Bi-Cohen-Macaulay graphs Jürgen Herzog Ahad Rahimi∗ Fachbereich Mathematik Universität Duisburg-Essen Essen, Germany Department of Mathematics Razi University Kermanshah, Iran [email protected] [email protected] Submitted: Sep 8, 2015; Accepted: Dec 9, 2015; Published: Jan 11, 2016 Mathematics Subject Classifications: 05E40, 13C14 Abstract In this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General biCohen-Macaulay graphs are classified up to separation. The inseparable bi-CohenMacaulay graphs are determined. We establish a bijection between the set of all trees and the set of inseparable bi-Cohen-Macaulay graphs. Keywords: Bi-Cohen–Macaulay, Bipartite and chordal graphs, Generic graphs, Inseparability 1 Introduction A simplicial complex ∆ is called bi-Cohen-Macaulay (bi-CM), if ∆ and its Alexander dual ∆∨ are Cohen-Macaulay. This concept was introduced by Fløystad and Vatne in [8]. In that paper the authors associated to each simplicial complex ∆ in a natural way a complex of coherent sheaves and showed that this complex reduces to a coherent sheaf if and only if ∆ is bi-CM. The present paper is an attempt to classify all bi-CM graphs. Given a field K and a simple graph on the vertex set [n] = {1, 2, . . . , n}, one associates with G the edge ideal IG of G, whose generators are the monomials xi xj with {i, j} an edge of G. We say that G is bi-CM if the simplicial complex whose Stanley-Reisner ideal coincides with IG is bi-CM. Actually, this simplicial complex is the so-called independence complex of G. Its faces are the independent sets of G, that is, subsets D of [n] with {i, j} ̸⊂ D for all edges {i, j} of G. This paper was written during the visit of the second author at Universität Duisburg-Essen, Campus Essen. He is grateful for its hospitality. ∗ the electronic journal of combinatorics 23(1) (2016), #P1.1 1 By its very definition, any bi-CM graph is also a Cohen-Macaulay graph (CM graph). A complete classification of all CM graphs is hopeless if not impossible. However, such a classification is given for bipartite graphs [10, Theorem 3.4] and for chordal graphs [11]. We refer the reader to the books [9] and [14] for a good survey on edge ideals and its algebraic and homological properties. Based on the classification of bipartite and chordal CM graphs, we provide in Section 3 a classification of bipartite and chordal bi-CM graphs, see Theorem 3 and Theorem 4. In Section 2 we first present various characterizations of bi-CM graphs. By using the Eagon-Reiner theorem [5], one notices that the graph G is bi-CM if and only if it is CM and IG has a linear resolution. Cohen-Macaulay ideals generated in degree 2 with linear resolution are of very special nature. They all arise as deformations of the square of the maximal ideal of a suitable polynomial ring. From this fact arise constraints on the number of edges of the graph and on the Betti numbers of IG . Though a complete classification of all bi-CM graphs seems to be again impossible, a classification of all bi-CM graphs up to separation can be given, and this is the subject of the remaining sections. A separation of the graph G with respect to the vertex i is a graph G′ whose vertex set is V (G) ∪ {i′ } having the property that G is obtained from G′ by identifying i with i′ and such that xi − xi′ is a non-zerodivisor modulo IG′ . The algebraic condition on separation makes sure that the essential algebraic and homological invariants of IG and IG′ are the same. In particular, G is bi-CM if and only if G′ is bi-CM. A graph which does not allow any separation is called inseparable, and an inseparable graph which is obtained by a finite number of separation steps from G is called an inseparable model of G. Any graph admits inseparable models and the number of inseparable models of a graph is finite. Separable and inseparable graphs from the view point of deformation theory have been studied in [1]. In Section 5 we determine all inseparable bi-CM graphs on n vertices. Indeed, in Theorem 11 it is shown that for any tree T on the vertex set [n] there exists a unique inseparable bi-CM graph GT determined by T , and any inseparable bi-CM graph is of this form. Furthermore, if G is an arbitrary bi-CM graph and T is the relation graph of the Alexander dual of IG , then GT is a separable model of G. For a bi-CM graph G, the Alexander dual J = (IG )∨ of IG is a Cohen-Macaulay ideal of codimension 2 with linear resolution. As described in [3], one attaches to any relation matrix of J a relation tree T . Replacing the entries in this matrix by distinct variables with the same sign, one obtains the so-called generic relation matrix whose ideals of 2minors JT and its Alexander dual has been computed in [13]. This theory is described in Section 4. The Alexander dual of JT is the edge ideal of graph, which actually is the graph GT mentioned before and which serves as a separable model of G. 2 Preliminaries and various characterizations of Bi-CM graphs In this section we recall some of the standard notions of graph theory which are relevant for this paper, introduce the bi-CM graphs and present various equivalent conditions of the electronic journal of combinatorics 23(1) (2016), #P1.1 2 a graph to be bi-CM. The graphs considered here will all be finite, simple graphs, that is, they will have no double edges and no loops. Furthermore we assume that G has no isolated vertices. The vertex set of G will be denoted V (G) and will be the set [n] = {1, 2, . . . , n}, unless otherwise stated. The set of edges of G we denote by E(G). A subset F ⊂ [n] is called a clique of G, if {i, j} ∈ E(G) for all i, j ∈ F with i ̸= j. The set of all cliques of G is a simplicial complex, denoted ∆(G). A subset C ⊂ [n] is called a vertex cover of G if C ∩ {i, j} ̸= ∅ for all edges {i, j} of G. The graph G is called unmixed if all minimal vertex covers of G have the same cardinality. This concept has an algebraic counterpart. We fix a field K and consider the ideal IG ⊂ S = K[x1 , . . . , xn ] which is generated by all monomials xi xj with {i, j} ∈ E(G). The ideal IG is called the edge ideal of G. Let C ⊂ [n]. Then the monomial prime ideal PC = ({xi : i ∈ C}) is a minimal prime ideal of IG if and only if C is a minimal vertex cover of G. Thus G is unmixed if and only if IG is unmixed in the algebraic sense. A subset D ⊂ [n] is called an independent set of G if D contains no set {i, j} which is an edge of G. Note that D is an independent set of G if and only if [n] \ D is a vertex cover. Thus the minimal vertex covers of G correspond to the maximal independent sets of G. The cardinality of a maximal independent set is called the independence number of G. It follows that the Krull dimension of S/IG is equal to c, where c is the independence number of G. The graph G is called bipartite if V (G) is the disjoint union of V1 and V2 such that V1 and V2 are independent sets, and G is called disconnected if V (G) is the disjoint union of W1 and W2 and there is no edge {i, j} of G with i ∈ W1 and j ∈ W2 . The graph G is called connected if it is not disconnected. A cycle C (of length r) in G is a sequence of edges {ik , jk } with k = 1, 2, . . . , r such that jk = ik+1 for k = 1, . . . , r − 1 and jr = i1 . A cord of C is an edge {i, j} of G with i, j ∈ {i1 , . . . , ir } and {i, j} is not an edge of C. The graph G is called chordal if each cycle of G of length ⩾ 4 has a chord. A graph which has no cycle and which is connected is called a tree. Now we recall the main concept we are dealing with in this paper. Let I ⊂ S be a ∩ squarefree monomial ideal. Then I = m prime j=1 Pj where each of the Pj is a monomial ∏ ∨ ideal of I. The ideal I which is minimally generated by the monomials uj = xi ∈Pj xi is called the Alexander dual of I. One has (I ∨ )∨ = I. In the case that I = IG , each Pj is generated by the variables corresponding to a minimal vertex cover of G. Therefore, (IG )∨ is also called the vertex cover ideal of G. According to [8] a squarefree monomial ideal I ⊂ S is called bi-Cohen-Macaulay (or simply bi-CM) if I as well as the Alexander dual I ∨ of I is a Cohen-Macaulay ideal. A graph G is called Cohen-Macaulay or bi-Cohen-Macaulay (over K) (CM or bi-CM for short), if IG is CM or bi-CM. One important result regarding the Alexander dual that will be used frequently in this paper is the Eagon-Reiner theorem which says that I is a Cohen-Macaulay ideal if and only if I ∨ has a linear resolution. Thus the Eagon-Reiner theorem implies that I is bi-CM if and only if I is a Cohen-Macaulay ideal with linear resolution. From this description it follows that a bi-CM graph is connected. Indeed, if the electronic journal of combinatorics 23(1) (2016), #P1.1 3 this is not the case, then there are induced subgraphs G1 , G2 ⊂ G such that V (G) is the disjoint union of V (G1 ) and V (G2 ). It follows that IG = IG1 + IG2 , and the ideals IG1 and IG2 are ideals in a different set of variables. Therefore, the free resolution of S/IG is obtained as the tensor product of the resolutions of S/IG1 and S/IG2 . This implies that IG has relations of degree 4, so that IG does not have a linear resolution. From now on we will always assume that G is connected, without further mentioning it. Proposition 1. Let K be an infinite field and G a graph on the vertex set [n] with independence number c. The following conditions are equivalent: (a) G is a bi-CM graph over K; (b) G is a CM graph over K, and S/IG modulo a maximal regular sequence of linear forms is isomorphic to T /m2T where T is the polynomial ring over K in n−c variables and mT is the graded maximal ideal of T . Proof. We only need to show that IG has a linear resolution if and only if condition (b) holds. Since K is infinite and since S/IG is Cohen-Macaulay of dimension c, there exists a regular sequence x of linear forms on S/IG of length c. Let T = S/(x). Then T is isomorphic to a polynomial ring in n − c variables. Let J be the image of IG in T . Then J is generated in degree 2 and has a linear resolution if and only if IG has linear resolution. Moreover, J is mT -primary. The only mT -primary ideals with linear resolution are the powers of mT . Thus, IG has a linear resolution if and only if J = m2T . Corollary 2. Let G be a graph on the vertex set [n] with independence number c. The following conditions are equivalent: (a) G is a bi-CM graph over K; (b) G is a CM graph over K and |E(G)| = ( n−c+1 2 ) ; (c) G is a CM graph over K and the number of minimal vertex covers of G is equal to n − c + 1; ( (d) βi (IG ) = (i + 1) n−c+1 i+2 ) for i = 0, . . . , n − c − 1. Proof. For the proof of the equivalent conditions we may assume that K is infinite and hence we may use Proposition 1. (a) ⇐⇒ (b): With the notation of proposition 1 we have J = m2T if and only if the ) ( n−c+1 . Since IG and J have the same number number of generators of J is equal to 2 of generators and since the number of generators of IG is equal to |E(G)|, the assertion follows. (b) ⇐⇒ (c): Since S/IG is Cohen-Macaulay, the multiplicity of S/IG is equal to the length ℓ(T /J) of T /J. On the other hand, the multiplicity is also the number of minimal prime ideals of IG which coincides with the number of minimal vertex covers of G. Thus the electronic journal of combinatorics 23(1) (2016), #P1.1 4 the length of T /J is equal to the number of minimal vertex covers of G. Since J = m2T if and only if ℓ(T /J) = n − c + 1, the assertion follows. (a) ⇒ (d): Note that βi (IG ) = βi (J) for all i. Since J is isomorphic to the ideal of 2-minors of the matrix ( ) y1 y2 . . . yn−c 0 0 y1 . . . yn−c−1 yn−c in the variables y1 , . . . , yn−c , the Eagon-Northcott complex ([4], [6]) provides a free resolution of J and the desired result follows. (d) ⇒ (a): It follows from the description of the Betti numbers of the ideal IG that proj dim S/IG = n − c. Thus, depth S/IG = c. Since ( dim)S/IG = c, it follows that IG is a , condition (b) is satisfied, and Cohen-Macaulay ideal. Since |E(G)| = β0 (IG ) = n−c+1 2 hence G is bi-CM, as desired. Finally we note that G is a bi-CM graph over K if and only if the vertex cover ideal of G is a codimension 2 Cohen-Macaulay ideal with linear relations. Indeed, let JG be the vertex cover ideal of G. Since JG = (IG )∨ , it follows from the Eagon-Reiner theorem JG is bi-CM if and only if IG is bi-CM. 3 The classification of bipartite and chordal bi-CM graphs In this section we give a full classification of the bipartite and chordal bi-CM graphs. Theorem 3. Let G be a bipartite graph on the vertex set V with bipartition V = V1 ∪ V2 where V1 = {v1 , . . . , vn } and V2 = {w1 , . . . , wm }. Then the following conditions are equivalent: (a) G is a bi-CM graph; (b) n = m and E(G) = {{vi , wj } : 1 ⩽ i ⩽ j ⩽ n}. Proof. (a) ⇒ (b): Since G is a bi-CM graph, it is in particular a CM-graph, and so n = m, and by [9, Theorem 9.1.13] there exists a poset P = {p1 , . . . , pn } such that G = G(P ). Here G(P ) is the bipartite graph on V = {v1 , . . . , vn , w1 , . . . , wn } whose edges are those 2-element subset {vi , wj } of V such that pi ⩽ pj . Thus IG = IG(P ) = HP∨ , where HP = ∩ (xi , yj ) pi ⩽pj is an ideal of S = K[{xi , yi }pi ∈P ], the polynomial ring in 2n variables over K. Since G is bi-CM, it follows that HP is Cohen–Macaulay, and hence proj dim S/HP = 2n − depth S/HP = 2n − dim S/HP = height HP = 2. Thus proj dim HP = 1, and hence, by [10, Corollary 2.2], the Sperner number of P , i.e., the maximum of the cardinalities of antichains of P equals 1. This implies that P is a chain, and this yields (b). (b) ⇒ (a): The graph G described in (b) is of the form G = G(P ) where P is a chain. By what is said in (a) ⇒ (b), it follows that G is bi-CM. the electronic journal of combinatorics 23(1) (2016), #P1.1 5 The following picture shows a bi-CM bipartite graph for n = 4. x1 • x2 • x3 • x4 • • y1 • y2 • y3 • y4 Figure 1: A bi-CM bipartite graph. Theorem 4. Let G be a chordal graph on the vertex set [n]. The following conditions are equivalent: (a) G is a bi-CM graph; (b) Let F1 , . . . , Fm be the facets of the clique complex of G. Then m = 1, or m > 1 and (i) V (G) = V (F1 ) ∪ V (F2 ) ∪ . . . ∪ V (Fm ), and this union is disjoint; (ii) each Fi has exactly one free vertex ji ; (iii) the restriction of G to [n] \ {j1 , . . . , jm } is a clique. Proof. Let In,d be the ideal generated by all squarefree monomials of degree d in S = ∨ K[x1 , . . . , xn ]. It is known (and easy to prove) that In,d = In,n−d+1 , and that all these ideals are Cohen-Macaulay, and hence all bi-CM. If m = 1, then IG = In,2 and the result follows. Now let m > 1. A bi-CM graph is a CM graph. The CM chordal graphs have been classified in [11]: they are the chordal graphs satisfying (b)(i). Thus for the proof of the theorem we may assume that (b)(i) holds and simply have to show that (b)(ii) and (b)(iii) are satisfied if and only if IG has a linear resolution. Let Pi be the monomial prime ideal generated by the variables xk with k ∈ V (Fi )\{ji }, and let G′ be subgraph of G whose edges do not belong to any Fi . It is shown in the proof of [11, Corollary 2.1] that there exists a regular sequence on S/IG such that after reduction modulo this sequence one obtains the ideal J ⊂ T where T is the polynomial ring on the variables xk with k ̸= ji for i = 1, . . . , m and where J = (P12 , . . . , Pm2 , IG′ ). (1) By Proposition 1, it follows that IG has a linear resolution if and only if J = m2T , where mT denotes the graded maximal ideal of T . So, now suppose first that IG has a linear resolution, and hence J = m2T . Suppose that some Fi has more than one free vertex, say Fi has the vertex k with k ̸= ji . Choose any the electronic journal of combinatorics 23(1) (2016), #P1.1 6 Ft different from Fi and let l ∈ Fj with l ̸= jt . Then xk and xl belong to T but xk xl ̸∈ J as can be seen from (1). This is a contradiction. Thus (b)(ii) follows. Suppose next that the graph G′′ which is the restriction of G to [n] \ {j1 , . . . , jm } is not a clique. Then there exist i, j ∈ V (G′′ ) such that {i, j} ̸∈ E(G′′ ). However, since all xk with k ∈ V (G′′ ) belong to T and since J = m2T , it follows xi xj ∈ J. Thus, by (1), xi xj ∈ Pk2 for some k or xi xj ∈ IG′ . Since (b)(ii) holds, this implies in both cases that {i, j} ∈ E(G′′ ), a contradiction. Thus (b)(iii) follows. Conversely, suppose (b)(ii) and (b)(iii) hold. We want to show that J = m2T . Let xi , xj ∈ T . We have to show that xi xj ∈ J. It follows from the description of J that x2k ∈ J for all xk ∈ T . Thus we may assume that i ̸= j. If {i, j} is not an edge of any Fk , then by definition it is an edge of G′ , and hence xi xj ∈ IG′ ⊂ J. On the other hand, if {i, j} is an edge of Fk for some k, then i, j ̸= ik , and hence xi xj ∈ Pk2 ⊂ J. Thus the desired conclusion follows. Let G be a chordal bi-CM graph as in Theorem 4(b) with m > 1. We call the complete graph G′′ which is the restriction of G to [n] \ {j1 , . . . , jm } the center of G. The following picture shows, up to isomorphism, all bi-CM chordal graphs whose center is the complete graph K4 on 4 vertices: • • • • • • • • • • • • • • • • • • • • • • • • • • • Figure 2: 4 Generic Bi-CM graphs As we have already seen in the first section, the Alexander dual J = IG∨ of the edge ideal of a bi-CM graph G is a Cohen–Macaulay ideal of codimension 2 with linear resolution. The ideal J may have several distinct relation matrices with respect to the unique minimal monomial set of generators of J. As shown in [3], one may attach to each of the relation matrices of J a tree as follows: let u1 , . . . , um be the unique minimal set of generators of J. Let A be one of the relation matrices of J. Because J has a linear resolution, the generating relations of J may be chosen all of the form xk ui − xl uj = 0. This implies that in each row of the (m − 1) × m-relation matrix A there are exactly two non-zero entries (which are variables with different signs). We call such relations, relations of binomial type. the electronic journal of combinatorics 23(1) (2016), #P1.1 7 Example 5. Consider the bi-CM graph G on the vertex set [5] and edges {1, 2} {2, 3}, {3, 1}, {2, 4}, {3, 4}, {4, 5} as displayed in Figure 3. x2 • x4 • x1 • x5 • • x3 Figure 3: The ideal J = IG∨ is generated by u1 = x2 x3 x4 , u2 = x1 x3 x4 , u3 = x2 x3 x5 and u4 = x1 x2 x4 . The relation matrices with respect to u1 , u2 , u3 and u4 are the matrices     x1 −x2 0 0   A1 = x5 0 −x4 0  , x1 0 0 −x3 and x1 −x2 0 0 x 0 −x 0  A2 =  .  5 4 0 x2 0 −x3 Coming back to the general case, one assigns to the relation matrix A the following graph Γ: the vertex set of Γ is the set V (Γ) = {1, 2, . . . , m}, and {i, j} is said to be an edge of Γ if and only if some row of A has non-zero entries for the ith- and jth-component. It is remarked in [3] and easy to see that Γ is a tree. This tree is in general not uniquely determined by G. In our Example 5 the relation tree of A1 is x3 • • x4 • x1 • x2 Figure 4: while the relation tree of A2 is Now let J be any codimension 2 Cohen-Macaulay monomial ideal with linear resolution. Then, as observed in Section 2, J ∨ = IG where G is a bi-CM graph. Now we follow the electronic journal of combinatorics 23(1) (2016), #P1.1 8 • x3 • x1 • x2 • x4 Figure 5: Naeem [13] and define for any given tree T on the vertex set [m] = {1, . . . , m} with edges e1 , . . . , em−1 the (m − 1) × m-matrix AT whose entries akl are defined as follows: we assign to the kth edge ek = {i, j} of T with i < j the kth row of AT by setting    xij , akl =  −xji ,  0, if l = i, if l = j, otherwise. (2) The matrix AT is called the generic matrix attached to the tree T . By the Hilbert-Burch theorem [2], the matrix AT is the relation matrix of the ideal JT of maximal minors of AT , and JT is a Cohen-Macaulay ideal of codimension 2 with linear resolution. We let GT be the graph such that IGT = J ∨ , and call GT the generic bi-CM graph attached to T . Our discussion so far yields Proposition 6. For any tree T , the graph GT is bi-CM. In order to describe the vertices and edges of GT , let i and j be any two vertices of the tree T . There exists a unique path P : i = i0 , i1 , . . . , ir = j from i to j. We set b(i, j) = i1 and call b(i, j) the begin of P , and set e(i, j) = ir−1 and call e(i, j) the end of P . It follows from [13, Proposition 1.4] that IGT is generated by the monomials xib(i,j) xje(i,j) . Thus the vertex set of the graph GT is given as V (GT ) = {(i, j), (j, i) : {i, j} is an edge of T }. In particular, {(i, k), (j, l)} is an edge of GT if and only if there exists a path P from i to j such that k = b(i, j) and l = e(i, j). In Example 5, let T1 and T2 be the relation trees of A1 and A2 , respectively. Then the generic matrices corresponding to these trees are and     x12 −x21 0 0  0 −x31 0  B1 = x13 , x14 0 0 −x41 x12 −x21 0 0  0 −x31 0  B2 = x13 . 0 x24 0 −x42 the electronic journal of combinatorics 23(1) (2016), #P1.1 9 • x12 • x24 • x21 • x42 x41 • • x31 • x14 x • 31 • x13 x12 • • x13 • x21 GT1 GT2 Figure 6: The generic graphs corresponding to the trees T1 and T2 are displayed in Figure 6. It follows from this description that ) GT has 2(m − 1) vertices. Since GT is bi-CM, ( n−c+1 , see Corollary 2. Here n − c is the degree of the the number of edges of GT is 2 ( ) generators of IG∨ which is m − 1. Hence GT has m2 edges. Among the edges of GT are in particular the m − 1 edges {(i, j), (j, i)} where {i, j} is an edge of T . Proposition 7. Let A be the relation matrix of a codimension 2 Cohen-Macaulay monomial ideal J with linear resolution, and assume that all the variables appearing in A are pairwise distinct. Let T be the relation tree of A. Then J is isomorphic to JT and J admits the unique relation tree, namely T . Proof. Since all variables appearing in A are pairwise distinct, we may rename the variables appearing in a binomial type relation and call them as in the generic matrix xij and xji . Then A becomes AT and this shows that J ∼ = JT . To prove the uniqueness of the relation tree, we first notice that the shifts in the multigraded free resolution of J are uniquely determined and independent of the particular choice of the relation matrix A. A possibly different relation matrix A′ can arise from A only be row operations with rows of the same multidegree. Let r1 , . . . , rl by rows of A with the same multidegree corresponding to binomial type relations, and fix a column j. Then the non-zero jth columns of each of the ri must be the same, up to a sign. Since we assume that the variables appearing in A are pairwise distinct, it follows that l = 1. In particular, there is, up to the order of the rows, only one relation matrix with rows corresponding to binomial type relations. This shows that T is uniquely determined. 5 Inseparable models of Bi-CM graphs In order to state the main result of this paper we recall the concept of inseparability introduced by Fløystad et al in [7], see also [12]. Let S = K[x1 , . . . , xn ] be the polynomial ring over the field K and I ⊂ S a squarefree monomial ideal minimally generated by the monomials u1 , . . . , um . Let y be an indeterminate over S. A monomial ideal J ⊂ S[y] is called a separation of I for the variable xi if the following holds: the electronic journal of combinatorics 23(1) (2016), #P1.1 10 (i) the ideal I is the image of J under the K-algebra homomorphism S[y] → S with y 7→ xi and xj 7→ xj for all j; (ii) xi as well as y divide some minimal generator of J; (iii) y − xi is a non-zero divisor of S[y]/J. The ideal I is called separable if it admits a separation, otherwise inseparable. If J is an ideal which is obtained from I by a finite number of separation steps, then we say that J specializes to I. If moreover, J is inseparable, then J is called an inseparable model of I. Each monomial ideal admits an inseparable model, but in general not only one. For example, the inseparable models of the powers of the graded maximal ideal of S have been considered by Lohne [12]. Forming the Alexander dual behaves well with respect to specialization and separation. Proposition 8. Let I ⊂ S be a squarefree monomial ideal. Then the following holds: (a) If J specializes to I, then J ∨ specializes to I ∨ . (b) The ideal I is separable if and only if I ∨ is separable. Proof. (a) It follows from [7, Proposition 7.2] that if L ⊂ S[y] is a monomial ideal such that y − xi is a regular element on S[y]/L with (S[y]/L)/(y − xi )(S[y]/L) ∼ = S/I, then ∨ ∨ ∨ ∼ y − xi is a regular element on S[y]/L and (S[y]/L )/(y −xi )(S[y]/L ) = S/I ∨ . Repeated applications of this fact yields the desired result. (b) We may assume that the ideal L as in (a) is a separation of I with respect to xi . Since (a) holds, it remains to show that y as well as xi divides some generator of L∨ . By assumption this is the case for L. Suppose that y does not divide any generator of L∨ . Then it follows from the definition of the Alexander dual that y also does not divide any generator of (L∨ )∨ . This is a contradiction, since L = (L∨ )∨ . Similarly it follows that xi divides some generator of L∨ . We now apply these concepts to edge ideals. Let G be a graph on the vertex set [n]. We call G separable if IG is separable, and otherwise inseparable. Let J be a separation of IG for the variable xi . Then by the definition of separation, J is again an edge ideal, say J = IG′ where G′ is a graph with one more vertex than G. The graph G is obtained from G′ by identifying this new vertex with the vertex i of G. Algebraically, this identification amounts to say that S/IG ∼ = (S ′ /IG′ )/(y − xi )(S ′ /IG′ ), where S ′ = S[y] and y − xi is a non-zerodivisor of S ′ /IG′ . In particular, it follows that IG and IG′ have the same graded Betti-numbers. In other words, all important homological invariants of IG and IG′ are the same. It is therefore of interest to classify all inseparable graphs. An attempt for this classification is given in [1]. Example 9. Let G be the triangle and G′ be the line graph displayed in Figure 7. Then IG′ = (x1 x2 , x1 x3 , x2 x4 ). Since Ass(IG′ ) = {(x1 , x2 ), (x1 , x4 ), (x2 , x3 )}, it follows that x3 − x4 is a non-zero divisor on S ′ /IG′ where S ′ = K[x1 , x2 , x3 , x4 ]. Moreover, the electronic journal of combinatorics 23(1) (2016), #P1.1 11 x3 • • x1 x3 • • x2 • x1 • x4 • x2 Figure 7: A triangle and its inseparable model (S ′ /IG′ )/(x3 − x4 )(S ′ /IG′ ) ∼ = S/IG . Therefore, the triangle in Figure 7 is obtained as a specialization from the line graph in Figure 7 by identifying the vertices x3 and x4 . We denote by G(i) the complementary graph of the restriction GN (i) of G to N (i) where N (i) = {j : {j, i} ∈ E(G)} is the neighborhood of i. In other words, V (G(i) ) = N (i) and E(G(i) ) = {{j, k} : j, k ∈ N (i) and {j, k} ̸∈ E(G)}. Note that G(i) is disconnected if and only if N (i) = A ∪ B, where A, B ̸= ∅, A ∩ B = ∅ and all vertices of A are adjacent to those of B. Here we will need the following result of [1, Theorem 3.1]. Theorem 10. The following conditions are equivalent: (a) The graph G is inseparable; (b) G(i) is connected for all i. Now we are ready to state our main result. Theorem 11. (a) Let T be a tree. Then GT is an inseparable bi-CM graph. (b) For any inseparable bi-CM graph G, there exists a unique tree T such that G ∼ = GT . (c) Let G be any bi-CM graph. Then there exists a tree T such that GT is an inseparable model of G. Proof. (a) By Corollary 6, GT is a bi-CM graph. In order to see that GT is inseparable we apply the criterion given in Theorem 10, and thus we have to prove that for each vertex (i, j) of GT and for each disjoint union N ((i, j)) = A ∪ B of the neighborhood of (i, j) for which A ̸= ∅ = ̸ B, not all vertices of A are adjacent to those of B. As follows from the discussion in Section 4, N ((i, j)) = {(k, l) : there exists a path from i to l, and j = b(i, l) and k = e(i, l)}. In particular, (j, i) ∈ N ((i, j)). Let N ((i, j)) = A ∪ B, as above. We may assume that (j, i) ∈ A. Since T is a tree, then there is no path from j to any l with (k, l) ∈ N ((i, j)), because otherwise we would have a loop in T . This shows that (j, i) is connected to no vertex in B, as desired. (b) Let A be a relation matrix of J = IG∨ and T the relation tree of A. The non-zero entries of A are variables with sign ±1. Say the kth row of A has the non-zero entries akik the electronic journal of combinatorics 23(1) (2016), #P1.1 12 and akjk with ik < jk . We may assume that the variable representing akik has a positive sign while that akjk has a negative sign, and that this is so for each row. We claim that the variables appearing in the non-zero entries of A are pairwise distinct. By Proposition 7 this then implies that T is the only relation tree of J and that G ∼ = GT . In order to prove the claim, we consider the generic matrix AT corresponding to T . Let S ′ be the polynomial ring over S in the variables xij and xji with {i, j} ∈ E(T ). For each k we consider the linear forms ℓk1 = xik jk − akik and ℓk2 = xjk ik − akjk . For example, for the matrix A2 in Example 5 the linear forms are ℓ11 = x12 − x1 , ℓ12 = x21 − x2 , ℓ21 = x13 − x5 , ℓ22 = x31 − x4 , ℓ31 = x24 − x2 and ℓ32 = x42 − x3 . We let ℓ be the sequence of linear form ℓ11 , ℓ12 , . . . , ℓm−1,1 , ℓm−1,2 in S ′ . Then we have (S ′ /JT S ′ )/(ℓ)(S ′ /JT S ′ ) ∼ = S/J. Since both ideals, J as well as JT , are Cohen-Macaulay ideals of codimension 2, it follows that ℓ is a regular sequence on S ′ /JT S ′ . Thus, assuming the variables appearing in the non-zero entries of A are not all pairwise distinct, we see that J is separable. Indeed, suppose that the variable xk appears at least twice in the matrix. Then we replace only one of the xk by the corresponding generic variable xij to obtain the matrix A′ . Let J ′ be the ideal of maximal minors of A′ . It follows from the above discussions that xij − xk is a regular element of S[xij ]/J ′ . In order to see that J ′ is a separation of J it remains to be shown that xij as well as xk appear as factors of generators of J ′ . Note that J ′ is a specialization of JT . The minors of AT which are the ∏ generators of JT are the monomials m+1 i=1 xib(i,j) for j = 1, . . . , m + 1, see [13, Proposition i̸=j 1.2]. From this description of the generators of JT it follows that all entries of AT appear as factors of generators of JT . Since J ′ is a specialization of JT , the same holds true for J ′ , and since xij as well as xk are entries of A′ , the desired conclusion follows. Now since we know that J is separable, Proposition 8(b) implies that G is separable as well. This is a contradiction. (c) Let A be a relation matrix of J = IG∨ and T the corresponding relation tree. As shown in the proof of part (b), JT specializes to J, and hence IGT specializes to IG , by Proposition 8(a). By part (a), the graph GT is inseparable. 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