Matching numbers and the regularity of the Rees algebra of an edge ideal

The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.


Introduction
In the study of powers of monomial ideals, the Rees algebra plays an important role. In the present paper we focus on the Rees algebra of the edge ideal of a finite simple graph.
Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring over K in the variables x 1 , . . . , x n . Furthermore, let G be a simple graph on the vertex set V (G) = [n] and edge set E(G). The edge ideal I = I(G) of G is the monomial ideal in S generated by the monomials x i x j with {i, j} ∈ E(G), and the edge ring K[G] is the K-algebra generated by the monomials x i x j ∈ I. The Rees algebra R(I) = s≥0 I s of I may be viewed as the edge ring of the graph G * with V (G * ) = [n + 1] and E(G * ) = E(G) ∪ {{i, n + 1} : i ∈ V (G)}. We are interested in bounding the regularity of R(I), because this provides information about the regularity of the powers of I. In our situation, computing regularity of R(I) amounts to compute the regularity of the edge ring of graphs of the form G * . This class of algebras are particular classes of toric rings. To our knowledge, there are essentially two methods available to compute the regularity of a toric ring A. The first method, which can always be applied, is to compute the multigraded Betti numbers of A by using the squarefree divisor complex (see [2,Proposition 1.1] and [4,Theorem 3.28]). In concrete cases, as discussed in Section 1, this allows to give lower bounds for the regularity, but in general it is hard to use. The second method can be applied, if A is Cohen-Macaulay, in which case one needs to compute the a-invariant of A. If in addition, A is normal, then following Danilov and Stanley [1,Theorem 6.3.5], the canonical module can computed which in particular gives us the a-invariant of A.
Let A be a toric ring generated by monomials in S. A K-subalgebra B of A is called a combinatorial pure subalgebra of A if there exists a subset T ⊂ [n] such that B = A ∩ K[x i : i ∈ T ]. In Section 1 we recall squarefree divisor complexes and use them to prove that if B ⊂ A is a combinatorial pure subring of A, then 2010 Mathematics Subject Classification. Primary 13H10; Secondary 13D02, 05E40. The second author was partially supported by JSPS KAKENHI 19H00637. β i,j (B) ≤ β i,j (A) for all i and j. Here the β i,j (M) denote the graded Betti numbers of a graded module M. This result implies in particular that if I is a monomial ideal generated in a single degree, then reg R(I) ≥ reg F (I), where F (I) is the fiber cone of I. We also use squarefree divisor complexes to give lower bounds of the regularity of the Rees algebra of I(G), when G is a disjoint union of edges. These results will be used in the next section. Now, we devote Section 2 to finding an upper bound and a lower bound of the regularity of the Rees algebra R(I(G)) of the edge ideal I(G) of a finite simple graph G in terms of the matching number of G and the induced matching number of G.
Recall that a matching of G is a subset M ⊂ E(G) such that e ∩ e ′ = ∅ for all e and e ′ belonging to M with e = e ′ . A matching M of G is called perfect if, for each i ∈ [n], there is e ∈ M with i ∈ e. An induced matching of G is a matching M of G such that if e and e ′ belong to M with e = e ′ , then there is no edge f ∈ E(G) with e ∩ f = ∅ and e ′ ∩ f = ∅. The maximal of matchings of G is called the matching number of G and is denoted by mat(G). The induced matching number of G, denoted by indmat(G), is the maximal cardinality of induced matchings of G.
It is known [ Theorem 4.2] that, when G is bipartite, one has reg R(I(G)) = mat(G).
Furthermore, if G has a perfect matching, then reg R(I(G)) = mat(G). Our proof heavily depends on the theory of normal edge polytopes created in [5]  On the other hand, it follows from Proposition 1.2 that, for any finite simple graph G, one has indmat(G) ≤ reg R(I(G)). We, however, very much believe that the inequality mat(G) ≤ reg R(I(G)) is valid for any finite simple graph G.

Combinatorial pure subrings and regularity
Let K be a field and A = K[u 1 , . . . , u m ] ⊂ S = K[x 1 , . . . , x n ] be the K-algebra minimally generated by the monomials u i = x a i in the polynomial ring S. Here for a = (a 1 , . . . , a n ) we denote by x a the monomial x a 1 1 x a 2 2 · · · x an n . The K-algebra A has a K-basis consisting of monomials x a . The set of exponents a appearing as exponents of the basis elements of A together with addition form a positive affine semigroup H ⊂ N n which is generated by a 1 , . . . , a n .
Given an element a ∈ H, we define the simplicial complex The simplicial complex ∆ a (A) is called the squarefree divisor complex of H (or of A) with respect to a.
We recall the following theorem from [2, Proposition 1.1] (see also [4, Theorem 3.28]) Theorem 1.1. For the multigraded Betti numbers of A one has Here H i (Γ; K) denotes the ith reduced simplicial homology of a simplicial complex Γ.
We demonstrate this theorem by a simple example.
j for some integers a j ≥ 0. Since supp(x a ) ⊂ T , the same holds true for all u j with a j > 0. It follows that j=1,...,m u a j j ∈ B, and hence F ∈ ∆ a (B). Corollary 1.3. Let A be a toric ring with all generators of same degree d, and let B be a combinatorial pure subring. Then A and B are naturally standard graded, and we have β i,j (B) ≤ β i,j (A) for all i and j.
Proof. For a = (a 1 , . . . , a n ) ∈ Z n we set |a| = i=1,... a i . Let H = {a : x a ∈ B} and H ′ = {a : x a ∈ A}. Then Proof. The Rees ring R(I) is isomorphic to K[G * ]. Since K[G] is a combinatorial pure subring of K[G * ], the assertion follows from Corollary 1.3.
In the next two proposition we consider the Rees algebra of a special graph which plays a role in the next section.
Proof of the claim: we first notice that ∆ a has the facets where the vertices of ∆ a are identified with the monomials u i and v j . Indeed, since u i j=1,...,2m j =2i−1,2i v j = x a for i = 1, . . . , m it follows that F 1 , . . . , F m are facets of ∆ a .
In order to prove that these are all the facets of ∆ a , we show that if F is a face of ∆ a , then F ⊂ F i for some i. Suppose {e i , e j } ⊂ F for some i = j. By symmetry we may assume that {e 1 , e 2 } ⊂ F . Then for all i, and by symmetry we may assume that f 2i−1 ∈ F for i = 1, . . . , m. Since u F divides x a it follows that |F | ≤ 2m − 1. By symmetry we may assume that f 2 and f 4 do not belong to F . Then x a /u F = x 2 x 4 i f 2i ∈F x 2i ∈ A. Hence, F ∈ ∆ a . It remains to consider the case that F ∩ {e 1 , . . . , e m } = {e i } for some i. By symmetry we may assume that i = 1. Suppose that f 1 ∈ F . Then u 1 f 1 = x 2 1 x 2 x m+1 divides x a , a contradiction. Thus f 1 ∈ F . Similarly, f 2 ∈ F . This shows that F ⊂ F 1 .
Next we notice that geometric realization |∆ a | of ∆ a is homotopic to the geometric realization of the simplicial complex Γ whose facets are We choose the standard geometric realization by identifying the vertices e 1 , . . . , e m , f 1 , . . . , f 2m of ∆ a with the standard unit vectors in R 3m .
Indeed, the homotopy is given by the affine maps ϕ t induced by e i → e ′ i = Here 0 ≤ t ≤ 1. We have ϕ 1 = id and |Γ| = ϕ 0 (|∆ a |), as desired. Now since |Γ| is homotopic to |∆ a |, we see that H m−2 (Γ) ∼ = H m−2 (∆ a ). Observe that |Γ| is homotopic to an (m − 2)-sphere, so that H m−2 (Γ) = 0. This concludes the proof of the proposition. Remarks 1.6. Let G be the sum of the graphs G 1 and G 2 . Assume that G has no isolated vertices and G 1 or G 2 has at least 2 edges. Considering several examples we come up with the following question: Is it true that reg R(I(G 1 + G 2 )) = reg R(I(G 1 )) + reg R(I(G 2 ))?
If this question has a positive answer, then Proposition 1.5 is just a very special case of this statement. Of course it is also a simple consequence of the theorem of Cid-Ruiz [3]. However, in order to keep this paper as self-contained as possible and also to demonstrate the use of squarefree divisor complexes, we included Proposition 1.5 to this paper.
With CoCoA we considered the case that G is the sum of two 3-cycles, and found reg R(I(G)) = 4, as expected. In the next section we show that reg R(I(G)) ≤ mat(G) + 1 if R(IG)) is normal. In our example with the two 3-cycles, R(I(G)) is not normal and mat(G) = 2. Therefore, the inequality reg R(I(G)) ≤ mat(G) + 1 is in general not valid if reg R(I(G)) is not normal.
If our question has a positive answer, then one has reg R(I(G)) = 2m if G is the sum of m 3-cycles. On the other hand, for this graph, mat(G) = m. This then gives a family of graphs for which reg R(I(G)) − mat(G) can be any positive integer.

Bounds for the regularity of the Rees algebra of an edge ideal
Let, as before, G be a finite simple graph on the vertex V (G) = [n] and S = K[x 1 , . . . , x n ] the polynomial ring in n variables over a field K. Let P G ⊂ R n be the edge polytope of G which is the convex hull of { e i + e j : {i, j} ∈ E(G) }, where e i is the ith unit coordinate vector of R n . Let A P G denote the Ehrhart ring of G, which is the toric ring in the n + 1 variables x 1 , . . . , x n , t whose K-basis consists of those monomials x a 1 1 . . . x a 1 1 t q , where 1 ≤ q ∈ Z, with (a 1 , . . . , a n ) ∈ qP G ∩ Z n ≥0 . We refer the reader to [5] for basic materials and fundamental results on edge polytopes and their Ehrhart rings. The Ehrhart ring A P G is normal and its canonical module is spanned by those monomials x a 1 1 . . . x a 1 1 t q with (a 1 , . . . , a n ) ∈ q(P G \ ∂P G ) ∩ Z n ≥0 . The edge ring K[G] is normal if and only if A P G is standard grading, i.e., A P G is generated by those monomials x a 1 1 . . . x a 1 1 t with (a 1 , . . . , a n ) ∈ P G ∩Z n ≥0 as an algebra over K. Thus in particular K[G] is normal if and only if K[G] is isomorphic to A P G . Furthermore, it is shown [5, Corollary 2.3] that K[G] is normal if and only if each connected component G ′ of G satisfies the odd cycle condition, that is, if C and C ′ are odd cycles of G ′ with V (C) ∩ V (C ′ ) = ∅, then there exist i ∈ V (C) and j ∈ V (C ′ ) with {i, j} ∈ E(G ′ ). In particular, if G is bipartite, then K[G] is normal.
We say that a finite subset L ⊂ E(G) is an edge cover of G if ∪ e∈L = [n]. Let µ(G) denote the minimal cardinality of edge covers of G. We now come to the main result of the present paper. It is known [3,Theorem 4.2] that, when G is bipartite, one has reg R(I(G)) = mat(G). Proof. (a) The "If" part is clear. We show now the "Only If" part. If, say, G 1 fails to satisfy the odd cycle condition, then G * also fails to satisfy the odd cycle condition. If, say, G 1 and G 2 are non-bipartite and if C i is an odd cycle of G i for i ∈ {1, 2}, then, even though G * is connected, there is no edge e ∈ E(G * ) with e ∩ V (C i ) = ∅ for i ∈ {1, 2}, as desired.
(b) We first prove the lower bound. In case mat(G) = 1, the graph G is a star graph which by assumption has at least 2 edges. Then R(I(G)) is not polynomial and therefore reg R(I(G)) ≥ 1 = mat(G). We now prove the upper bound. The highlight of the proof is to estimate the positive integer (Case I) Suppose that G is connected and non-bipartite. Each vertex i ∈ [n + 1] of P G * is regular. It then follows from [5, Theorem 1.7] that, if (a 1 , . . . , a n , a n+1 ) belongs to q(P G * \ ∂P G * ) ∩ Z n+1 , then each a i > 0. By using Lemma 2.1 one has q 0 ≥ µ(G) = (n + 1) − mat(G * ) ≥ (n + 1) − (mat(G) + 1).
(Case II) Suppose that G is disconnected and non-bipartite. Let G 1 , . . . G c be the connected components of G, where G 1 is non-bipartite and where each of i be the decomposition of V (G i ). Let (a 1 , . . . , a n , a n+1 ) belong to q(P G * \ ∂P G * ) ∩ Z n+1 . Since each i ∈ [n] is regular, one has a i > 0 for each i ∈ [n]. Since T = V 2 ∪ · · · ∪ V c is fundamental with N G (T ) = V ′ 2 ∪ · · · ∪ V ′ c ∪ {n + 1}, it follows that a n+1 > 0. Hence, as in (Case I), one has reg R(I(G)) ≤ mat(G) + 1, as required.
It would, of course, be of interest to characterize finite simple graphs G with reg R(I(G)) = mat(G). On the other hand, if G is a 5-cycle, then R(I(G)) is normal, reg R(I(G)) = 3 and mat(G) = 2.
When K[G] is non-normal, instead of [6, Theorem 3.3], we can enjoy the merit of combinatorial pure subrings (Corollary 1.3).
Proposition 2.5. Let G be an arbitrary finite simple graph. Then one has indmat(G) ≤ reg R(I(G)).
We, however, believe that the lower bound inequality mat(G) ≤ reg R(I(G)) is valid for any finite simple graph G.