STANLEY DEPTH OF POWERS OF THE EDGE IDEAL OF A FOREST

M. R. POURNAKI, S. A. SEYED FAKHARI, AND S. YASSEMI

Abstract. Let 𝕂 be a field and S = 𝕂[x_{1}, …, x_{n}] be the polynomial ring in n variables over the field 𝕂. Let G be a forest with p connected components G_{1}, …, G_{p} and let I = I(G) be its edge ideal in S. Suppose that d_{i} is the diameter of G_{i}, 1 ≤ i ≤ p, and consider d = max{d_{i} | 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the quantity max \{ ⌈\frac{d−t+2}{3}⌉ + p − 1, p\} is a lower bound for depth(S/I^{t}). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/I^{t}). By combining this inequality with Burch’s inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.

1. Introduction

Let 𝕂 be a field and S = 𝕂[x_{1}, …, x_{n}] be the polynomial ring in n variables over the field 𝕂. Let I be a monomial ideal in S. We denote the 𝕂-linear subspace of S generated by all monomials which do not belong to I by I^{c}. Therefore S = I ⊕I^{c}, and S/I ≅ I^{c} as 𝕂-linear spaces. Let u ∈ S be a monomial and Z ⊆ {x_{1}, …, x_{n}}. The 𝕂-subspace u𝕂[Z] whose basis consists of all monomials uv with v ∈ 𝕂[Z] is called a Stanley space of dimension |Z|. Here, as usual, |Z| denotes the number of elements of Z. A decomposition D of I^{c} as a finite direct sum of Stanley spaces is called a Stanley decomposition of S/I. The minimum dimension of a Stanley space in D is called the Stanley depth of D and is denoted by sdepth(D). The quantity

sdepth(S/I) := max {sdepth(D) | D is a Stanley decomposition of S/I}

is called the Stanley depth (sdepth for short) of S/I. For a reader-friendly introduction to sdepth, we refer the reader to [5]. Stanley [9] made a conjecture on Stanley decompositions of ℤ^{n}-graded modules. In the special case when the ℤ^{n}-graded module is S/I, where I is a monomial ideal of S, the conjecture says that sdepth(S/I) ≥ depth(S/I). A monomial ideal I of S is called Stanley if it satisfies Stanley’s conjecture.

Let G be a forest with p connected components G_{1}, …, G_{p} and let I = I(G) be its edge ideal in S. Suppose that d_{i} is the diameter of G_{i}, 1 ≤ i ≤ p, and consider d = max{d_{i} | 1 ≤ i ≤ p}. Morey has shown that for every t ≥ 1, the inequality

2000 Mathematics Subject Classification. Primary: 13C15, 05E99; Secondary: 13C13.

Key words and phrases. Edge ideal, Monomial ideal, Stanley depth, Stanley conjecture.

The research of M. R. Pournaki and S. Yassemi was in part supported by a grant from IPM (No.

90130073 and No. 90130214).

depth(S/I^{t}) ≥ max \{ ⌈\frac{d − t + 2}{3}⌉ + p − 1, p\}

holds true. Assuming Stanley’s conjecture and combining it with Morey’s inequality, one should expect that for every t ≥ 1,

sdepth(S/I^{t}) ≥ max \{ ⌈\frac{d − t + 2}{3}⌉ + p − 1, p\}.

In this paper, we prove the latter inequality. By combining this inequality with Burch’s inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.

We remark that the proofs of the main theorem and its extension are minor modifications of the original proofs for depth, replacing depth by sdepth and the depth lemma by [7, Lemma 2.2]. Nevertheless, here we give complete proofs in order to make the treatment clear and self-contained.

2. The main results

Let G be a graph with vertex-set V (G) = {v_{1}, …, v_{n}}. For a vertex v_{i}, the neighbor set of v_{i} is N(v_{i}) = {v_{j} | v_{i}v_{j} is an edge of G}. The vertex v_{i} is called a leaf if N(v_{i}) is a singleton and is called isolated if N(v_{i}) is empty. There are two types of operations performed on a graph that produce smaller, related graphs that are referred to as minors of G. The one used here is the deletion, denoted by G \ v, which is formed by removing the vertex v from the vertex-set of G and deleting any edge in G that contains v. Paths are special types of graphs. For n ≥ 2, a path P_{n} of length n−1 is a set of n distinct vertices x_{1}, …, x_{n} together with n − 1 edges x_{i}x_{i+1} for 1 ≤ i ≤ n − 1. The diameter of a connected graph is the maximum distance between any two vertices. Here, the distance between two vertices is the minimum length of a path connecting the vertices. Thus if the diameter of a graph G is d, then there exist vertices u and v of G and a path P_{d+1} of length d connecting u and v such that no path of length less than d exists between u and v. Such a path will be referred to as a path realizing the diameter of G.

Proposition 2.1. Let P_{n} be a path and I(P_{n}) be its edge ideal. Then we have

sdepth(S/I(P_{n})) ≥ ⌈\frac{n}{3}⌉ .

Proof. For n with n ≤ 3, the assertion holds by [3, Theorem 1.4]. We therefore assume that n ≥ 4. Consider the following short exact sequence:

0 ⟶ S/(I(P_{n}) : x_{n−1}) ⟶ S/I(P_{n}) ⟶ S/(I(P_{n}), x_{n−1}) ⟶ 0.

Since (I(P_{n}) : x_{n−1}) = (I(P_{n−3}), x_{n−2}, x_{n}), by induction and [6, Theorem 1.1] we obtain that

sdepth(S/(I(P_{n}) : x_{n−1})) = sdepth(S′[x_{n−1}]/I(P_{n−3}))

= sdepth(S′/I(P_{n−3})) + 1

≥ ⌈\frac{n − 3}{3}⌉ + 1 = ⌈\frac{n}{3}⌉ ,

where S′ = 𝕂[x_{1}, x_{2}, …, x_{n−3}]. Since (I(P_{n}), x_{n−1}) = (I(P_{n−2}), x_{n−1}), similarly we obtain that

sdepth(S/(I(P_{n}), x_{n−1})) = sdepth(S′[x_{n−2}]/I(P_{n_{2}})) + 1 ≥ ⌈\frac{n − 3}{3}⌉ + 1.

Now [7, Lemma 2.2] implies that sdepth(S/I(P_{n})) ≥ ⌈\frac{n}{3}⌉, as required. □

By applying Proposition 2.1 and [4, Lemma 2.8] we conclude the following corollary.

Corollary 2.2. The edge ideal of every path is a Stanley ideal.

The sdepth formula for a path given by Proposition 2.1 can be extended to a formula that gives a lower bound for the sdepth of a tree. Note that since the diameter is the maximum distance between vertices, a path realizing the diameter of a tree must connect two leaves of the tree, where a leaf is a vertex with a unique neighbor.

Proposition 2.3. Let G be a tree of diameter d and I = I(G) its edge ideal. Then we have

sdepth(S/I) ≥ ⌈\frac{d + 1}{3}⌉ .

Proof. If d ≤ 2, then ⌈\frac{d+1}{3}⌉ = 1, and so the result follows from [3, Theorem 1.4]. We therefore suppose that d ≥ 3. In this case, the proof is essentially the proof of [4, Proposition 2.9], with depth replaced by sdepth and the depth lemma replaced by [7, Lemma 2.2]. We provide the proof for completeness.

Let u and v be the vertices of G such that the distance between u and v is d, and let P_{d+1} be a path connecting u and v that realizes the diameter of G. Then u is a leaf, and so we may consider y as the unique neighbor of u. Therefore we obtain that (I, y) = (J, y), where J is the edge ideal of the minor G′ of G formed by deleting y. Note that the diameter of G′ is at least d − 2 and also that u is isolated in G′. Thus if S′ is the polynomial ring formed by deleting u, then sdepth(S/(I, y)) = sdepth(S′[u]/(J, y)). Now [6, Theorem 1.1] implies that sdepth(S/(I, y)) = sdepth(S′/(J, y))+1, and again by [6, Theorem 1.1] and using induction we conclude that

sdepth(S/(I, y)) ≥ ⌈\frac{d − 2 + 1}{3}⌉ + 1 ≥ ⌈\frac{d + 1}{3}⌉ .

Now consider (I : y) = (L, N(y)), where L is the ideal of the minor G′′ of G formed by deleting the variables in N(y). Let S′′ be the polynomial ring formed by deleting the variables in y ∪N(y). Then the diameter of G′′ is at least d−3 and y is an isolated vertex, and so

sdepth(S/(I : y)) = sdepth(S′′[y]/L) ≥ ⌈\frac{d − 3 + 1}{3}⌉ + 1 = ⌈\frac{d + 1}{3}⌉

by induction and [6, Theorem 1.1].

We now conclude the proposition by applying [7, Lemma 2.2] to the following short exact sequence:

0 ⟶ S/(I : y) ⟶ S/I ⟶ S/(I, y) ⟶ 0.
□

The goal of this paper is to use graph invariants to provide lower bounds on the sdepth of the powers of the edge ideal of a forest. When the graph is a tree or forest, a lower bound for the sdepth of any power will be given in Theorem 2.7. Since the proof makes repeated use of applying [7, Lemma 2.2] to a pair of sequences, we first state and prove a lemma for simplification.

Lemma 2.4. Let G be a graph and I = I(G) be its edge ideal. If z_{1} and z_{2} are vertices of G and for some s ≥ 0, sdepth(S/(I^{t} : z_{1}z_{2})) ≥ s, sdepth(S/(I^{t}, z_{1})) ≥ s and sdepth(S/((I^{t} : z_{1}), z_{2})) ≥ s, then sdepth(S/I^{t}) ≥ s.

Proof. The proof is essentially the proof of [4, Lemma 3.1] with depth replaced by sdepth and the depth lemma replaced by [7, Lemma 2.2]. We provide the proof for completeness. Applying [7, Lemma 2.2] to the short exact sequence

0 ⟶ S/(I^{t} : z_{1}z_{2}) ⟶ S/(I^{t} : z_{1}) ⟶ S/((I^{t} : z_{1}), z_{2}) ⟶ 0

yields that sdepth(S/(I^{t} : z_{1})) ≥ s. Now again applying [7, Lemma 2.2] to the short exact sequence

0 ⟶ S/(I^{t} : z_{1}) ⟶ S/I^{t} ⟶ S/(I^{t}, z_{1}) ⟶ 0

implies that sdepth(S/I^{t}) ≥ s. □

As a first step toward determining the sdepth of powers of edge ideals of trees and forests, we may now determine a lower bound on the sdepth of the powers of the ideal of a path.

Proposition 2.5. Let P_{n} be a path and I(P_{n}) be its edge ideal. If n ≥ 2 and t ≥ 1, then we have

sdepth(S/I(P_{n})^{t}) ≥ max \{ ⌈\frac{n − t + 1}{3}⌉ , 1\}.

Proof. Note that by [3, Theorem 1.4] and [8, Theorem 5.9], for n ≥ 2 and t ≥ 1 we have sdepth(S/I(P_{n})^{t}) ≥ 1. We therefore prove that

sdepth(S/I(P_{n})^{t}) ≥ ⌈\frac{n − t + 1}{3}⌉ .

We use induction on n and t to prove this latter inequality. Note that by [3, Theorem 1.4] and [8, Theorem 5.9] the result holds for n ≤ 3 and for all t. Also, by Proposition 2.1, the result holds for t = 1 and for all n. We therefore assume that n ≥ 4 and t ≥ 2. By this assumption, the proof is the same as the proof of [4, Proposition 3.2], with depth replaced by sdepth, the depth lemma replaced by [7, Lemma 2.2] and [4, Lemma 3.1] replaced by Lemma 2.4. □

The following lemma is an immediate consequence of [7, Theorem 3.1].

Lemma 2.6. Let G = G_{1} ∪ G_{2} be the disjoint union of two graphs G_{1} and G_{2}. Then we have

sdepth(S/I(G)) ≥ sdepth(S′/I(G_{1})) + sdepth(S′′/I(G_{2})),

where S′ is the polynomial ring over |V (G_{1})| variables, S′′ is the polynomial ring over |V (G_{2})| variables and S is the polynomial ring over |V (G)| variables.

We are now ready to prove the main theorem of this paper. Note that isolated vertices will not be considered as connected components of G.

Theorem 2.7. Let G be a forest with p connected components G_{1}, …, G_{p} and let I = I(G) be its edge ideal in S. Suppose that d_{i} is the diameter of G_{i}, 1 ≤ i ≤ p, and consider d = max{d_{i} | 1 ≤ i ≤ p}. Then for every t ≥ 1 we have

sdepth(S/I^{t}) ≥ max \{ ⌈\frac{d − t + 2}{3}⌉ + p − 1, p\}.

Proof. To prove the theorem we use induction on t and n, where n is the number of nonisolated vertices of G. Without loss of generality, we may assume d = d_{1}. For t = 1 and p = 1 the result follows from Proposition 2.3. For t = 1 and any p ≥ 2 the result follows from Lemma 2.6 and Proposition 2.3. Thus the result holds for t = 1 and every n. Now assume that t ≥ 2. If n = 2, then G = P_{2}, d = 1, and the result holds for all t by Proposition 2.5. Assume n ≥ 3. In this case, the proof is essentially the proof of [4, Theorem 3.4], with depth replaced by sdepth, the depth lemma replaced by [7, Lemma 2.2] and [4, Lemma 3.1] replaced by Lemma 2.4. We provide the proof for completeness.

Fix a path P_{d+1} in G realizing the diameter, let x_{1} be an endpoint of this path (and thus a leaf of G), let y be its unique neighbor, and let N(y) = {x_{1}, …, x_{r}} be the neighbors of y. Note that r ≥ 1 and r is finite. Note also that by [4, Lemma 3.3], at most one of the x_{i}’s is not a leaf. Without loss of generality, assume that x_{i} is a leaf for 1 ≤ i < r. Let I_{j} be the ideal of the minor of G formed by deleting x_{1}, …, x_{j} for every 1 ≤ j ≤ r. Let S_{j} = 𝕂[x_{j+1}, …, x_{n−1}, y] be the subring of S excluding x_{1}, …, x_{j} and let S′_{j} = 𝕂[x_{j+1}, …, x_{n−1}]. Note that for every j, I_{j} ⊆ S_{j} is the edge ideal of a graph involving fewer than n vertices. To use Lemma 2.4 to find the sdepth of S/I^{t}, the sdepths of three ideals must be checked. For simplification, let

s = max \{ ⌈\frac{d − t + 2}{3}⌉ + p − 1, p\}.

Since x_{1} is a leaf, by [4, Lemma 2.10], (I^{t} : x_{1}y) = I^{t−1}, and so by induction on t we have

sdepth(S/(I^{t} : x_{1}y)) ≥ max \{ ⌈\frac{d − (t − 1) + 2}{3}⌉ + p − 1, p\} ≥ s.

To find the sdepth of the second ideal, note that (I^{t}, y) = (J^{t}, y), where J is the edge ideal of the minor G′ of G formed by deleting y. Then G′ is again a forest with fewer than n vertices, it has at least p − 1 connected components, and the generators of J lie in S′_{1}. Thus by induction, sdepth(S′_{1}/J^{t}) ≥ p − 1, and so we have

sdepth(S/(I^{t}, y)) = sdepth(S_{1}[x_{1}]/(J^{t}, y)) ≥ p − 1 + 1 = p.

Suppose d ≤ 3. Then ⌈\frac{d−t+2}{3}⌉ ≤ 1 for every t ≥ 2, and thus we have s = p and sdepth(S/(I^{t}, y)) ≥ s.

For d > 3, note that the number of connected components of J is at least p, since G_{2}, …, G_{p} and d − 2 ≥ 1 edges of P_{d+1} survive in G′. This also implies that the maximal diameter d′ of a connected component of G′ is at least d − 2. Thus by induction on n, we conclude that

sdepth(S_{1}[x_{1}]/(J^{t}, y)) ≥ max \{ ⌈\frac{d − 2 − t + 2}{3}⌉ + p − 1, p\} + 1 ≥ s.

Thus for every d we have sdepth(S/(I^{t}, y)) ≥ s.

Now consider ((I^{t} : y), x_{1}). By [4, Lemma 2.5] we have ((I^{t} : y), x_{1}) = ((I_{1}^{t} : y), x_{1}), where I_{1} is as defined above. Thus

sdepth(S/((I^{t} : y), x_{1})) = sdepth(S_{1}/(I_{1}^{t} : y)).

Note that if r = 1, then d = 1 and s = p as above. Also, for r = 1, (I_{1}^{t} : y) = I_{1}^{t} and I_{1} ⊆ S′_{1} is the edge ideal of a forest with fewer than n vertices and p − 1 connected components. Thus

sdepth(S/((I^{t} : y), x_{1})) = sdepth(S_{1}/I_{1}^{t}) = sdepth(S′_{1}[y]/I_{1}^{t}) ≥ p − 1 + 1 = p,

and so for r = 1 we have sdepth(S/((I^{t} : y), x_{1})) ≥ s.

Now suppose that r ≥ 2. In this case d ≥ 2, since x_{1}, y, x_{r} are all vertices included in P_{d+1}. To find sdepth(S_{1}/(I_{1}^{t} : y)), we use reverse induction on r. First, consider (I_{r}^{t} : y) = I_{r}^{t}. Note that the generators of I_{r}^{t} all lie in S′_{r}, and so sdepth(S_{r}/I_{r}^{t}) = sdepth(S′_{r}[y]/I_{r}^{t}) = sdepth(S′_{r}/I_{r}^{t}) + 1. If d ≤ 3, then I_{r} ⊆ S′_{r} corresponds to a graph with at least p − 1 connected components, and so by induction on n we have sdepth(S′_{r}/I_{r}^{t}) ≥ p − 1 and therefore depth(S_{r}/I_{r}^{t}) ≥ p = s. If d ≥ 4, then I_{r} corresponds to a graph with diameter at least d − 3 with at least p connected components, and so by induction on n we have

sdepth(S′_{r}/I_{r}^{t}) ≥ max \{ ⌈\frac{d − 3 − t + 2}{3}⌉ + p − 1, p\},

and thus

sdepth(S_{r}/(I_{r}^{t} : y)) = sdepth(S_{r}/I_{r}^{t}) = sdepth(S′_{r}/I_{r}^{t}) + 1 ≥ s.

Now assume that sdepth(S_{j}/(I_{j}^{t} : y)) ≥ s for some 2 ≤ j ≤ r. Consider the short exact sequence

0 ⟶ S_{j−1}/(I_{j−1}^{t} : yx_{j}) ⟶ S_{j−1}/(I_{j−1}^{t} : y) ⟶ S_{j−1}/((I_{j−1}^{t} : y), x_{j}) ⟶ 0.

By [4, Lemma 2.5] we have ((I_{j}^{t} : y), x_{j}) = ((I_{j−1}^{t} : y), x_{j}) and therefore by assumption we obtain that sdepth(S_{j−1}/((I_{j−1}^{t} : y), x_{j})) = sdepth(S_{j}/(I_{j}^{t} : y)) ≥ s. By [4, Lemma 2.10], (I_{j−1}^{t} : yx_{j}) = I_{j−1}^{t−1}, since x_{j} is a leaf for j < r and y is a leaf when j = r. The diameter of the graph associated to I_{j−1} is at least d − 1, and so

sdepth(S_{j−1}/(I_{j−1}^{t} : yx_{j})) = sdepth(S_{j−1}/I_{j−1}^{t−1})

≥ max \{ ⌈\frac{d − 1 − (t − 1) + 2}{3}⌉ + p − 1, p\} = s

by induction. Thus by [7, Lemma 2.2] we have sdepth(S_{j−1}/(I_{j−1}^{t} : y)) ≥ s. Hence by reverse induction, sdepth(S_{1}/(I_{1}^{t} : y)) ≥ s.

Since sdepth(S/((I^{t} : y), x_{1})) = sdepth(S_{1}/(I_{1}^{t} : y)), as above we conclude that sdepth(S/((I^{t} : y), x_{1})) ≥ s. Thus by applying Lemma 2.4 with z_{1} = y and z_{2} = x_{1}, we obtain that sdepth(S/I^{t}) ≥ s, as required. □

Let 𝕂 be a field and S = 𝕂[x_{1}, …, x_{n}] be the polynomial ring in n variables over the field 𝕂, and let I ⊂ S be a monomial ideal. A classical result by Burch [2] so that

min_{t} depth(S/I^{t}) ≤ n − ℓ(I),

where ℓ(I) is the analytic spread of I, that is, the dimension of R(I)/𝔪R(I), where R(I) = ⨁_{n=0}^{∞} I^{n} = S[It] ⊆ S[t] is the Rees ring of I and 𝔪 = (x_{1}, …, x_{n}) is the maximal ideal of S. By a theorem of Brodmann [1], depth(S/I^{t}) is constant for large t. We call this constant value the limit depth of I, and denote it by lim_{t→∞} depth(S/I^{t}). Brodmann improved the Burch inequality by showing that

lim_{t→∞} depth(S/I^{t}) ≤ n − ℓ(I).

Corollary 2.8. Let G be a forest. Then there exists a positive integer k_{0} such that for every k ≥ k_{0}, the ideal I^{k} is Stanley.

Proof. One can easily check that for every forest G, the analytic spread of I(G) is equal to n−p, where p is the number of connected components of G (see also [10, Page 50]). Therefore by the minimum value given in [4, Theorem 3.4] and the argument above, depth(S/I^{t}) = p, for sufficiently large k. Now Theorem 2.7 completes the proof. □

3. Appendix: A generalization of the main theorem

The proof of Theorem 2.7 depends heavily on the existence of a vertex y such that at most one neighbor is not a leaf. A careful examination of [4, Lemma 3.3] guarantees that any tree with diameter d ≥ 3 will contain at least two such vertices that are not themselves leaves, namely the neighbors of the two leaves of a path realizing the diameter. Call a vertex v of G a near leaf of G if v is not a leaf and N(v) contains at most one vertex that is not a leaf. Let q denote the number of near leaves of G. Then the bound given in Theorem 2.7 can be strengthened using essentially the same proof. However, a strengthening of Proposition 2.3 is needed.

Lemma 3.1. Let G be a tree with diameter d ≥ 1 and q near leaves, and let I = I(G) be its edge ideal. Then we have

sdepth(S/I) ≥ ⌈\frac{d + q − 1}{3}⌉ .

Proof. For q ≤ 2 the lemma holds by Proposition 2.3, and so we assume that q ≥ 3. Note that for a connected graph, if two near leaves are adjacent, then we obtain that d = 3 and q = 2, since all other vertices must be leaves. Thus for q ≥ 3, no neighbor of a near leaf is a near leaf. Let P_{d+1} be a path realizing the diameter of G with vertices x_{1}, x_{2}, …, x_{d+1}. In this stage, we continue the proof as the proof of [4, Lemma 3.6], with depth replaced by sdepth and the depth lemma replaced by [7, Lemma 2.2]. □

Corollary 3.2. Let G be a forest with p connected components G_{1}, …, G_{p} and let I = I(G) be its edge ideal in S. Suppose that d_{i} is the diameter of G_{i}, 1 ≤ i ≤ p, and consider d = max{d_{i} | 1 ≤ i ≤ p}. If q is the number of near leaves of a component of diameter d, then for every t ≥ 1 we have

sdepth(S/I^{t}) ≥ max \{ ⌈\frac{d − t + q}{3} ⌉ + p − 1, p\}.

Proof. The proof is essentially the proof of [4, Corollary 3.7], with depth replaced by sdepth and [4, Lemma 3.1] replaced by Lemma 2.4. We provide the proof for completeness.

As before, assume that d = d_{1} and q is the number of near leaves of G_{1}. When counting near leaves in this proof, only those in minors of G_{1} will be considered. For simplification, let

s = max \{ ⌈\frac{d − t + q}{3} ⌉ + p − 1, p\}.

If d ≤ 3, then q ≤ 2, and so the result holds by Theorem 2.7. When t = 1, the result follows from Lemma 3.1, [3, Theorem 1.4] and Lemma 2.6. Fix the notation as in Theorem 2.7 and assume q ≥ 3 and d ≥ 4. Then (I^{t} : x_{1}y) = I^{t−1} as before, and

sdepth(S/(I^{t} : x_{1}y)) ≥ max \{ ⌈\frac{d − t + 1 + q}{3} ⌉ + p − 1, p\} ≥ s

by induction on t. Now (I^{t}, y) = (J^{t}, y), where J is the edge ideal of G′. Note that the diameter of G′ is at least d − 2, G′ has at least q − 1 near leaves, and x_{1} is an isolated vertex. Thus by induction on n,

sdepth(S/(I^{t}, y)) ≥ max \{ ⌈\frac{d − 2 − t + q − 1}{3}⌉ + p − 1, p\} + 1 ≥ s,

as before.

As in the proof of Theorem 2.7, since d ≥ 4, we have r ≥ 2 and (I_{r}^{t} : y) = I_{r}^{t}. Note that the diameter of the graph corresponding to I_{r} is at least d − 3. Also note that because x_{r} lies on a path of maximal length and has distance two from a leaf on that path, every path that connects x_{r} to a leaf and does not contain any other vertex in P_{d+1} must have length at most two. Thus any near leaves on such a path must be directly adjacent to x_{r}.

Let a be the number of near leaves adjacent to x_{r} but not on P_{d+1}. Then the graph of I_{r} has p + a connected components, and the number of near leaves of I_{r} in the connected component containing P_{d−2} is at least q − a − 1 when d ≥ 5, since y is also no longer a near leaf. If d = 4, the number of near leaves of I_{r} is 0 and a = q − 2 since every near leaf is adjacent to x_{r}, including both that lie on P_{d+1}. Hence, for d = 4, the minor associated to I_{r} consists of P_{2}, together with a additional connected components and at least one isolated vertex. Thus

\begin{aligned}sdepth(S_{r}/I_{r}^{t}) & ≥ 1 + a + p − 1 + 1 ≥ q + p − 1 \\ & ≥ ⌈\frac{q + 3}{3}⌉ + p − 1 = ⌈\frac{d + q − 1}{3}⌉ + p − 1 ≥ s \end{aligned}

for d = 4 and q ≥ 3.

Assume that d = 5. Hence, either the connected component containing P_{d−2} contains two near leaves, and thus has diameter at least 3 = d − 2, or a ≥ 1. In the first case, by induction,

sdepth(S_{r}/I_{r}^{t}) = sdepth(S′_{r}/I_{r}^{t}) + 1

≥ max \{ ⌈\frac{d − 2 − t + q − 1}{3}⌉ + p − 1, p\} + 1 ≥ s.

In the second case, a ≥ 1, and so

sdepth(S_{r}/I_{r}^{t}) = sdepth(S′_{r}/I_{r}^{t}) + 1

≥ max \{ ⌈\frac{d − 3 − t + q − a − 1}{3}⌉ + p − 1, p\} + a + 1 ≥ s.

If d > 5, then d − 2 ≥ 4, and so either P_{d−2} contains two near leaves, one of which was not a near leaf of I, or P_{d−2} is not maximal, and so the diameter of I_{r} is at least d − 2. In the first case, the number of near leaves is q − a and the diameter is at least d − 3, and so

sdepth(S_{r}/I_{r}^{t}) = sdepth(S′_{r}/I_{r}^{t}) + 1

≥ max \{ ⌈\frac{d − 3 − t + q − a}{3}⌉ + p − 1, p\} + a + 1 ≥ s.

In the second case, the number of near leaves is q − a − 1 and the diameter is at least d − 2, so again

sdepth(S_{r}/I_{r}^{t}) = sdepth(S′_{r}/I_{r}^{t}) + 1

≥ max \{ ⌈\frac{d − 2 − t + q − a − 1}{3}⌉ + p − 1, p\} + a + 1 ≥ s.

Now by reverse induction on j, using the proof of Theorem 2.7, we have

sdepth(S_{1}/(I_{1}^{t} : y)) = sdepth(S/((I^{t} : y), x_{1})) ≥ s.

As in Theorem 2.7, the result now follows from Lemma 2.4. □

Acknowledgment

The authors would like to thank the referee for a careful reading of the paper and for valuable comments.

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M. R. Pournaki, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

 E-mail address: pournaki@ipm.ir
 URL: http://math.ipm.ac.ir/pournaki/

S. A. Seyed Fakhari, Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran.

 E-mail address: fakhari@ipm.ir
 URL: http://math.ipm.ac.ir/fakhari/

S. Yassemi, School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran, and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran.

 E-mail address: yassemi@ipm.ir
 URL: http://math.ipm.ac.ir/yassemi/