Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs
Seyed Amin Seyed Fakhari, Kosuke Shibata, Naoki Terai, Siamak Yassemi
aa r X i v : . [ m a t h . A C ] M a r COHEN-MACAULAY EDGE-WEIGHTED EDGE IDEALS OF VERYWELL-COVERED GRAPHS
SEYED AMIN SEYED FAKHARI, KOSUKE SHIBATA, NAOKI TERAI,AND SIAMAK YASSEMI
Abstract.
We characterize unmixed and Cohen-Macaulay edge-weighted edge idealsof very well-covered graphs. We also provide examples of oriented graphs which haveunmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge idealof their underlying graph is Cohen-Macaulay. This disproves a conjecture posed byPitones, Reyes and Toledo. Introduction
In this article, a graph means a simple graph without loops, multiple edges, andisolated vertices. Let G be a graph with the vertex set V ( G ) = { x , . . . , x n } and withthe edge set E ( G ). Suppose w : E ( G ) −→ Z > is an edge weight on G . We write G w for the pair ( G, w ) and call it an edge-weighted graph. Let S = K [ x , . . . , x n ] be thepolynomial ring in n variables over a field K . The (edge-weighted) edge ideal of anedge-weighted graph G w was introduced in [12] and it is defined as I ( G w ) = (cid:0) ( x i x j ) w ( x i x j ) | x i x j ∈ E ( G ) (cid:1) , (by abusing the notation, we identify the edges of G with quadratic squarefree mono-mials of S ). Paulsen and Sather-Wagstaff [12] studied the primary decomposition ofthese ideals. They also investigated unmixedness and Cohen-Macaulayness of theseideals, in the case that G is a cycle, tree or a complete graph. The aim of this paperis to continue this study. In Section 3, we characterize unmixed and Cohen-Macaulayproperties of the edge-weighted edge ideals of very well-covered graphs (see Section2 for the definition of very well-covered graphs). Our results can be seen as general-izations of the results concerning the Cohen-Macaulay property of usual edge idealsof very well-covered graphs (see e.g., [2, 3, 4, 8]). For other aspects of ring-theoreticstudy for very well-covered graphs, see e.g., [1, 9, 10, 16].Another kind of generalization of edge ideals is considered in [7, 13, 14]. In-deed, Pitones, Reyes and Toledo [13] introduced the vertex-weighted edge ideal ofan oriented graph as follows. Let D = ( V ( D ) , E ( D )) be an oriented graph with V ( D ) = { x , . . . , x n } , and let w : V ( D ) −→ Z > be a vertex-weighted on D . Set Mathematics Subject Classification.
Primary 05C75, Secondary 05C90, 13H10, 55U10.
Key words and phrases.
Cohen-Macaulay graph, Edge-weighted graph, Unmixed ideal, Weightededge ideal. w j := w ( x j ). The vertex-weighted edge ideal of D is defined as I ( D ) = (cid:0) x i x ω j j | ( x i , x j ∈ E ( D ) (cid:1) . Pitones, Reyes and Toledo proposed the following conjecture.
Conjecture 1.1. [13, Conjecture 53]
Let D be a vertex-weighted oriented graph andlet G be its underlying graph. If I ( D ) is unmixed and S/I ( G ) is Cohen-Macaulay,then S/I ( D ) is Cohen-Macaulay. In Section 4, we provide counterexamples for this conjecture.We close this introduction by mentioning that unmixed and Cohen-Macaulay prop-erties of vertex-weighted edge ideals of vertex-weighted oriented very well-coveredgraphs are studied by Pitones, Reyes and Villarreal [14].2.
Preliminaries
In this section, we provide the definitions and basic facts which will be used in thenext sections. We refer to [5] and [18] for detailed information.Let G be a graph with the vertex set V ( G ) = { x . . . , x n } and with the edge set E ( G ). For every integer 1 ≤ i ≤ n , the degree of x i , denoted by deg G x i , is the numberof edges of G which are incident to x i . For F ⊂ E ( G ) we denote ( V ( G ) , E ( G ) \ F )by G − F . For a family F of 2-element subsets of V ( G ) the graph ( V ( G ) , E ( G ) ∪ F )is denoted by G + F . A subset C ⊂ V ( G ) is a vertex cover of G if every edge of G is incident with at least one vertex in C . A vertex cover C of G is called minimal ifthere is no proper subset of C which is a vertex cover of G . A subset A of V ( G ) iscalled an independent set of G if no two vertices of A are adjacent. An independentset A of G is maximal if there exists no independent set which properly includes A .Observe that C is a minimal vertex cover of G if and only if V ( G ) \ C is a maximalindependent set of G . A subset M ⊆ E ( G ) is a matching if e ∩ e ′ = ∅ , for every pair ofedges e, e ′ ∈ M . If every vertex of G is incident to an edge in M , then M is a perfectmatching of G . A graph G without isolated vertices is said to be very well-covered if | V ( G ) | is an even integer and every maximal independent subset of G has cardinality | V ( G ) | / G is called Cohen-Macaulay if S/I ( G ) is a Cohen-Macaulay ring. Here, I ( G ) is the edge ideal of G , which is defined as I ( G ) = (cid:0) x i x j | x i x j ∈ E ( G ) (cid:1) . An ideal I ⊂ S is unmixed if the associated primes of S/I have the same height. It iswell known that I is unmixed if S/I is a Cohen-Macaulay ring. A graph G is called unmixed if the minimal vertex covers of G have the same size. It can be easy seenthat G is an unmixed graph if and only if I ( G ) is an unmixed ideal. Also, note thatheight I ( G ) is equal to the smallest size of vertex covers of G .We introduce polarization according to [17]. Let I be a monomial ideal of S = K [ x , . . . , x n ] with minimal generators u , . . . , u m , where u j = Q ni =1 x a i,j i , 1 ≤ j ≤ m . EIGHTED EDGE IDEAL 3
For every i with 1 ≤ i ≤ n , let a i = max { a i,j | ≤ j ≤ m } , and suppose that T = K [ x , x , . . . , x a , x , x , . . . , x a , . . . , x n , x n , . . . , x na n ]is a polynomial ring over the field K . Let I pol be the squarefree monomial ideal of T with minimal generators u pol1 , . . . , u pol m , where u pol j = Q ni =1 Q a i,j k =1 x ik , 1 ≤ j ≤ m .The monomial u pol j is called the polarization of u j , and the ideal I pol is called the polarization of I . It is well known that polarization preserves the height of ideal.Moreover, I is an unmixed ideal if and only if I pol is an unmixed ideal.Finally, we recall the concept of Serre’s condition. Let I be a monomial ideal of S .For a positive integer k , the ring S/I satisfies the Serre’s condition ( S k ) ifdepth( S/I ) p ≥ min { dim( S/I ) p , k } for every p ∈ Spec(
S/I ). Lemma 2.1. [15, Lemma 3.2.1]
The following two conditions are equivalent.(1)
S/I satisfies the Serre’s condition ( S k ) .(2) For every integer i with ≤ i < dim S/I , the inequality dim Ext n − iS ( S/I, S ) ≤ i − k holds, where the dimension of zero module is defined to be −∞ . Edge-weighted edge ideal of very well-covered graphs
In this section, we study the unmixed and Cohen-Macaulay properties of edge-weighted edge ideal of very well-covered graphs. We first recall some known factsabout the structure of very well-covered graphs.
Lemma 3.1. [6]
Let G be a very well-covered graph. Then G has a perfect matching. By the above lemma, we may assume that the vertices of the very well-coveredgraph G are labeled such that the following condition is satisfied.(*) V ( G ) = X ∪ Y , X ∩ Y = ∅ , where X = { x , . . . , x h } is a minimal ver-tex cover of G and Y = { y , . . . , y h } is a maximal independent set of G such that { x y , . . . , x h y h } ⊂ E ( G ).Following the notations of condition (*), for the rest of this section, we set S = K [ x , . . . , x h , y , . . . , y h ]. For later use, we recall the following characterization of verywell-covered graphs. Proposition 3.2. [3, 11]
Let G be a graph with h vertices, which are not isolated.Assume that the vertices of G are labeled such that the condition (*) is satisfied. Then G is very well-covered if and only if the following hold.(i) If z i x j , y j x k ∈ E ( G ) , then z i x k ∈ E ( G ) for distinct indices i , j and k and for z i ∈ { x i , y i } .(ii) If x i y j ∈ E ( G ) , then x i x j / ∈ E ( G ) . SEYED AMIN SEYED FAKHARI, KOSUKE SHIBATA, NAOKI TERAI, AND SIAMAK YASSEMI
We are now ready to state and prove the first main result of this paper, whichcharacterizes edge-weighted very well-covered graphs with unmixed edge ideals.
Theorem 3.3.
Let G be a very well-covered graph with h vertices and let w be anedge weight on G . Moreover, assume that the vertices of G are labeled in such a waythat the condition (*) is satisfied. Then I ( G w ) is unmixed if and only if the followinghold.(i) If x i z j ∈ E ( G ) , then w ( x i z j ) ≤ w ( x i y i ) and w ( x i z j ) ≤ w ( x j y j ) for distinctindices i, j , and for any vertex z j ∈ { x j , y j } .(ii) If z i x j and y j x k are edges of G , then w ( z i x k ) ≤ w ( z i x j ) and w ( z i x k ) ≤ w ( y j x k ) for distinct indices i, j, k and for z i ∈ { x i , y i } , or for distinct indices j, i = k and for z i = y i .Proof. Set J := I ( G w ) pol .Suppose I ( G w ) is unmixed. Then J is an unmixed ideal of height h . In particular,for every integer i with 1 ≤ i ≤ h , any minimal prime of J contains exactly onevariable whose first index is i . We first prove condition (i). Assume that x i z j ∈ E ( G ).Set a := w ( x i z j ) and b := w ( x i y i ). As x i x i · · · x ia z j z j · · · z ja ∈ J, there is a minimal prime p of J with x ia ∈ p . By contradiction, suppose a > b . Itfollows from x i x i · · · x ib y i y i · · · y ib ∈ J that at least one of the variable x i , x i , . . . , x ib , y i , y i , . . . , y ib belongs to p . There-fore, p contains two variables with first index i , which is a contradiction. Hence, a ≤ b .Now, set c := w ( x j y j ) and suppose a > c . As x i x i · · · x ia z j z j · · · z ja ∈ J, there is a minimal prime p of J with z ja ∈ p . Also, it follows from x j x j · · · x jc y j y j · · · y jc ∈ J that at least one of the variable x j , x j , . . . , x jc , y j , y j , . . . , y jc belongs to p . There-fore, p contains two variables with first index j , which is a contradiction. Hence, a ≤ c .Next, we prove condition (ii). Assume that z i x j and y j x k ∈ E ( G ). Since G isunmixed, it follows from Proposition 3.2 that z i x k ∈ E ( G ) (this is trivially true, if i = k and for z i = y i ). Set d := w ( z i x k ), e := w ( z i x j ), f := w ( y j x k ). Suppose d > e .Since w ( z i x k ) = d , it follows that z i z i · · · z i ( d − x k x k · · · x kf / ∈ J. Thus, there is a minimal prime p of J with z i z i · · · z i ( d − x k x k · · · x kf / ∈ p . EIGHTED EDGE IDEAL 5
Hence, neither of the variables z i , z i , . . . , z i ( d − , x k , x k , . . . , x kf belongs to p . Thenwe deduce from z i z i · · · z ie x j x j · · · x je , y j y j · · · y jf x k x k · · · x kf ∈ J that x js , y jt ∈ p , for some positive integers s and t . This is a contradiction, as nominimal prime of J can contain both of x js and y jt . Thus, d ≤ e .Suppose d > f . Since z i z i · · · z ie x k x k · · · x k ( d − / ∈ J, there is a minimal prime p of J which contains neither of the variables z i , z i , . . . , z ie , x k , x k , . . . , x k ( d − . It follows from z i z i · · · z ie x j x j · · · x je , y j y j · · · y jf x k x k · · · x kf ∈ J that x jℓ , y jr ∈ p , for some positive integers ℓ and r . This is again a contradiction.Therefore, d ≤ f .We now prove the reverse implication. Suppose conditions (i) and (ii) hold andassume by contradiction that I ( G w ) is not unmixed. Hence, J is not an unmixedideal. Thus, there is a minimal prime p of J such that x jp , y jq ∈ p , for some integers j, p, q ≥
1. As above set c := w ( x j y j ).Assume p > c . Since p is a minimal prime of J , there is i = j and z i ∈ { x i , y i } suchthat z i x j ∈ E ( G ) and w ( z i x j ) ≥ p . Then by (i), we have c ≥ w ( z i x j ) ≥ p > c , whichis a contradiction. Hence p ≤ c .Suppose q > c . Since p is a minimal prime of J , there is k = j such that y j x k ∈ E ( G )with w ( y j x k ) ≥ q . Then by (i) we have c ≥ w ( y j x k ) ≥ q > c , which is a contradiction.Therefore, q ≤ c .Since p is a minimal prime of J , there is ℓ = j and z ℓ ∈ { x ℓ , y ℓ } such that z ℓ x j ∈ E ( G ) with α := w ( z ℓ x j ) ≥ p and z ℓ , z ℓ , . . . , z ℓα p . Similarly,, there is r = j such that y j x r ∈ E ( G ) with β := w ( y j x r ) ≥ q and x r , x r , . . . , x rβ p . By Proposition 3.2, z ℓ x r ∈ E ( G ). Set γ := w ( z ℓ x r ). It follows from condition (ii) that γ ≤ α and γ ≤ β . Thus, z ℓ , z ℓ , . . . , z ℓγ , x r , x r , . . . , x rγ / ∈ p . This contradicts z ℓ z ℓ · · · z ℓγ x r x r · · · x rγ ∈ J. Hence, I ( G w ) is an unmixed ideal. (cid:3) SEYED AMIN SEYED FAKHARI, KOSUKE SHIBATA, NAOKI TERAI, AND SIAMAK YASSEMI
Remark 3.4.
Let G be a very well-covered graph and let w be an edge weight,such that I ( G w ) is an unmixed ideal. Assume that the vertices of G are labeled insuch a way that the condition (*) is satisfied. It follows from Theorem 3.3 that if x i y j , x j y i ∈ E ( G ), then w ( x i y i ) = w ( x j y j ) = w ( x i y j ) = w ( x j y i ).Our next goal is to provide a combinatorial characterization for Cohen-Macaulaynessof edge-weighted edge ideal of very well-covered graphs. First we summarize the knownresults concerning the Cohen-Macaulay property of a (non-weighted) very well-coveredgraph. Lemma 3.5. [3]
Let G be an unmixed graph with h vertices, which are not isolated,and assume that the vertices of G are labeled such the condition (*) is satisfied. If G is a Cohen-Macaulay graph then there exists a suitable simultaneous change oflabeling on both { x i } hi =1 and { y i } hi =1 (i.e., we relabel ( x i , . . . , x i h ) and ( y i , . . . , y i h ) as ( x , . . . , x h ) and ( y , . . . , y h ) at the same time), such that x i y j ∈ E ( G ) implies i ≤ j . Hence, for a Cohen-Macaulay very well-covered graph G satisfying the condition(*), we may assume that(**) x i y j ∈ E ( G ) implies i ≤ j .Now we recall Cohen-Macaulay criterion for very well-covered graphs. See also [2, 4]for different characterizations. Theorem 3.6. [3]
Let G be a graph with h vertices, which are not isolated andassume that the vertices of G are labeled such that the conditions (*) and (**) aresatisfied. Then the following conditions are equivalent:(1) G is Cohen-Macaulay;(2) G is unmixed;(3) The following conditions hold:(i) If z i x j , y j x k ∈ E ( G ) , then z i x k ∈ E ( G ) for distinct indices i, j, k and for z i ∈ { x i , y i } ;(ii) If x i y j ∈ E ( G ) , then x i x j / ∈ E ( G ) . In order to study the Cohen-Macaulay property of edge-weighted edge ideal of verywell-covered graphs, we introduce an operator which allows us to construct a newweighted very well-covered graph from a given one.Let G w be a weighted very well-covered graph with n = 2 h vertices and assumethat the vertices of G are labeled such that the condition (*) is satisfied. For any i ∈ [ h ] := { , . . . , h } , set N i := { k ∈ [ h ] : x k y i ∈ E ( G ) } \ { i } , and define the base graph O i ( G ) as follows O i ( G ) := G − { x k y i : k ∈ N i } + { x k x i : k ∈ N i } . EIGHTED EDGE IDEAL 7
Now we define the weight w ′ on O i ( G ) by w ′ ( e ) = (cid:26) w ( x k y i ) if e = x k x i , k ∈ N i w ( e ) otherwise.Finally, we set O i ( G w ) := O i ( G ) w ′ . We are now ready to prove the second main result of this paper.
Theorem 3.7.
Let G be a Cohen-Macaulay very well-covered graph and let w be anedge weight on G . Then the following conditions are equivalent.(1) I ( G w ) is an unmixed ideal.(2) S/I ( G w ) is a Cohen-Macaulay ring.Proof. The implication (2) = ⇒ (1) is well known. So, we prove (1) implies (2). As G is a Cohen-Macaulay very well-covered graph, we may assume that conditions (*)and (**) are satisfied. In particular, | V ( G ) | = 2 h . It follows from unmixedness of I ( G w ) that the height of every associated prime of S/I ( G w ) is h . Thus, for every p ∈ Ass
S/I ( G w ) and for every integer k with 1 ≤ k ≤ h , exactly one of x k and y k belongs to p .We use induction on m := P hi =1 deg G y i . For m = h , the assertion follows from[12, Theorem 5.7]. Hence, suppose m > h . Then there exists an integer k with1 ≤ k ≤ h such that deg y k ≥
2. By contradiction, assume that
S/I ( G w ) is notCohen-Macaulay. Set G ′ w ′ = O k ( G w ). Using Theorem 3.3, one can easily check that I ( G ′ w ′ ) is an unmixed ideal. By induction hypotheses S/I ( G ′ w ′ ) is Cohen-Macaulay.Therefore, ( S/I ( G w )) / ( x k − y k ) ∼ = ( S/I ( G ′ w ′ )) / ( x k − y k )is Cohen-Macaulay. Since S/I ( G w ) is not Cohen-Macaulay, x k − y k is not regular on S/I ( G w ). Hence, x k − y k ∈ [ p ∈ Ass
S/I ( G w ) p . Thus, there exists an associated prime ideal p of S/I ( G w ) such that x k − y k ∈ p .Consequently, x k , y k ∈ p . This is a contradiction and proves that S/I ( G w ) is Cohen-Macaulay. (cid:3) It is well known (and easy to prove) that every unmixed bipartite graph is verywell-covered. Hence, as an immediate consequence of Theorem 3.7, we obtain thefollowing corollary.
Corollary 3.8.
Let G be a Cohen-Macaulay bipartite graph and let w be an edgeweight on G . Then the following conditions are equivalent.(1) I ( G w ) is an unmixed ideal.(2) S/I ( G w ) is a Cohen-Macaulay ring. SEYED AMIN SEYED FAKHARI, KOSUKE SHIBATA, NAOKI TERAI, AND SIAMAK YASSEMI Examples
Let D be a vertex-weighted oriented graph and let G be its underlying graph. Aswe mentioned in Section 1, Pitones, Reyes and Toledo conjectured that S/I ( D ) isCohen-Macaulay, if I ( D ) is unmixed and S/I ( G ) is Cohen-Macaulay (see Conjecture1.1). The following example shows that the assertion of Conjecture 1.1 is not true. Example 4.1.
Let K be a field with char( K ) = 0 and let D be the oriented graphwith vertex set V ( D ) = { x , . . . , x } and edge set E ( D ) = (cid:8) ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) , ( x , x ) (cid:9) . Consider the weight functions w ( x i ) = (cid:26) i = 112 if i = 11 , and w ( x i ) = (cid:26) i = 72 if i = 7 . For i = 1 ,
2, let D i be the vertex-weighted oriented graph obtained from D by consid-ering the weight function w i . Then I ( D ) =( x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x ) , and I ( D ) =( x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x ) . Let G be the underlying graph of D . The edge ideal I ( G ) of G comes from the tri-angulation of the real projective plane (see for example [18, Exercise 6.3.65]). It isknown that S/I ( G ) is Cohen-Macaulay. However, for i = 1 , Macaulay2 computa-tion shows that I ( D i ) is unmixed but not Cohen-Macaulay, disproving Conjecture 1.1.We show that S/I ( D ) satisfies the Serre’s condition ( S ) condition, while S/I ( D )does not. Using Macaulay2 we know that depth
S/I ( D i ) = 2 for i = 1 ,
2. Since for i = 1 ,
2, dim
S/I ( D i ) = 3, the quotient ring S/I ( D i ) satisfies ( S ) condition if andonly if dim Ext S ( S/I ( D i ) , S ) = dim Ext − S ( S/I ( D i ) , S ) ≤ − , by Lemma 2.1. With Macaulay2 , one can check that dim Ext S ( S/I ( D ) , S ) = 0 anddim Ext S ( S/I ( D ) , S ) = 1. EIGHTED EDGE IDEAL 9
The following example provide counterexamples for the edge-weighted version ofConjecture 1.1.
Example 4.2.
Let K be a field with char( K ) = 0 and let G be the same graph as inExample 4.1. Consider the following weighted edge ideals. I ( G w ) =( x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x ) .I ( G w ) =( x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x , x x ) . Then
S/I ( G ) is Cohen-Macaulay. However, Macaulay2 computation shows that I ( G w ) is unmixed, but S/I ( G w ) does not satisfy the Serre’s condition ( S ). Onthe other hand, Macaulay2 computation shows that I ( G w ) is unmixed and S/I ( G w )satisfies the Serre’s condition ( S ) condition, but it is not Cohen-Macaulay. Acknowledgment
This work was partially supported by JSPS Grant-in Aid for Scientific Research(C) 18K03244.
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Seyed Amin Seyed Fakhari, School of Mathematics, Statistics and Computer Sci-ence, College of Science, University of Tehran, Tehran, Iran.
E-mail address : [email protected] Kosuke Shibata, Department of Mathematics, Faculty of Science, Okayama Uni-versity, Kita-ku Okayama 700–8530, Japan.
E-mail address : [email protected] Naoki Terai, Department of Mathematics, Faculty of Science, Okayama Univer-sity, Kita-ku Okayama 700–8530, Japan.
E-mail address : [email protected] Siamak Yassemi, School of Mathematics, Statistics and Computer Science, Collegeof Science, University of Tehran, Tehran, Iran.
E-mail address ::