Herzog, Hibi and Ohsugi conjecture for trees
aa r X i v : . [ m a t h . A C ] M a y SYMBOLIC POWERS OF COVER IDEALS OF CERTAIN GRAPHS
AJAY KUMAR AND RAJIV KUMAR
Abstract.
Let S = k [ x , . . . , x n ] be a polynomial ring, where k is a field, and G be asimple graph on n vertices. Let J ( G ) ⊂ S be the cover ideal of G . In this article, we solve aconjecture due to Herzog, Hibi and Ohsugi for trees which states that powers of cover idealsof trees are componentwise linear. Also, we show that if G is a unicyclic vertex decomposablegraph unless it contains C or C , then symbolic powers of J ( G ) are componentwise linear. Introduction
Let k be a field and S = k [ x , . . . , x n ] be a polynomial ring, n ∈ N > , where N > denotesthe set of positive integers. Set N = N > ∪ { } . Let G be a simple graph with vertex set V ( G ) = { x , . . . , x n } and edge set E ( G ) = {{ x i , x j } : x i , x j ∈ V ( G ) } . Then one can associatean edge ideal I ( G ) ⊂ S to G generated by all monomials x i x j such that { x i , x j } ∈ E ( G ). TheAlexander dual of I ( G ), i.e., J ( G ) = I ( G ) ∨ = T { x i ,x j }∈ E ( G ) h x i , x j i , is called the cover ideal of G . A graph G is said to be vertex decomposable/shellable if its independent complex∆( G ) has this property. A graph G is called (sequentially) Cohen-Macaulay, if the quotientring S/I ( G ) is (sequentially) Cohen-Macaulay. For a graph G , the following implications areknown: vertex decomposable = ⇒ shellable = ⇒ sequentially Cohen-Macaulay . Eagon and Reiner [4] showed that a graph is Cohen-Macaulay if and only if its vertex coverideal has a linear resolution. More generally in [10], Herzog and Hibi proved that a graph issequentially Cohen-Macaulay if and only if its vertex cover ideal is componentwise linear (seeDefinition 2.6). R¨omer [18] observed that if Char( k )= 0, then the multiplicity Conjecturedue to Herzog, Huneke and Srinivasan holds for componentwise linear ideals. In 2009, thisconjecture was solved by Boij-S¨oderberg [1] and Eisenbud-Schreyer [5]. Thus, one wouldlike to find some classes of ideals having componentwise linear resolution. In particular, onemay be interested in finding some combinatorial conditions on certain combinatorial objects(simplicial complex, graph) such that the corresponding associated ideals have (component-wise linear) linear resolution. Authors in [8] proved that the vertex cover ideals of a chordalgraph are componentwise linear. In [12], Herzog, Hibi and Ohsugi studied powers of vertexcover ideals of graphs and stated the following conjecture for powers of the vertex cover idealof a chordal graph. Conjecture 1.1.
Let G be a chordal graph. Then powers of the cover ideal of G arecomponentwise linear. Date : May 19, 2020.2010
Mathematics Subject Classification.
Primary 13C14, 13D02, 05E40.
Key words and phrases.
Componentwise linear, vertex decomposable, regularity, sequentially Cohen-Macaulay, cover ideal, symbolic powers.
There is very little progress made in this conjecture except for very few classes like gen-eralized star graphs, Cohen-Macaulay chordal graphs, (see [12, 15]). Authors in [7] showthat the second power of the vertex cover ideal of a path is componentwise linear. Theyalso ask a question that whether powers of the vertex cover ideal of a chordal graph havelinear quotients or not? This question is a stronger version of the conjecture stated above.For some particular cases the above conjecture has been studied by various authors (see[6, 15, 16, 19]).Authors in [9] show that for a bipartite graph J ( G ) ( k ) = J ( G ) k , where J ( G ) ( k ) denotes the k th symbolic power of J ( G ). Thus, to study Conjecture 1.1 for bipartite chordal graphs, onecan consider symbolic powers of the vertex cover ideal. For a given graph G , Fakhari [20]introduced a new graph G ( k ), and showed that the polarization of J ( G ) ( k ) is the cover idealof G ( k ). He studied a relationship between algebraic properties, e.g., Cohen-Macaulayness,very well covered, of graphs G and G ( k ) . As a consequence, he observed that if G is aCohen-Macaulay and very well covered graph, then symbolic powers of the cover ideal of G have linear quotients, and hence are componentwise linear.The problem of finding the regularity of edge ideals and cover ideals has been extensivelystudied since the last decade. For a graded ideal I of a ring S , it is well known that reg( I s )is a linear function of s for s ≫
0, i.e. there exist non-negative integers a, b and s suchthat reg( I s ) = as + b for all s ≥ s (see [2, 13]). Although the constant a is given by themaximum degree of minimal generators of I , no explicit formula for b and s is known. Theproblem of computing the bounds for the regularity of (symbolic) powers of the vertex coverideal of a graph has been studied by many researchers (see [6, 14, 19, 20, 21]).In this article, our main focus is to address Conjecture 1.1 for trees. For this, we studythe vertex decomposable property of the graph G ( k ) . In Example 3.7, we see that G ( k ) isnot vertex decomposable when G is a chordal graph. For a given graph G , we introducea new construction G ( k t ) which generalizes the construction of G ( k ), and this helps us tounderstand the vertex decomposability of G ( k ). Further, we prove that G ( k t ) is vertexdecomposable when G is a tree. Hence we solve Conjecture 1.1 for trees. For a givenvertex decomposable graph G , we observe that G ( k t ) need not be vertex decomposable (seeExamples 3.5 and 3.6).We now give a brief overview of this paper. In Section 2, we introduce basic notions ofgraph theory and commutative algebra. In Section 3, we settle Conjecture 1.1 for trees,which is a main result of this article (see Theorem 3.3).In Section 4, we prove that for a vertex decomposable unicyclic graph G with cycle C n ,where n = 3 , G ( k ) is vertex decomposable, (see Theorem 4.3). Further, we know that theregularity of a componentwise linear ideal can be determined by the maximum degree of itsminimal generators. As a consequence, we find the regularity of symbolic powers of vertexcover ideals of some classes of vertex decomposable graphs.2. Preliminaries
Definition 2.1. A simplicial complex ∆ on the vertex set V is a collection of subsets of V which satisfy the following:i) For x ∈ V , { x } ∈ ∆.ii) If F ∈ ∆ and F ′ ⊂ F , then F ′ ∈ ∆.An element of ∆ is called a face of ∆ and a maximal face of ∆ with respect to inclusion iscalled a facet of ∆. YMBOLIC POWERS OF COVER IDEALS OF CERTAIN GRAPHS 3
Definition 2.2.
Let ∆ be a simplicial complex on the vertex set V .a) For a face F of ∆, the deletion of F , denoted as the del ∆ ( F ), is a simplicial complexwhich is defined as del ∆ ( F ) = { H ∈ ∆ : H ∩ F = φ } . b) Let F ∈ ∆. Then the link of F , denoted as lk ∆ ( F ), is a simplicial complex which isdefined as lk ∆ ( F ) = { H ∈ ∆ : H ∪ F ∈ ∆ , H ∩ F = φ } . c) A simplicial complex ∆ is said to be vertex decomposable if it is either a simplex or elsehas some vertex x such thati) del ∆ ( x ) and lk ∆ ( x ) are vertex decomposable, andii) no face of lk ∆ ( x ) is a facet of ∆ \ { x } .A vertex x which satisfies Condition (ii) is called a shedding vertex .d) A simplicial complex ∆ is called shellable if there exists a linear order F , . . . , F r of allfacets of ∆ such that for all 1 ≤ i < j ≤ r , there exist x ∈ F j \ F i and s ∈ { , . . . , j − } with F j \ F s = { x } . Definition 2.3.
Let G be a simple graph with the vertex set V ( G ) and the edge set E ( G ).i) A subset A ⊂ V ( G ) is called a vertex cover of G , if A ∩ { x, y } 6 = ∅ for any { x, y } ∈ E ( G )and it is called minimal if for any a ∈ A , A \ a is not a vertex cover of G .ii) A subset C of V ( G ) is called an independent set of G if { x, y } / ∈ E ( G ) for any x, y ∈ C .The collection ∆( G ) of all independent sets of G is a simplicial complex on the vertexset V ( G ), called the independent complex of G .iii) Let x ∈ V ( G ). Then an edge { x, y } obtained by adding a new vertex y at x is called a whisker of G .iv) Let K ⊂ V ( G ). Then by G \ K , we mean the induced subgraph of G on V ( G ) \ K .For a vertex x ∈ V ( G ), the open neighborhood of x in G is defined as N G ( x ) = { y ∈ G : { x, y } ∈ E ( G ) } , and N G [ x ] = N G ( x ) ∪ { x } is called the closed neighborhood of x in G .A graph G is said to be a vertex decomposable graph (resp. shellable ) if the independentcomplex ∆( G ) is vertex decomposable (resp. shellable). Thus the definition of a vertexdecomposable simplicial complex translates to a vertex decomposable graph (see [22]) asfollowing. Definition 2.4.
A graph G is said to be vertex decomposable if it has no edges or there isa vertex x in G such thati) G \ { x } and G \ N G [ x ] are vertex decomposable, andii) for every independent set C in G \ N G [ x ], there exists some y ∈ N G ( x ) such that C ∪ { y } is independent in G \ { x } . Definition 2.5.
Let M be a finitely generated Z -graded S -module.i) Then β Si,j ( M ) = (dim k (Tor Si ( M, k )) j is called the ( i, j ) th graded Betti number of M .ii) The regularity of M , denoted as reg( M ), is defined asreg( M ) = max { j − i : β Si,j ( M ) = 0 } . iii) A module M is said to have linear resolution , if for some integer d , β i,i + b = 0 for all i and every b = d . A. KUMAR AND R. KUMAR iv) A module M is called sequentially Cohen-Macaulay if there is a finite filtration of graded S -modules 0 = M ⊂ M ⊂ · · · ⊂ M t = M such that all M i /M i − are Cohen-Macaulay,and the Krull dimensions of their quotients satisfydim( M /M ) < dim( M /M ) < · · · < dim( M t /M t − ) . A graph G is called a sequentially Cohen-Macaulay over k if S/I ( G ) is sequentially Cohen-Macaulay. Definition 2.6.
Let I be a graded ideal of S . Then I
Let I be a squarefree monomial ideal of S . Then S/I is sequentially Cohen-Macaulay if and only if I ∨ is componentwise linear. Definition 2.8.
Let I be a monomial ideal of S . Then I is said to have linear quotients if there is an ordering u , . . . , u r of minimal generators of I such that h u , . . . , u i − i : h u i i isgenerated by a subset of { x , . . . , x n } for all i .The following lemma shows that the concept of linear quotient is very useful to determineif an ideal has a linear resolution. Lemma 2.9.
Let I be a monomial ideal of S generated in same degree. Then I has a linearresolution if it has linear quotients. Definition 2.10.
Let I be a squarefree monomial ideal in S with irredundant primarydecomposition I = p ∩ · · · ∩ p r , where p i is an ideal generated by some variables in S . Thenfor s ∈ N > , the s th symbolic power of I , denoted by I ( s ) , is defined as follows: I ( s ) = p s ∩ · · · ∩ p sr . The concept of polarization is a very useful tool to convert a monomial ideal into a square-free monomial ideal.
Definition 2.11.
Let u = Q ni =1 x m i i be a monomial. Then a polarization of u in T is thesquarefree monomial e u = Q ni =1 Q m i j =1 x ij , where T = k [ x , x , . . . , x , x , . . . , x n , x n , . . . ].If I is a monomial ideal of S generated by monomials u , . . . , u m , then the squarefree mono-mial ideal e I = h e u , . . . , e u m i ⊂ T is called the polarization of I .3. Powers of vertex cover ideals of trees
Fakhari [20] introduces a new construction of graphs to obtain a Cohen-Macaulay verywell covered graph from a given one. Motivated from his idea, we introduce a constructionto obtain a vertex decomposable graph from a given tree or unicyclic graph with cycle C n , n = 3 , Construction : Let G be a simple graph with the vertex set V = { x , . . . , x n } and edges E ( G ) = { e , . . . , e t } . Let p ∈ N > . Then for an edge e = { x i , x j } , we define a graph e ( p ) withvertices V ( e ( p )) = { x s,a : s ∈ { i, j } , ≤ a ≤ p } and edge set E ( e ( p )) = {{ x i,l , x j,m } : l + m ≤ p + 1 } . By convention, we set e (0) to be an isolated graph on vertices V ( e (0)) = { x i, , x j, } .Consider an ordered tuple k t = ( k , . . . , k t ) ∈ N t . Define a graph G ( k t ) = G ( k , . . . , k t ) onnew vertices V ( G ( k t )) = ∪ ti =1 V ( e i ( k i )) and the edge set E ( G ( k t )) = ∪ ti =1 E ( e i ( k i )). YMBOLIC POWERS OF COVER IDEALS OF CERTAIN GRAPHS 5
Remark 3.1.
Let G be a tree on the vertex set V ( G ) = { x , . . . , x n } and the edge set E ( G ) = { e , . . . , e n − } .i) Let x a ∈ V ( G ) be a vertex of degree 1 and x b be the unique neighbour of x a . Without theloss of generality, we assume that N G ( x b ) = { x a = x b , x b , . . . , x b r } with deg( x b q ) = 1 , for 0 ≤ q ≤ s and s ≤ r . Let e b q = { x b , x b q } , where q ∈ { , . . . , r } . Further, assume thatthe edges incident with a vertex x b q are e i q − s − +1 , . . . , e i q − s for q ∈ { s + 1 , . . . , r } , where i = 0.ii) Set k = max { k b , . . . , k b r } . By deleting a vertex x b, from G ( k n − ) and identifying x b,j with x b,j − for all 1 ≤ j ≤ k , we get G ( k n − ) \ { x b, } ≃ G ( k , . . . , k ′ b , . . . , k ′ b r , . . . , k n − ) , where k ′ b i = max { , k b i − } . Lemma 3.2.
Let G be a tree on the vertex set V ( G ) = { x , . . . , x n } and the edge set E ( G ) = { e , . . . , e n − } with notations as in Remark 3.1. Then G ( k n − ) \ N G ( k n − ) [ x b, ] isisomorphic to ( G \ N G [ x b ]) ( b k b ) ∪ r [ q = s +1 i q − s [ j q = i q − s − +1 e j q ( k j q − k b q ) , where b k b is obtained from k b by deleting components k b i corresponding to deleted edges, and e j q ( k j q − k b q ) = ∅ for k j q − k b q ≤ .Proof. First observe that N G ( k n − ) [ x b, ] = r S q =0 { x b q ,l : 1 ≤ l ≤ k b q } ∪ { x b, } . For a fixed q ∈ { s + 1 , . . . , r } , we have the following cases:Case 1: If k j q − k b q ≤
0, then clearly G ( k n − ) \ N G ( k n − ) [ x b, ] = ( G \ N G [ x b ]) ( b k b ) ∪{ x b, , . . . , x b,k } .Case 2: If k j q − k b q >
0, then the result follows by identifying x b q ,k bq + p = x b q ,p for all1 ≤ p ≤ k j q − k b q . (cid:3) Theorem 3.3.
Let G be a tree on the vertex set V ( G ) = { x , . . . , x n } and the edge set E ( G ) = { e , . . . , e n − } with notations as in Remark 3.1. Then G ( k n − ) is a vertex decom-posable graph, where k n − ∈ N n − .Proof. We may assume that n >
1. Now we use the induction on N = n − P i =1 k i . If N = 0, then G ( k n − ) is a collection of isolated vertices, and hence vertex decomposable. We thereforesuppose that N >
0. Observe that N G ( k n − ) [ x a,k b ] ⊂ N G ( k n − ) [ x b, ]. Thus, by [3, Lemma4.2], it is enough to show that G ( k n − ) \ { x b, } and G ( k n − ) \ N G ( k n − ) [ x b, ] are vertexdecomposable. By Lemma 3.2 and induction on N , it follows that G ( k n − ) \ N G ( k n − ) [ x b, ]is vertex decomposable. Also, G ( k n − ) \ { x b, } is vertex decomposable follows from Remark3.1 and induction on N . (cid:3) We denote G ( k ) = G ( k , . . . , k t ) with k i = k for all i . Using the concept of polarizaton ofa monomial ideal, Fakhari in [20] showed that J ( G ( k )) = ^ J ( G ) ( k ) , where k ∈ N > . In thefollowing, we prove Conjecture 1.1 for trees. Corollary 3.4.
Let G be a tree. Then for every k ∈ N > , J ( G ) k has linear quotients, andhence it is componentwise linear.Proof. In view of Theorem 3.3, G ( k ) is vertex decomposable, and hence shellable. Now using[11, Theorem 8.2.5], J ( G ( k )) = ^ J ( G ) ( k ) has linear quotients. By [20, Lemma 3.5], J ( G ) ( k ) A. KUMAR AND R. KUMAR has a linear quotient, and hence by [11, Theorem 8.2.15] componentwise linear. Now, theresult follows from [9, Corrolary 2.6]. (cid:3)
For k t ∈ N t , we have proved that G ( k t ) is vertex decomposable if G is forest. Thus, it isnatural to ask what are other classes of graphs for which G ( k t ) is vertex decomposable. Inthe following examples, we show that this is not true for bipartite vertex decomposable andchordal graphs. Example 3.5.
Let G be a graph on the vertex set V ( G ) = { x , x , x , x , x } and theedge set E ( G ) = { e = { x , x } , e = { x , x } , e = { x , x } , e = { x , x } , e = { x , x }} asshown in the figure. Then the fact N G [ x ] ⊂ N G [ x ] implies that x is a shedding vertex of G . Also, it is easy to see that G \ { x } and G \ N G [ x ] are both vertex decomposable, andhence G is a vertex decomposable graph. But G (1 , , , ,
2) is not a vertex decomposablegraph. To verify this note that x is the unique shedding vertex of G (1 , , , , G (1 , , , , \ { x } is a cycle C which is not vertex decomposable. x x x x x G G (1 , , , , x , x , x , x , x , x , x , x , Example 3.6.
Let G be a graph on the vertex set V ( G ) = { x , x , x , x , x } and the edgeset E ( G ) = { e = { x , x } , e = { x , x } , e = { x , x } , e = { x , x } , e = { x , x } , e = { x , x }} . Then G is a chordal graph. Proceeding as in the above example one can checkthat G (1 , , , , ,
1) is not a vertex decomposable graph. x x x x x G G (1 , , , , , x , x , x , x , x , x , x , x , The following example illustrate the fact that if G is a vertex decomposable graph, then G ( k ) need not be vertex decomposable. Example 3.7.
Let G be a graph on the vertex set V ( G ) = { x , x , x , x } and the edge set E ( G ) = { e = { x , x } , e = { x , x } , e = { x , x } , e = { x , x } , e = { x , x }} . Then G isa vertex decomposable graph but G ( ) is not a vertex decomposable graph. YMBOLIC POWERS OF COVER IDEALS OF CERTAIN GRAPHS 7 Powers of vertex cover ideals of unicyclic graphs
From Example 3.5, we observe that if G is a unicyclic graph on n vertices, then G ( k n ) neednot be vertex decomposable. On the other hand, if G is a unicyclic vertex decomposablegraph with cycle C n , n = 3 ,
5, then we show that G ( k ) is a vertex decomposable graph (seeTheorem 4.3 ). The following results will be useful in proving the main result of this section. Lemma 4.1 (Selvaraja, [19]) . Let G be a graph and { x , . . . , x m } ⊂ V ( G ) . Set ψ = G,ψ i = ψ i − \ { x i } , φ i = ψ i − \ N ψ i − [ x i ] for all ≤ i ≤ m . Then G is a vertex decomposablegraph if it satisfies the following: i) x i is a shedding vertex of ψ i − for all ≤ i ≤ m , ii) φ i is vertex decomposable for all ≤ i ≤ m , and iii) ψ m is vertex decomposable. In [17], Mohammadi, Kiani and Yassemi gives a complete description of vertex decompos-able unicyclic graphs which is noted in the following lemma.
Lemma 4.2 (Mohammadi, Kiani and Yassemi, [17]) . Let G be a unicyclic graph with cycle C n , n = 3 , . Then G is vertex decomposable if and only if atleast one whisker is attached to C n . Now we prove the main theorem of this section.
Theorem 4.3.
Let G be a unicyclic graph with cycle C n , n = 3 , . If G is vertex decom-posable, then G ( k ) is also vertex decomposable.Proof. Using Lemma 4.2, there exists a whisker attached to C n . Further, let { x a , x b } be awhisker attached to C n at a vertex x b . We set ψ = G ( k ) , ψ i = ψ i − \ { x b,i } , φ i = ψ i − \ N ψ i − [ x b,i ]for all 1 ≤ i ≤ k. In order to prove the theorem, first we show that x b,i is a shedding vertex in ψ i − for1 ≤ i ≤ k . Since x b is the only vertex which is adjacent to x a in G , It follows from thedefinition of G ( k ) that a vertex y is adjacent to x a,k − i +1 if and only if y = x b,j for some j with k − i + 1 + j ≤ k + 1. This implies that N G ( k ) ( x a,k − i +1 ) = { x b,j : 1 ≤ j ≤ i } . From thedefinition of ψ i − = G ( k ) \ { x b,j : 1 ≤ j ≤ i − } , it follows that x b,i is the only vertex whichis adjacent to x a,k − i +1 in ψ i − . Thus, using [3, Lemma 4.2], we get x b,i is a shedding vertexof ψ i − for all 1 ≤ i ≤ k .Now, we show that φ i is vertex decomposable for 1 ≤ i ≤ k . From the definition of ψ i − ,observe that N ψ i − ( x b,j ) ⊂ N ψ i − ( x b,i ) for all j ≥ i . This implies that, for j > i , x b,j is anisolated vertex in φ i . Set H = G \{ x b } , H = G \ N G [ x b ] and assume that | E ( H ) | = r . Notethat φ is isomorphic to H ( k ) with some isolated vertices, and hence vertex decomposableby Theorem 3.3. For i ≥
2, identify x l,j with x l,j − k − i in φ i for all x l ∈ N G ( x b ). Now, afterdeleting isolated vertices, we get φ i = H ( k i , . . . , k i r ), where k i j ≤ k . Since H is forest, byTheorem 3.3, we get that φ i is vertex decomposable.Now, using Lemma 4.1, it is remaining to show that ψ k is vertex decomposable. Thisfollows from Theorem 3.3 and the fact that ψ k = H ( k ), and H is a forest. (cid:3) As an immediate consequence, we get the following result.
Corollary 4.4.
Let G be a unicyclic vertex decomposable graph with cycle C n , n = 3 , .Then symbolic powers of J ( G ) are componentwise linear A. KUMAR AND R. KUMAR
Proof.
For k ≥
1, by Theorem 4.3, G ( k ) is vertex decomposable, and hence sequentiallyCohen-Macaulay. Using Lemma 2.7, we get that J ( G ( k )) is componentwise linear. Since ^ J ( G ) ( k ) = J ( G ( k )), J ( G ) ( k ) is componentwise linear. (cid:3) For a graph G , we denote deg( J ( G )) is the maximum degree of minimal monomial gener-ators of J ( G ). As an application of Corollary 3.4 and Corollary 4.4, we have the following: Theorem 4.5. i) Let G be a tree. Then reg( J ( G ) s ) = s deg( J ( G )) for s ≥ . ii) Let G be a unicyclic vertex decomposable graph with cycle C n , n = 3 , . Then reg( J ( G ) ( s ) ) = s deg( J ( G )) for s ≥ . References [1] M. Boij and J. S¨oderberg. Graded betti numbers of cohen–macaulay modules and the multiplicityconjecture.
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Indian Institute of Technology Jammu, India.
E-mail address : [email protected] The LNM Institute of Information Technology, Jaipur, India.
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