,cn2cn1),Bj), so we have Sn/((In,Bn2):b2)(An3/Pn3)K(Cn3/¯Pn3)KK[b1,b2], by [18,Theorem 2.2.21], it follows that depth(Sn/((In,Bn2):b2))=depth(An3/Pn3)+depth(Cn3/¯Pn3)+2. By Lemma 2.9, depth(An3/Pn3)=n33=depth(Cn3/¯Pn3). Thus we have

    depth(Sn/((In,Bn2):b2))=n33+n33+2.

    Also (In,Bn1)=(an1an2,,a2a1,a1b1,b1c1,c1c2,c2c3,,cn2cn1,Bn1), so we have Sn/(In,Bn1)¯Dn1/I(P2n1). Thus by Lemma 2.9, it follows that depth(Sn/(In,Bn1))=2n13. 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)n1. For n5, as bnIn, by Corollary 2.7, we have depth(Sn/In)depth(Sn/(In:bn)). Since Sn/(In:bn)Sn1/In1[bn], the induction hypothesis and Lemma 2.6 yield depth(Sn/(In:bn))n11+1=n1.

    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)n1 by using induction on n. For n5, as bnIn, from Proposition 2.8, we get sdepth(Sn/In)sdepth(Sn/(In:bn)). As Sn/(In:bn)Sn1/In1[bn], by induction and Lemma 2.6, it follows that sdepth(Sn/(In:bn))n11+1=n1. 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)){cn1cn,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 In and In respectively. The minimal sets of monomial generators of In and In are G(In)=G(In){cn1cn,bncn} and G(In)=G(In){c1cn+1,b1cn+1}. First, we find bounds for depth and Stanley depth of the cyclic modules Sn/In and Sn/In, where Sn=Sn[cn] and Sn=Sn[cn,cn+1].

    Figure 3.  From left to right, supergraphs U6 and V6 of L(L6) respectively.

    Proposition 3.4. Let n2. Then n2depth(Sn/In),sdepth(Sn/In)n.

    Proof. If n=2, then by using CoCoA, we obtain depth(Sn/In)=sdepth(Sn/In)=2. For n3, we first prove that n2depth(Sn/In) by using induction on n. For this, we assume the following short exact sequence

    0Sn/(In:cn)cnSn/InSn/(In,cn)0. (3.4)
    Here (In:cn)=(n2i=1{aibi,bici,aibi+1,bi+1ci}n3i=1{aiai+1,cici+1},an1an2,an1bn1,bn,cn1),

    so we have Sn/(In:cn)Sn1/In1[cn]. By using induction and Lemma 2.6,

    depth(Sn/(In:cn))n12+1=n+12.
    As (In,cn)=(n1i=1{aibi,bici,aibi+1,bi+1ci}n2i=1{aiai+1,cici+1},cn)=(In,cn),

    so we obtain Sn/(In,cn)Sn/In. By Theorem 3.2, it follows that depth(Sn/(In,cn))n2. Therefore by applying Depth Lemma on the exact sequence (3.4), we get depth(Sn/In)n2. Now we prove depth(Sn/In)n by using induction on n. For n3, as cnIn, from Corollary 2.7, we have depth(Sn/In)depth(Sn/(In:cn)). Since Sn/(In:cn)Sn1/In1[cn], by induction and Lemma 2.6, depth(Sn/In)n1+1=n. It remains to show the result for Stanley depth. For n3, by using induction on n, and by applying Lemma 2.4 on the exact sequence (3.4), we get sdepth(Sn/In)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 n2, we have that n2depth(Sn/In),sdepth(Sn/In)n+1.

    Proof. If n=2, then by using CoCoA, we obtain depth(Sn/In)=sdepth(Sn/In)=2, and for n=3, depth(Sn/In)=sdepth(Sn/In)=3. For n4, we first prove that depth(Sn/In)n2 by using induction on n. Let us consider the following short exact sequence

    0Sn/(In:cn)cnSn/InSn/(In,cn)0. (3.5)
    As (In,cn)=(n1i=1{aibi,bici,aibi+1,bi+1ci}n2i=1{aiai+1,cici+1},c1cn+1,b1cn+1,cn),

    so we have that Sn/(In,cn)Sn/In. Therefore by Proposition 3.4, it follows that

    depth(Sn/(In,cn))n2.
    Let T=(In:cn)=(n2i=1{aibi,bici,aibi+1,bi+1ci}n3i=1{aiai+1,cici+1},an1an2,bn,an1bn1,c1cn+1,b1cn+1,cn1)=(In1,an1an2,an1bn1,bn,cn1).

    Now consider another short exact sequence

    0Sn/(T:an1)an1Sn/TSn/(T,an1)0, (3.6)
    (T:an1)=(n3i=1{aibi,bici,aibi+1,bi+1ci}n4i=1{aiai+1,cici+1},bn2cn2,cn2cn3,bn1,c1cn+1,b1cn+1,bn,cn1,an2)=(In2,bn,cn1,an2,bn1),

    so we have Sn/(T:an1)Sn2/In2[an1,cn]. Thus induction on n and Lemma 2.6 give that depth(Sn/(T:an1))n22+2=n2+1. As (T,an1)=(In1,an1,bn,cn1), which implies Sn/(T,an1)Sn1/In1. By Proposition 3.4 and Lemma 2.6, we obtain depth(Sn/(T,an1))n12+1=n+12. Therefore by applying Depth Lemma on the exact sequences (3.5) and (3.6), we get depth(Sn/In)n2. Now we prove depth(Sn/In)n+1. We show this by induction on n. For n4, as an1cnIn, from Corollary 2.7, we have

    depth(Sn/In)depth(Sn/(In:an1cn)).

    Since Sn/(In:an1cn)Sn2/In2[an1,cn], by induction and Lemma 2.6, depth(Sn/In)n2+1+2=n+1. It remains to prove the result for Stanley depth. For n4, by using induction on n, and by applying Lemma 2.4 on the exact sequences (2.10) and (2.11) we get sdepth(Sn/In)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 n3. Then n2depth(¯Sn/Jn)n1, and n2sdepth(¯Sn/Jn)n.

    Proof. For 3n4, 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 n5, 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)=(n1i=1{aibi,bici,aibi+1,bi+1ci}n2i=1{aiai+1,cici+1},c1cn,cn1cn,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)=(n1i=1{aibi,bici,aibi+1,bi+1ci}n2i=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)=(n2i=2{aibi,bici,aibi+1,bi+1ci}n3i=2{aiai+1,cici+1},a1a2,a1b2,an1an2,an,an1bn1,b1,bn,c1,cn1)=(In2,an,b1,bn,c1,cn1),

    so we get ¯Sn/(U:cn)Sn2/In2[cn]. Thus by Proposition 3.5 and Lemma 2.6 we have

    depth(¯Sn/(U:cn))n22+1=n2.
    Let V=(Jn:an)=(n2i=2{aibi,bici,aibi+1,bi+1ci}n3i=2{aiai+1,cici+1},c1b2,c1c2,cn1cn2,cn1bn1,c1cn,cncn1,a1,an1,b1,bn).

    Now consider the following short exact sequence

    0¯Sn/(V:cn)cn¯Sn/V¯Sn/(V,cn)0, (3.9)
    (V:cn)=(n2i=2{aibi,bici,aibi+1,bi+1ci}n3i=2{aiai+1,cici+1},c1,cn1,a1,an1,b1,bn)=(In2,c1,cn1,a1,an1,b1,bn),

    so we have ¯Sn/(V:cn)Sn2/In2[an,cn]. Thus by Theorem 3.2 and Lemma 2.6, we have

    depth(¯Sn/(V:cn))n22+2=n2+1.
    As (V,cn)=(n2i=2{aibi,bici,aibi+1,bi+1ci}n3i=2{aiai+1,cici+1},c1b2,c1c2,cn1cn2,cn1bn1,a1,an1,b1,bn,cn),

    so we have that ¯Sn/(V,cn)Sn2/In2. By Proposition 3.5 and Lemma 2.6, we obtain depth(¯Sn/(V,cn))n22+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)n1. For n5, as ancnJn, from Corollary 2.7, it follows that depth(¯Sn/Jn)depth(¯Sn/(Jn:ancn)). Since ¯Sn/(Jn:ancn)Sn2/In2[an,cn], by Theorem 3.2 and Lemma 2.6, we have depth(¯Sn/Jn)n21+2=n1.

    It remains to show the result for Stanley depth. For n5, 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 n2, we have that dim(Sn/In)n.

    Proof. Let E={a1,a2,,an1,c1,c2,,cn1} 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 1in1 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 1in1 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)2n2. Since Sn is a polynomial ring of dimension 3n2, which implies that dim(Sn/In)3n2(2n2)=n.

    Proposition 3.9. Let n3. 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)n1, 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.



    [1] R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math., 68 (1982), 175-193. doi: 10.1007/BF01394054
    [2] B. Ichim, L. Katthän, J. J. Moyano-Fernández, The behavior of Stanley depth under polarization, J. Combin. Theory, Ser. A, 135 (2015), 332-347. doi: 10.1016/j.jcta.2015.05.005
    [3] A. M. Duval, B. Goeckneker, C. J. Klivans, et al. A non-partitionable Cohen-Macaulay simplicial complex, Adv. Math., 299 (2016), 381-395. doi: 10.1016/j.aim.2016.05.011
    [4] J. Herzog, T. Hibi, The depth of powers of an ideal, J. Algebra, 291 (2005), 534-550. doi: 10.1016/j.jalgebra.2005.04.007
    [5] J. Herzog, M. Vladoiu, X. Zheng, How to compute the Stanley depth of a monomial ideal, J. Algebra, 322 (2009), 3151-3169. doi: 10.1016/j.jalgebra.2008.01.006
    [6] C. Biro, D. M. Howard, M. T. Keller, et al. Interval partitions and Stanley depth, J. Combin. Theory, Ser. A, 117 (2010), 475-482. doi: 10.1016/j.jcta.2009.07.008
    [7] M. Cimpoeas, Stanley depth of squarefree Veronese ideals, An. St. Univ. Ovidius Constanta, 21 (2013), 67-71.
    [8] M. Ishaq, Upper bounds for the Stanley depth, Comm. Algebra, 40 (2012), 87-97. doi: 10.1080/00927872.2010.523642
    [9] M. Ishaq, M. I. Qureshi, Upper and lower bounds for the Stanley depth of certain classes of monomial ideals and their residue class rings, Comm. Algebra, 41 (2013), 1107-1116. doi: 10.1080/00927872.2011.630708
    [10] M. T. Keller, Y. Shen, N. Streib, et al. On the Stanley Depth of Squarefree Veronese Ideals, J. Algebr. Comb., 33 (2011), 313-324. doi: 10.1007/s10801-010-0249-1
    [11] B. Ichim, L. Katthän, J. J. Moyano-Fernández, How to compute the Stanley depth of a module, Math. Comput., 86 (2016), 455-472. doi: 10.1090/mcom/3106
    [12] B. Ichim, L. Katthän, J. J. Moyano-Fernández, LCM lattices and Stanley depth: a first computational approach, Exp. Math., 25 (2016), 46-53. doi: 10.1080/10586458.2015.1005257
    [13] B. Ichim, L. Katthän, J. J. Moyano-Fernández, Stanley depth and the lcm-lattice, J. Combin. Theory, Ser. A, 150 (2017), 295-322. doi: 10.1016/j.jcta.2017.03.005
    [14] Z. Iqbal, M. Ishaq, M. Aamir, Depth and Stanley depth of the edge ideals of square paths and square cycles, Comm. Algebra, 46 (2018), 1188-1198. doi: 10.1080/00927872.2017.1339068
    [15] Z. Iqbal, M. Ishaq, Depth and Stanley depth of the edge ideals of the powers of paths and cycles, An. St. Univ. Ovidius Constanta, In press. Available from: http://arxiv.org/pdf/1710.05996.pdf.
    [16] M. R. Pournaki, S. A. Seyed Fakhari, S. Yassemi, Stanley depth of powers of the edge ideals of a forest, P. Am. Math. Soc., 141 (2013), 3327-3336. doi: 10.1090/S0002-9939-2013-11594-7
    [17] J. Herzog, T. Hibi, Monomial Ideals, Springer-Verlag London Limited, 2011.
    [18] R. H. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics 238, Marcel Dekker, New York, 2001.
    [19] J. A. Bondy, U. S. R. Murty, Graph Theory with applications, Springer, 2008.
    [20] F. Harary, Graph theory, Addison-Wesley, Reading, MA, 1969.
    [21] A. Rauf, Depth and Stanley depth of multigraded modules, Comm. Algebra, 38 (2010), 773-784. doi: 10.1080/00927870902829056
    [22] M. Cimpoeas, Several inequalities regarding Stanley depth, Romanian Journal of Mathematics and Computer Science, 2 (2012), 28-40.
    [23] S. Morey, Depths of powers of the edge ideal of a tree, Comm. Algebra, 38 (2010), 4042-4055. doi: 10.1080/00927870903286900
    [24] A. Stefan, Stanley depth of powers of path ideal}. Available from: http://arxiv.org/pdf/1409.6072.pdf.
    [25] M. Cimpoeas, On the Stanley depth of edge ideals of line and cyclic graphs, Romanian Journal of Mathematics and Computer Science, 5 (2015), 70-75.
    [26] L. Fouli, S. Morey, A lower bound for depths of powers of edge ideals, J. Algebr. Comb., 42 (2015), 829-848. doi: 10.1007/s10801-015-0604-3
    [27] G. Rinaldo, An algorithm to compute the Stanley depth of monomial ideals, Le Matematiche, 63 (2008), 243-256.
  • This article has been cited by:

    1. Zahid IQBAL, Depth and Stanley depth of the edge ideals of the strong product of some graphs, 2020, 2651-477X, 1, 10.15672/hujms.638033
    2. Malik Muhammad Suleman Shahid, Muhammad Ishaq, Anuwat Jirawattanapanit, Khanyaluck Subkrajang, Depth and Stanley depth of the edge ideals of multi triangular snake and multi triangular ouroboros snake graphs, 2022, 7, 2473-6988, 16449, 10.3934/math.2022900
  • Reader Comments
  • © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5544) PDF downloads(1149) Cited by(2)

Figures and Tables

Figures(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog