Stable value of depth of symbolic powers of edge ideals of graphs

Let $G$ be a simple graph on $n$ vertices. We introduce the notion of bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that $$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le \operatorname{bc}(G),$$ where $I(G)$ denotes the edge ideal of $G$ and $S = \mathrm{k}[x_1, \ldots, x_n]$ is a standard graded polynomial ring over a field $\mathrm{k}$. We further compute the depth of symbolic powers of edge ideals of several classes of graphs, including odd cycles and whisker graphs of complete graphs to illustrate the cases where the above inequality becomes equality.


Introduction
Let I be a homogeneous ideal in a standard graded polynomial ring S = k[x 1 , . . ., x n ] over a field k.While the depth function of powers of I is convergent by the result of Brodmann [1979], the depth function of symbolic powers of I is more exotic.Nguyen and N. V. Trung [2019] proved that for every positive eventually periodic function f : ‫ގ‬ → ‫ގ‬ there exists an ideal I such that depth S/I (s)  = f (s) for all s ≥ 1, where I (s) denotes the s-th symbolic power of I .On the other hand, when I is a squarefree monomial ideal, by the result of Hoa et al. [2017] and Varbaro [2011], lim s→∞ depth S/I (s)  = min{depth S/I (s) where ℓ s (I ) is the symbolic analytic spread of I .Nonetheless, given a squarefree monomial ideal I , computing the stable value of depth of symbolic powers of I is a difficult problem even in the case of edge ideals of graphs.
Let us now recall the notion of the edge ideals of graphs.Let G be a simple graph with the vertex set V (G) = {1, . . ., n} and edge set E(G).The edge ideal of G, denoted by I (G), is the squarefree monomial ideal generated by x i x j where {i, j} is an edge of G. Trung [2016] showed that lim s→∞ depth S/I (G) s equals the number of bipartite connected components of G, and that depth S/I (G) s stabilizes when it reaches the limit depth.By the results in [Nguyen and Vu 2019;Hà et al. 2020], we may assume that G is a connected graph when considering the depth of (symbolic) powers of the edge ideal of G.In this case, the result of Trung [2016] can be written as (1-1) lim s→∞ depth S/I (G) s = 1 if G is bipartite, 0, otherwise, and the stabilization index of depth of powers of I (G), denoted by dstab(I (G)), is the smallest exponent s such that depth S/I (G) s equals the limit depth of powers.
Since we expect that the depth functions of symbolic powers of edge ideals are nonincreasing, this property should hold for symbolic powers of I (G) as well.Hien, Lam, and Trung [2024] characterized graphs for which lim s→∞ depth S/I (G) (s)  = 1 and proved that the stabilization index of depth of symbolic powers in this case is also the smallest exponent s such that depth S/I (G) (s)  = 1.For a general nonbipartite graph G, we do not know the value lim s→∞ depth S/I (G) (s) .
In this paper, we introduce the notion of bipartite connectivity of G and show that this is tightly connected to the stable value of depth of symbolic powers of I (G).Let B(G) denote the set of maximal induced bipartite subgraphs H of G, i.e., for any v ∈ V (G) \ V (H ), the induced subgraph of G on V (H ) ∪ {v} is not bipartite.Note that H might contain isolated vertices.Since H is maximal, it contains at least one edge.Then we define bc(G) = min{c(H ) | H ∈ B(G)} and call it the bipartite connectivity number of G, where c(H ) is the number of connected components of H .With this notation, the result of Hien et al. [2024] can be stated as lim s→∞ depth S/I (G) (s)  = 1 if and only if bc(G) = 1, i.e., there exists an induced connected bipartite subgraph H of G such that H dominates G.In this paper, we generalize this result and prove: Theorem 1.1.Let G be a simple graph.Then lim s→∞ depth S/I (G) (s)  ≤ bc(G).
In contrast to (1-1), we show that the limit depth of symbolic powers of I (G) could be any positive number even when G is a connected graph.
Proposition 1.2.Let n ≥ 2 be a positive number and W n = W (K n ) be the whisker graph on the complete graph on n vertices.Then, bc(W n ) = n − 1 and We also note that the inequality in Theorem 1.1 could be strict as given in the following example.
Example 1.3.Let W be the graph obtained by gluing two whiskers at the vertices of a 3-cycle.Then bc(W ) = 3 while depth S/I (W ) (s)  = Nonetheless, if we cluster the isolated points in a maximal bipartite subgraph H of G by the bouquets in G then we obtain a finer invariant of G that gives the stable value of depth of symbolic powers.More precisely, assume that where H i are connected components of H with at least one edge and p 1 , . . ., p t are isolated points in H .We say that p i 1 , . . ., p i u are clustered if there exists a the restricted bipartite connectivity number of G.It is easy to see that for the graph W in Example 1.3, we have bc ′ (W ) = 2.We conjecture that: Conjecture 1.4.Let G be a simple graph.Then lim s→∞ depth S/I (G) (s)  = bc ′ (G).
We verify this conjecture for whisker graphs of complete graphs.
Theorem 1.5.Let a = (a 1 , . . ., a n ) ∈ ‫ގ‬ n and W a be the graph obtained by gluing a i leaves to the vertex i of a complete graph K n .Assume that a i ≥ 1 for all i = 1, . . ., n.
Then bc ′ (W a ) = n − 1 and Finally, we compute the depth of symbolic powers of edge ideals of odd cycles by extending our argument in [Minh et al. 2023].This shows that the bound for the index of depth stability of symbolic powers of I given in [Hien et al. 2024] is sharp.
Theorem 1.6.Let I (C n ) be the edge ideal of a cycle of length n = 2k + 1 ≥ 5. Then In particular, sdstab(I (C n )) = n − 2, where sdstab(I ) is the index of depth stability of symbolic powers of I .
We structure the paper as follows.In Section 2, we set up the notation and provide some background.In Section 3, we prove Theorem 1.1 and compute the depth of symbolic powers of edge ideals of whisker graphs of complete graphs.In Section 4, we prove Theorem 1.6.

Preliminaries
In this section, we recall some definitions and properties concerning depth, graphs and their edge ideals, and the symbolic powers of squarefree monomial ideals.The interested readers are referred to [Bruns and Herzog 1993] for more details.
Throughout the paper, we denote by S = k[x 1 , . . ., x n ] a standard graded polynomial ring over a field k.Let m = (x 1 , . . ., x n ) be the maximal homogeneous ideal of S.
Depth.For a finitely generated graded S-module L, the depth of L is defined to be where H i m (L) denotes the i-th local cohomology module of L with respect to m.We have the following estimates on depth along short exact sequences (see [Bruns and Herzog 1993, Proposition 1.2.9]).
We make repeated use of the following two results in the sequence.The first one is [Rauf 2010, Corollary 1.3].The second one is [Caviglia et al. 2019, Theorem 4.3].
Lemma 2.2.Let I be a monomial ideal and f a monomial such that f / ∈ I .Then depth S/I ≤ depth S/(I : f ).
Finally, we also use the following simple result.
(2) It follows from (1) and the fact that depth(R 2 /(x a+1 , . . ., Depth of Stanley-Reisner rings.Let be a simplicial complex on the vertex set V ( ) = [n] = {1, . . ., n}.For a face F ∈ , the link of F in is the subsimplicial complex of defined by For each subset F of [n], let x F = i∈F x i be a squarefree monomial in S. We now recall the Stanley-Reisner correspondence.
Definition 2.5.For a squarefree monomial ideal I , the Stanley-Reisner complex of I is defined by For a simplicial complex , the Stanley-Reisner ideal of is defined by The Stanley-Reisner ring of is k[ ] = S/I .
Definition 2.6.The q-th reduced homology group of with coefficients over k, denoted H q ( ; k) is defined to be the q-th homology group of the augmented oriented chain complex of over k.
From the Hochster's formula, we deduce that: Lemma 2.7.Let be a simplicial complex.Then . By Hochster's formula [Bruns and Herzog 1993, Theorem 5.3.8], the conclusion follows.□ We will also use the following nerve theorem from [Borsuk 1948].First, we recall the definition of the nerve complex.Assume that the set of maximal facets of is A = {A 1 , . . ., A r }.The nerve complex of , denoted by N ( ) is the simplicial complex on the vertex set Theorem 2.8.Let be a simplicial complex.Then for all integer i, we have Graphs and their edge ideals.Let G denote a finite simple graph over the vertex set V (G) = [n] = {1, 2, . . ., n} and the edge set E(G).The edge ideal of G is defined to be For simplicity, we often write . By abuse of notation, we also call x i a vertex of G and x i x j ∈ I (G) an edge of G.
A path P n of length n − 1 is the graph on [n] whose edges are {i, i For a vertex x ∈ V (G), let the neighborhood of x be the subset A leaf is a vertex of degree 1.The unique edge attached to a leaf is called a leaf edge.Denote d G (x) the number of nonleaf edges incident to x.
Projective dimension of edge ideals of weakly chordal graphs.A graph G is called weakly chordal if G and its complement do not contain an induced cycle of length at least 5.The projective dimension of edge ideals of weakly chordal graphs can be computed via the notion of strongly disjoint families of complete bipartite subgraphs, introduced in [Kimura 2016].For a graph G, we consider all families of (noninduced) subgraphs B 1 , . . ., B g of G such that: (1) Each B i is a complete bipartite graph for 1 ≤ i ≤ g.
(3) There exist an induced matching e 1 , . . ., e g of G for each Such a family is termed a strongly disjoint family of complete bipartite subgraphs.We define where the maximum is taken over all the strongly disjoint families of complete bipartite subgraphs B 1 , . . ., B g of G.We have the following result of Nguyen and Vu [Nguyen and Vu 2016, Theorem 7.7].
Theorem 2.9.Let G be a weakly chordal graph with at least one edge.Then We now use it to compute the depth of the edge ideals of whisker graphs of complete graphs.
Lemma 2.10.Let a = (a 1 , . . ., a n ) ∈ ‫ގ‬ n and W a be the graph obtained by gluing a i leaves to the vertex i of a complete graph K n .Assume that Proof.From the definition of W a , it is clear that W a is a chordal graph.For simplicity of notation, we assume that For any edges e 1 , e 2 of W a , we have . Now, assume that y i, j ∈ U 1 for some i, j.Then x i ∈ U 2 and y k,l / ∈ V (B) for any k ̸ = i since B is a complete bipartite graph.Hence, |V (B)| ≤ n + a i .Therefore, for any complete bipartite subgraph B of W a , we have The conclusion follows from the Auslander-Buchsbaum formula.□ Symbolic powers of edge ideals.Let I be a squarefree monomial ideal in S with the irreducible decomposition The s-th symbolic power of I is defined by By the proof of [Kimura et al. 2018, Theorem 5.2], we have: Lemma 2.11.Assume that e is a leaf edge of G. Then for all s ≥ 2 we have I (G) (s)  : e = I (G) (s−1) .In particular, depth S/I (G) (s) is a nonincreasing function.We also have the following simple result that will be used later.

Stable value of depth of symbolic powers of edge ideals
In this section, we prove that the stable value of depth of symbolic powers of edge ideals is at most the bipartite connectivity number of G.We assume that We first introduce some notation.Let H be a connected bipartite graph with the partition V (H ) = X ∪ Y .The bipartite completion of H , denoted by H is the complete bipartite graph K X,Y .Now, assume that where H 1 , . . ., H c are connected components of H with at least one edge, and p 1 , . . ., p t are isolated points of H . Then the bipartite completion of H is defined by where H is the bipartite completion of H .
Proof.Since variables corresponding to isolated points do not appear in I (H ), we may assume that H does not have isolated points.Assume that where H i are connected components of H with at least one edge.Let a i = d(H i ).
Note that x a i is equal to the product of nonleaf edges of H i , hence Then we have f m x a ∈ I (H ) s+1 for some m > 0. Thus, we must have f m i x a i ∈ I (H i ) s i +1 for some i.Hence, we may assume that H is connected.The conclusion then follows from [Trung 2016, Lemma 3.1]   ∈ I (G) (s+1) .We claim that (3-1) By Lemma 3.1, it is sufficient if we prove that x j ∈ I (G) (s+1) : x a for all j ∈ V (G)\V (H ).Since the induced subgraph of G on { j}∪ H is not bipartite, there must exist a connected component, say H 1 of H such that the induced subgraph of G on V (H 1 ) ∪ { j} has an odd cycle.Let G 1 be the induced subgraph of G on H 1 ∪ { j}.Let j, 1, . . ., 2k be an induced odd cycle in G 1 .Then j=1 e j is a product of k edges of H 1 .By the definition of a, we have x a 1 equals the products of all edges of H 1 .In other words, we have By Lemma 2.2 and equation (3-1), we deduce that For any t ≥ s+1, let x c = x a •e t−s−1 where e is an arbitrary edge of H . Then we have x c / ∈ I (G) (t) and I (G) (t) : x c ⊇ I (G) (s+1) : x a .Hence, depth S/I (G) (t)  ≤ c(H ) for all t ≥ s + 1.The conclusion follows.□ Definition 3.3.Let G be a simple graph.Denote by B(G) the set of all maximal induced bipartite subgraphs of G.The bipartite connectivity number of G is defined by bc We are now ready for the proof of Theorem 1.1.
Proof of Theorem 1.1.The conclusion follows immediately from the definition and Lemma 3.2.□ We now prove Proposition 1.2 giving an example of connected graphs for which the above inequality is equality and that the limit depth of symbolic powers of I (G) could be any positive number.
Proof of Proposition 1.2.We may assume that V (W n ) = {x 1 , . . ., x n , y 1 , . . ., y n } and Let H be a maximal bipartite subgraph of W n .Then y 1 , . . ., y n ∈ H and H contains at most two vertices in {x 1 , . . ., x n }.By the maximality of H , we deduce that H must be the induced subgraph of W n on {y 1 , . . ., y n } ∪ {x i , x j } for some i ̸ = j.Hence, c(H By Lemma 2.11, depth S/I (W n ) (s) is nonincreasing.Furthermore, we have Hence, depth S/I (W n ) (2) ≤ n − 1.It remains to prove that depth S/I (W n ) (s)  ≥ n − 1 for all s ≥ 2. We prove by induction on n and s the following statement.Let Since m k , the maximal homogeneous ideal of S k , is not an associated prime of I k , depth S k /I (s)  k ≥ 1 for all k.Thus, we may assume that s ≥ 2 and n ≥ k ≥ 3.By Lemma 2.3, By Lemma 2.11, k .Thus, by induction, it suffices to prove that The conclusion follows from induction on k and Lemma 2.1.□ The inequality in Theorem 1.1 might be strict.We will now define a finer invariant of G which we conjecture to be equal to the stable value of depth of symbolic powers of I (G).
We need a preparation lemma to prove Theorem 1.5.Lemma 3.5.Let a = (a 1 , . . ., a n ) ∈ ‫ގ‬ n be such that a i ≥ 1 for all i = 1, . . ., n.Let W a be a graph whose vertex set and edge set are V (W a ) = {x 1 , . . ., x n , y 1,1 , . . ., y 1,a 1 , . . ., y n,1 , . . ., y n,a n }, Proof.For simplicity of notation, we set We also denote I = I (W a ) and For each C ⊆ V (W a ), let m C = x∈C x be a monomial in Since W a is a chordal graph, by [Sullivant 2008, Theorem 3.10], we have The cliques C 1 , . . ., C t are not necessarily distinct.In W a , C ⊆ V (W a ) is a clique if and only if either C = {x i , y i, j } for some i = 1, . . ., t and j = 1, . . ., a i or C ⊆ X .In particular, (x n) for all edges e of W a and (x 1 y 1, j 1 ) for all j 1 , . . ., j n such that 1 ≤ j ℓ ≤ a ℓ .Hence, We now prove by induction on n the reverse containment The base case n = 2 is clear.Thus, assume that n ≥ 3. Let C 1 , . . ., C t be cliques of W a such that t i=1 It suffices to prove that M/gcd(M, f ) ∈ J .Since |C i | ≤ n for all i = 1, . . ., t, we must have t ≥ 2. We have two cases.
Case 1. C i ∩ Y = ∅ for all i = 1, . . ., t.In this case, we have M ∈ I (K n ) (n) .By Lemma 2.12, we deduce that M/gcd(M, f Case 2. C i ∩ Y ̸ = ∅ for some i ∈ {1, . . ., t}.Since Y is the set of leaves, we deduce that |C i | = 2.For simplicity, we assume that In particular, we deduce that x 1 y 1,1 | M/gcd(M, f ).Hence, M/gcd(M, f ) ∈ J .Thus, we may assume that , where a ′ = (a 2 , . . ., a n ) and W a ′ is the whisker graph obtained by gluing a i leaves to the vertex i of the complete graph on {2, . . ., n}.Since y 1,1 does not divide f , we deduce that y By induction on n, the conclusion follows.□ Proof of Theorem 1.5.We may assume that a 1 ≥ a 2 ≥ • • • ≥ a n ≥ 1.We keep the notations as in Lemma 3.5.For ease of reading, we divide the proof into several steps.
Step 1. bc ′ (W a ) = n − 1.As in the proof of Proposition 1.2, we deduce that a maximal induced bipartite subgraph H of W a is an induced subgraph of W a on Y ∪ {x i , x j } for some i ̸ = j.For such H , we have c(H ) = |a| − (a i + a j ) + 1 but c ′ (H ) = n − 1 as {y ℓ,1 , . . ., y ℓ,a ℓ } can be clustered into a bouquet in G for all ℓ = 1, . . ., n.Thus, bc(W a ) = a 3 + • • • + a n + 1 and bc ′ (W a ) = n − 1.
Step 2. depth S/I (W a ) (s) ≥ n − 1 for all s ≥ 1 and all a such that a i ≥ 1 for i = 1, . . ., n.
Therefore, the Stanley-Reisner complex (J ) of J has exactly n facets Hence, F 1 ∩ • • • ∩ F n = ∅ and for any j, we have Therefore, the nerve complex of (J ) is isomorphic to the n − 2-sphere.By Theorem 2.8, H n−2 ( (J ); k) ̸ = 0.By Lemma 2.7, the conclusion follows.□ Remark 3.6.
(1) The notion of maximal bipartite subgraphs of a graph has been studied by many researchers as early as in [Erdős 1965;Malle 1982].They are interested in finding the maximum number of edges of a maximal bipartite subgraph of G.
(2) In general, the problem of finding a maximum induced bipartite subgraph of a graph is NP-complete [Lewis and Yannakakis 1980].Nonetheless, we do not know if the problem of computing the bipartite connectivity number or restricted bipartite connectivity number is NP-complete.
But depth S/I (s) = 1 for all s ≥ 2, regardless of the characteristic of the base field k.
(2) By the result of Trung [2016], the stable value of depth of powers of edge ideals of graphs does not depend on the characteristic of the base field k.If Conjecture 1.4 holds, the stable value of depth of symbolic powers of edge ideals also does not depend on the characteristic of the base field k.This is in contrast to the asymptotic behavior of the regularity of (symbolic) powers of edge ideals as [Minh and Vu 2022, Corollary 5.3] shows that the linearity constant of the regularity function of (symbolic) powers of edge ideals of graphs might depend on the characteristic of the base field k.

Depth of symbolic powers of edge ideals of cycles
In this section, we compute the depth of symbolic powers of edge ideals of cycles.The purpose of this is twofold.First, together with Proposition 1.2, this gives the first classes of nonbipartite graphs where one computes explicitly the depth of symbolic powers of their edge ideals.Second, this shows that the stabilization index of depth of symbolic powers of G is tightly connected to the stabilization index of depth of powers of maximal induced bipartite subgraphs of G.
We fix the following notation.Let S = k[x 1 , . . ., x n ] and C n be a cycle of length n.For each i = 1, . . ., n − 1, we denote e .
We recall the following results (Lemmas 3.4,3.10,3.11,andWe now establish some preparation results.Proof.Let p 1 , . . ., p t be the associated primes of I .Then Since p i is generated by k +1 variables for all i = 1, . . ., t, we have p s i : f = p s−k−1 i .Hence, I (s)  : f = I (s−k−1) = I s−k−1 since s ≤ 2k − 1. □ We are now ready for the proof of Theorem 1.6.
It suffices to prove the following two claims.
Claim 1.For any nonempty subgraph H of C n , we have depth S/(I (s) + I (H )) ≥ ϕ(n, s).
Proof of Claim 1.Since k + 1 ≤ s ≤ 2k − 1, by (4-1), we have that For any nonempty subgraph H of C n , we have f ∈ I (H ).Therefore, we have I (s) + I (H ) = I s + I (H ).The conclusion follows from Lemma 4.2.Proof of Claim 2. Let J = I (s)  + I (H ) + (x n−2 x n−1 x n ) and e = x n−2 x n−1 .Note that J + (e) can be expressed as I (s)  + I (H 1 ) for some subgraph H 1 of C n and J : e = I (P n−1 ) s−1 + I (H ′ ) + (x n ) where H ′ is a subgraph of P n−1 .The claim follows from Lemma 2.3, Claim 1, and Lemma 4.1.The conclusion follows.□ Remark 4.7.For cycles C 2k of even length, by the result of Simis, Vasconcelos, and Villarreal [1994], I (C 2k ) (s)  = I (C 2k ) s for all s ≥ 1.The depth of powers of the edge ideal of C 2k has been computed in [Minh et al. 2023, Theorem 1.1].
and [Minh et al. 2022,  Lemma 2.19].□Now,assume that H is a maximal induced bipartite subgraph of G, that is, for any p t } where H i are connected components of H with at least one edge and p 1 , . . ., p t are isolated points of H . Then c(H ) = c + t is the number of connected components of H .We have ≥ |E(H )| + 1, where c(H ) is the number of connected components of H . Proof.Assume that H = H 1 ∪ • • • ∪ H c ∪ { p 1 , . . ., p t } where H 1 , . . ., H c are connected components of H with at least one edge and p 1 , . . ., p t are isolated points of H . Let b = d(H ) and x a = x b • (e | e is a leaf edge of H ). Then x a is the product of edges of H . Let s = |a| 2 = |E(H )|.By [Minh et al. 2022, Corollary 2.7], x a / p t } be a maximal induced bipartite subgraph of G where H 1 , ..., H c are connected components of H with at least one edge and p 1 , ..., p t are isolated points.We say that { p i 1 , ...,p i u } are clustered if there exists v ∈ V (G) \ V (H ) such that the induced subgraph of G on {v, p i 1 , . .., p i u } is a bouquet.Let bou G (H )be the smallest number b such that the set { p 1 , . . ., p t } can be clustered into b bouquets in G.We call c ′ (H ) = c + bou G (H ) the number of restricted connected components of H . Definition 3.4.Let G be a simple graph.The restricted bipartite connectivity number of G is defined by bc Lemma 4.3.Assume that I = I (C n ) and t ≤ n − 2. Then depth S/(I t : (e 2 • • • e t )) ≤ ϕ(n, t).Theorem 4.4.Let I (C n ) be the edge ideal of a cycle of length n ≥ 5. Then depth(S/I (C n n ) t + I (H )) ≥ ϕ(n, t).Lemma 4.2.Let H be a nonempty subgraph of C n .Then for t ≥ 2, we have thatdepth S/(I (C n ) t + I (H )) ≥ ϕ(n, t).