aa r X i v : . [ m a t h . A C ] D ec BINOMIAL EDGE IDEALS OF SMALL DEPTH
M. ROUZBAHANI MALAYERI, S. SAEEDI MADANI, D. KIANI
Abstract.
Let G be a graph on [ n ] and J G be the binomial edge ideal of G inthe polynomial ring S = K [ x , . . . , x n , y , . . . , y n ]. In this paper we investigatesome topological properties of a poset associated to the minimal primary decom-position of J G . We show that this poset admits some specific subposets which arecontractible. This in turn, provides some interesting algebraic consequences. Inparticular, we characterize all graphs G for which depth S/J G = 4. Introduction
Binomial edge ideals were introduced in 2010 by Herzog, Hibi, Hreinsd´ottir, Kahleand Rauh in [12] and independently by Ohtani in [19]. Let G be a simple graph onthe vertex set [ n ] and the edge set E ( G ). Let S = K [ x , . . . , x n , y , . . . , y n ] be thepolynomial ring over 2 n variables where K is a field. Then the binomial edge ideal associated to the graph G denoted by J G is an ideal in S which is defined as follows: J G = ( x i y j − x j y i : { i, j } ∈ E ( G ) , ≤ i < j ≤ n ) . This class of ideals could be interpreted as a natural generalization of the well-studiedso-called determinantal ideal of the (2 × n )-matrix X = (cid:20) x · · · x n y · · · y n (cid:21) . The study of algebraic properties as well as numerical invariants of binomial edgeideals has attracted a considerable attention in the meantime, see e.g. [1, 3, 5, 8, 9,10, 14, 16, 17, 18, 20, 21, 22].One of the interesting homological invariants associated to binomial edge idealsis depth. Let H i m ( S/J G ) denote the i th local cohomology module of S/J G supportedon the irrelevant maximal ideal m = ( x , . . . , x n , y , . . . , y n ). Then we havedepth S/J G = min { i : H i m ( S/J G ) = (0) } . While computing the depth of the binomial edge ideal of a graph is hard in general,there have been several attempts to get some interesting results in this direction forsome special families of graphs. Moreover, some lower and upper bounds for thedepth of binomial edge ideal of graphs have been obtained by several authors whichwill be briefly discussed in the sequel.Let C n denote the cycle on n vertices. In [24] it was shown that depth S/J C n = n ,for n >
3. Also, in [8] the authors showed that depth
S/J G = n + 1, for a connected Mathematics Subject Classification.
Key words and phrases.
Binomial edge ideals, depth, Hochster type formula, meet-contractible. lock graph G . Later in [15] the authors computed the depth of a wider class ofgraphs which are called generalized block graphs. In [17], a nice formula was givenfor the depth of the join product of two graphs G and G . Roughly speaking, thejoin product of two graphs G and G , denoted by G ∗ G , is the graph which isobtained from the union of G and G by joining all the vertices of G to verticesof G , (the precise definition is given in Section 2).In [3] the authors gave an upper bound for the depth of the binomial edge idealof a graph in terms of some graphical invariants. Indeed, they showed that for anon-complete connected graph G , depth S/J G ≤ n − κ ( G ) + 2, where κ ( G ) denotesthe vertex connectivity of G .There is also a lower bound for depth S/J G . Indeed, let cd( J G , S ) denote thecohomological dimension of J G in S , namely, cd( J G , S ) = max { i : H iJ G ( S ) = (0) } .Now a result in [11] due to Faltings implies that(1) cd( J G , S ) ≤ n − j n − J G k , whenever J G = (0). On the other hand, since S/J G is a cohomologically full ringby a result in [6], (see [7] for the definition of cohomologically full rings), by [7] wehave that(2) depth S/J G ≥ n − cd( J G , S ) . Hence, by (1) and (2), we get the following lower bound for the depth of
S/J G :(3) depth S/J G ≥ j n − J G k . In this paper we apply some results and techniques from the topology of posets tostudy the depth of binomial edge ideals. We are interested in studying binomial edgeideals of small depth. Specifically, we characterize all graphs G with depth S/J G = 4.This is based on a Hochster type decomposition formula for the local cohomologymodules of binomial edge ideals provided recently by `Alvarez Montaner in [1].This paper is organized as follows. In Section 2, we fix the notation and reviewsome definitions and some known facts that will be used throughout the paper.In Section 3, we associate a poset to the binomial edge ideal of a graph. Then, westate in Theorem 3.6 the Hochster type formula for the local cohomology modulesof binomial edge ideals arised from [1, Theorem 3.9].Section 4 is devoted to extract some topological properties like contractibility ofsome specific subposets of the poset associated to binomial edge ideals which isintroduced in Definition 3.1.In Section 5, in Theorem 5.2, we supply a lower bound for the depth of binomialedge ideals. In particular, we show that depth S/J G ≥
4, where G is a graph with atleast three vertices. Then, by using the aforementioned lower bound and also by theprovided ingredients in Section 4, we characterize all graphs G with depth S/J G = 4,in Theorem 5.3. . Preliminaries
In this section we review some notions and facts that will be used throughoutthe paper. In this paper all graphs are assumed to be simple (i.e. with no loops,directed and multiple edges).Let G be a graph on [ n ] and T ⊆ [ n ]. A subgraph H of G on the vertex set T iscalled an induced subgraph of G , whenever for any two vertices i, j ∈ T such that { i, j } ∈ E ( G ), one has { i, j } ∈ E ( H ). Moreover, by G − T , we mean the inducedsubgraph of G on the vertex set [ n ] \ T . A vertex i ∈ [ n ] is said to be a cut vertex of G whenever G − { i } has more connected components than G . We say that T has cutpoint property for G , whenever each i ∈ T is a cut vertex of the graph G − ( T \{ i } ).In particular, the empty set ∅ , has cut point property for G . We denote by C ( G ),the family of all subsets T of [ n ] which have the cut point property for G . Namely, C ( G ) = { T ⊆ [ n ] : T has cut point property for G } . Let G and G be two graphs on the disjoint vertex sets V ( G ) and V ( G ),respectively. Then by the join product of G and G , denoted by G ∗ G , we meanthe graph on the vertex set V ( G ) ∪ V ( G ) and the edge set E ( G ) ∪ E ( G ) ∪ {{ u, v } : u ∈ V ( G ) and v ∈ V ( G ) } . Let G be a graph and T ⊆ [ n ]. Assume that G , . . . , G c G ( T ) are the connectedcomponents of G − T . Let e G , . . . , e G c G ( T ) be the complete graphs on the vertex sets V ( G ) , . . . , V ( G c G ( T ) ), respectively, and let P T ( G ) = ( x i , y i ) i ∈ T + J e G + · · · + J e G cG ( T ) . Then it is easily seen thatheight P T ( G ) = n − c G ( T ) + | T | . Also, in [12, Corollary 3.9], it was shown that P T ( G ) is a minimal prime ideal of J G if T has cut point property for G . Moreover, it was proved in [12, Corollary 2.2]that J G is a radical ideal. So, J G = T T ∈C ( G ) P T ( G ).Let ∆ be a simplicial complex. Recall that the 1 -skeleton graph of ∆ is thesubcomplex of ∆ consisting of all of the faces of ∆ which have cardinality at most 2.The simplicial complex ∆ is said to be connected if its 1-skeleton graph is connected.Let ( P , ) be a poset. Recall that the order complex of P , denoted by ∆( P ), isthe simplicial complex whose facets are the maximal chains in P . If P is an emptyposet, then we consider ∆( P ) = {∅} , i.e. the empty simplicial complex.3. Hochster type formula
In this section we focus on a Hochster type formula for the local cohomologymodules of binomial edge ideals recently provided by `Alvarez Montaner in [1]. Firstwe need to recall the definition of a poset associated to the binomial edge ideal of agraph G from [1]. et I be an ideal in the polynomial ring S and I = q ∩ · · · ∩ q t be a not necessarilyminimal decomposition for the ideal I . Now, the set of all possible sums of ideals inthis decomposition forms a poset ordered by the reverse inclusion and is denoted by P I . In the special case, we use the notation P G , instead of P J G , for the poset arisedfrom the minimal primary decomposition of J G .Now, the following, is the definition of a poset associated to the binomial edgeideal of a graph G which was introduced in [1, Definition 3.3].Let G be a graph. Associated to J G is the following poset which is denoted by A G : The ideals contained in A G are the prime ideals in P G , the prime ideals in theposets P I arised from the minimal primary decomposition of every non prime ideal I in P G and the prime ideals that are obtained by repeating this procedure everytime a non prime ideal is discovered.Note that for some technical goals that will be discussed later, we need to consideranother poset associated to binomial edge ideals. Indeed, the importance of our newposet will be exhibited when we study the topological properties of some specificsubposets of it, see Lemma 4.1, Theorem 4.4 and also Remark 4.5.Now, inspired by the `Alvarez Montaner’s definition, we define a new poset asso-ciated to the binomial edge ideal of a graph G as follows: Definition 3.1.
Let G be a graph on [ n ] and J G = T T ∈C ( G ) P T ( G ) be the minimal pri-mary decomposition of J G . Associated to this decomposition, we consider the poset( Q G , ) ordered by reverse inclusion which is made up of the following elements: • the prime ideals in the poset P G , • the prime ideals in the posets P I , arised from the following type of decom-positions: I = q ∩ q ∩ · · · ∩ q t ∩ ( q + P ∅ ( G )) ∩ ( q + P ∅ ( G )) ∩ · · · ∩ ( q t + P ∅ ( G )) , where I ’s are the non-prime ideals in the poset P G and q , q , . . . , q t are theminimal prime ideals of I , and • the prime ideals that we obtain repeatedly by this procedure every time thatwe find a non-prime ideal.It is worth mentioning here that the process of the construction of our poset Q G terminates after a finite number of steps just like the construction process of theposet A G . Indeed, as we will see in Corollary 3.5, every element q in the poset Q G is of the form P T ( H ) for some graph H on the vertex set [ n ] and some T ⊆ [ n ].Moreover, the main difference between our construction of the poset Q G and theconstruction of the poset A G , is that our construction involves a special decomposi-tion of the form I = q ∩ q ∩· · ·∩ q t ∩ ( q + P ∅ ( G )) ∩ ( q + P ∅ ( G )) ∩· · ·∩ ( q t + P ∅ ( G )) forthe non-prime ideals I , in spite of using the minimal primary decomposition for theideals I in the construction of A G . It turns out that A G is a subposet of the poset Q G . Now, a natural question might be whether the posets Q G and A G coincide ingeneral or not. The following example aims to show that, in general, Q G and A G do not coincide. ere, K n denotes the complete graph on [ n ]. Example 3.2.
Let G be the graph shown in Figure 1. One could see that C ( G ) = { T : T ⊆ { , , , }} . Now, since( x , y , x , y , x , y , x , y , x , y , x , y ) ∈ Min( P { , } ( G ) + P { , } ( G )) , we have that q = ( x , y , x , y , x , y , x , y , x , y , x , y ) + J K −{ , , , , , } ∈ Q G .However, q / ∈ A G . Indeed, assume on contrary that q ∈ A G . Now let A = { P T ( G ) , . . . , P T s ( G ) } be a subset of the maximal elements of the poset A G withthe property that I = P si =1 P T i ( G ) creates the element q in the process of construc-tion of the poset A G . We call the set A with such property, a predecessor for theelement q . Now, regarding the described structure of the minimal prime ideals of J G , and also by using the fact that the vertices 5 and 10 are adjacent in the graph K − { , , , , , } , one could easily check that P { } ( G ) ∈ A and P { } ( G ) ∈ A .This implies that I = ( x j , y j : j ∈ ∪ si =1 T i ) + J L , where L is the join product of two isolated vertices 5 and 10, with the completegraph on the vertex set { , , , , , } . So, I = q ∩ q is the minimal primarydecomposition for I , where q = ( x j , y j : j ∈ ∪ si =1 T i ) + ( x k , y k : k ∈ { , , , , , } )and q = ( x j , y j : j ∈ ∪ si =1 T i ) + P ∅ ( L ) . Now, since q + q is a prime ideal and q / ∈ { q , q , q + q } , we get a contradictionwith the fact that A is a predecessor for q .123 45 67 8910 Figure 1.
A graph G for which A G is a proper subposet of Q G .Now, to complete our discussion, we give an example of a graph G for which Q G = A G . This example was appeared in [1, Example 3.2]. Example 3.3.
Let G be the path on 5 vertices illustrated in Figure 2. Thus, theminimal prime ideals of J G are: P ∅ ( G ) , P { } ( G ) , P { } ( G ) , P { } ( G ) , P { , } ( G ) . o, it was mentioned in [1, Example 3.2] that the elements of the poset P G are: P ∅ ( G ) , P { } ( G ) , P { } ( G ) , P { } ( G ) , P { , } ( G ) , ( x , y , f , f , f , f , f , f ) , ( x ,y , f , f , f , f , f , f ) , ( x , y , x , y , f ) , ( x , y , x , y , f ) , ( x , y , f , f , f ,f , f , f ) , ( x , y , x , y , f ) , ( x , y , x , y , f ) , ( x , y , x , y , f , f ) , ( x , y , x ,y , x , y ) , ( x , y , x , y , f , f , f ) , ( x , y , x , y , f , f , f ) , ( x , y , x , y , f ,f , f ) , ( x , y , x , y , x , y , f ) . Note that I = ( x , y , x , y , f , f ), is the only non-prime ideal in the poset P G .Moreover, I = q ∩ q is the minimal primary decomposition for I , where q = ( x , y , x , y , f , f , f )and q = ( x , y , x , y , x , y ) . Now, according to the construction of the poset Q G , we need to consider the followingtype of decomposition for I :(4) I = q ∩ q ∩ ( q + P ∅ ( G )) ∩ ( q + P ∅ ( G )) . On the other hand, since q + P ∅ ( G ) = q and q + P ∅ ( G ) = q + q , the elementsof the poset P I , arised from the decomposition in (4) are exactly q , q and q + q .Therefore, the constructions of the posets Q G and A G imply that Q G = A G .1 2 3 4 5 Figure 2.
A graph G for which Q G = A G .Now, we are going to state the Hochster type formula for the local cohomologymodules of binomial edge ideals arised from [1, Theorem 3.9]. First, we need topresent the following proposition:Here, Min( J ), denotes the set of minimal prime ideals of an ideal J . Proposition 3.4.
Let G i be a graph on [ n ] and T i ⊆ [ n ] for each ≤ i ≤ k . Let J = P ki =1 P T i ( G i ) and q ∈ Min( J ) . Then q = P T ( H ) , for some graph H on [ n ] andsome T ⊆ [ n ] .Proof. For each 1 ≤ i ≤ k , we have P T i ( G i ) = ( x s , y s : s ∈ T i ) + J e G i + · · · + J e G icGi ( Ti ) .Let H , . . . , H ℓ be the connected components of the graph G = ( S ki =1 S c Gi ( T i ) j =1 f G ij ) − S ki =1 T i . One could easily see that(5) J = ( x s , y s : s ∈ k [ i =1 T i ) + J H + · · · + J H ℓ . ow, by [13, Problem 7.8, part (ii)], there exist U , . . . , U ℓ with U i ∈ C ( H i ) foreach 1 ≤ i ≤ ℓ , such that q = ( x s , y s : s ∈ S ki =1 T i ) + P ℓi =1 P U i ( H i ). Let T =( S ki =1 T i ) ∪ ( S ℓi =1 U i ). Now we take H to be the graph which is obtained fromthe graph L = ( S ℓi =1 S c Hi ( U i ) j =1 f H ij ), by adding those elements of [ n ] which do notbelong to the vertex set of the graph L , as isolated vertices. Then it follows that q = P T ( H ). (cid:3) Note that the above proposition together with the construction of the poset Q G imply the next corollary that will be crucial throughout the paper: Corollary 3.5.
Let G be a graph on [ n ] . Then, every element q in the poset Q G isof the form P T ( H ) , for some graph H on [ n ] and some T ⊆ [ n ] . Before stating the Hochster type formula, we need to fix some notation:Let 1 Q G be a terminal element that we add to the poset Q G . Then for every q ∈ Q G , by the interval ( q, Q G ), we mean the subposet { z ∈ Q G : q (cid:22) z (cid:22) Q G } of the poset Q G .Now, we are ready to state the Hochster type formula for the local cohomologymodules of binomial edge ideals based on [1, Theorem 3.9]. Moreover, we would liketo mention that since the poset Q G which we consider in this paper is different fromthe poset considered by `Alvarez Montaner in [1], we provide a proof for this formula.However, the proof is similar to the one that was proposed in [1, Theorem 3.9]. Theorem 3.6.
Let G be a graph on [ n ] and Q G be the poset associated to J G . Thenwe have the K -isomorphism H i m ( S/J G ) ∼ = M q ∈Q G H d q m ( S/q ) ⊕ M i,q , where d q = dim S/q and M i,q = dim K e H i − d q − (( q, Q G ); K ) .Proof. Let q ∈ Q G . By Corollary 3.5, we have that q = P T ( H ), for some graph H on [ n ] and some T ⊆ [ n ]. Now, the same method that was used for the proof of [1,Proposition 3.7] implies that Q G is a subset of a distributive lattice of ideals of S .Moreover, since S/q ∼ = N c H ( T ) i =1 S i /J f H i , where S i = K [ x j , y j : j ∈ V ( H i )], we havethat S/q is a Cohen-Macaulay domain, by [1, Theorem 2.2]. Therefore, the poset Q G fulfills all the required conditions in [1, Theorem 2.4]. Now, since the minimalityof the decompositions of non-prime ideals I appear in Definition 3.1 does not matterin [2, Theorem 5.22], the result follows by [1, Theorem 2.4]. (cid:3) Topology of the subposets of the poset associated to binomialedge ideals
In this section we investigate some topological properties of some specific sub-posets of the poset Q G associated to the binomial edge ideal of a graph G .The following lemma plays a vital role in our proofs. emma 4.1. Let G be a graph on [ n ] . Then q + P ∅ ( G ) ∈ Q G , for every q ∈ Q G .Proof. By the construction of the poset Q G , it is enough to show that q + P ∅ ( G ) is aprime ideal. By Corollary 3.5, we have that q = P T ( H ) for some graph H on [ n ] andsome T ⊆ [ n ]. First we assume that G is connected. Therefore, q + P ∅ ( G ) = ( x i , y i : i ∈ T ) + J K n − T , where K n − T denotes the complete graph on [ n ] \ T . Therefore, q + P ∅ ( G ) = P T ( K n ), which implies that q + P ∅ ( G ) is prime.Next assume that G is a disconnected graph with the connected components G , . . . , G r . We claim that every connected component of the graph H is containedin the graph G i , for some 1 ≤ i ≤ r . From this claim it will then follow that q + P ∅ ( G ) = ( x i , y i : i ∈ T ) + P ri =1 J K ni − T , where n i = | V ( G i ) | , for every 1 ≤ i ≤ r .Then q + P ∅ ( G ) = P T ( ∪ ri =1 K n i ), which is a prime ideal.To prove the claim, by the construction of the poset Q G and also by virtue ofCorollary 3.5 and using (5) repeatedly, we may assume that q ∈ Min( J + · · · + J ℓ ),where J i ∈ Min( J G ), for every 1 ≤ i ≤ ℓ . On the other hand, by [13, Prob-lem 7.8, part (ii)], we have that J i = P rj =1 P T ij ( G j ) for each 1 ≤ i ≤ ℓ , where T ij ∈ C ( G j ), for every 1 ≤ j ≤ r . Therefore, P ℓi =1 J i = P rj =1 P ℓi =1 P T ij ( G j ). This,together with [13, Problem 7.8, part (ii)], Corollary 3.5 and (5) imply that the con-nected components of the graph H should be contained in the connected componentsof the graph G , and hence the claim follows. (cid:3) The following definition is devoted to recall the concept of meet-contractibilityfor posets.
Definition 4.2.
A poset P is said to be meet-contractible if there exists an element α ∈ P such that α has a meet with every element β ∈ P .We say that a poset is contractible if its order complex is contractible.We use the following lemma to study the topology of some subposets of the poset Q G . Lemma 4.3. ([4, Theorem 3.2], see also [23, Proposition 2.4])
Every meet-contractibleposet is contractible.
The following theorem is the main theorem of this section.
Theorem 4.4.
Let G be a graph on [ n ] and assume that m ∈ Q G . Then ( m , Q G ) is a contractible poset.Proof. Let P = ( m , Q G ). By Lemma 4.3, it is enough to show that P is a meet-contractible poset.Clearly, P ∅ ( G ) ∈ P . Let q ∈ P . We show that P ∅ ( G ) and q have a meet in P .By Lemma 4.1, we have q + P ∅ ( G ) ∈ Q G . On the other hand, by Corollary 3.5 q + P ∅ ( G ) $ m , since q $ m and P ∅ ( G ) does not contain any variable. This impliesthat q + P ∅ ( G ) ∈ P .Now we claim that q + P ∅ ( G ) is the meet of q and P ∅ ( G ). Clearly, q + P ∅ ( G ) q and q + P ∅ ( G ) P ∅ ( G ). Now suppose that there exists q ′ ∈ P with q ′ q and ′ P ∅ ( G ). So, q ′ q + P ∅ ( G ). This means that q + P ∅ ( G ) is the meet of q and P ∅ ( G ), and hence the claim follows. Therefore, P is a meet-contractible poset. (cid:3) Remark 4.5.
Let b m = P T ( H ), where H is an arbitrary graph on [ n ] with | T | = n −
1, where n ≥
2. Then with the same argument that we used in the proof ofTheorem 4.4, one checks that the result of Theorem 4.4 still holds, if we replace b m with m .Now, as a consequence of Theorem 4.4 and the above remark, we get the followingcorollary which will be used in the next section. Corollary 4.6.
Let G be a graph on [ n ] with n ≥ . Let q be an element of theposet Q G such that q ∈ { m , b m } . Then we have M i,q = 0 , for every i . Characterization of binomial edge ideals of small depth
In this section, as an application of the results provided in the previous section,we characterize all binomial edge ideals J G , for which depth S/J G = 4. First, westate the the following remark that enables us to simplify the proofs in this section. Remark 5.1.
Let G be a graph on [ n ]. Then d q = 1, for every q ∈ Q G . Indeed,suppose on the contrary that d q = 1. By Corollary 3.5 we have that q = P T ( H ), forsome T ⊆ [ n ] and some graph H on [ n ]. Now d q = 1 implies that either | T | = n and c H ( T ) = 1, or | T | = n − c H ( T ) = 0, where both of them are clearly impossible.Now, we supply a lower bound for the depth of binomial edge ideals that will beused later in our characterization. Theorem 5.2.
Let G be a graph on [ n ] . Then depth S/J G ≥ r + n X i =1 r i ( i + 1) , where r is the number of non-complete connected components of G and r i is thenumber of complete connected components of G of size i , for every ≤ i ≤ n .Proof. Let G , . . . , G ω be the connected components of G . So, S/J G ∼ = N ωi =1 S i /J G i ,where S i = K [ x j , y j : j ∈ V ( G i )]. Therefore, depth S/J G = P ωi =1 depth S i /J G i .Then, by [8, Theorem 1.1], the result follows if we show that depth S/J G ≥ n ≥ H i m ( S/J G ) = 0, for all i with 0 ≤ i ≤
3. Let q ∈ Q G , d q = dim S/q and M i,q = dim K e H i − d q − (( q, Q G ); K ).By Theorem 3.6, it is enough to show that M i,q = 0, for i = 0 , , , q = P T ( H ), for some graph H on [ n ] andsome T ⊆ [ n ]. Now, we consider the following cases:Let i = 0. If d q >
0, the assertion is clear. So we assume that d q = 0. We haveheight P T ( H ) = n − c H ( T ) + | T | = 2 n . This implies that | T | − c H ( T ) = n . So that q = m , since | T | = n and c H ( T ) = 0. Now the result follows, since the order complexof the poset ( q, Q G ) is not empty. et i = 1. If d q ≥
2, then the assertion is obvious. So, by Remark 5.1 wemay assume that d q = 0. Then we get q = m , and hence the result follows byCorollary 4.6.Let i = 2. If d q ≥
3, then the assertion is clear. In addition, in Remark 5.1 weshowed that d q = 1. So assume that d q ∈ { , } . If d q = 0, then q = m . So thatthe result follows again by Corollary 4.6. Next suppose that d q = 2. Then we have | T | − c H ( T ) = n −
2. This implies that | T | = n − c H ( T ) = 1. Therefore, P ∅ ( G ) ∈ ( q, Q G ), and hence the result follows since the order complex of the poset( q, Q G ) is non-empty.Let i = 3. If d q ≥
4, then the assertion holds. Moreover, the discussion of thecases d q = 0 , , d q = 3. In this case we have that | T | = n − c H ( T ) = 1. So without lossof generality we may assume that q = ( x , . . . , x n − , y , . . . , y n − ) + ( x n − y n − x n y n − ) . Now, we have that P ∅ ( G ) $ q , since n ≥
3. Therefore, the order complex of theposet ( q, Q G ) is not empty, and hence the desired result follows. (cid:3) Note that the aforementioned lower bound in Theorem 5.2, recovers the statedbound in (3) in Section 1. Indeed, by the notation that we used in Theorem 5.2,and by putting t = P ni =1 r i , it is not difficult to see that j n − J G k ≤ r + t + 1 ≤ r + 2 t ≤ r + n X i =1 r i ( i + 1) , for every graph G with at least a non-complete connected component. Moreover, wewould like to remark that the given lower bound in Theorem 5.2 is sharp. For exam-ple, consider G to be the graph depicted in Figure 3. Then we have depth S/J G = 4,by [17, Theorem 4.4]. Figure 3.
A complete bipartite graph G with depth S/J G = 4.Now we are ready to state our main theorem. Here, we denote by 2 K , the graphconsisting of two isolated vertices. Theorem 5.3.
Let G be a graph on [ n ] with n ≥ . Then the following are equiva-lent: (a) depth S/J G = 4 . (b) G = G ′ ∗ K , for some graph G ′ . roof. ( a ) ⇒ ( b ): Assume that G = G ′ ∗ K , for any graph G ′ . We show thatdepth S/J G ≥
5. By the definition of depth and by Theorem 5.2, it suffices to showthat H m ( S/J G ) = 0.We keep using the notation that we used in Theorem 5.2. Let q ∈ Q G , d q =dim S/q and M ,q = dim K e H − d q (( q, Q G ); K ). By Theorem 3.6, the result followsonce we show that M ,q = 0. Notice that Corollary 3.5 implies that q = P T ( H ), forsome graph H on [ n ] and some T ⊆ [ n ]. Note also that if d q ≥
5, then the assertionis obvious. On the other hand, as we discussed in Remark 5.1, we have d q = 1. Sowe consider the following cases:Let d q = 0. So q = m , since | T | = n and c H ( T ) = 0. Therefore, M ,q = 0, byCorollary 4.6.Let d q = 2. We have | T | − c H ( T ) = n −
2. This implies that | T | = n − c H ( T ) = 1. Now the result follows by Corollary 4.6.Let d q = 3. We need to show that the order complex of the poset ( q, Q G ) isconnected. Note that since height q = n − c H ( T ) + | T | = 2 n −
3, we have | T | = n − c H ( T ) = 1, and hence we may assume that q = ( x , . . . , x n − , y , . . . , y n − ) + ( x n − y n − x n y n − ) . Now suppose that q , q ∈ ( q, Q G ) and q = q . By Corollary 3.5, we have that q = P T ( H ) and q = P T ( H ), for some graphs H and H on [ n ] and some T , T ⊆ [ n ]. Moreover, we have that T , T ⊆ { , . . . , n − } , since q , q ∈ ( q, Q G ).Now without loss of generality we may assume that T ⊆ { , . . . , n − } and T $ { , . . . , n − } , since q = q . We first assume that T $ { , . . . , n − } . Thus wehave q + P ∅ ( G ) ∈ ( q, Q G ) and q + P ∅ ( G ) ∈ ( q, Q G ), by Lemma 4.1. So we get thepath q , q + P ∅ ( G ) , P ∅ ( G ) , q + P ∅ ( G ) , q in the 1-skeleton graph of the order complex of the poset ( q, Q G ).Next assume that T = { , . . . , n − } . Now since the set of maximal elementsof the poset Q G coincides with the set of minimal prime ideals of J G , there exists U ∈ C ( G ) such that P U ( G ) ⊆ q . We show that U $ { , . . . , n − } . Indeed,otherwise by [12, Corollary 3.9] we get G = L ∗ K , where L = G − { n − , n } . Thiscontradicts the fact that G = G ′ ∗ K , for any graph G ′ . So, P U ( G ) $ q . On theother hand, P U ( G ) + P ∅ ( G ) ∈ ( q, Q G ) by Lemma 4.1. Therefore, we get the path q , P U ( G ) , P U ( G ) + P ∅ ( G ) , P ∅ ( G ) , q + P ∅ ( G ) , q in the 1-skeleton graph of the order complex of the poset ( q, Q G ).Therefore, the order complex of the poset ( q, Q G ) is connected, as desired.Now let d q = 4. Therefore, | T | − c H ( T ) = n −
4. So, the following cases occur:First assume that | T | = n − c H ( T ) = 2. Therefore, we may assume that q = ( x , . . . , x n − , y , . . . , y n − ). Then by a similar method that we used in the lastpart of the case d q = 3, there exists W ∈ C ( G ) such that P W ( G ) $ q . This meansthat P W ( G ) ∈ ( q, Q G ), and hence ( q, Q G ) is non-empty.Next assume that | T | = n − c H ( T ) = 1. So, we may assume that q = ( x , . . . , x n − , y , . . . , y n − , x n − y n − − x n − y n − , x n − y n − x n y n − , x n − y n − x n y n − ) . ow, we have P ∅ ( G ) ∈ ( q, Q G ), since n ≥
4. Thus, ( q, Q G ) is non-empty.Therefore, M ,q = dim K e H − (( q, Q G ); K ) = 0.( b ) ⇒ ( a ): Assume that G = G ′ ∗ K , for some graph G ′ . Now, if G ′ is acomplete graph then, the result follows from [17, Theorem 3.9]. So, assume that G ′ is not complete. Therefore, by Theorem 5.2, we have depth S ′ /J G ′ ≥
4, where S ′ = K [ x i , y i : i ∈ V ( G ′ )]. This, together with [17, Theorem 4.3] and [17, Theorem 4.4],imply the result. (cid:3) Acknowledgments:
The authors would like to thank Josep `Alvarez Montanerand the anonymous referee for their useful comments. The authors would also like tothank Institute for Research in Fundamental Sciences (IPM) for financial support.The research of the second author was in part supported by a grant from IPM (No.99130013). The research of the third author was in part supported by a grant fromIPM (No. 99050212).
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Mohammad Rouzbahani Malayeri, Department of Mathematics and Computer Sci-ence, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran
Email address : [email protected] Sara Saeedi Madani, Department of Mathematics and Computer Science, Amirk-abir University of Technology (Tehran Polytechnic), Tehran, Iran, and School ofMathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
Email address : [email protected] Dariush Kiani, Department of Mathematics and Computer Science, AmirkabirUniversity of Technology (Tehran Polytechnic), Tehran, Iran, and School ofMathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
Email address : [email protected]@aut.ac.ir