On the depth of binomial edge ideals of graphs
Mohammad Rouzbahani Malayeri, Sara Saeedi Madani, Dariush Kiani
aa r X i v : . [ m a t h . A C ] J a n ON THE DEPTH OF BINOMIAL EDGE IDEALS OF GRAPHS
M. ROUZBAHANI MALAYERI, S. SAEEDI MADANI, D. KIANI
Abstract.
Let G be a graph on the vertex set [ n ] and J G be the associatedbinomial edge ideal in the polynomial ring S = K [ x , . . . , x n , y , . . . , y n ]. In thispaper we investigate about the depth of binomial edge ideals. More precisely, wefirst establish a combinatorial lower bound for the depth of S/J G based on somegraphical invariants of G . Next, we combinatorially characterize all binomial edgeideals J G with depth S/J G = 5. To achieve this goal, we associate a new poset M G to the binomial edge ideal of G , and then elaborate some topological propertiesof certain subposets of M G in order to compute some local cohomology modulesof S/J G . Introduction
Over the last two decades, the study of ideals with combinatorial origins has beenan appealing trend in commutative algebra. One of the most well-studied types ofsuch ideals which has attracted special attention in the literature is the binomialedge ideal of a graph.Let G be a graph on [ n ] and S = K [ x , . . . , x n , y , . . . , y n ] be the polynomial ringover a field K . Then, the binomial edge ideal associated to G , denoted by J G , is theideal in S generated by all the quadratic binomials of the form f ij = x i y j − x j y i ,where { i, j } ∈ E ( G ) and 1 ≤ i < j ≤ n . This class of ideals was introduced in2010 by Herzog, Hibi, Hreinsd´ottir, Kahle and Rauh in [10], and independently byOhtani in [16], as a natural generalization of determinantal ideals, as well as idealsgenerated by adjacent 2-minors of a 2 × n -matrix of indeterminates.Since then, many researchers have studied the algebraic properties and homolog-ical invariants of binomial edge ideals. The main goal is to understand how theinvariants of the associated graph are reflected in the algebra of the ideal, and viceversa. Indeed, it has been proved that there exists a mutual interaction betweenalgebraic properties of binomial edge ideals and combinatorial properties of the un-derlying graphs, see e.g., [1, 3, 5, 6, 7, 8, 9, 11, 12, 15, 18, 19, 21, 22] for some effortsin this direction.One of the homological invariants associated to binomial edge ideals which is noteasy to compute, is depth . Recall thatdepth S/J G = min { i : H i m ( S/J G ) = 0 } , Mathematics Subject Classification.
Key words and phrases.
Binomial edge ideals, depth, diameter, Hochster type formula, meet-contractible. here H i m ( S/J G ) denotes the i th local cohomology module of S/J G with support atthe maximal homogeneous ideal m = ( x , . . . , x n , y , . . . , y n ) of S .Notice that unlike some other homological invariants associated to binomial edgeideals, like the Castelnuovo-Mumford regularity, which has been studied exten-sively, little is known about the depth of binomial edge ideals. Moreover, sincethe depth is in general dependent on the characteristic of the base field, findingsome characteristic-free results about the depth of such homogeneous ideals are ofgreat interest. The first important result about the depth of binomial edge idealsappeared in [6], where the authors showed that depth S/J G = n + 1, where G isa connected block graph. Afterwards, in [24] the depth of the binomial edge idealof cycles was computed, and later in [20] a lower bound was given for the depth ofbinomial edge ideals of unicyclic graphs. In [3], Banerjee and N´u˜nez-Betancourt es-tablished a nice combinatorial upper bound for the depth of binomial edge ideals interms of the vertex connectivity of the underlying graph. Indeed, for a non-completeconnected graph G , they showed that(1) depth S/J G ≤ n − κ ( G ) + 2 , where κ ( G ) denotes the vertex connectivity of G .It is worth mentioning here that to the best of our knowledge, beside the abovecombinatorial upper bound for the depth of binomial edge ideals, there is no com-binatorial lower bound for the depth of S/J G . So, as the first main result of thispaper, we supply the following combinatorial lower bound for the depth of binomialedge ideals: Theorem A (Theorem 3.5) . Let G be a graph on [ n ] . Then depth S/J G ≥ ξ ( G ) . In particular, if G is connected, then depth S/J G ≥ f ( G ) + diam( G ) . Here, ξ ( G ) = f ( G ) + d ( G ) where f ( G ) denotes the number of free vertices (or simplicial vertices) of the graph G , and d ( G ) denotes the sum of the diameters of theconnected components of G , and the number of the isolated vertices of G . Moreover,diam( G ) denotes the diameter of G .In order to prove Theorem A, we first introduce the concept of d -compatiblemaps. Such maps are defined from the set of all graphs to the set of non-negativeintegers that admit certain properties (see Definition 3.1). Then, this concept willbe exploited to establish a general lower bound for the depth of binomial edge ideals.We also provide a combinatorial d -compatible map, which yields the above lowerbound. We also show that our lower bound is best possible in the sense that thereare graphs G for which depth S/J G = ξ ( G ), see Figure 1.Now, we would like to mention another motivation of this paper. In [18], theauthors of the present paper, provided a general lower bound for the depth of bino-mial edge ideals. Indeed, they showed that depth S/J G ≥
4, where G is a connectedgraph with at least three vertices. Moreover, they gave an explicit characterization f the graphs G for which depth S/J G = 4. More precisely, they showed that forgraphs G with more than three vertices, depth S/J G = 4 if and only if G = G ′ ∗ K ,for some graph G ′ , where G ′ ∗ K denotes the join product of a graph G ′ and twoisolated vertices denoted by 2 K .Now, it is natural to ask about a combinatorial characterization of binomial edgeideals of higher depths. In this paper, by using a wonderful Hochster type formula forthe local cohomolgy modules of binomial edge ideals provided by `Alvarez Montanerin [1], we give such a characterization. Indeed, we prove the following characteriza-tion of the graphs G for which depth S/J G = 5, see Definition 5.3 for the requirednotation. Theorem B (Theorem 5.4) . Let G be a graph on [ n ] with n ≥ . Then the followingstatements are equivalent: (a) depth S/J G = 5.(b) G is a D -type graph.The proof of Theorem B involves some topological results that we obtain in thispaper about some specific subposets of a poset associated to binomial edge idealswhich are indeed of independent interest.The organization of this paper is as follows. In Section 2, we fix the notation andreview some facts and definitions that will be used throughout the paper.In Section 3, toward providing some lower bounds for the depth of binomial edgeideals, in Definition 3.1, we introduce a concept which is named as d -compatiblemaps. Such maps are defined from the set of all graphs to the set of non-negativeintegers with some specific properties. Then, in Theorem 3.3, by using the aforemen-tioned concept a general lower bound is given for the depth of binomial edge ideals.In addition, after providing a combinatorial d -compatible map in Theorem 3.4, acombinatorial lower bound is given for the depth of binomial edge ideals in Theo-rem 3.5. This bound together with a result from [18], provide a modified version ofthe bound given in Theorem 3.5.In Section 4, following the poset theoritical as well as the topological approachesused in [1] and [18], we associate a new poset to binomial edge ideals in Definition 4.1.Then, we state in Theorem 4.3, the Hochster type formula for the local cohomologymodules of binomial edge ideals arised from [1, Theorem 3.9].In Section 5, we use the Hochster type formula provided in Section 4 to character-ize all graphs G for which depth S/J G = 5, in Theorem 5.4. To prove our character-ization we need to provide several auxiliary ingredients. In particular, Theorem 5.8which studies the vanishing of the zeroth and the first reduced cohomology groupsof some subposets of the associated poset to binomial edge ideals, plays a crucialrole in the proof of Theorem 5.4.2. Preliminaries
In this section we recall some notions and known facts that are used in this paper.
Graph theory.
Throughout the paper, all graphs are assumed to be simple (i.e.with no loops, directed and multiple edges). Let G be a graph on the vertex set [ n ] nd T ⊆ [ n ]. A subgraph H of G on the vertex set T is called an induced subgraph of G , whenever for any two vertices u, v ∈ T , one has { u, v } ∈ E ( H ) if { u, v } ∈ E ( G ).Now, by G − T , we mean the induced subgraph of G on the vertex set [ n ] \ T . Inthe special case, when T = { v } , we use the notation G − v instead of G − { v } , forsimplicity.Let v ∈ [ n ]. Denoted by N G ( v ), is the set of all vertices of G which are adjacent to v . We say that v is a free vertex (or simplicial vertex) of G , if the induced subgraphof G on the vertex set N G ( v ) is a complete graph. Moreover, a vertex which isnot free, is called a non-free (or non-simplicial) vertex. We use f ( G ) and iv ( G ) todenote the number of free vertices and the number of non-free vertices of a graph G , respectively.A vertex v ∈ [ n ] is said to be a cut vertex of G whenever G − v has more connectedcomponents than G . Moreover, we say that T has cut point property for G , whenevereach v ∈ T is a cut vertex of the graph G − ( T \{ v } ). Particularly, the empty set ∅ ,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 with the edge set E ( G ) ∪ E ( G ) ∪ {{ u, v } : u ∈ V ( G ) and v ∈ V ( G ) } . Primary decomposition of binomial edge ideals.
Let G be a graph on[ n ] and T ⊆ [ n ]. Let also G , . . . , G c G ( T ) be the connected components of G − T ,and e G , . . . , e G c G ( T ) be the complete graphs on the vertex sets V ( G ) , . . . , V ( G c G ( T ) ),respectively. Let P T ( G ) = ( x v , y v ) v ∈ T + J e G + · · · + J e G cG ( T ) . Then, by [10, Theorem 3.2], it is known that J G = T T ⊆ [ n ] P T ( G ). Moreover, in [10,Corollary 3.9], all the minimal prime ideals of J G were determined. Indeed, it wasshown that P T ( G ) ∈ Min( J G ) if and only if T ∈ C ( G ), where C ( G ) = { T ⊆ [ n ] : T has cut point property for G } . Finally, the following useful formula could be easily verified:ht P T ( G ) = n − c G ( T ) + | T | . Poset topology.
Let ∆ be a simplicial complex. Then, by the 1 -skeleton graphof ∆ we mean the subcomplex of ∆ consisting of all the faces of ∆ which havecardinality 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 . We say that P is a connected poset if its order complex ∆( P ) is connected. Similarly, we say that P is contractible if ∆( P ) is contractible. If P is an empty poset, then we consider∆( P ) = {∅} , i.e. the empty simplicial complex. ayer-Vietoris sequence. Let ∆ be a simplicial complex and v ∈ V (∆). Recallthe following three subcomplexes of ∆ that will be used in this paper. • star ∆ ( v ) = { σ ∈ ∆ : σ ∪ { v } ∈ ∆ } ; • del ∆ ( v ) = { σ ∈ ∆ : v σ } ; • link ∆ ( v ) = { σ ∈ ∆ : v σ and σ ∪ { v } ∈ ∆ } .Let ∆ = star ∆ ( v ) and ∆ = del ∆ ( v ). Then ∆ ∪ ∆ = ∆ and ∆ ∩ ∆ = link ∆ ( v ),and we have the Mayer-Vietoris sequence: · · · → H i (link ∆ ( v ); K ) → H i (star ∆ ( v ); K ) ⊕ H i (del ∆ ( v ); K ) → H i (∆; K ) → H i − (link ∆ ( v ); K ) → · · · Moreover, we have the reduced version · · · → e H (link ∆ ( v ); K ) → e H (star ∆ ( v ); K ) ⊕ e H (del ∆ ( v ); K ) → e H (∆; K ) → , provided that ∆ ∩ ∆ = {∅} .3. A combinatorial lower bound for the depth of binomial edgeideals
Our main goal in this section is to establish some lower bounds for the depth ofbinomial edge ideals. We first introduce the concept of d - compatible maps which aredefined from the set of all graphs to the set of non-negative integers N with somedesirable properties. Then, considering this concept we give a general lower boundfor the depth of binomial edge ideals. We also provide a combinatorial d -compatiblemap to obtain a combinatorial lower bound for the depth of such ideals as well.We first introduce a graph which plays an important role in the proof of the maintheorem of this section. Let G be a graph on [ n ] and v ∈ [ n ]. Associated to thevertex v , there is a graph, denoted by G v , with the vertex set V ( G ) and the edgeset E ( G ) ∪ {{ u, w } : { u, w } ⊆ N G ( v ) } . Note that by the definition, it is clear that v is a free vertex of the graph G v , and N G ( v ) = N G v ( v ).Now we are ready to define the notion of a d -compatible map as follows: Definition 3.1.
Let G be the set of all graphs. A map ψ : G −→ N is called d-compatible , if it satisfies the following conditions:(a) if G = ˙ ∪ ti =1 K n i , where n i ≥ ≤ i ≤ t , then ψ ( G ) ≤ t + P ti =1 n i ;(b) if G = ˙ ∪ ti =1 K n i , then there exists v ∈ V ( G ) such that(1) ψ ( G − v ) ≥ ψ ( G ), and(2) ψ ( G v ) ≥ ψ ( G ), and(3) ψ ( G v − v ) ≥ ψ ( G ) − iv ( G ) denotes thenumber of non-free vertices of a graph G . Lemma 3.2. [13, Lemma 3.4]
Let G be a graph and v be a non-free vertex of G .Then, max { iv ( G v ) , iv ( G − v ) , iv ( G v − v ) } < iv ( G ) . he following theorem provides a general lower bound for the depth of binomialedge ideals. Theorem 3.3.
Let G be a graph on [ n ] and ψ be a d -compatible map. Then depth S/J G ≥ ψ ( G ) . Proof.
Our proof is based on the technique applied in the proof of [6, Theorem 1.1].So, we may omit some details here. We prove the assertion by using inductionon iv ( G ). If iv ( G ) = 0, then G is a disjoint union of complete graphs, that is, G = ˙ ∪ ti =1 K n i , where n i ≥ ≤ i ≤ t . We have depth S/J G = t + P ti =1 n i ,by [6, Theorem 1.1]. On the other hand, by condition ( a ) of Definition 3.1 we have ψ ( G ) ≤ t + P ti =1 n i , so that the assertion holds in this case. Now, we assumethat iv ( G ) >
0. Let v ∈ [ n ] be the vertex with the properties of condition ( b ) ofDefinition 3.1.Let Q = T T ⊆ [ n ] v / ∈ T P T ( G ) and Q = T T ⊆ [ n ] v ∈ T P T ( G ). We have that Q = J G v , Q =( x v , y v ) + J G − v and also Q + Q = ( x v , y v ) + J G v − v . Therefore, the short exactsequence 0 −→ SJ G −→ SJ G v ⊕ S v J G − v −→ S v J G v − v −→ S v = K [ x i , y i : i ∈ [ n ] \{ v } ].Now, the well-known depth lemma implies that(2) depth S/J G ≥ min { depth S/J G v , depth S v /J G − v , depth S v /J G v − v + 1 } . Moreover, by Lemma 3.2, induction hypothesis and by Definition 3.1 part ( b ), wehave(3) depth S/J G v ≥ ψ ( G v ) ≥ ψ ( G ) , (4) depth S v /J G − v ≥ ψ ( G − v ) ≥ ψ ( G )and(5) depth S v /J G v − v ≥ ψ ( G v − v ) ≥ ψ ( G ) − . So, (2) together with (3), (4), and (5) imply the result. (cid:3)
Now, we are going to provide a combinatorial d -compatible map. Before that, weneed to recall the concept of diameter of a connected graph. Let G be a connectedgraph on [ n ], u and v be two vertices of G . Then, by the distance between the vertices u and v in G , which is denoted by d G ( u, v ), we mean the length of a shortest pathconnecting u and v in G . Now, the diameter of G , denoted by diam( G ), is definedas diam( G ) = max { d G ( u, v ) : u, v ∈ V ( G ) } . We call a shortest path between two vertices u and v of G with d G ( u, v ) = diam( G ),an LSP in G . et G be a graph on [ n ] with the connected components G , . . . , G t . Then we set d ( G ) := i ( G ) + t X i =1 diam( G i ) , where i ( G ) denotes the number of isolated vertices of G .Now, in the next theorem, we provide a d -compatible map given by ξ ( G ). Here, f ( G ) denotes the number of free vertices of G . Theorem 3.4.
The map ξ : G −→ N defined by ξ ( G ) = f ( G ) + d ( G ) , for every G ∈ G is d -compatible.Proof. Let G ∈ G . First assume that G = ˙ ∪ ti =1 K n i , where n i ≥ ≤ i ≤ t .Clearly we have f ( G ) = P ti =1 n i and d ( G ) = t , and hence ξ ( G ) = t + P ti =1 n i .Therefore, we just need to show that ξ satisfies condition ( b ) of Definition 3.1. Todo so, without loss of generality we assume that G is a non-complete connectedgraph. Therefore, we have d ( G ) = diam( G ). For convenience, we set d = d ( G ).Notice that d ≥
2, since G is not a complete graph.First we show that there exists v ∈ [ n ] such that ξ ( G − v ) ≥ ξ ( G ). It is clear thatfor any non-free vertex v , we have f ( G − v ) ≥ f ( G ). If there exists a non-free vertex v ∈ [ n ] such that v does not belong to the vertex set of some LSP in G , then v iscertainly a vertex with the desired property. So, we may assume that every non-freevertex of G belongs to the vertex set of every LSP in G . Let P : v , v , . . . , v d +1 bean arbitrary LSP in G . Notice that v i is a non-free vertex of G for every 2 ≤ i ≤ d .Also, every vertex v of G which is not on the path P is a free vertex of G . Theseimply that for every 2 ≤ j ≤ d , the vertices v j − and v j +1 belong to two differentconnected components of the graph G − v j . Indeed, assume on contrary that thereexists a path P ′ : v j − , u , . . . , u t , v j +1 in the graph G − v j . Without loss of generalitywe may assume that P ′ is an induced path in G − v j . This clearly implies that { u , . . . , u t } ⊆ V ( P ) \ { v j } . So, we have that u = v j − , u = v j − , . . . , u t = v j − t − .Thus, { v j − t − , v j +1 } ∈ E ( G ), a contradiction.Now, let 2 ≤ j ≤ d , and { n j , v j } ∈ E ( G ), where n j ∈ [ n ] \ V ( P ). By a sim-ilar argument one could see that d G − v j +1 ( n j , v ) = j , if { n j , v j − } / ∈ E ( G ), and d G − v j ( n j , v ) = j −
1, if { n j , v j − } ∈ E ( G ). Also, we have d G − v j − ( n j , v d +1 ) = d − j +2,if { n j , v j +1 } / ∈ E ( G ), and d G − v j ( n j , v d +1 ) = d − j + 1, if { n j , v j +1 } ∈ E ( G ). We usethese facts throughout the proof.Now, assume that there exists 2 ≤ i ≤ d such that N G ( v i ) ∩ N G ( v i +1 ) = ∅ . Let j = min { i : 2 ≤ i ≤ d, N G ( v i ) ∩ N G ( v i +1 ) = ∅} . We claim that v j is a vertex withthe desired property. To prove the claim, we distinguish the following cases:First assume that j >
3. Let n j ∈ N G ( v j ) ∩ N G ( v j +1 ). We have d G − v j ( n j , v d +1 ) = d − j + 1. Now, if v j − is a free vertex of G − v j , we have f ( G − v j ) ≥ f ( G ) + 1.On the other hand, we have that d ( G − v j ) ≥ j − d G − v j ( n j , v d +1 ) = d −
1, sincethe vertices v j − and v j +1 are contained in two connected components of the graph G − v j . So, we get f ( G − v j ) + d ( G − v j ) ≥ f ( G ) + 1 + d − f ( G ) + d , as desired.Now, if v j − is a non-free vertex of G − v j , then, there exists n j − ∈ N G − v j ( v j − ) uch that n j − is not on the path P . Since j − ≥
2, by the choice of j , we havethat { n j − , v j − } / ∈ E ( G ). This implies that d G − v j ( v , n j − ) = j −
1, and hence d ( G − v j ) ≥ j − d G − v j ( n j , v d +1 ) = d , which proves the claim in this case.Next suppose that j = 3. Let n ∈ N G ( v ) ∩ N G ( v ). If v is a free vertex of G − v , then the claim follows by the same argument of the previous case. If v is a non-free vertex of G − v , there exists an induced path P ′′ : α, v , β in G − v .Therefore d ( G − v ) ≥ d G − v ( n , v d +1 ) = d . This implies the claim.Finally, suppose that j = 2, and let n ∈ N G ( v ) ∩ N G ( v ). Then d G − v ( n , v d +1 ) = d −
1. Therefore, since v and v belong to two connected components of G − v , weget d ( G − v ) ≥ d − d , as desired.Now, we assume that N G ( v i ) ∩ N G ( v i +1 ) = ∅ , for every 2 ≤ i ≤ d . We show that v is a vertex with the desired property. First, notice that either v is a free vertexof G − v , or there exists a vertex n such that { n , v } ∈ E ( G ) and { n , v } / ∈ E ( G ).This implies that either f ( G − v ) ≥ f ( G ) + 1 which completes the proof in thiscase or d G − v ( n , v d +1 ) = d − G satisfyconditions (2) and (3) of Definition 3.1. Suppose that v is an arbitrary non-freevertex of G .First we prove that ξ ( G v ) ≥ ξ ( G ). Notice that f ( G v ) ≥ f ( G ) + 1, by Lemma 3.2.Therefore, the result follows if we show that d ( G v ) ≥ d −
1. Let α and β be twovertices of G with d G ( α, β ) = d . It suffices to show that d G v ( α, β ) ≥ d −
1. Assumeon contrary that there exists a path P : α = u , u , . . . , u ℓ +1 = β in G v with ℓ ≤ d − P is an induced path in G v . Nowwe consider the fallowing cases:First assume that v ∈ V ( P ). This clearly implies that v = α or v = β , since v isa free vertex of G v . Therefore, P is a path in G . So, we have that d G ( α, β ) ≤ d − v / ∈ V ( P ). Now, since P is an induced path in G v and since P isnot a path in G , we get | N G v ( v ) ∩ V ( P ) | = 2. Therefore, N G v ( v ) ∩ V ( P ) = { u i , u i +1 } for some 1 ≤ i ≤ ℓ . Now, the path P ′ : α = u , . . . , u i , v, u i +1 , . . . , u ℓ +1 = β is a pathin G between α and β with the length at most d −
1, which is a contradiction.So, we have d G v ( α, β ) ≥ d −
1, as desired.Finally, we show that ξ ( G v − v ) ≥ ξ ( G ) −
1. To prove this, it is enough to showthat d ( G v − v ) ≥ d −
1, since f ( G v − v ) ≥ f ( G ) by Lemma 3.2. Let P : v , v , . . . , v d +1 be an LSP in G . We consider the following cases:First assume that v = v and v = v d +1 . We show that d G v − v ( v , v d +1 ) ≥ d − d G v − v ( v , v d +1 ) ≤ d −
2. Therefore, there exists a path P ′ in G v − v between the vertices v and v d +1 with the length at most d −
2. We mayalso assume that P ′ is an induced path in G v − v . Since v is a free vertex of G v andsince P ′ is not a path in G , the vertex v has exactly two adjacent neighbours in G on the path P ′ . This implies that d G ( v , v d +1 ) ≤ d −
1, a contradiction. ext without loss of generality we assume that v = v . Now, the result followsif we show that d G v − v ( v , v d +1 ) ≥ d −
1. Assume on contrary that there exists aninduced path P ′′ : v , u , . . . , u r = v d +1 in G v − v between the vertices v and v d +1 with the length r , where r ≤ d −
2. Now, we have either N G v ( v ) ∩ V ( P ′′ ) = { v } or N G v ( v ) ∩ V ( P ′′ ) = { v , u } . Therefore, we get a path in G between v and v d +1 ,with the length at most d −
1, and hence d G ( v , v d +1 ) ≤ d −
1, a contradiction.So, we have d ( G v − v ) ≥ d −
1, as desired. (cid:3)
Now, combining of Theorem 3.3 and Theorem 3.4, we get the following combina-torial lower bound for the depth of binomial edge ideals.
Theorem 3.5.
Let G be a graph on [ n ] . Then depth S/J G ≥ ξ ( G ) . In particular, if G is connected, then depth S/J G ≥ f ( G ) + diam( G ) . We would like to remark that the lower bound in Theorem 3.5 could be tight. Forinstance, let G be the graph illustrated in Figure 1. Then f ( G ) = 8 and d ( G ) = 4,and hence depth S/J G ≥ ξ ( G ) = 12, by Theorem 3.5. On the other hand, the upperbound given in (1) implies that depth S/J G ≤ | V ( G ) | + 1 = 12, since κ ( G ) = 1.Therefore, we get depth S/J G = 12. Figure 1.
A graph G with depth S/J G = ξ ( G ) = 12.Now, combining Theorem 3.5 together with [18, Theorem 5.2] yields the followingbound for the depth of binomial edge ideals. Corollary 3.6.
Let G be a graph on [ n ] with n ≥ . Then depth S/J G ≥ max { , ξ ( G ) } . Moreover, in [18, Theorem 5.3], it was shown that for graphs G with more thanthree vertices, depth S/J G = 4 if and only if G = G ′ ∗ K for some graph G ′ .Therefore, we have: Corollary 3.7.
Let G be a graph on [ n ] with n ≥ and G = G ′ ∗ K for everygraph G ′ . Then depth S/J G ≥ max { , ξ ( G ) } . s we saw in Figure 1, there are graphs G for which depth S/J G = ξ ( G ) > G for which depth S/J G = 5 while ξ ( G ) <
5. For instance,let G be the graph shown in Figure 2. We have ξ ( G ) = 2. However, we will seein Theorem 5.4 that depth S/J G = 5. Our main goal in the rest of this paper is tocharacterize all graphs G with depth S/J G = 5.4. A poset associated to binomial edge ideals and a Hochster typeformula
In this section, continuing the topological approach from [1] and [18], we studythe local cohomology modules of binomial edge ideals. To this end, we first associatea new poset, adopted to our needs, to binomial edge ideals as follows.Let I be an ideal in the polynomial ring S and I = q ∩ · · · ∩ q t be an arbitrarydecomposition for the ideal I . Then, by the poset R I , ordered by the reverse inclu-sion, we mean the poset of all possible sums of ideals in this decomposition, definedin [2, Example 2.1]. Now, we know that J G = T T ⊆ [ n ] P T ( G ). We use R G , instead of R J G , to denote the poset arised from the above decomposition of J G .Now, we define another poset associated to the binomial edge ideal of a graph G . Definition 4.1.
Let G be a graph on [ n ]. Associated to the decomposition J G = T T ⊆ [ n ] P T ( G ), we consider the poset ( M G , ), ordered by reverse inclusion, which ismade up of the following elements: • the prime ideals in the poset R G , • the prime ideals in the posets R 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 R 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.Note that using the non-minimal primary decomposition J G = T T ⊆ [ n ] P T ( G ) inDefinition 4.1, turns the poset M G to be different from the posets A G and Q G ,considered by the authors in [1] and [18], respectively. We also notice that thesignificance of this new definition will be demonstrated in the proof of Theorem 5.4and Theorem 5.8. Furthermore, the following lemma, which is a direct consequenceof [18, Proposition 3.4], guaranties that the process of the construction of the poset M G terminates after a finite number of steps just like the construction process ofthe poset A G as well as the poset Q G . Lemma 4.2.
Let G be a graph on [ n ] . Then every element q of the poset M G is ofthe form P T ( H ) , for some graph H on [ n ] and some T ⊆ [ n ] . Now, by applying Lemma 4.2 and thanks to the flexibility for the decomposition ofthe ideals in [2, Theorem 5.22], the following Hochster type decomposition formula or the local cohomology modules of binomial edge ideals is established by using thesame argument that was applied in the proof of [1, Theorem 3.9]. We first need tofix a notation before stating the formula.Let 1 M G be a terminal element that we add to the poset M G . Then, recall thatfor every q ∈ M G , by the interval ( q, M G ), one means the subposet { z ∈ M G : q (cid:22) z (cid:22) M G } . Theorem 4.3. (see [1, Theorem 3.9] and [18, Theorem 3.6].)
Let G be a graph on [ n ] . Then we have the K -isomorphism H i m ( S/J G ) ∼ = M q ∈M 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, M G ); K ) . Note that the above theorem suggests an interesting and also a wonderful methodto study the depth of binomial edge ideals. Indeed, instead of working directly withthe minimal graded free resolution of
S/J G , which is almost an intractable task, wemay elaborate the topological properties of the subposets ( q, M G ) of M G . In thisapproach beside the algebraic tools, we also employ the topological tools.5. Combinatorial characterization of some binomial edge ideals interms of their depth
At the end of Section 3, we provided a lower bound for the depth of binomial edgeideals. We showed that for a graph G with more than three vertices, depth S/J G ≥ max { , ξ ( G ) } , if G = G ′ ∗ K for every graph G ′ . Now, our main goal in this sectionis to give a combinatorial characterization of graphs G with depth S/J G = 5. Forthis aim, we need to study some topological properties of some certain subposets of M G in order to compute some local cohomology modules of S/J G .First we need to state the following definition. Definition 5.1.
Let T ⊆ [ n ] with | T | = n −
2. Associated to T , we introduce afamily of graphs on [ n ], denoted by G T , such that for each G ∈ G T , there exist twonon-adjacent vertices u and w of G with u, w ∈ [ n ] \ T , and three disjoint subsetsof T , say V , V and V with V , V = ∅ and S i =0 V i = T , such that the followingconditions hold:(1) N G ( u ) = V ∪ V and N G ( w ) = V ∪ V .(2) { v , v } ∈ E ( G ), for every v ∈ V and every v ∈ V . Remark 5.2.
Given three arbitrary graphs G , G and G on disjoint sets of vertices V , V and V , respectively, where V , V = ∅ , we can construct a graph in the family G T with T = S i =0 V i . Note that the vertices in V can be adjacent to some verticesin V and V .An explicit example of a graph G for which G ∈ G T for some T ⊆ V ( G ) with | T | = | V ( G ) | − efore stating the main theorem of this section we need to introduce a family ofgraphs that is essential in our characterization. In the following, 3 K denotes thegraph consisting of three isolated vertices. Definition 5.3.
Let G be a graph on [ n ] with G = G ′ ∗ K for any graph G ′ . Wesay that G is a D -type graph, if one of the following conditions holds:(1) G ∈ G T for some T ⊆ [ n ] with | T | = n − G = H ∗ K , for some graph H ;(3) there exists T ∈ C ( G ) with | T | = n − c G ( T ) = 2. u w Figure 2. A D -type graph G with G ∈ G T , where T = V ( G ) \ { u, w } .Now, we are ready to state the main result of this section which is an explicitcharacterization of graphs G with depth S/J G = 5. Theorem 5.4.
Let G be a graph on [ n ] with n ≥ . Then the following statementsare equivalent: (a) depth S/J G = 5 . (b) G is a D -type graph. To prove the above theorem we need to prepare several auxiliary ingredients.First, we state the following lemma that follows with the same argument as in theproof of [18, Lemma 4.1].
Lemma 5.5.
Let G be a graph on [ n ] . Then q + P ∅ ( G ) ∈ M G , for every q ∈ M G . We also need to recall a concept from the literature of topology of posets.
Definition 5.6.
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 .The following lemma clarifies the importance of the notion of meet-contractibleposets. Lemma 5.7. ([4, Theorem 3.2], see also [23, Proposition 2.4])
Every meet-contractibleposet is contractible.
In the following theorem that is crucial in the proof of Theorem 5.4, we discussthe vanishing of the zeroth and the first reduced cohomology groups of the subposetsassociated to the elements of M G , which are of the form P T ( H ) for some graph H on [ n ] and some T ⊆ [ n ] with | T | = n − heorem 5.8. Let G be a graph on [ n ] and q ∈ M G , where q = P T ( H ) for somegraph H on [ n ] and some T ⊆ [ n ] with | T | = n − . (a) If c H ( T ) = 2 , then ( q, M G ) is connected if and only if G / ∈ G T . (b) If c H ( T ) = 1 , then e H (( q, M G ); K ) = 0 if and only if G / ∈ G T .Proof. Without loss of generality we assume that T = { , . . . , n − } .( a ) We have q = ( x , . . . , x n − , y , . . . , y n − ), since c H ( T ) = 2. Also, { n − , n } / ∈ E ( G ). Indeed, assume on contrary that { n − , n } ∈ E ( G ). Since q ∈ M G , thereexists U ∈ C ( G ) such that P U ( G ) ⊆ q . It follows that U ⊆ T . On the other hand, n − n are two adjacent vertices of G − U . So, we get f n − ,n ∈ P U ( G ) ⊆ q , acontradiction.Now, assume that G / ∈ G T . We show that ( q, M G ) is a connected poset. Weproceed in the following steps:Let L = N G ( n − ∩ N G ( n ), L = N G ( n ) \ N G ( n − L = N G ( n − \ N G ( n ).We also let L = { i ∈ T : i / ∈ N G ( n − ∪ N G ( n ) } . Set X = { P T \{ α } ( G ) : α ∈ L ∪ L ∪ L } . One has X ⊆ ( q, M G ), since { n − , n } / ∈ E ( G ) and α / ∈ L for every α ∈ L ∪ L ∪ L . Step 1:
Let q ′ ∈ ( q, M G ). We claim that there exists P T \{ α } ( G ) ∈ X such thatthere is a path between q ′ and P T \{ α } ( G ) in the 1-skeleton graph of ( q, M G ).By Lemma 4.2 we have that q ′ = P T ′ ( H ′ ), for some graph H ′ on [ n ] and some T ′ ⊆ [ n ]. Now, there exists U ∈ C ( G ) such that P U ( G ) ⊆ q ′ . It follows that U ⊆ T ′ ( T , since q ′ ∈ ( q, M G ). Now we consider the following cases:First assume that T \ U ⊆ L . Therefore, the vertices n − n are isolated in G − U . This implies that P U ( G ) ⊆ P T \{ α } ( G ), for every α ∈ T \ U .Next assume that T \ U * L . Let α ∈ ( T \ U ) \ L . Clearly, α / ∈ L . Indeed,otherwise we get f n − ,n ∈ P U ( G ), a contradiction. So, without loss of generality weassume that α ∈ L . It follows that there is no path between vertices α and n − G − U . Therefore, we have that P U ( G ) ⊆ P T \{ α } ( G ).Thus, the claim follows from both above cases. Step 2:
Assume that L = ∅ . Let α ∈ L and q ′ ∈ ( q, M G ). We show that thereexists a path between q ′ and P T \{ α } ( G ) in the 1-skeleton graph of ( q, M G ).By Step 1, there exists P T \{ β } ( G ) ∈ X such that there is a path between q ′ and P T \{ β } ( G ) in the 1-skeleton graph of ( q, M G ). Moreover, since α ∈ L , it is notdifficult to see that P T \{ α } ( G )+ P T \{ α,β } ( G ) ∈ ( q, M G ) and P T \{ β } ( G ) ⊇ P T \{ α,β } ( G ).Therefore, we have the path P T \{ α } ( G ) , P T \{ α } ( G ) + P T \{ α,β } ( G ) , P T \{ α,β } ( G ) , P T \{ β } ( G ) , in the 1-skeleton graph of ( q, M G ). This implies that ( q, M G ) is connected. So, forthe rest of the proof we may assume that L = ∅ . Step 3:
Now assume that α, β ∈ L , and α < β . We claim that there existsa path in the 1-skeleton graph of ( q, M G ) between P T \{ α } ( G ) and P T \{ β } ( G ), (thesituation in the case α, β ∈ L is similar).We have that P T \{ α } ( G ) = ( x i , y i : i ∈ T \ { α } ) + ( f α,n ) , nd P T \{ β } ( G ) = ( x i , y i : i ∈ T \ { β } ) + ( f β,n ) . So, we get P T \{ α,β } ( G ) = ( x i , y i : i ∈ T \ { α, β } ) + ( f α,n , f β,n , f α,β ) . Therefore, we get the path P T \{ α } ( G ) , P T \{ α,β } ( G ) , P T \{ β } ( G )in the 1-skeleton graph of ( q, M G ), as desired. Step 4:
Let α ∈ L and β ∈ L . We show that there exists a path in the1-skeleton graph of ( q, M G ) between the vertices P T \{ α } ( G ) and P T \{ β } ( G ).First assume that { α, β } / ∈ E ( G ). It follows that P T \{ α,β } ( G ) = ( x i , y i : i ∈ T \ { α, β } ) + ( f β,n − , f α,n ) . Therefore, we get the path P T \{ α } ( G ) , P T \{ α,β } ( G ) , P T \{ β } ( G ) , as desired.Next assume that { α, β } ∈ E ( G ). Now, there exist vertices t ∈ L and t ∈ L ,such that { t , t } / ∈ E ( G ). Indeed, if no such vertices exist, then by putting u = n and w = n − V = L , V = L and V = L in Definition 5.1, we get G ∈ G T , since { n − , n } / ∈ E ( G ) and L = ∅ . Therefore, we get a contradiction.Now, by Step 3, there is a path between P T \{ α } ( G ) and P T \{ t } ( G ), and also a pathbetween P T \{ β } ( G ) and P T \{ t } ( G ), in the underlying graph of ( q, M G ). On theother hand, since { t , t } / ∈ E ( G ), the argument that we used in the first part ofthis step yields a path between P T \{ t } ( G ) and P T \{ t } ( G ). Therefore, we get a pathbetween P T \{ α } ( G ) and P T \{ β } ( G ) in the 1-skeleton graph of ( q, M G ), as desired. Step 5:
Now suppose that q , q ∈ ( q, M G ) and q = q . Then, by Step 1, thereexist P T \{ α } ( G ) , P T \{ β } ( G ) ∈ X such that there is a path between q and P T \{ α } ( G ),and a path between q and P T \{ β } ( G ) in the 1-skeleton graph of ( q, M G ). Moreover,we have a path between P T \{ α } ( G ) and P T \{ β } ( G ), by Steps 3 and 4. Therefore, weget a path between q and q in the 1-skeleton graph of ( q, M G ). Thus, ( q, M G ) isconnected.For the converse, assume that G ∈ G T . Therefore, there exist three disjointsubsets V , V and V of T , such that the conditions of Definition 5.1 hold. Now,let q = P V ∪ V ( G ), and q = P V ∪ V ( G ). We have that q , q ∈ ( q, M G ). Now, weclaim that there is no path between q and q in the 1-skeleton graph of ( q, M G ),and then we conclude the result.Notice that { V ∪ V , V ∪ V } ⊆ C ( G ). Moreover, we have T ′ ∈ { V ∪ V , V ∪ V } ,for every T ′ ∈ C ( G ) such that n − / ∈ T ′ and n / ∈ T ′ . Indeed, one could see that T ′ ⊇ V ∪ V or T ′ ⊇ V ∪ V , for every T ′ ∈ C ( G ) such that n − / ∈ T ′ and n / ∈ T ′ .Without loss of generality we can assume that T ′ ⊇ V ∪ V . Now we claim that T ′ = V ∪ V . Assume on contrary that there exists v ∈ T ′ ∩ V . This implies that v is not a cut vertex of G − ( T ′ \ { v } ), a contradiction to the fact that T ′ ∈ C ( G ).Therefore, we have either q ′ ⊇ q or q ′ ⊇ q , for every q ′ ∈ ( q, M G ). Now, assume n contrary that there exists a path ℓ : q , q ′ , . . . , q ′ t , q , between q and q in the1-skeleton graph of ( q, M G ). Moreover, we may assume that ℓ is an induced pathbetween the vertices q and q . Now, we have that t ≥
2. Indeed, t = 1 implies that q ′ ⊇ q and q ′ ⊇ q , and hence q ′ = q , a contradiction.Now, if q ′ ⊆ q ′ , then q ′ ⊇ q . This clearly contradicts the minimality of thepath ℓ . So, we have q ′ ⊇ q ′ . On the other hand, we have q ′ ⊇ q , since q ′ + q .Therefore, q ′ ⊇ q , a contradiction to the minimality of ℓ .So, there is no path between two vertices q and q in the 1-skeleton graph of( q, M G ), as desired.( b ) Clearly we have q = ( x , . . . , x n − , y , . . . , y n − ) + ( f n − ,n ), since c H ( T ) = 1.Let q ′ = ( x , . . . , x n − , y , . . . , y n − ).First assume that G / ∈ G T . We show that e H (( q, M G ); K ) = 0. We consider thefollowing cases:Case 1: Assume that q ′ / ∈ ( q, M G ). We claim that ( q, M G ) is a meet-contractibleposet. Then the result follows by Lemma 5.7.Clearly, P ∅ ( G ) ∈ ( q, M G ). Now, let q ∈ ( q, M G ) such that q = P ∅ ( G ). ByLemma 4.2, we have that q = P T ( H ), for some graph H on [ n ] and some T ⊆ [ n ].Also, we have that T $ T , since q ′ / ∈ ( q, M G ). Therefore, q + P ∅ ( G ) ∈ ( q, M G ),by Lemma 5.5. Moreover, it is observed that q + P ∅ ( G ) is the meet of the elements q and P ∅ ( G ). Therefore, ( q, M G ) is a meet-contractible poset, as desired.Case 2: Assume that q ′ ∈ ( q, M G ). Let ∆ = ∆( q, M G ), ∆ = star ∆ ( q ′ ) and∆ = del ∆ ( q ′ ). One has del ∆ ( q ′ ) = ∆(( q, M G ) \ { q ′ } ). Then by a similar argumentas in Case 1, it follows that ∆ is a contractible simplicial complex. On the otherhand ∆ ∩ ∆ = link ∆ ( q ′ ) = ∆( q ′ , M G ), since q ′ is a minimal element in the poset( q, M G ).Now, first assume that ∆ ∩ ∆ = {∅} . So, clearly we have H (link ∆ ( q ′ ); K ) = 0.Now, since ∆ is a cone and ∆ is contractible, the Mayer-Vietoris sequence · · · → H (star ∆ ( q ′ ); K ) ⊕ H (del ∆ ( q ′ ); K ) → H (∆; K ) → H (link ∆ ( q ′ ); K ) → · · · implies the result.Next assume that ∆ ∩ ∆ = {∅} . Hence, the reduced Mayer-Vietoris sequence · · · → e H (star ∆ ( q ′ ); K ) ⊕ e H (del ∆ ( q ′ ); K ) → e H (∆; K ) → e H (link ∆ ( q ′ ); K ) → · · · is induced. On the other hand, q ′ ∈ M G . Thus, By Lemma 4.2, we have q ′ = P T ′ ( H ′ ), for some graph H ′ on [ n ] and some T ′ ⊆ [ n ]. Now, it is easily seen that T ′ = T . Also, c H ′ ( T ′ ) = 2, since otherwise we get q ′ = q , a contradiction. Therefore,since G / ∈ G T , by part ( a ) we have e H (link ∆ ( q ′ ); K ) = e H (∆( q ′ , M G ); K ) = 0, andhence by using the latter Mayer-Vietoris sequence we get the result.Now, for the converse, assume that G ∈ G T . We show that e H (( q, M G ); K ) = 0.It is clear by Definition 5.1 that q ′ = P T ( G ), and hence q ′ ∈ ( q, M G ). Moreover,Definition 5.1 again implies that ( q ′ , M G ) is a non-empty poset. Indeed, by thenotation of Definition 5.1 we have P V ∪ V ( G ) , P V ∪ V ( G ) ∈ ( q ′ , M G ). Now by keeping he same notation and also by the same argument as in Case 2 in above, the exactsequence 0 → e H (∆; K ) → e H (link ∆ ( q ′ ); K ) → c G ( T ) = 2, by Definition 5.1. Therefore,since G ∈ G T , part ( a ) implies that e H (link ∆ ( q ′ ); K ) = 0, and hence the resultfollows. (cid:3) Finally, we need to state the following remark that will be used in the proof ofTheorem 5.4. The proof of this remark can be verified by taking a precise look atthe proof of [18, Theorem 4.4].
Remark 5.9.
Let G be a graph on [ n ] with n ≥ . Let b m = P T ( H ) , where H is a graph on [ n ] with | T | = n − . Then the posets ( m , M G ) and ( b m , M G ) arecontractible. Now we are ready to prove the main theorem of this section.Proof of Theorem 5.4: ( a ) ⇒ ( b ) Assume that G is not a D -type graph. We showdepth S/J G = 5. Notice that if G = G ′ ∗ K for some graph G ′ , then depth S/J G =4, by [18, Theorem 5.3]. So, we may assume that G = G ′ ∗ K , for any graph G ′ . Now, by the definition of depth and by Theorem 4.3, it is enough to show that M ,q = dim K e H − d q (( q, M G ); K ) = 0, for all q ∈ M G .Let q be an arbitrary element of the poset M G . We have q = P T ( H ) for somegraph H on [ n ] and some T ⊆ [ n ], by Lemma 4.2. If d q ≥
6, then obviously we havethe result. Moreover, it is easily seen that there is no q ′ ∈ M G such that d q ′ = 1.Therefore, we assume that d q ∈ { , , , , } . Now, we consider the following cases:Let d q ∈ { , } . So, ht q ∈ { n − , n } . This implies that | T |− c H ( T ) ∈ { n − , n } .Therefore, without loss of generality, we assume that either q = m or q = b m .Now, the result follows, since the posets ( m , M G ) and ( b m , M G ) are contractible byRemark 5.9.Let d q = 3. Then | T | − c H ( T ) = n −
3, and hence | T | = n − c H ( T ) = 1.Now, by Definition 5.3, G / ∈ G T , since G is not a D -type graph. Thus, we have M ,q = dim K e H (( q, M G ); K ) = 0, by Theorem 5.8 part ( b ).Let d q = 4. It follows that | T | − c H ( T ) = n −
4. So, we have either | T | = n − c H ( T ) = 2, or | T | = n − c H ( T ) = 1.First assume that | T | = n − c H ( T ) = 2. Therefore, Theorem 5.8 part ( a )implies that M ,q = dim K e H (( q, M G ); K ) = 0, since G / ∈ G T .Next assume that | T | = n − c H ( T ) = 1. Without loss of generality weassume that T = { , . . . , n − } . Therefore, q = ( x , . . . , x n − , y , . . . , y n − ) + ( f n − ,n − , f n − ,n , f n − ,n ) . Let q , q ∈ ( q, M G ) and q = q . Then, there exist T , T ∈ C ( G ) such that q ⊇ P T ( G ) and q ⊇ P T ( G ). Moreover, we have T , T ⊆ T , since q , q ∈ ( q, M G ).Now, we distinguish the following cases: irst assume that T , T $ T . Therefore, by Lemma 5.5, we have that q + P ∅ ( G ) ∈ ( q, M G ) and q + P ∅ ( G ) ∈ ( q, M G ), since q + P ∅ ( G ) $ q and q + P ∅ ( G ) $ q . So,we get the path q , P T ( G ) , P T ( G ) + P ∅ ( G ) , P ∅ ( G ) , P ∅ ( G ) + P T ( G ) , P T ( G ) , q in the 1-skeleton graph of the order complex of the poset ( q, M G ).Next assume that T = T or T = T . If T = T , then we get the path q , P T ( G ) = P T ( G ) , q , as desired. So, without loss of generality, we assume that T = T and T $ T . One has c G ( T ) ∈ { , } , since T ∈ C ( G ). Notice that if c G ( T ) = 2,then Definition 5.3 implies that G is a D -type graph, a contradiction. So, we have c G ( T ) = 3. Furthermore, there exist two vertices α ∈ T and β ∈ { n − , n − , n } ,such that { α, β } / ∈ E ( G ). Indeed, otherwise we get G = H ∗ K , where H = G − { n − , n − , n } , which is a contradiction with Definition 5.3. Now, one couldsee that P T ( G ) + P T \{ α } ( G ) = ( x , . . . , x n − , y , . . . , y n − ) + ( f i,j ) , where i < j and i, j ∈ { n − , n − , n } \ { β } . Therefore, we have that P T ( G ) + P T \{ α } ( G ) ∈ ( q, M G ). So, we get the path q P T ( G ) P T ( G ) + P T \{ α } ( G ) P T \{ α } ( G ) P T \{ α } ( G ) + P ∅ ( G ) P ∅ ( G ) P ∅ ( G ) + P T ( G ) P T ( G ) q between q and q in the 1-skeleton graph of the order complex of the poset( q, M G ).Therefore, it follows from the both cases that ( q, M G ) is connected, and hencethe desired result follows.Let d q = 5. It follows that | T | − c H ( T ) = n −
5. So, we have that either | T | = n − c H ( T ) = 2, or | T | = n − c H ( T ) = 1. Now the result follows once we showthat ( q, M G ) is a non-empty poset.First suppose that | T | = n − c H ( T ) = 2. Assume on contrary that ( q, M G )is an empty poset, i.e. q ∈ Min( J G ). Then, q = P T ′ ( G ) for some T ′ ∈ C ( G ). Itfollows that T ′ = T and c G ( T ′ ) = c H ( T ) = 2. Therefore, Definition 5.3 implies that G is a D -type graph, a contradiction.Next suppose that | T | = n − c H ( T ) = 1. Therefore, we have P ∅ ( G ) ⊆ q .Moreover, the assumption n ≥ T = ∅ , and hence P ∅ ( G ) ∈ ( q, M G ),as desired.( b ) ⇒ ( a ) Assume that G is a D -type graph. Then, Corollary 3.7 implies thatdepth S/J G ≥
5. Therefore, the result follows if we show that depth
S/J G ≤
5. Todo so, we consider the following cases:First assume that G ∈ G T , for some T ⊆ [ n ] with | T | = n −
2. Let q = P T ( G ).We have q ∈ M G and by Definition 5.1 we have c G ( T ) = 2. Thus, Theorem 5.8part ( a ) implies that ( q, M G ) is not connected. Since d q = 4, we have that M ,q = im K e H (( q, M G ); K ) = 0. Therefore, we get H m ( S/J G ) = 0, by Theorem 4.3. Thisyields that depth S/J G ≤
5, as desired.Next assume that G = H ∗ K , for some graph H . If H is a complete graph,then the result follows by [14, Theorem 3.9]. If H is not complete, then it followsfrom [14, Theorem 4.3] and [14, Theorem 4.4] that depth S/J G ≤
5, as desired.Finally, suppose that there exists T ∈ C ( G ) such that | T | = n − c G ( T ) = 2.Let q = P T ( G ). Then d q = 5. Now, the maximality of the element q in the poset M G implies that the poset ( q, M G ) is empty, and hence M ,q = dim K e H − (( q, M G ); K ) =0. Therefore, H m ( S/J G ) = 0, by Theorem 4.3. This implies that depth S/J G ≤ Acknowledgments:
The authors would like to thank Institute for Researchin Fundamental Sciences (IPM) for financial support. The research of the secondauthor was in part supported by a grant from IPM (No. 99130113). The researchof the third author was in part supported by a grant from IPM (No. 99050211).
References [1] J. `Alvarez Montaner,
Local cohomology of binomial edge ideals and their generic initial ideals ,Collect. Math. 71 (2019), 331-348.[2] J. `Alvarez Montaner, A. F. Boix, S. Zarzuela,
On some local cohomology spectral sequences ,Int. Math. Res. Not. 19 (2020), 6197-6293.[3] A. Banerjee, L. L. N´u˜nez-Betancourt,
Graph connectivity and binomial edge ideals , Proc.Amer. Math. Soc. 145 (2017), 487-499.[4] A. Bj¨orner, J. W. Walker,
A homotopy complementation formula for partially ordered sets ,European J. Combin. 4 (1983), 11-19.[5] D. Bolognini, A. Macchia, F. Strazzanti,
Binomial edge ideals of bipartite graphs , EuropeanJ. Combin. 70 (2018), 1-25.[6] V. Ene, J. Herzog, T. Hibi,
Cohen-Macaulay binomial edge ideals , Nagoya Math. J. 204 (2011),57-68.[7] V. Ene, G. Rinaldo, N. Terai,
Licci binomial edge ideals , J. Combin. Theory Ser. A. 175 (2020),105278, 23pp.[8] V. Ene, G. Rinaldo, N. Terai,
Powers of binomial edge ideals with quadratic Gr¨obner bases ,(2020), arXiv:2009.08341v2.[9] V. Ene, A. Zarojanu,
On the regularity of binomial edge ideals , Math. Nachr. 288(1) (2015),19-24.[10] J. Herzog, T. Hibi, F. Hreinsd´ottir, T. Kahle, J. Rauh,
Binomial edge ideals and conditionalindependence statements , Adv. Appl. Math. 45 (2010), 317-333.[11] A. V. Jayanthan, A. Kumar, R. Sarkar,
Regularity of powers of quadratic sequences withapplications to binomial ideals , J. Algebra. 564 (2020), 98-118.[12] D. Kiani, S. Saeedi Madani,
The Castelnuovo-Mumford regularity of binomial edge ideals , J.Combin. Theory Ser. A. 139 (2016), 80-86.[13] A. Kumar,
Regularity bound of generalized binomial edge ideal of graphs , J. Algebra. 546(2020), 357-369.[14] A. Kumar, R. Sarkar,
Depth and extremal Betti number of binomial edge ideals , Math. Nachr.293(9) (2020), 1746-1761.[15] K. Matsuda, S. Murai,
Regularity bounds for binomial edge ideals , J. Commutative Algebra.5(1) (2013), 141-149.
16] M. Ohtani,
Graphs and ideals generated by some 2-minors , Comm. Algebra. 39 (2011), 905-917.[17] M. Rouzbahani Malayeri, S. Saeedi Madani, D. Kiani,
Regularity of binomial edge ideals ofchordal graphs , Collect. Math. (2020), https://doi.org/10.1007/s13348-020-00293-3.[18] M. Rouzbahani Malayeri, S. Saeedi Madani, D. Kiani,
Binomial edge ideals of small depth , J.Algebra. 572 (2021), 231-244.[19] M. Rouzbahani Malayeri, S. Saeedi Madani, D. Kiani,
A proof for a conjecture on the regularityof binomial edge ideals , (2020), arXiv:2007.09959v1.[20] R. Sarkar,
Binomial edge ideals of unicyclic graphs , (2020), arXiv:1911.12677v2.[21] S. Saeedi Madani, D. Kiani,
Binomial edge ideals of graphs , Electronic J. Combin. 19(2)(2012), ♯ P44.[22] S. Saeedi Madani, D. Kiani,
Binomial edge ideals of regularity 3 , J. Algebra. 515 (2018),157-172.[23] J. W. Walker,
Homotopy type and Euler characteristic of partially ordered sets , European J.Combin. 2 (1981), 373-384.[24] Z. Zahid, S. Zafar,
On the Betti numbers of some classes of binomial edge ideals , The ElectronicJournal of Combinatorics. 20(4) (2013), ♯ P37.
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