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

1.   Introduction

Let $S : = K[x_{1}, \dots, x_{n}]$ be a polynomial ring over a field $K,$ and $A$ be a finitely generated $\mathbb{Z}^{n}$-graded $S$-module. Let $a\in A$ be a homogeneous element and $X\subset \{x_1, x_2, \dots, x_n\}.$ We denote by $aK[X]$ the $K$-subspace of $A$ generated by all elements $ab$ where $b$ is a monomial in $K[X]$. The $\mathbb{Z}^{n}$-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

The number ${\rm{sdepth}}(\mathcal{D}) = \min\{|X_{i}|: i = 1, \ldots, r\}$ is called the Stanley depth of $\mathcal{D}.$ Let ${\rm{sdepth}}(A) = \max \{ {\rm{sdepth}}({\mathcal D}):{\mathcal D}\;\;{\rm{is}}\;\;{\rm{a}}\;\;{\rm{Stanley}}\;\;{\rm{decomposition}}\;\;{\rm{of}}\;\;A\} ,$ then ${\rm{sdepth}}(A)$ is called the Stanley depth of $A.$ Stanley conjectured in ^[1] that ${\rm{sdepth}}(A)\geq {\rm{depth}}(A)$ for any ${\Bbb Z}^n$-graded $S$-module $A$. Let $I\subset J\subset S$ 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 $\mathfrak{m}$ 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 $\mathfrak{m};$ 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 $\mathbb{Z}^{n}$-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.

2.   Definitions and notation

Let $G$ be a graph having vertex set $V(G) = \{a_1, a_2, \dots, a_n\}$ and edge set $E(G)$, then the edge ideal $I(G) = (x_{i}x_{j} : \{a_i, a_j\}\in E(G))$ associated with $G$ is a squarefree monomial ideal of $S$. If $G$ is a graph on vertices $\{a_1, a_2, \dots, a_n\}$, then $G$ is said to be a path if $E(G) = \{\{a_i, a_{i+1}\}:i\in[n-1]\}$ and $G$ is called a cycle if $E(G) = \{\{a_i, a_{i+1}\}:i\in[n-1]\}\bigcup\{\{a_1, a_n\}\}.$ We use the notations $P_n$ and $C_n$ 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 $e\in E(G),$ $e\cap B \neq\emptyset$ and $B$ is minimal with respect to this property, that is for any proper subset $B'$ of $B,$ then there exists an edge $e\in E(G)$ with $e\cap B' = \emptyset.$ A prime ideal $Q$ is a minimal prime of an ideal $I$ if $I \subset Q$ and if $Q'$ is a prime ideal with $I \subset Q'\subset Q,$ 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 $\alpha(G): = \{ {\rm{min}}|B|:B\;{\rm{is}}\;{\rm{a}}\;{\rm{minimal}}\;{\rm{vertex}}\;{\rm{cover}}\;{\rm{of}}\;G\} ,$ then $\alpha(G) = {\rm{height}}(I).$ 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 ${{\rm{d}}_G}(a, b)$. If no such path exists between $a$ and $b$, then $d_{G}(a, b) = \infty$. The diameter of a connected graph $G$ is ${\rm{diam}}(G): = \max\{{{\rm{d}}_G}(a, b):a, b\in V(G)\}.$ 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,

Definition 2.3. ^[19] The Cartesian product of two graphs $H_1$ and $H_2,$ is a graph, represented by $H_1\Box H_2,$ which has vertex set $V(H_1)\times V (H_2)$(the Cartesian product of sets), and for $(v_1, u_1), (v_2, u_2) \in V(H_1\Box H_2)$, $(v_1, u_1)(v_2, u_2)\in E(H_1 \Box H_2)$, if either

● $v_1v_2 \in E(H_1)$ and $u_1 = u_2$ or

● $v_1 = v_2$ and $u_1u_2\in E(H_2).$

If $n\geq 2$, then the Cartesian product of $P_{2}$ and $P_{n}$ is called ladder graph. We denote this graph by $\mathcal{L}_n$ that is $\mathcal{L}_n: = P_2\Box P_n$. For $n\geq 3$, the Cartesian product of $P_2$ and $C_n$ is said to be a circular ladder graph. We denote this graph by $\mathcal{CL}_n$ that is $\mathcal{CL}_n: = P_2\Box C_n$. Clearly $|V(P_2)\times V (P_n)| = |V(P_2)\times V (C_n)| = 2n,$ thus we have $|V(\mathcal{L}_n)| = |V(\mathcal{CL}_n)| = 2n.$ The graph $\mathcal{L}_n$ has four vertices of degree 2 and $2n-4$ vertices of degree 3 so by using Lemma 2.2, we have $|E(\mathcal{L}_n)| = 3n-2.$ By definition of line graph, it follows that $|E(\mathcal{L}_n)| = |V(L(\mathcal{L}_n))| = 3n-2.$ If $n = 2$, then $\mathcal{L}_n\cong L(\mathcal{L}_n)$. Let $n\geq3$, the graph $L(\mathcal{L}_n)$ has two vertices of degree 2, four vertices of degree 3 and $3n-8$ vertices of degree 4, by Lemma 2.2 we have $|E(L(\mathcal{L}_n))| = 6n-8.$ Similarly, one can show that $|E(\mathcal{CL}_n)| = |V(L(\mathcal{CL}_n))| = 3n$, and $|E(L(\mathcal{CL}_n))| = 6n$. 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

be a short exact sequence of ${\Bbb Z}^{n}$-graded $S$-modules. Then

Lemma 2.5. (Depth Lemma) If $0\rightarrow M \rightarrow N \rightarrow P \rightarrow 0$ is a short exact sequence of modules over a local ring $S$, or a Noetherian graded ring with $S_0$ local, then

1. ${\rm{depth}} (N) \geq \min\{{\rm{depth}}(P), {\rm{depth}}(M)\}$.

2. ${\rm{depth}} (M) \geq \min\{{\rm{depth}}(N), {\rm{depth}}(P)+1\}$.

3. ${\rm{depth}} (P) \geq \min\{{\rm{depth}}(M)-1, {\rm{depth}}(N)\}$.

Lemma 2.6. ([5], Lemma 3.6]). Let $J\subset S$ be a monomial ideal, and $\hat{S} = S[x_{n+1}, x_{n+2}, \dots, x_{n+r}]$ a polynomial ring of $n+r$ variables then

Corollary 2.7. ([21], Corollary 1.3]). Let $J\subset S$ be a monomial ideal. Then ${\rm{depth}} (S/(J : v))\geq {\rm{depth}}(S/J)$ for all monomials $v\notin J.$

Proposition 2.8. ([22], Proposition 2.7]). Let $J\subset S$ be a monomial ideal. Then ${\rm{sdepth}} (S/(J : v))\geq {\rm{sdepth}}(S/J)$ for all monomials $v\notin J.$

Let $\lceil t\rceil,$ $t\in Q$, 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 $n\geq2,$ then ${\rm{depth}}(S/I(P_n)) = \lceil\frac{n}{3}\rceil.$

Stefan proved a similar result for Stanley depth.

Lemma 2.10. ([24], Lemma 4]). Let $n\geq2,$ then ${\rm{sdepth}}(S/I(P_n)) = \lceil\frac{n}{3}\rceil.$

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 $n\geq3,$ then ${\rm{depth}}(S/I(C_n)) = \lceil\frac{n-1}{3}\rceil.$

Theorem 2.12. ([25], Theorem 1.9]). Let $n\geq3,$ then

$(1)$ ${\rm{sdepth}}(S/I(C_n)) = \lceil\frac{n-1}{3}\rceil$, if $n\equiv 0, 2\, (\mod 3)$.

$(2)$ ${\rm{sdepth}}(S/I(C_n))\leq \lceil\frac{n}{3}\rceil$, if $n\equiv 1\, (\mod 3)$.

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 $d = {\rm{diam}}(G),$ then

3.   Results and discussions

In this section, we find some bounds for depth and Stanley depth of the cyclic modules associated to the line graphs of $\mathcal{L}_{n}$ and $\mathcal{CL}_{n}$. We denote the edge ideals of the line graphs of $\mathcal{L}_{n}$ and $\mathcal{CL}_{n}$ with $I_{n}$ and $J_{n}$ respectively. We label the vertices of the line graphs of $\mathcal{L}_{n}$ and $\mathcal{CL}_{n}$ by using three sets of variables $\{a_1, a_2, \dots, a_{n}\},$ $\{b_1, b_2, \dots, b_n\}$ and $\{c_1, c_2, \dots, c_{n}\},$ see Figures 1 and 2. Let $S_n: = K[a_1, a_2, \dots, a_{n-1}, b_1, b_2, \dots, b_n, c_1, c_2, \dots, c_{n-1}]$ and $\overline{S}_{n} = S_n[a_{n}, c_{n}]$ be the rings of polynomials in these variables over the field $K.$ Then $I_{n}$ and $J_{n}$ are squarefree monomial ideals of $S_n$ and $\overline{S}_{n}$ respectively. With the labeling as shown in Figures 1 and 2, we have:

where $\mathcal{G}(I_{n})$ and $\mathcal{G}(J_{n})$ stand for the minimal sets of monomial generators of monomial ideals $I_{n}$ and $J_{n}$ respectively.

Lemma 3.1. For $2\leq n\leq 4$ we have that ${\rm{depth}}(S_n/I_{n}) = {\rm{sdepth}}(S_n/I_{n}) = n-1.$

Proof. If $n = 2,$ then $\mathcal{G}(I_{2}) = \{a_{1}b_{1}, b_{1}c_{1}, a_{1}b_{2}, b_{2}c_{1}\},$ which is a minimal generating set of the edge ideal of $C_4$. Thus by Proposition 2.11 it follows that ${\rm{depth}}(S_2/I_{2}) = 1.$

If $n = 3,$ then $\mathcal{G}(I_{3}) = \mathcal{G}(I_{2})\bigcup\{a_{2}b_{2}, b_{2}c_{2}, a_{2}b_{3}, b_{3}c_{2}, a_{1}a_{2}, c_{1}c_{2}\}.$ Consider the following short exact sequence

Here $(I_{3}:b_{2}) = (a_1, a_2, c_1, c_2)$, so we have $S_3/(I_{3}:b_{2})\cong K[b_{1}, b_{2}, b_{3}],$ thus ${\rm{depth}}(S_3/(I_{3}:b_{2})) = 3.$ Also $(I_{3}, b_{2}) = (a_{1}b_{1}, b_{1}c_{1}, c_{1}c_{2}, c_{2}b_{3}, b_{3}a_{2}, a_{2}a_{1}, b_{2})$, so we have $S_3/(I_{3}, b_{2})\cong K[a_{1}, a_{2}, b_1, b_3, c_{1}, c_{2}]/(a_{1}b_{1}, b_{1}c_{1}, c_{1}c_{2}, c_{2}b_{3}, b_{3}a_{2}, a_{2}a_{1})\cong K[a_{1}, a_{2}, b_1, b_3, c_{1}, c_{2}]/I(C_6),$ by Proposition 2.11, we have ${\rm{depth}}(S_3/(I_{3}, b_{2})) = 2.$ By using Depth lemma on the exact sequence (3.1), we obtain ${\rm{depth}}(S_3/I_{3})\geq2.$ For the upper bound, since $b_{3}\not\in I_{3}$, by Corollary 2.7, we get ${\rm{depth}}(S_{3}/I_{3})\leq {\rm{depth}}(S_{3}/(I_{3}:b_{3})).$ As $(I_{3}:b_{3}) = (a_2, c_2, I_{2})$, thus $S_3/(I_{3}:b_{3})\cong S_{2}/I_{2}[b_{3}],$ by Lemma 2.6, it follows that ${\rm{depth}}(S_3/(I_{3}:b_{3}))\leq {\rm{depth}}(S_2/I_{2})+1 = 1+1 = 2.$ Hence ${\rm{depth}}(S_3/I_{3}) = 2.$ If $n = 4,$ then $\mathcal{G}(I_{4}) = \mathcal{G}(I_{3})\bigcup\{a_{3}b_{3}, b_{3}c_{3}, a_{3}b_{4}, b_{4}c_{3}, a_{2}a_{3}, c_{2}c_{3}\}.$ Consider the following short exact sequence

Here $(I_{4}:b_{3}) = (I_{2}, a_2, c_2, a_3, c_3)$, so we have $S_4/(I_{4}:b_{3})\cong S_{2}/I_{2}[b_{3}, b_{4}],$ thus Lemma 2.6 yields ${\rm{depth}}(S_4/(I_{4}:b_{3})) = {\rm{depth}}(S_2/I_{2})+2 = 1+2 = 3.$ Let $T: = (I_{4}, b_{3}) = (I_{2}, a_{2}b_{2}, b_{2}c_{2}, a_{1}a_{2}, a_{2}a_{3}, c_{1}c_{2}, c_{2}c_{3}, a_3b_4, c_3b_4, b_3).$ Again consider the following short exact sequence

Here $(T:b_{2}) = (a_1, a_2, c_1, c_2, b_{3}, a_3b_4, c_3b_4)$, so we have $S_4/(T:b_{2})\cong K[a_3, b_4, c_3]/(a_3b_4, c_3b_4)[b_1, b_2],$ by Lemmas 2.6 and 2.9, ${\rm{depth}}(S_4/(T:b_{2})) = 1+2 = 3.$ Also $(T, b_{2}) = (a_{1}b_{1}, b_{1}c_{1}, c_{1}c_{2}, c_{2}c_{3}, c_3b_4, b_4 a_3, a_{3}a_{2}, a_{2}a_{1}, b_2, b_3)$, which implies that

thus Proposition 2.11 gives that ${\rm{depth}}(S_4/(T, b_{2})) = 3.$ By applying [15, Lemma 3.1] on the exact sequences (3.2), and (3.3), we have ${\rm{depth}}(S_4/I_{4}) = 3.$

For Stanley depth, if $n = 2,$ then by Theorem 2.12, we have ${\rm{sdepth}}(S_2/I_{2})\leq1.$ Also, we use ^[5] to show that there exist Stanley decompositions of desired Stanley depth.

Thus we have ${\rm{sdepth}}(S_2/I_{2}) = 1.$ If $n = 3,$ then by applying Lemmas 2.4, 2.10, and Theorem 2.12 on the exact sequences (3.1), we have ${\rm{sdepth}}(S_3/I_{3})\geq2.$ For upper bound, since $b_{3}\not\in I_{3}$, by Proposition 2.8, we get ${\rm{sdepth}}(S_{3}/I_{3})\leq {\rm{sdepth}}(S_{3}/(I_{3}:b_{3})).$ As $(I_{3}:b_{3}) = (a_2, c_2, I_{2})$, thus $S_3/(I_{3}:b_{3})\cong S_{2}/I_{2}[b_{3}],$ by Lemma 2.6, it follows that ${\rm{sdepth}}(S_3/(I_{3}:b_{3}))\leq {\rm{sdepth}}(S_2/I_{2})+1 = 1+1 = 2.$ Hence ${\rm{sdepth}}(S_3/I_{3}) = 2.$ 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 ${\rm{sdepth}}(S_4/I_{4})\geq3.$ For upper bound, since $b_{4}\not\in I_{4}$, by Proposition 2.8, we get ${\rm{sdepth}}(S_{4}/I_{4})\leq {\rm{sdepth}}(S_{4}/(I_{4}:b_{4})).$ As $(I_{4}:b_{4}) = (a_3, c_3, I_{3})$, thus $S_4/(I_{4}:b_{4})\cong S_{3}/I_{3}[b_{4}],$ by Lemma 2.6, it follows that ${\rm{sdepth}}(S_4/(I_{4}:b_{4}))\leq {\rm{sdepth}}(S_3/I_{3})+1 = 2+1 = 3.$ Hence ${\rm{sdepth}}(S_4/I_{4}) = 3.$ This completes the proof.

Let $1\leq k\leq n-1$ and $A_{k}: = K[a_{n-1}, a_{n-2}, \dots, a_{n-k}], C_{k}: = K[c_{n-1}, c_{n-2}, \dots, c_{n-k}], D_{k}: = A_{k}\otimes_K C_{k}$ and

$\overline{D}_{k}: = D_{k}\otimes_K K[b_{1}]$ be the subrings of $S_n.$ Let $B_{0}: = (0),$ $B_{j}: = (b_{n}, b_{n-1}, \dots, b_{n-j+1}),$ for $1\leq j\leq n$, and for $3\leq j\leq n-2$, $\mathcal{P}_{j-1}: = (a_{n-j+1}a_{n-j+2}, a_{n-j+2}a_{n-j+3}, \dots, a_{n-2}a_{n-1})$ and $\overline{\mathcal{P}}_{j-1} = (c_{n-j+1}c_{n-j+2}, \dots, c_{n-2}c_{n-1})$ are the squarefree monomial ideals of $S_n.$ In the following theorem, we give some bounds for depth and Stanley depth of $S_n/I_{n}.$

Theorem 3.2. For $n\geq 2$ we have that $\lceil\frac{n}{2}\rceil\leq{\rm{depth}}(S_n/I_{n}), {\rm{sdepth}}(S_n/I_{n})\leq n-1$.

Proof. If $2\leq n\leq 4$, then the result follows by Lemma 3.1. For $n\geq5$, first we prove that $\lceil\frac{n}{2}\rceil\leq{\rm{depth}}(S_n/I_{n})\leq n-1$ using induction on $n$. For $0\leq j\leq n-2$, consider the family of short exact sequences

(1) If $j = 0$, then $(I_{n}:b_{n}) = (I_{n-1}, a_{n-1}, c_{n-1})$, so we have $S_n/(I_{n}:b_{n})\cong S_{n-1}/I_{n-1}[b_{n}],$ the induction hypothesis and Lemma 2.6 give that ${\rm{depth}}(S_n/(I_{n}:b_{n}))\geq\lceil\frac{n-1}{2}\rceil+1 = \lceil\frac{n+1}{2}\rceil.$

(2) If $j = 1$, then $((I_{n}, B_{1}):b_{n-1}) = (I_{n-2}, a_{n-1}, a_{n-2}, c_{n-1}, c_{n-2}, B_{1}),$ so we obtain $S_n/((I_{n}, B_{1}):b_{n-1})\cong S_{n-2}/I_{n-2}[b_{n-1}],$ by induction and Lemma 2.6, it follows that ${\rm{depth}}(S_n/((I_{n}, B_{1}):b_{n-1}))\geq\lceil\frac{n-2}{2}\rceil+1 = \lceil\frac{n}{2}\rceil.$

(3) If $j = 2$, then $((I_{n}, B_{2}):b_{n-2}) = (I_{n-3}, a_{n-2}, a_{n-3}, c_{n-2}, c_{n-3}, B_{2}),$ so we have $S_n/((I_{n}, B_{2}):b_{n-2})\cong S_{n-3}/I_{n-3}[a_{n-1}, b_{n-2}, c_{n-1}],$ the induction hypothesis and Lemma 2.6 give that ${\rm{depth}}(S_n/((I_{n}, B_{2}):b_{n-2}))\geq\lceil\frac{n-3}{2}\rceil+3 = \lceil\frac{n+1}{2}\rceil+1.$

(4) If $3\leq j\leq n-3$, then $((I_{n}, B_{j}):b_{n-j}) =$$({I_{n - (j + 1)}},({a_{n - j + 1}}{a_{n - j + 2}},{a_{n - j + 2}}{a_{n - j + 3}}, \ldots ,{a_{n - 2}}{a_{n - 1}}),({c_{n - j + 1}}{c_{n - j + 2}},{c_{n - j + 2}}{c_{n - j + 3}}, \ldots ,{c_{n - 2}}{c_{n - 1}}),$${a_{n - j}},{a_{n - (j + 1)}},{c_{n - j}},{c_{n - (j + 1)}},{B_j})$, that further implies

By [18], Theorem 2.2.21], we have

By Lemma 2.9, we get ${\rm{depth}}(A_{j-1}/\mathcal{P}_{j-1}) = \lceil\frac{j-1}{3}\rceil = {\rm{depth}}(C_{j-1}/\overline{\mathcal{P}}_{j-1})$ and by induction on $n$, ${\rm{depth}}(S_{n-(j+1)}/I_{n-(j+1)})\geq\lceil\frac{n-(j+1)}{2}\rceil.$ Thus we have

(5) If $j = n-2$, then

$((I_{n}, B_{n-2}):b_{2}) = \Big((a_{3}a_{4}, a_{4}a_{5}, \dots, a_{n-2}a_{n-1}), a_{n-j}, a_{n-(j+1)}, c_{n-j}, c_{n-(j+1)}, (c_{3}c_{4}, \dots, c_{n-2}c_{n-1}), B_{j}\Big),$ so we have $S_n/((I_{n}, B_{n-2}):b_{2})\cong (A_{n-3}/\mathcal{P}_{n-3})\otimes_{K} (C_{n-3}/ \overline{\mathcal{P}}_{n-3})\otimes_{K}K[b_1, b_{2}],$ by [18,Theorem 2.2.21], it follows that ${\rm{depth}}(S_n/((I_{n}, B_{n-2}):b_{2})) = {\rm{depth}}(A_{n-3}/\mathcal{P}_{n-3})+{\rm{depth}}(C_{n-3}/ \overline{\mathcal{P}}_{n-3})+2.$ By Lemma 2.9, ${\rm{depth}}(A_{n-3}/\mathcal{P}_{n-3}) = \lceil\frac{n-3}{3}\rceil = {\rm{depth}}(C_{n-3}/ \overline{\mathcal{P}}_{n-3})$. Thus we have

Also $(I_{n}, B_{n-1}) = (a_{n-1}a_{n-2}, \dots, a_{2}a_{1}, a_{1}b_{1}, b_{1}c_{1}, c_{1}c_{2}, c_{2}c_{3}, \dots, c_{n-2}c_{n-1}, B_{n-1}),$ so we have $S_n/(I_{n}, B_{n-1})\cong \overline{D}_{n-1}/I(\mathcal{P}_{2n-1}).$ Thus by Lemma 2.9, it follows that ${\rm{depth}}(S_n/(I_{n}, B_{n-1})) = \lceil\frac{2n-1}{3}\rceil.$ 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 ${\rm{depth}}(S_n/I_{n})\leq n-1$. For $n\geq 5$, as $b_{n}\not\in I_{n}$, by Corollary 2.7, we have ${\rm{depth}}(S_{n}/I_{n})\leq {\rm{depth}}(S_{n}/(I_{n}:b_{n})).$ Since $S_n/(I_{n}:b_{n})\cong S_{n-1}/I_{n-1}[b_{n}],$ the induction hypothesis and Lemma 2.6 yield ${\rm{depth}}(S_n/(I_{n}:b_{n}))\leq n-1-1+1 = n-1.$

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 ${\rm{sdepth}}(S_n/I_{n})\leq n-1$ by using induction on $n$. For $n\geq 5$, as $b_{n}\not\in I_{n}$, from Proposition 2.8, we get ${\rm{sdepth}}(S_{n}/I_{n})\leq {\rm{sdepth}}(S_{n}/(I_{n}:b_{n})).$ As $S_n/(I_{n}:b_{n})\cong S_{n-1}/I_{n-1}[b_{n}],$ by induction and Lemma 2.6, it follows that ${\rm{sdepth}}(S_n/(I_{n}:b_{n}))\leq n-1-1+1 = n-1.$ This finishes the proof.

Remark 3.3. Clearly ${\rm{diam}}(L(\mathcal{L}_{n})) = n$, then by Theorems 2.13 we have ${\rm{depth}}(S_{n}/I_{n}), {\rm{sdepth}}(S_{n}/I_{n}) \geq\lceil\frac{n+1}{3}\rceil.$ Our Theorem 3.2 shows ${\rm{depth}}(S_{n}/I_{n}), {\rm{sdepth}}(S_{n}/I_{n}) \geq\lceil\frac{n}{2}\rceil$. 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 $\overline{S}_{n}/J_{n},$ we consider two supergraphs $U_n$ and $V_n$ of $L(\mathcal{L}_{n}).$ The vertex and edge sets of $U_n$ are $V(U_n) = V(L(\mathcal{L}_n))\cup\{c_{n}\}$ and $E(U_n) = E(L(\mathcal{L}_n))\cup\{c_{n-1}c_{n}, b_{n}c_{n}\}$ respectively. The vertex and edge sets of $V_n$ are $V(V_n) = V(U_n)\cup\{c_{n+1}\}$ and $E(V_n) = E(U_n)\cup\{c_{1}c_{n+1}, b_{1}c_{n+1}\}$ respectively. For examples of $U_n$ and $V_n,$ see Figure 3. We denote the edge ideals of $U_n$ and $V_n$ with $I_{n}^{\ast}$ and $I_{n}^{\ast\ast}$ respectively. The minimal sets of monomial generators of $I_{n}^{\ast}$ and $I_{n}^{\ast\ast}$ are $\mathcal{G}(I_{n}^{\ast}) = \mathcal{G}(I_{n})\bigcup\big\{c_{n-1}c_{n}, b_{n}c_{n}\big\}$ and $\mathcal{G}(I_{n}^{\ast\ast}) = \mathcal{G}(I_{n}^{\ast})\bigcup\big\{c_{1}c_{n+1}, b_{1}c_{n+1}\big\}.$ First, we find bounds for depth and Stanley depth of the cyclic modules $S^{\ast}_{n}/I_{n}^{\ast}$ and $S^{\ast\ast}_{n}/I_{n}^{\ast\ast}$, where $S^{\ast}_{n} = S_{n}[c_{n}]$ and $S^{\ast\ast}_{n} = S_{n}[c_{n}, c_{n+1}].$

Proposition 3.4. Let $n\geq 2$. Then $\lceil\frac{n}{2}\rceil\leq {\rm{depth}}(S^{\ast}_{n}/I_{n}^{\ast}), {\rm{sdepth}}(S^{\ast}_{n}/I_{n}^{\ast})\leq n$.

Proof. If $n = 2,$ then by using CoCoA, we obtain ${\rm{depth}}(S^{\ast}_{n}/I_{n}^{\ast}) = {\rm{sdepth}}(S^{\ast}_{n}/I_{n}^{\ast}) = 2.$ For $n\geq 3$, we first prove that $\lceil\frac{n}{2}\rceil\leq{\rm{depth}}(S^{\ast}_{n}/I_{n}^{\ast})$ by using induction on $n$. For this, we assume the following short exact sequence

so we have $S^{\ast}_{n}/(I_{n}^{\ast}:c_{n})\cong S^{\ast}_{n-1}/I_{n-1}^{\ast}[c_n].$ By using induction and Lemma 2.6,

so we obtain $S^{\ast}_{n}/(I_{n}^{\ast}, c_{n})\cong S_{n}/I_{n}$. By Theorem 3.2, it follows that ${\rm{depth}}(S^\ast_{n}/(I_{n}^{\ast}, c_{n}))\geq \lceil\frac{n}{2}\rceil.$ Therefore by applying Depth Lemma on the exact sequence (3.4), we get ${\rm{depth}} (S^\ast_{n}/I_{n}^{\ast})\geq\lceil\frac{n}{2}\rceil.$ Now we prove ${\rm{depth}}(S^{\ast}_{n}/I_{n}^{\ast})\leq n$ by using induction on $n$. For $n\geq 3$, as $c_{n}\not\in I_{n}^{\ast}$, from Corollary 2.7, we have ${\rm{depth}}(S^{\ast}_{n}/I_{n}^{\ast})\leq {\rm{depth}}(S^{\ast}_{n}/(I_{n}^{\ast}:c_{n})).$ Since $S^{\ast}_{n}/(I_{n}^{\ast}:c_{n})\cong S^{\ast}_{n-1}/I_{n-1}^{\ast}[c_{n}]$, by induction and Lemma 2.6, ${\rm{depth}}(S^{\ast}_{n}/I_{n}^{\ast})\leq n-1+1 = n.$ It remains to show the result for Stanley depth. For $n\geq 3$, by using induction on $n,$ and by applying Lemma 2.4 on the exact sequence (3.4), we get ${\rm{sdepth}}(S^{\ast}_{n}/I_{n}^{\ast})\geq \lceil\frac{n}{2}\rceil.$ 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 $n\geq 2$, we have that $\lceil\frac{n}{2}\rceil\leq {\rm{depth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast}), {\rm{sdepth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast})\leq n+1$.

Proof. If $n = 2,$ then by using CoCoA, we obtain ${\rm{depth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast}) = {\rm{sdepth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast}) = 2,$ and for $n = 3,$ ${\rm{depth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast}) = {\rm{sdepth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast}) = 3.$ For $n\geq 4$, we first prove that ${\rm{depth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast})\geq\lceil\frac{n}{2}\rceil$ by using induction on $n$. Let us consider the following short exact sequence

so we have that $S^{\ast\ast}_{n}/(I_{n}^{\ast\ast}, c_{n})\cong S^{\ast}_{n}/I_{n}^{\ast}$. Therefore by Proposition 3.4, it follows that

Now consider another short exact sequence

so we have $S^{\ast\ast}_{n}/(T:a_{n-1})\cong S^{\ast\ast}_{n-2}/I_{n-2}^{\ast\ast}[a_{n-1}, c_n].$ Thus induction on $n$ and Lemma 2.6 give that ${\rm{depth}}(S^{\ast\ast}_{n}/(T:a_{n-1}))\geq\lceil\frac{n-2}{2}\rceil+2 = \lceil\frac{n}{2}\rceil+1.$ As $(T, a_{n-1}) = (I_{n-1}^{\ast}, a_{n-1}, b_{n}, c_{n-1}),$ which implies $S^{\ast\ast}_{n}/(T, a_{n-1})\cong S^{\ast}_{n-1}/I_{n-1}^{\ast}$. By Proposition 3.4 and Lemma 2.6, we obtain ${\rm{depth}}(S^{\ast\ast}_{n}/(T, a_{n-1}))\geq \lceil\frac{n-1}{2}\rceil+1 = \lceil\frac{n+1}{2}\rceil.$ Therefore by applying Depth Lemma on the exact sequences (3.5) and (3.6), we get ${\rm{depth}} (S^{\ast\ast}_{n}/I_{n}^{\ast\ast})\geq\lceil\frac{n}{2}\rceil.$ Now we prove ${\rm{depth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast})\leq n+1$. We show this by induction on $n$. For $n\geq 4$, as $a_{n-1}c_{n}\not\in I_{n}^{\ast\ast}$, from Corollary 2.7, we have

Since $S^{\ast\ast}_{n}/(I_{n}^{\ast\ast}:a_{n-1}c_{n})\cong S^{\ast\ast}_{n-2}/I_{n-2}^{\ast\ast}[a_{n-1}, c_{n}]$, by induction and Lemma 2.6, ${\rm{depth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast})\leq n-2+1+2 = n+1.$ It remains to prove the result for Stanley depth. For $n\geq 4$, by using induction on $n,$ and by applying Lemma 2.4 on the exact sequences (2.10) and (2.11) we get ${\rm{sdepth}}(S^{\ast\ast}_{n}/I_{n}^{\ast\ast})\geq \lceil\frac{n}{2}\rceil.$ 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 $n\geq 3$. Then $\lceil\frac{n}{2}\rceil\leq {\rm{depth}}(\overline{S}_{n}/J_{n})\leq n-1,$ and $\lceil\frac{n}{2}\rceil\leq {\rm{sdepth}}(\overline{S}_{n}/J_{n})\leq n.$

Proof. For $3\leq n\leq 4$, by using CoCoA, (for sdepth we use SdepthLib:coc ^[27]), ${\rm{depth}}(\overline{S}_{3}/J_{3}) = {\rm{sdepth}}(\overline{S}_{3}/J_{3}) = 2,$ ${\rm{depth}}(\overline{S}_{4}/J_{4}) = {\rm{sdepth}}(\overline{S}_{4}/J_{4}) = 3.$ Now for $n\geq 5,$ we first show that ${\rm{depth}}(\overline{S}_{n}/J_{n})\geq\lceil\frac{n}{2}\rceil$. Let us consider the following short exact sequence

Now assume another short exact sequence

so we obtain $\overline{S}_{n}/(U, c_{n})\cong S_{n}/I_{n}$. Thus Theorem 3.2 gives that ${\rm{depth}}(\overline{S}_{n}/(U, c_{n}))\geq \lceil\frac{n}{2}\rceil.$

so we get $\overline{S}_{n}/(U:c_{n})\cong S^{\ast\ast}_{n-2}/I_{n-2}^{\ast\ast}[c_n].$ Thus by Proposition 3.5 and Lemma 2.6 we have

Now consider the following short exact sequence

so we have $\overline{S}_{n}/(V:c_{n})\cong S_{n-2}/I_{n-2}[a_{n}, c_n].$ Thus by Theorem 3.2 and Lemma 2.6, we have

so we have that $\overline{S}_{n}/(V, c_{n})\cong S^{\ast\ast}_{n-2}/I_{n-2}^{\ast\ast}$. By Proposition 3.5 and Lemma 2.6, we obtain ${\rm{depth}}(\overline{S}_{n}/(V, c_{n}))\geq \lceil\frac{n-2}{2}\rceil+1 = \lceil\frac{n}{2}\rceil.$ Therefore by applying Depth Lemma on the exact sequences (3.7), (3.8) and (3.9), we get ${\rm{depth}} (\overline{S}_{n}/J_{n})\geq\lceil\frac{n}{2}\rceil.$ Now we prove ${\rm{depth}}(\overline{S}_{n}/J_{n})\leq n-1$. For $n\geq 5$, as $a_{n}c_{n}\not\in J_{n}$, from Corollary 2.7, it follows that ${\rm{depth}}(\overline{S}_{n}/J_{n})\leq {\rm{depth}}(\overline{S}_{n}/(J_{n}:a_{n}c_{n})).$ Since $\overline{S}_{n}/(J_{n}:a_{n}c_{n})\cong S_{n-2}/I_{n-2}[a_{n}, c_{n}]$, by Theorem 3.2 and Lemma 2.6, we have ${\rm{depth}}(\overline{S}_{n}/J_{n})\leq n-2-1+2 = n-1.$

It remains to show the result for Stanley depth. For $n\geq 5$, by applying Lemma 2.4 on the exact sequences (3.7), (3.8) and (3.9), we get that ${\rm{sdepth}}(\overline{S}_{n}/J_{n})\geq \lceil\frac{n}{2}\rceil.$ 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 ${\rm{diam}}(L(\mathcal{CL}_{n})) = \lceil \frac{n+1}{2}\rceil$, then by Theorems 2.13, we have ${\rm{depth}}(\overline{S}_{n}/J_{n}), {\rm{sdepth}}(\overline{S}_{n}/J_{n})\geq\lceil\frac{n+2}{6}\rceil.$ Our Theorem 3.6 shows that ${\rm{depth}}(\overline{S}_{n}/J_{n}), {\rm{sdepth}}(\overline{S}_{n}/J_{n})\geq \lceil\frac{n}{2}\rceil$. Thus we find a much better lower bound for depth and Stanley depth for these classes of edge ideals.

Proposition 3.8. For $n\geq2,$ we have that $\dim(S_{n}/I_{n})\geq n.$

Proof. Let $E = \{a_{1}, a_{2}, \dots, a_{n-1}, c_{1}, c_{2}, \dots, c_{n-1}\}$ be a subset of vertex set $V(L(\mathcal{L}_n)).$ The set $E$ is a vertex cover because it covers all the edges. Now if we remove $a_{i}$ for some $1\leq i\leq n-1$ from set $E$ then the resulting set is not a vertex cover because the edges $a_{i}b_i$ and $a_{i}b_{i+1}$ will not covered. Similarly by removing $c_{i}$ for some $1\leq i\leq n-1$ from set $E$ then the resulting set is not a vertex cover because the edges $c_{i}b_i$ and $c_{i}b_{i+1}$ will not covered. This shows that the set $E$ forms a minimal vertex cover of $I_{n}$. Thus we have ${\rm{height}}(I_{n})\leq 2n-2$. Since $S_{n}$ is a polynomial ring of dimension $3n-2$, which implies that $\dim(S_{n}/I_{n})\geq3n-2-(2n-2) = n.$

Proposition 3.9. Let $n\geq3.$ Then $\dim(\overline{S}_{n}/J_{n})\geq n.$

Proof. As in the Proposition 3.8, one can shows in a similar way that the set $F = \{a_{1}, a_{2}, \dots, a_{n}, c_{1}, c_{2}, \dots, c_{n}\}$ forms a minimal vertex cover of $J_{n}$, therefore ${\rm{height}}(J_{n})\leq2n.$ As $\overline{S}_{n}$ is a polynomial ring of dimension $3n$, thus $\dim(\overline{S}_{n}/J_{n})\geq n.$

Remark 3.10. By Theorem 3.2 and 3.6 we have that ${\rm{depth}} (S_n/I_n), {\rm{depth}} (\overline{S}_n/J_n)\leq n-1$, and by Proposition 3.8 and 3.9 we have $\dim (S_n/I_n), \dim (\overline{S}_n/J_n)\geq n$. Thus graphs $L(\mathcal{L}_n)$ and $L(\mathcal{CL}_n)$ are not Cohen-Macaulay.

Conflict of interest

The authors declare that there is no conflicts of interest in this paper.