Induced matchings and the v-number of graded ideals

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when $G$ is very well-covered, or $G$ has a simplicial partition, or $G$ is well-covered connected and contain neither $4$- nor $5$-cycles. In all these cases the v-number of $I(G)$ is a lower bound for the regularity of the edge ring of $G$. We classify when the upper bound holds when $G$ is a cycle, and classify when all vertices of a graph are shedding vertices to gain insight on $W_2$-graphs.


Introduction
Let S = K[t 1 , . . . , t s ] = ∞ d=0 S d be a polynomial ring over a field K with the standard grading and let I be a graded ideal of S. A prime ideal p of S is an associated prime of S/I if (I : f ) = p for some f ∈ S d , where (I : f ) is the set of all g ∈ S such that gf ∈ I. The set of associated primes of S/I is denoted by Ass(I) and set of maximal elements of Ass(I) with respect to inclusion is denoted by Max(I). The v-number of I, denoted v(I), is the following invariant of I that was introduced in [8] to study the asymptotic behavior of the minimum distance of projective Reed-Muller-type codes [8,Corollary 4.7]: v(I) := min{d ≥ 0 | ∃ f ∈ S d and p ∈ Ass(I) with (I : f ) = p}.
One can define the v-number of I locally at each associated prime p of I:   The formulas of parts (a) and (b) give an algorithm to compute the v-number number using Macaulay2 [17] (Example 5.1, Procedure A.1).
The v-number of non-graded ideals was used in [26] to compute the regularity index of the minimum distance function of affine Reed-Muller-type codes [26,Proposition 6.2]. In this case, one considers the vanishing ideal of a set of affine points over a finite field.
For certain classes of graded ideals v(I) is a lower bound for reg(S/I), the regularity of the quotient ring S/I (Definition 2.1), see [8,21,31]. There are examples of ideals where v(I) > reg(S/I) [21]. It is an open problem whether v(I) ≤ reg(S/I)+ 1 holds for any squarefree monomial ideal. Upper and lower bounds for the regularity of edge ideals and their powers are given in [2,4,9,20,22,30,41], see Section 2. Using the polarization technique of Fröberg [15], we give an upper bound for the regularity of a monomial ideal I in terms of the dimension of S/I and the exponents of the monomials that generate I (Proposition 3.4).
Let G be a graph with vertex set V (G) and edge set E(G). If V (G) = {t 1 , . . . , t s }, we can regard each vertex t i as a variable of the polynomial ring S = K[t 1 , . . . , t s ] and think of each edge {t i , t j } of G as the quadratic monomial t i t j of S. The edge ideal of G is the squarefree monomial ideal of S defined as I(G) := (t i t j | {t i , t j } ∈ E(G)).
This ideal, introduced in [38], has been studied in the literature from different perspectives, see [16,19,33,36,40] and the references therein. We use induced matchings of G to compare the v-number of I(G) with the regularity of S/I(G) for certain families of graphs.
A subset C of 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 minimal if each proper subset of C is not a vertex cover of G. A subset A of V (G) is called stable if no two points in A are joined by an edge. Note that a set of vertices A is a (maximal) stable set of G if and only if V (G) \ A is a (minimal) vertex cover of G. The stability number of G, denoted by β 0 (G), is the cardinality of a maximum stable set of G and the covering number of G, denoted α 0 (G), is the cardinality of a minimum vertex cover of G. For use below we introduce the following two families of stable sets: According to [21,Theorem 3.5], F G ⊂ A G and the v-number of I(G) is given by v(I(G)) = min{|A| : A ∈ A G }.
The v-number of I(G) is a combinatorial invariant of G that has been used to characterize the family of W 2 -graphs (see the discussion below before Corollary 4.13). We can define the v-number of a graph G as v(G) := v(I(G)) and study v(G) from the viewpoint of graph theory.
A set P of pairwise disjoint edges of G is called a matching.
An induced matching of a graph G is a matching P = {e 1 , . . . , e r } of G such that the only edges of G contained in r i=1 e i are e 1 , . . . , e r . The matching number of G, denoted β 1 (G), is the maximum cardinality of a matching of G and the induced matching number of G, denoted im(G), is the number of edges in the largest induced matching.
The graph G is well-covered if every maximal stable set of G is of the same size and G is very well-covered if G is well-covered, has no isolated vertices, and |V (G)| = 2α 0 (G). The class of very well-covered graphs includes the bipartite well-covered graphs without isolated vertices [35,39] and the whisker graphs [36, p. 392] (Lemma 2.7). A graph without isolated vertices is very well-covered if and only if G is well-covered and β 1 (G) = α 0 (G) (Proposition 2.8). One of the properties of very well-covered graphs that will be used to show the following theorem is that they can be classified using combinatorial properties of a perfect matching as was shown by Favaron [ We come to one of our main results.
Theorem 4.3. Let G be a very well-covered graph and let P = {e 1 , . . . , e r } be a perfect matching of G. Then, there is an induced submatching P ′ of P and D ∈ A G such that D ⊂ V (P ′ ) and |e D| = 1 for each e ∈ P ′ . Furthermore v(I(G)) ≤ |P ′ | = |D| ≤ im(G) ≤ reg(S/I(G)). Let G be a graph and let W G be its whisker graph (Section 2). As a consequence we recover a result of [21] showing that the v-number of I(W G ) is bounded from above by the regularity of the quotient ring K[V (W G )]/I(W G ) (Corollary 4.4). The independent domination number of G, denoted by i(G), is the minimum size of a maximal stable set [1, Proposition 2]: A cycle of length s is denoted by C s . The inequality v(I(G)) ≤ reg(S/I(G)) of Theorem 4.3 is false if we only assume that G is a well-covered graph, since the cycle C 5 is a well-covered graph, but one has im(C 5 ) = 1 < 2 = v(I(C 5 )). We prove that C 5 is the only cycle where the inequality v(I(C s )) ≤ im(C s ) fails.
Theorem 4.6. Let C s be an s-cycle and let I(C s ) be its edge ideal. Then, v(I(C s )) ≤ im(C s ) if and only if s = 5. If ] for some simplicial vertex v. A graph G is simplicial if every vertex of G is either simplicial or is adjacent to a simplicial vertex of G.
If A is a stable set of a graph G, H i is a complete subgraph of G for i = 1, . . . , r and A {V (H i )} r i=1 is a partition of V (G), then reg(S/I(G)) ≤ r [41, Theorem 2]. We consider a special type of partitions of V (G) that allow us to link A G with induced matchings of G. A graph G has a simplicial partition if G has simplexes H 1 , . . . , H r , such that {V (H i )} r i=1 is a partition of V (G). Our next result shows that v(I(G)) ≤ im(G) if G has a simplicial partition. Theorem 4.9. Let G be a graph with simplexes H 1 , . . . , H r , such that {V (H i )} r i=1 is a partition of V (G). If G has no isolated vertices, then there is D = {y 1 , . . . , y k } ∈ A G , and there are simplicial vertices x 1 , . . . , x k of G and integers 1 ≤ j 1 < · · · < j k ≤ r such that As a consequence, using a result of Finbow, Hartnell and Nowakowski that classifies the connected well-covered graphs without 4-and 5-cycles [14, Theorem 1.1] (Theorem 2.9), we show two more families of graphs where the v-number is a lower bound for the regularity. A graph G belongs to class W 2 if |V (G)| ≥ 2 and any two disjoint stable sets A 1 , A 2 are contained in two disjoint maximum stable sets B 1 , B 2 with |B i | = β 0 (G) for i = 1, 2. A graph G is in W 2 if and only if G is well-covered, G \ v is well-covered for all v ∈ V (G) and G has no isolated vertices [25,Theorem 2.2]. A graph G without isolated vertices is in W 2 if and only if v(I(G)) = β 0 (G) [21,Theorem 4.5]. As an application we recover the only if implication of this result (Corollary 4.13). Using that a graph G without isolated vertices is in W 2 if and only if G is well-covered and A G = F G [21,Theorem 4.3], by Proposition 4.11, we recover the fact that a graph G without isolated vertices is in W 2 if and only if G is well-covered and every v ∈ V (G) is a shedding vertex [25,Theorem 3.9]. For other characterizations of graphs in W 2 see [25,37] and the references therein.
In Section 5 we show examples illustrating some of our results. In particular in Example 5.3 we compute the combinatorial and algebraic invariants of the well-covered graphs C 7 and T 10 that are depicted in Figure 1. These two graphs occur in the classification of connected wellcovered graphs without 4-and 5-cycles [14, Theorem 1.1] (Theorem 2.9). A related result is the characterization of well-covered graphs of girth at least 5 given in [13].
For all unexplained terminology and additional information, we refer to [10,18] for the theory of graphs and [16,19,40] for the theory of edge ideals and monomial ideals.

Preliminaries
In this section we give some definitions and present some well-known results that will be used in the following sections. To avoid repetitions, we continue to employ the notations and definitions used in Section 1.
Definition 2.1. [11] Let I ⊂ S be a graded ideal and let F be the minimal graded free resolution of S/I as an S-module: The Castelnuovo-Mumford regularity of S/I (regularity of S/I for short) is defined as The integer g, denoted pd(S/I), is the projective dimension of S/I.

Theorem 2.2. [5] If a graph G is well-covered and is not complete, then
If G is a graph, then β 1 (G) ≤ α 0 (G). We say that G is a Kőnig graph if β 1 (G) = α 0 (G). This notion can be used to classify very well-covered graphs (Proposition 2.8). If G is a graph without 3-, 5-, and 7-cycles or G is a Kőnig graph, then G is well-covered if and only if G is very well-covered.
Remark 2.5. Let P be a perfect matching of a graph G with property (P). Note that if {b, b ′ } is in P and a ∈ V (G), then {a, b} and {a, b ′ } cannot be both in E(G) because G has no loops. In other words G has no triangle containing an edge in P .
Theorem 2.6. [12, Theorem 1.2] The following conditions are equivalent for a graph G: (1) G is very well-covered.
(2) G has a perfect matching with property (P).
(3) G has a perfect matching, and each perfect matching of G has property (P).
Let G be a graph with vertex set V (G) = {t 1 , . . . , t s } and let U = {u 1 , . . . , u s } be a new set of vertices. The whisker graph or suspension of G, denoted by W G , is the graph obtained from G by attaching to each vertex t i a new vertex u i and a new edge as a device to study the numerical invariants and properties of graphs and edge ideals.
Lemma 2.7. Let G be a graph without isolated vertices. The following hold.
Proof. (a) A bipartite well-covered graph without isolated vertices has a perfect matching P that satisfies property (P) [39, Theorem 1.1]. Thus, by Theorem 2.6, G is very well-covered.
⇐) Assume that G is well-covered and β 1 (G) = α 0 (G). Let P = {e 1 , . . . , e r } be a matching of G with r = β 1 (G). We need only show that |V (G)| = 2α 0 (G). Clearly |V (G)| is greater than or equal to 2α 0 (G) because r i=1 e i ⊂ V (G). We proceed by contradiction assuming that As v is not an isolated vertex of G, there is a minimal vertex cover C of G that contains v. As G is well-covered one has that |C| = α 0 (G) = r. Since e i C = ∅ for i = 1, . . . , r and v ∈ C, we get |C| ≥ r + 1, a contradiction. 3. The v-number of a graded ideal

We say that a graph G is in the family
S d be a polynomial ring over a field K with the standard grading and let I be a graded ideal of S. In this section we show a formula for the v-number of I that can be used to compute this number using Macaulay2 [17]. To avoid repetitions, we continue to employ the notations and definitions used in Sections 1 and 2. . . , g r } be a minimal generating set of (I : p)/I such that g i is a homogeneous polynomial for all i. As (I : f ) = p, one has f = 0 and f ∈ (I : p). Then, we can choose homogeneous polynomials . Therefore, using that all g i 's are in (I : p), one has the inclusions Hence p = (I : Theorem 3.2. Let I ⊂ S be a graded ideal and let p ∈ Ass(I). The following hold.
(a) If G = {g 1 , . . . , g r } is a homogeneous minimal generating set of (I : p)/I, then v p (I) = min{deg(g i ) | 1 ≤ i ≤ r and (I : We give a direct proof of the next result that in particular relates the v-number of a Cohen-Macaulay monomial ideal I ⊂ S with that of (I, h), where h ∈ S 1 and (I : h) = I.  Proof. To show this inequality we use the polarization technique due to Fröberg (see [28] and [40, p. 203]). To polarize I we use the set of new variables A result of Beintema [3] shows that a zero-dimensional monomial ideal is Gorenstein if and only if it is a complete intersection. The next result classifies the complete intersection property using the regularity.
. Conversely assume that reg(S/I) is equal to s i=1 (d i − 1). We proceed by contradiction assuming that m > s. Then the exponents of the monomial t dm = t c 1 1 · · · t cs s satisfy c i ≤ d i − 1 for i = 1, . . . , s because t dm ∈ G(I). The regularity of S/I is the largest integer d ≥ 0 such that (S/I) d = (0) [11,Proposition 4.14]. Pick a monomial t a = t a 1 1 · · · t as s such that t a ∈ S d \ I and d = s i=1 (d i − 1). Then, a i ≤ d i − 1 for i = 1, . . . , s because t a is not in I, and consequently a i = d i − 1 for i = 1, . . . , s. Hence, t a = t δ t dm for some δ ∈ N s , a contradiction.

Induced matchings and the v-number
In this section we show that the induced matching number of a graph G is an upper bound for the v-number of I(G) when G is very well-covered, or G has a simplicial partition, or G is well-covered connected and contain neither 4-nor 5-cycles. We classify when the upper bound holds when G is a cycle, and classify when all vertices of a graph G are shedding vertices, we use this to gain insight on W 2 -graphs. To avoid repetitions, we continue to employ the notations and definitions used in Sections 1 and 2.  Proof. To show the first part we use induction on |P |. If r = 1, we set P ′ = P = {e 1 } and D = {x 1 }, where e 1 = {x 1 , y 1 }. Assume r > 1. We set e r = {x, x ′ }, G 1 := G \ {x, x ′ } and P 1 := P \ {e r }. By Theorem 2.6, P satisfies the property (P). Then, P 1 satisfies the property (P) as well. Thus, by Theorem 2.6, G 1 is very well-covered with a perfect matching P 1 . Hence, by induction hypothesis, there is an induced submatching P ′ 1 of P 1 and D 1 ∈ A G 1 such that D 1 ⊂ V (P ′ 1 ) and |e D 1 | = 1 for each e ∈ P ′ 1 . Consequently, N G 1 (D 1 ) is a minimal vertex cover of G 1 . We will consider two cases: e r N G (D 1 ) = ∅ and e r N G (D 1 ) = ∅.
Case (I): Assume that e r N G (D 1 ) = ∅. Thus, we may assume that there is Therefore, by Lemma 4.2, D 1 ∈ A G , so this case follow by making D = D 1 and P ′ = P ′ 1 . Case (II): Assume that e r N G (D 1 ) = ∅. We set D 2 := V (P ′ 1 ) \ D 1 , then D 2 is a stable set of G 1 and also of G, since P ′ 1 is an induced matching of G 1 and also of G. One has the inclusion (4.1) , and x ∈ e r N G (D 1 ), a contradiction. We claim that , since P satisfies property (P), a contradiction, since D 2 is a stable set of G. Hence, |e r N G (D 2 )| ≤ 1 and we may assume If the intersection is non-empty, by Eq. (4.1), we can pick z in D 2 N G (x ′ ), then {z, x ′ } ∈ E(G) and x ′ ∈ N G (D 2 ), a contradiction to Eq. (4.2). Therefore, by Eq. (4.1), we obtain the inclusion Thus, the edge set Q := {e ∈ P ′ 1 | e A 2 = ∅} {e r }, is an induced matching, since P ′ 1 is an induced matching. Setting i.e., D 3 = (D 1 V (Q)) {x}, we get |f D 3 | = 1 for each f ∈ Q, since |e D 1 | = 1 for each e ∈ P ′ 1 . Note that D 3 is a stable set of G, since D 1 is a stable set and {x} N G (D 1 ) = ∅. Now, take e ∈ E(G). We will prove that e N G (

then by Eq. (4.3) and the inclusion
and P satisfies property (P). Thus, z ∈ e N G (x) ⊂ e N G (D 3 ). Hence, N G (D 3 ) is a vertex cover and, by Lemma 4.2, D 3 ∈ A G . Therefore, this case follows by making P ′ = Q and D = D 3 . This completes the induction process.
Proof. We proceed by induction on r. If r = 1, then V (H 1 ) = V (G) and there is a simplicial vertex ] is a complete graph with at least two vertices. Picking y 1 ∈ N G [x 1 ], y 1 = x 1 , one has {x 1 } ∈ A G and {x 1 , y 1 } is an induced matching. Now assume that r > 1. We set G 1 := G \ V (H r ). Note that H 1 , . . . , H r−1 are simplexes of G 1 (Lemma 4.8) and {V (H i )} r−1 i=1 is a partition of V (G 1 ). Then, by induction hypothesis, there is D 1 = {y 1 , . . . , y k ′ } ∈ A G 1 , and there are simplicial vertices x 1 , . . . , x k ′ of G 1 and integers ). This proves that N G (D 1 ) is a vertex cover of G. Hence, by Lemma 4.2, D 1 ∈ A G and, noticing that P 1 is an induced matching of G, this case follows by making D = D 1 and P = P 1 .
Case (II): Assume that there is y ∈ V (H r )\{x} such that y / ∈ N G (D 1 ). Then, D 2 := D 1 {y} is a stable set of G. Also, N G (D 2 ) is a vertex cover of G, since N G 1 (D 1 ) is a vertex cover of G 1 , H r is a complete subgraph of G, and V (H r ) ⊂ N G [y]. Thus, by Lemma 4.2, D 2 is in A G . We set x k ′ +1 := x, y k ′ +1 := y and H j k ′ +1 := H r . Then, {x k ′ +1 , y k ′ +1 } ∈ E(H r ) and , for i = 1, . . . , k ′ + 1 . Therefore, this case follows by making D = D 2 and P = P 2 .
The equality |D| = |P | is clear. The inequality v(I(G)) ≤ |D| follows from Theorem 4.1 and |P | ≤ im(G) is clear by definition of im(G). Finally, the inequality im(G) ≤ reg(S/I(G)) follows directly from Theorem 2.10. Proof. Assume that G is simplicial. Let {z 1 , . . . , z ℓ } be the set of all simplicial vertices of G.
Next assume that G is connected and contain neither 4-nor 5-cycles. Then, by Theorem 2.9, G ∈ {C 7 , T 10 } or G ∈ F. The cases G = C 7 or G = T 10 are treated in Example 5.3 (cf. Theorem 4.6). If G ∈ F, then there exists {x 1 , . . . , k} is a partition of V (G). In particular G is simplicial and the asserted inequalities follow from the first part of the proof. (1) Every vertex of G is a shedding vertex.
∈ D, a contradiction, since a ∈ N G (D) and A ′ {a} is a stable set. Hence, A ′ is a maximal stable set of G \ x. Therefore x is not a shedding vertex of G, a contradiction.
(2) ⇒ (1): By contradiction, suppose there is x ∈ V (G) such that x is not a shedding vertex. Thus, there is a maximal stable set A of G \ x such that A ⊂ V (G) \ N G [x] . Then, C := V (G \ x) \ A is a minimal vertex cover of G \ x and A {x} is a stable set of G. So, A / ∈ F G . Since C is a minimal vertex cover of G \ x, we have that for each z ∈ C, there is z ′ ∈ V (G \ x) \ C = A such that {z, z ′ } ∈ E(G). Consequently, C ⊂ N G (A). Furthermore, if a ∈ N G (x), then a ∈ G \ x and a / ∈ A. Thus, a ∈ N G (A), since A is a maximal stable set of G \ x. Hence, N G (x) ⊂ N G (A). This implies, N G (A) is a vertex cover of G, since C ⊂ N G (A). Therefore, by Lemma 4.2, A ∈ A G , a contradiction since A / ∈ F G . Proof. Let v be a vertex of G. We may assume that G is not a complete graph. Let A be a stable set of . We proceed by contradiction assuming that A is a maximal stable set of G \ v. Then, as G and G \ v are well-covered, we get According to [32,Theorem 5], the graph G v is in W 2 and β 0 (G v ) = β 0 (G) − 1. In particular G v is well-covered and β 0 (G v ) = β 0 (G) − 1 (cf. Theorem 2.2). But A is a stable set of G v and |A| = β 0 (G), a contradiction. Proof. By Theorem 4.1, there is D ∈ A G such that v(I) = |D|. Since G is a W 2 -graph, by Lemma 4.12, every vertex of G is a shedding vertex. Thus, by Proposition 4.11, D ∈ F G , i.e., D is a maximal stable set of G. Furthermore, G is well-covered, since G is a W 2 -graph. Hence, |D| = β 0 (G). Therefore, v(I) = β 0 (G).
The graph G defined by the generators of this ideal is well-covered and not very well-covered, α 0 (G) = 4, and v(I) = im(G) = reg(S/I) = 1.
The neighbor set of A = {t 1 , t 4 } in T 10 is N T 10 (A) = {t 2 , t 3 , t 5 , t 7 , t 8 , t 10 } and N T 10 (A) is a minimal vertex cover of T 10 , that is, A ∈ A T 10 .  Example 5.4. Let G be the graph consisting of two disjoint 3-cycles with vertices x 1 , x 2 , x 3 and y 1 , y 2 , y 3 . Take two disjoint independent sets of G, say A 1 = {x 1 } and A 2 = {y 1 }, to verify that G is a graph in W 2 note that B 1 = {x 1 , y 2 } and B 2 = {y 1 , x 2 } are maximum independent sets of G containing A 1 and A 2 and |B i | = β 0 (G) = 2.