Squarefree powers of edge ideals of forests

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.


Introduction
Let G be a finite simple graph with the vertex set V (G) = {x 1 , . . . , x n } and the edge set E(G). Let k be a field and let S = k[x 1 , . . . , x n ] be the polynomial ring in n variables over k. The edge ideal of G, denoted by I(G), is the monomial ideal generated by x i x j such that {x i , x j } ∈ E(G). Computation of Castelnuovo-Mumford regularity of edge ideals and their powers is a challenging problem in commutative algebra which has led to extensive literature.
Matchings in graphs appeared in the context of bounding or computing regularity. For example, it is well known that the regularity reg(I(G)) of edge ideal of G is bounded below by indm(G) + 1 [13] and above by mat(G) + 1 [8] where indm(G) and mat(G) denote respectively the induced matching number and the matching number of the graph G. It is also known that such lower bound is attained when G is a chordal graph [8]. These bounds were generalized to powers of edge ideals in [2,3]. In particular, for any positive integer k, the following inequalities hold: 2k + indm(G) − 1 ≤ reg(I(G) k ) ≤ 2k + mat(G) − 1.
The authors of [3] proved that the lower bound is attained when G is a forest and it was conjectured in [1] that such lower bound should be also attained by chordal graphs.
In this article, we investigate squarefree powers of edge ideals. The kth squarefree power I(G) [k] of edge ideal of a graph G is generated by the squarefree monomials in the kth ordinary power I(G) k . If k > mat(G), then I(G) [k] = (0). The study of squarefree powers was initiated in [4] and continued in [5]. Our motivation to study such powers is twofold. Firstly, thanks to the Restriction Lemma (Lemma 2.5) the regularity of I(G) [k] is bounded above by that of I(G) k . This suggests that squarefree powers might be useful in the study of ordinary powers. For instance, if the kth squarefree power does not have linear resolution, then the kth ordinary power cannot have linear resolution either. The second part of our motivation comes from the fact that the generators of I(G) [k] correspond to the matchings in G of size k. This makes a close connection between squarefree powers of edge ideals and the theory of matchings in graphs.
In this article, we introduce the concept of k-admissable matching of a graph. A matching M is called k-admissable if there exists a partition of M that satisfy certain conditions, see Definition 3.2. A 1-admissable matching is the same as an induced matching. Therefore, k-admissable matchings can be seen as generalization of induced matchings. The k-admissable matching number of G, denoted by aim(G, k), is the maximum size of a kadmissable matching. Our first main result (Theorem 4.7) gives an upper bound for the regularity of squarefree powers of edge ideals of forests: In Theorem 4.10 we show that the upper bound above is attained when k = 2: If G is a forest with mat(G) ≥ 2, then reg(I(G) [2] ) = aim(G, 2) + 2.
Our second main result (Theorem 5.10) gives a complete classification of forests G for which I(G) [k] has linear resolution: As a consequence of the above theorem, we show that for any forest G and 1 ≤ k < mat(G), if I(G) [k] has linear resolution, then I(G) [k+1] has linear resolution as well.

Definitions and notations.
Let G be a finite simple graph with the vertex set V (G) and the edge set E(G). Given a vertex x in G, we say y is a neighbor of x if {x, y} ∈ E(G). We denote the set of all neighbors of x by N G (x). We set A vertex of degree 0 is called an isolated vertex. A vertex of degree 1 is called a leaf. A complete graph on n vertices is denoted by K n .
We say H is a subgraph of G if V (H) ⊆ V (G) and E(H) ⊆ E(G). A subgraph H of G is called an induced subgraph if for any two vertices x, y in H, {x, y} ∈ E(H) if and only if {x, y} ∈ E(G). For any U ⊆ V (G), the induced subgraph of G on U is the graph with the vertex set U and the edge set {{x, y} : x, y ∈ U and {x, y} ∈ E(G)}. For any U ⊆ V (G), we denote by G − U the induced subgraph of G on V (G) \ U .
A graph G is called connected if any two vertices of G are connected by a path in G. A maximal connected subgraph of G is called a connected component of G. We say G is a forest if G has no cycle subgraphs. A connected forest is called a tree.
A matching of G is a collection of edges which are pairwise disjoint. The matching number of G, denoted by mat(G), is defined by Clearly, indm(G) ≤ mat(G) for any graph G. An induced matching of size 2 is called a gap. If {e 1 , e 2 } is a gap in G, we say the edges e 1 and e 2 form a gap in G. A matching M of G is called a perfect matching if for every vertex x of G, there is an edge e ∈ M such that x ∈ e.
For any positive integer n, we denote {1, . . . , n} by [n]. Let G be a graph with the vertex set V (G) = {x 1 , . . . , x n }. Let k be a field and let S = k[x 1 , . . . , x n ] be the polynomial ring in n variables over k. The edge ideal of G, denoted by I(G), is the monomial ideal defined by By abuse of notation, we will use an edge e = {x i , x j } of G interchangeably with the monomial x i x j . For any 1 ≤ k ≤ mat(G), we define the kth squarefree power of the edge ideal of G by I(G)

2.2.
Background. In this section, we collect some results that will be useful to prove our results. The following lemma shows the existence of a certain kind of leaf in forests. The authors of [4] proved the surprising result that the highest non-vanishing squarefree power of an edge ideal has linear resolution. This result will be crucial in the proof of Theorem 4.7.  Let I ⊂ S be a monomial ideal, and let F be its minimal multigraded free S-resolution. Let G(I) denote the minimal set of monomial generators of I. Furthermore, let m be a monomial. We set is a subcomplex of F and the minimal multigraded free resolution of I ≤m .
We will use the following consequence of Lemma 2.5: The following result is well-known, see for example Lemma 3.1 in the survey article [7].
Lemma 2.7. For any homogeneous ideal I ⊂ S and any homogeneous element m ∈ S of degree d the short exact sequence yields the following regularity bound for I:

k-admissable matchings
In this section, we define k-admissable matching of a graph, and we make some observations about their properties. Definition 3.1. For any positive integers k and n, we call a sequence (a 1 , . . . , a n ) of integers a k-admissable sequence if the following conditions are satisfied: (1) a i ≥ 1 for each i = 1, . . . , n (2) a 1 + · · · + a n ≤ n + k − 1.
Definition 3.2. Let G be a graph with matching number mat(G). Let M be a matching of G. For any 1 ≤ k ≤ mat(G) we say M is k-admissable matching if there exists a sequence M 1 , . . . , M r of non-empty subsets of M such that In such case, we say M = M 1 ∪ · · · ∪ M r is a k-admissable partition of M for G. Definition 3.3. The k-admissable matching number of a graph G, denoted by aim(G, k), is defined by Remark 3.4. For any graph G, one can deduce the following properties of k-admissable matchings from the definition.
(1) A matching M of G is 1-admissable if and only if M is an induced matching of G.
Moreover, if G is a forest, then aim(G, mat(G)) = mat(G). It is not hard to see that the induced matching number of G is 3. Therefore aim(G, 1) = 3.
Remark 3.7. If the sequence (a 1 , . . . , a n ) is k-admissable, then so is (a 1 , . . . , a n , 1). Lemma 3.8. Let M be a k-admissable matching of a graph G. Then any non-empty subset of M is also a k-admissable matching of G.
Proof. Let M = M 1 ∪ · · · ∪ M r be a k-admissable partition of M for G. Then the sequence (|M 1 |, . . . , |M r |) is k-admissable. Therefore |M | ≤ r + k − 1. Let us assume that |M | > 1 since otherwise M is the only non-empty subset of itself. It suffices to show that for any N ⊆ M with |N | = |M | − 1, the matching N is k-admissable. Without loss of generality,

Upper bounds for squarefree powers of edge ideals of forests
In this section, we provide a sharp upper bound for reg(I(G) [k] ) where G is a forest, in terms of k-admissable matching number of G. A key idea of our method is to work with a special type of vertex in a forest, which we define below.
Definition 4.1. Let G be a forest with a leaf x and its unique neighbor y. We say x is a distant leaf if y has at most one neighbor whose degree is greater than 1. In this case, we say {x, y} is a distant edge. Proof. If G has no vertex of degree at least 2, then G consists of union of some isolated vertices and K 2 's. In such case, every edge is a distant edge. Otherwise, the result follows from Lemma 2.1.   Suppose that x does not divide e 1 . . . e k . Then uy is divisible by e 1 . . . e k . Since {e 1 , . . . , e k } is a matching, we may assume that y does not divide e 2 . . . e k . Hence u is divisible by e 2 . . . e k and the result follows as in the previous case.   Proof. We use induction on |V (G)| + k. First note that if k = 1, then the statement follows from Theorem 2.3 as aim(G, 1) = indm(G). Also, if k = mat(G), then by Remark 3.4 we have aim(G, mat(G)) = mat(G) and the result follows from Theorem 2.4. Therefore, let us assume that 2 ≤ k < mat(G).
Indeed, for each 0 ≤ i ≤ r − 1, since mat(G i ) = mat(G) and mat( which proves Eq. (1). We can apply Eq. (1) and Lemma 2.7 successively to eliminate x 1 and its duplicates as follows.
Then it suffices to show that the maximum in the above inequality is at most aim(G, k) + k. Note that by Lemma 4.3 we have

Remark 2.2, induction assumption and (4) of Remark 3.4 imply
. , x r , z} for some vertex z of degree greater than 1. We will consider these cases separately.
We will now show that the maximum in (2) is at most aim(G, k) + k − 1 which will complete the proof. Observe that by Remark 2.
Observe that Remark 2.2 implies reg((I r : (y)) + (z)) = reg(I(G − {y, z}) [k] ). We may assume that the matching number of G − {y, z} is at least k since otherwise the proof is immediate. By induction, we have Therefore, it remains to show that aim(G − {y, z}, k) + 1 ≤ aim(G, k). Indeed, keeping Remark 3.7 in mind, any k-admissable matching of G − {y, z} can be extended to a kadmissable matching of G by adding the edge {x 1 , y}. Proof. We may assume that mat(G) ≥ 2 as the statement is vacuously true otherwise. If M is 1-admissable matching, then the result follows from Lemma 4.8. So, let us assume that M is not 1-admissable. Let Since M is not an induced matching of G, without loss of generality, we may assume that {x 2 , x 3 } is an edge of G. Claim: G has exactly r + 1 edges. Proof of the claim: Let M = M 1 ∪ · · · ∪ M q be a 2-admissable partition of M for G. By condition (3) of Definition 3.2 we may assume that both {x 1 , x 2 } and {x 3 , x 4 } are in M 1 . Since the sequence (|M 1 |, . . . , |M q |) is 2-admissable, we have |M 1 | + · · · + |M q | ≤ q + 1. On the other hand, since |M 1 | ≥ 2 and |M i | ≥ 1 for all i ≥ 2, we obtain |M 1 | = 2 and |M i | = 1 for each i ≥ 2. The claim then follows from conditions (3) and (5) of Definition 3.2 together with the fact that M is a perfect matching of G.
We now give a formula for the regularity of I(G) [2] when G is a forest. Theorem 4.10. If G is a forest with mat(G) ≥ 2, then reg(I(G) [2] ) = aim(G, 2) + 2.
In particular, Theorem 4.10 gives a lower bound for the regularity of second squarefree power of edge ideal of any graph.
A graph G that satisfies indm(G) = mat(G) is called a Cameron-Walker graph. Such graphs were studied from a commutative algebra point of view in [11]. The following proposition shows that the upper bound in Theorem 4.7 is sharp. Using the structural classification of Cameron-Walker graphs [11], for any given positive integer m, one can construct a Cameron-Walker tree G with indm(G) = mat(G) = m. Figure 1 illustrates an example with m = 2.
Based on the results of this section and Macaulay2 [6] computations, we expect that the upper bound in Theorem 4.7 would give the exact formula for the regularity of squarefree powers of edge ideals of forests. Thus, we propose the following conjecture.

Characterization of squarefree powers with linear resolutions
In this section, we will classify forests G such that I(G) [k] has linear resolution. From Theorem 2.3 it follows that I(G) k has linear resolution if and only if indm(G) = 1 when G is a forest. So, for ordinary powers, such characterization does not depend on k, and the class of forests with induced matching number equal to one is rather small. On the other hand, we will see that linearity of resolution of I(G) [k] depends on both the forest G and the integer k.
Let us briefly recall some definitions about simplicial complexes. A simplicial complex ∆ on a finite vertex set V (∆) is a collection of subsets of V (∆) such that if F ∈ ∆, then every subset of F is also in ∆. Each element of ∆ is called a face of ∆. If F is a maximal face of ∆ with respect to inclusion, then we say F is a facet of ∆. We write ∆ = F 1 , . . . , F r if F 1 , . . . , F r are all the facets of ∆. We say ∆ is connected if for every pair of vertices u and v there exists a sequence F 1 , . . . , F s of facets of ∆ such that u ∈ F 1 , v ∈ F s and F i ∩ F i+1 = ∅ for each i = 1, . . . , s − 1.
Notation 5.2. Let m = x α 1 1 . . . x αn n be a monomial in k[x 1 , . . . , x n ]. To ease the notation, the monomial m and the multidegree (α 1 , . . . , α n ) will be used interchangeably. Moreover, if m = x i 1 . . . x i k is squarefree, we will denote the set {x i 1 , . . . , x i k } by m. Proof. By definition of the upper-Koszul simplicial complex, it is clear that each m/m i corresponds to a face of K m (I). Moreover, m/m i corresponds to a maximal face since m i is a minimal monomial generator. Lastly, if u is a monomial that corresponds to a face of K m (I), then m/u ∈ I. Then there is a monomial v such that m/u = vm i for some i ∈ [t]. This implies that the face u is contained in the facet m/m i .
The following lemma is well-known in graph theory.
Lemma 5.4. Let G be a graph with connected components G 1 , . . . , G r . Then G has a perfect matching if and only if G i has a perfect matching for each i ∈ [r].
Lemma 5.5. Let G be a graph which has a perfect matching. Then for any vertex x of G, the graph G − {x} has no perfect matching.
Proof. If a graph has perfect matching, then it has even number of vertices.
Lemma 5.6. Let G be a graph with connected components G 1 , . . . , G r where r ≥ 2. Suppose that G has a perfect matching. If x ∈ V (G 1 ) and y ∈ V (G 2 ), then G − {x, y} has no perfect matching.
Proof. By Lemma 5.4 each G i has a perfect matching. Let U 1 , . . . , U t be the connected components of G 1 − {x} and V 1 , . . . , V s be the connected components of G 2 − {y}. Then the connected components of G − {x, y} are U 1 , . . . , U t , V 1 , . . . , V s , G 3 , . . . , G r . By Lemma 5.5 the graph G 1 − {x} has no perfect matching. Then by Lemma 5.4 there exists U j which has no perfect matching. Since U j is also a connected component of G − {x, y}, it follows that G − {x, y} has no perfect matching.
Notation 5.7. If M = {e 1 , . . . , e k } is a matching of G, then we will write u M for the squarefree monomial e 1 . . . e k = x i ∈e, e∈M x i . Lemma 5.8. Let G be a graph with a k-admissable perfect matching M . Let M = M 1 ∪ · · · ∪ M r be a k-admissable partition of M for G. Using Notation 5.7 let x|u M i and y|u M j for some vertices x and y with i = j. Then x and y are in different connected components of G.
Proof. If {a, b} is an edge of G, then since M is a perfect matching, a|u Mp and b|u Mq for some p and q. Since M is k-admissable, we get p = q. Therefore there is no path in G that connects x and y.