Regularity of powers of edge ideals: from local properties to global bounds

Let $I = I(G)$ be the edge ideal of a graph $G$. We give various general upper bounds for the regularity function $\text{reg} I^s$, for $s \ge 1$, addressing a conjecture made by the authors and Alilooee. When $G$ is a gap-free graph and locally of regularity 2, we show that $\text{reg} I^s = 2s$ for all $s \ge 2$. This is a slightly weaker version of a conjecture of Nevo and Peeva. Our method is to investigate the regularity function $\text{reg}I^s$, for $s \ge 1$, via local information of $I$.


Introduction
During the last few decades, studying the regularity of powers of homogeneous ideals has evolved to be a central research topic in algebraic geometry and commutative algebra. This research program began with a celebrated theorem, proved independently by Cutkosky-Herzog-Trung [9] and Kodiyalam [24], which stated that for a homogeneous ideal I in a standard graded algebra over a field, the regularity function reg I s is asymptotically a linear function (see also [3,33]). Though despite much effort from many researchers, this asymptotic linear function is far from being well understood. In this paper, we investigate this regularity function for edge ideals of graphs. We shall explore several classes of graphs for which this regularity function can be explicitly described or bounded in terms of combinatorial data of the graphs. This problem has been studied recently by many authors (cf. [1,2,4,5,6,12,13,21,22,23,27,30]).
Our initial motivation for this paper is the general belief that global conclusions often could be derived from local information. Particularly, local conditions on an edge ideal I (i.e., conditions on reg(I : x), for x ∈ V (G)) should give a global understanding of the function reg I s , for s ≥ 1. Our motivation furthermore comes from the following conjectures (see [5,28,29]), which provide a general upper bound for the regularity function of edge ideals, and describe a special class of edge ideals whose powers (at least 2) all have linear resolutions.
Our aim is to investigate Conjectures A and B using the local-global principle. Finding general upper bounds for reg I(G) s has received a special interest and generated a large number of papers during the last few years. This partly thanks to a general lower bound for reg I(G) s given in [6]; particularly, if ν(G) denotes the induced matching number of G then, for any s ≥ 1, we have reg I(G) s ≥ 2s + ν(G) − 1. (1.1) Our first main result gives a weaker general upper bound for reg I(G) s than that of Conjecture A. The motivation of this result comes from an upper bound for the regularity of I(G) given by Adam Van Tuyl and the last author, namely reg I(G) ≤ β(G) + 1, where β(G) denotes the matching number of G (see [16]). We prove the following theorem.
Theorem 3.4. Let G be a graph with edge ideal I = I(G), and let β(G) be its matching number. Then, for all s ≥ 1, we have reg I s ≤ 2s + β(G) − 1.
A graph G is said to be locally of regularity at most r − 1 if reg(I(G) : x) ≤ r − 1 for all vertex x in G. Note that, by [8,Proposition 4.9], if G is locally of regularity at most r − 1 then reg I(G) ≤ r. In the local-global spirit, we reformulate Conjecture A to a slightly weaker conjecture as follows.
Conjecture A . Let G be a simple graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1, for some r ≥ 2. Then, for any s ≥ 1, we have reg I s ≤ 2s + r − 2.
Our next main result proves Conjecture A for gap-free graphs.
Theorem 4.2. Let G be a simple graph with edge ideal I = I(G). Suppose that G is gap-free and locally of regularity at most r − 1, for some r ≥ 2. Then, for any s ≥ 1, we have It is an easy observation that if I(G) s has a linear resolution for some s ≥ 1 then G must be gap-free. Conjecture B serves as a converse statement to this observation, and has remained intractable. By applying the local-global principle, we prove a weaker statement, in which the condition reg I = 3 is replaced by the condition that G is locally linear (i.e., locally of regularity at most 2). Our main result toward Conjecture B is stated as follows.
Theorem 4.5. Let G be a simple graph with edge ideal I = I(G). Suppose that G is gap-free and locally linear. Then for all s ≥ 2, we have reg I s = 2s.
As a consequence of Theorem 4.5, we quickly recover a result of Banerjee, which showed that if G is gap-free and cricket-free then I(G) s has a linear resolution for all s ≥ 2 (see Corollary 4.6).
We end the paper by exhibiting an evidence for Conjecture A at the first nontrivial value of s, i.e., s = 2, for all graphs.
Theorem 5.1. Let G be a graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1. Then, for any edge e ∈ E(G), reg(I 2 : e) ≤ r. Particularly, this implies that reg(I 2 ) ≤ r + 2.
Our paper is structured as follows. In the next section we give necessary notation and terminology. The reader who is familiar with previous work in this research area may want to proceed directly to Section 3. In Section 3, we discuss general upper bound for the regularity function, aiming toward Conjecture A. Theorem 3.4 is proved in this section. In Section 4, we focus further on gap-free graphs, investigating both Conjectures A and B using the local-global principle. Theorems 4.2 and 4.5 are proved in this section. We end the paper with Section 5, proving Theorem 5.1 and discussing briefly how an effective bound on the regularity of I(G) 2 may give us information on the regularity of the second symbolic power I(G) (2) . This gives a glimpse into future work on the regularity function of symbolic powers of edge ideals.

Preliminaries
In this section, we collect notations and terminology used in the paper. For unexplained notions, we refer the reader to standard texts [7,11,18,26,31,34].
Graph Theory. Throughout the paper, G shall denote a finite simple graph with vertex set V (G) and edge set E(G). A subgraph G of G is called induced if for any two vertices u, v in G , uv ∈ E(G ) ⇔ uv ∈ E(G). For a subset W ⊆ V (G), we shall denote by G W the induced subgraph of G over the vertices in W , and denote by G − W the induced subgraph of G on V (G) \ W . When W = {w} consists of a single vertex, we also write G − w for G − {w}. The complement of a graph G, denoted by G c , is the graph on the same vertex set V (G) in which uv ∈ E(G c ) ⇔ uv ∈ E(G).
Definition 2.1. Let G be a graph.
(1) A walk in G is a sequence of (not necessarily distinct) vertices x 1 , x 2 , . . . , x n such that x i x i+1 is an edge for all i = 1, 2, . . . , n. A circuit is a closed walk (i.e., when x 1 ≡ x n ). (2) A path in G is a walk whose vertices are distinct (except possibly the first and the last vertices).
(3) A cycle in G is a closed path. A cycle consisting of n distinct vertices is called an n-cycle and often denoted by C n . (4) An anticycle is the complement of a cycle.
A graph in which there is no induced cycle of length greater than 3 is called a chordal graph. A graph whose complement is chordal is called a co-chordal graph.
Definition 2.2. Let G be a graph.
(1) A matching in G is a collection of disjoint edges. The matching number of G, denoted by β(G) is the maximum size of a matching in G.
(2) An induced matching in G is a matching C such that the induced subgraph of G over the vertices in C does not contain any edge other than those already in C. The induced matching number of G, denoted by ν(G), is the maximum size of an induced matching in G. For any integer n, K n denotes the complete graph over n vertices (i.e., there is an edge connecting any pair of vertices). For any pair of integers m and n, K m,n denotes the complete bipartite graph; that is, a graph with a bipartition (U, V ) of the vertices such that |U | = m, |V | = n and E(K m,n ) = {uv | u ∈ U, v ∈ V }. (1) A graph isomorphic to K 1,3 is called a claw. A graph without any induced claw is called a claw-free graph. (2) A graph isomorphic to the graph with vertex set {w 1 , w 2 , w 3 , w 4 , w 5 } and edge set {w 1 w 3 , w 2 w 3 , w 3 w 4 , w 3 w 5 , w 4 w 5 } is called a cricket. A graph without any induced cricket is called a cricket-free graph.
Observation 2.5. A claw-free graph is cricket-free.
Notation 2.6. Let G be a graph, let u, v ∈ V (G), and let e = xy ∈ E(G).
(1) The set of vertices incident to u, the neighborhood of u, is denoted by The set of vertices incident to an endpoint of e, the neighborhood of e, is denoted by . An edge is called a leaf or a whisker if any of its vertices has degree exactly 1. (4) The distance between u and v, denoted by d(u, v), is the fewest number of edges that must be traversed to travel from u to v in G.
We can naturally extend these notions to get N G (W ), N G [W ], N G (E) and N G [E] for a subset of the vertices W ⊆ V (G) or a subset of the edges E ⊆ E(G). Definition 2.7. Let G be a graph.
(1) A collection W of the vertices in G is called an independent set if there is no edge connecting two vertices in W . (2) The independent complex of G, denoted by ∆(G), is the simplicial complex whose faces are independent sets of G.
Commutative Algebra. Let G be a simple graph over the vertices V (G) = {x 1 , . . . , x n }. By abusing notation, we shall identify the vertices of G with the variables in a polynomial ring S = k[x 1 , . . . , x n ], where k is any infinite field. Particularly, we shall use uv to denote both the edge uv in G and the monomial uv in S (the choice would be obvious from the context).
Castelnuovo-Mumford regularity is the invariant being investigated in this paper. We shall give a definition most suitable for our context. Definition 2.9. Let S be a standard graded polynomial ring over a field k. The regularity of a finitely generated graded S module M , written as reg M , is given by For a graph G, we shall use reg I(G) and reg G interchangeably. The following simple bound is often used without references.
A standard use of short exact sequences yields the following result, which we shall also often use.
Lemma 2.11. Let I ⊆ S be a monomial ideal, and let m be a monomial of degree d. Then Moreover, if m is a variable appearing in I, then reg I is equal to one of the right-hand-side terms.
Definition 2.12. Let r ∈ N. A graph G is said to be locally of regularity ≤ r if for every vertex x ∈ V (G), we have reg(I(G) : x) ≤ r. A graph which is locally of regularity ≤ 2 is called locally linear.
Auxiliary Results. We next recall a few results that are useful for our purpose.
We shall make use of the following characterization for edge ideals of graphs with linear resolutions. This characterization was first given in topological language by Wegner [35] and later, independently, by Lyubeznik [25] and Fröberg [14] in monomial ideals language. In the study of powers of edge ideals, Banerjee developed the notion of even-connection and gave an important inductive inequality in [4]. This inductive method has proved to be quite powerful, which we shall make use of often.
Theorem 2.14. For any finite simple graph G and any s ≥ 1, let the set of minimal monomial generators of I(G) s be {m 1 , ...., m k }, then The ideal (I(G) s+1 : m) in Theorem 2.14 and its generators are understood via the following notion of even-connection.
Two vertices u and v (u may be the same as v) are said to be even-connected with respect to an s-fold product e 1 · · · e s where e i 's are edges of G, not necessarily distinct, if there is a path p 0 p 1 · · · p 2k+1 , k ≥ 1 in G such that: It turns out that (I(G) s+1 : m) is generated by monomials in degree 2.

General Upper Bounds for Regularity Function
The aim of this section is to give a weaker general upper bound for reg I(G) s than that of Conjecture A.
The heart of many studies on regularity of powers of edge ideals is to understand the colon ideal J = I(G) s : e 1 . . . e s−1 in making use of Banerjee's inductive method, Theorem 2.14. We start by examining a local property for J.
for some t ≤ s, and a subcollection {f 1 , . . . , f t−1 } of {e 2 , . . . , e s−1 }. Moreover, in this case, the graph associated to the polarization of Proof. (1) It follows from Theorem 2.16 that J is obtained by adding to I quadratic generators uv, where u and v are even-connected in G with respect to e 1 . . . e s−1 . If e 1 is an isolated edge then clearly, by definition, the even-connected path between u and v does not contain e 1 . Thus, uv ∈ I s−1 : e 2 . . . e s−1 and (1) is proved.
(2) It can be seen that if w ∈ N G [{e 1 , . . . , e s−1 }] then w is not in any even-connected path with respect to e 1 . . . e s−1 . Thus, even-connected paths with respect to e 1 . . . e s−1 between two vertices that are not in N G [w] are even-connected path with respect to e 1 . . . e s−1 in would be divisible by u ∈ J : w and, thus, subsumed into the ideal (u u ∈ N G [w]). Therefore, (2) follows.
Moreover, for any u ∈ N G (w), u and w are even-connected with respect to e 1 . . . e s−1 , and so uw ∈ J, i.e., u ∈ (J : w). Thus, we have the inclusion To prove the other inclusion, let us analyse the minimal generators of (J : w) more closely. Consider any uv ∈ J, where u and v are even-connected with respect to e 1 . . . e s−1 . If v ≡ w (similarly if u ≡ w) then u ∈ N G (w). If u, v ≡ w, but v ∈ N G (w) (similarly if u ∈ N G (w)), then uv is subsumed in the ideal (u u ∈ N G (w)).
, which are even-connected with respect to e 1 . . . e s−1 . Observe that if the even-connected path between u and v contains e 1 then, by considering a subpath of this path, either u and w or v and w are even-connected with respect to e 1 . . . e s−1 (see Figures 1 and 2). That is, either u or v is in N G (w), and so uv is again subsumed in the ideal (u u ∈ N G (w)). Therefore, we may assume that u and v are even-connected with respect to a subcollection {f 1 , . . . , f t−1 } of {e 2 , . . . , e s−1 }. That is, u v f j w w even-connected even-connected even-connected   (1) for any G ∈ F, reg I(G) ≤ f (G); and (2) for any nonempty graph G ∈ F and each non-isolated vertex w ∈ V (G), Then, for any G ∈ F and any s ≥ 1, we have Proof. Fix a graph G ∈ F and let I = I(G). If f (G) ≤ 2 then the result is immediate from [19]. Assume that f (G) ≥ 3. Then the condition on By Theorem 2.14 and the hypothesis that reg I(G) ≤ f (G), it suffices to show that for any collection of edges e 1 , . . . , e s−1 in G (not necessarily distinct), we have reg(I s : e 1 . . . e s−1 ) ≤ f (G). This, by successively applying Lemma 2.11 with (J, w 1 , . . . , w i ) and w i+1 , implies that The assertion now follows by utilizing Lemma 2.11 with J and w 1 .
Based on the known upper bound for reg I(G), given in [16], one can take f (G) in Theorem 3.3 to be the matching number of a graph and obtain the following interesting bound for the regularity function. Proof. Let F be the family of all simple graphs. Then F clearly is a hierarchy. Let f (G) = β(G) + 1 for all G ∈ F. It is easy to see that: (1) reg I(G) ≤ f (G) by [16]; and (2) For any non-isolated vertex w in G, clearly β(G − w) ≤ β(G), and we can always add an edge incident to w to any matching of G − N G [w] to get a bigger matching, and so f ( Hence, the statement follows from Theorem 3.3. A particular interesting application of Theorem 3.4 is for the class of Cameron-Walker graphs introduced in [10]. These are graphs for which ν(G) = β(G). See [20] for a further classification of Cameron-Walker graphs. Proof. The conclusion is an immediate consequence of Theorem 3.4 noting that ν(G) = β(G) if G is a Cameron-Walker graph.
It is known, by the main theorem of [19], that if I(G) has a linear resolution then so does I(G) s for any s ∈ N. Thus, the first nontrivial case of Conjecture A is for those graphs G such that G is locally linear and reg I(G) > 2. Recall that by [8,Proposition 4.9], in this case, we necessarily have reg I(G) = 3. Theorem 3.3 allows us to settle Conjecture A for this class of graphs.
Theorem 3.6. Let G be a graph with edge ideal I = I(G). Suppose that G is locally linear. Then for all s ≥ 1, we have reg I s ≤ 2s + reg I − 2 ≤ 2s + 1.
Proof. Let F be the family of locally linear graphs (including those whose edge ideals have linear resolutions). Define f : F −→ N by f (G) = reg I(G) for all G ∈ F. By the definition and Lemma 2.10, the edge ideal of any proper induced subgraph of G ∈ F has a linear resolution. Thus, F is a hierarchy and f satisfies conditions of Theorem 3.3. The conclusion now follows from that of Theorem 3.3.
Example 3.7. Let G be a graph such that G c is triangle-free (see, for example, Figure 4). It can be seen that for any x ∈ V (G), G − N G [x] is a complete graph (and, thus, is of regularity 2). Therefore, G is a locally linear graph.

Regularity Function of Gap-free Graphs
In this section, we focus on gap-free graphs, investigating both Conjectures A and B. We start with a stronger version of [4,Lemma 6.18]. The proof is almost the same as that given in [4,Lemma 6.18] x 1 x 2 x 3 x 4 x 5 x 6 Figure 4. A graph whose complement is triangle-free (1) u, v ∈ W ; and (2) this path is of the longest possible length with respect to condition (1).
is obtained by adding isolated vertices to an induced subgraph of Proof. By Theorem 2.16, uv ∈ G −W . Consider any other edge u v ∈ G \G with u , v ∈ W . Then, there is an even-connected path u = q 0 , . . . , q 2l+1 = v in G with respect to e 1 . . . e s−1 for some 1 ≤ l ≤ k.
If there exist i and j such that p 2i+1 p 2i+2 and q 2j+1 q 2j+2 are the same edge in G then by combining these two even-connected paths, either u or v will be even-connected to u. That is, either u or v will become an isolated vertex in G − W − N G [u]. We may assume that the two even-connected path between u, v and u , v do not share any edge.
Consider p 1 p 2 and q 1 q 2 . Since these two edges do not form a gap in G, they must be connected. Let us now explore different possibilities for this connection.
If p 1 ≡ q 1 then u and v are even-connected with respect to e 1 . . . e s−1 , and so v becomes an isolated vertex in G − W − N G [u]. If p 1 ≡ q 2 (similarly for the case that p 2 ≡ q 1 ) then u and u are even-connected with respect to e 1 . . . e s−1 , and so u becomes an isolated vertex If p 1 q 1 ∈ E(G) then combining the two even-connected paths between u, v and u , v and the edge p 1 q 1 , we get an even-connected path between v and v that is of length > k, a contradiction. If p 1 q 2 ∈ E(G) (similarly for the case that p 2 q 1 ∈ E(G)) then by combining the two even-connected paths between u, v and u , v and the edge p 1 q 2 , we have an even connected path between u and v that is of length > k, a contradiction.
Thus, in any case, either u or v will becomes an isolated vertex in G − W − N G [u]. That is, any edge in G \ G will reduce to an isolated vertex in G − W − N G [u]. The statement is proved.
Our next main result establishes Conjecture A for gap-free graphs.
Theorem 4.2. Let G be a graph with edge ideal I = I(G) and let r ≥ 3 be an integer. Assume that G is gap-free and locally of regularity ≤ r − 1. Then, for all s ∈ N, we have reg I s ≤ 2s + r − 2.
Proof. By [8, Proposition 4.9], we have reg I ≤ r. By Theorem 2.14, it suffices to show that for any collection of edges e 1 , . . . , e s−1 (not necessarily distinct) in G, we have reg(I s : e 1 . . . e s−1 ) ≤ r.
Let G be the graph associated to the polarization of J = I s : e 1 . . . e s−1 . It follows from Lemma 2.11 that, for any vertex x ∈ G , It remains to consider reg(G − u). Let u and v be even-connected in G with respect to e 1 . . . e s−1 such that u , v ∈ G −u and there is an even-connected path u = q 0 , . . . , q 2l+1 = v in G with respect to e 1 . . . e s−1 such that l is the maximum possible length. By using Lemma 4.1 again, we can deduce that reg Thus, by applying (4.1) to the graph G − u, it suffices to show that reg(G − {u, u }) ≤ r.
We can continue in this fashion until all edges in G \ G are examined, i.e., we obtain a collection W ⊆ V (G) such that G − W = G − W , and reduce the problem to showing that reg(G − W ) = reg(G − W ) ≤ r. This is obviously true by Lemma 2.10 and the fact that reg G ≤ r. The theorem is proved.
We shall now shift our attention to Conjecture B. We begin by an improved statement of [8, Corollary 6.5]. Lemma 4.3. Let G be a gap-free and cricket-free graph. Then G is locally linear.
Proof. We may assume that G contains no isolated vertices. By Theorem 2.13, it suffices to is an induced subgraph of G, it is gap-free and cannot have any induced anticycle of length 4.
Let y be a neighbor of x. Since G is gap-free, {x, y} and {w 1 , w 3 } cannot form a gap. Thus, these edges must be connected in G. That is, either {y, w 1 } or {y, w 3 } (or both) must be an edge in G.
Suppose that {y, w 1 } and {y, w 3 } are both edges in G. Then, by considering edges {x, y} and {w 2 , w n } in G, either {y, w 2 } or {y, w n } must be an edge in G. If {y, w 2 } is an edge, then the induced subgraph on {x, y, w 1 , w 2 , w 3 } is a cricket in G, a contradiction. Otherwise, {y, w n } ∈ E(G). Since {x, y} and {w 2 , w n−1 } cannot form a gap in G, we must have {y, w n−1 } ∈ E(G). Thus, the induced subgraph on {x, y, w 1 , w n−1 , w n } is a cricket in G, a contradiction.
If {y, w 1 } ∈ E(G) and {y, w 3 } ∈ E(G) (similarly for the case {y, w 1 } ∈ E(G) and {y, w 3 } ∈ E(G)), then {y, w n } must be an edge in G; otherwise, {x, y} and {w 3 , w n } form a gap in G. By considering {x, y} and {w 2 , w n−1 }, either {y, w 2 } or {y, w n−1 } must be an edge in G. If {y, w 2 } ∈ E(G), then the induced subgraph on {x, y, w 1 , w 2 , w n } is a cricket in G, a contradiction. Otherwise, {y, w n−1 } ∈ E(G), and the induced subgraph on {x, y, w 1 , w n−1 , w n } is a cricket in G, a contradiction.
There are examples for locally linear gap-free graphs for which the regularity could be either 2 or 3 (see Figure 5).
x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 5 C 4 C 5 Figure 5. Locally linear gap-free graphs with regularity 2 and 3 (respectively) On the other hand, note that if G is not gap-free, then ν(G) ≥ 2 =⇒ reg I(G) ≥ 3. Thus, if, in addition, I(G) is locally linear, then we have reg I(G) = 3 by [8,Proposition 4.9]. Figure 6 depicts such a graph.
x 1 x 2 x 3 x 4 x 5 Figure 6. A graph that is not gap-free but locally linear with regularity 3 We are now ready to state our main result toward Conjecture B. In this result, we establish the conclusion of Conjecture B replacing the condition that reg I(G) = 3 by the condition that G is locally linear. That is, the graph G associated to the ideal J = I s : e 1 . . . e s−1 is a co-chordal graph.
By [4,Lemma 6.14], G is also gap-free, and so G does not contain an anticycle of length 4. Suppose that W = {w 1 . . . w n }, for n ≥ 5, is such that G [W ] is an induced anticycle of G . It follows from [4, Lemma 6.15] that G[W ] is an induced anticycle of G.
Let e 1 = ab. We shall consider different possibilities for the relative position of a and b with respect W .
If a, b ∈ W , say a ≡ w 1 and b ≡ w i (for i = 1), then since {w 1 , w 2 }, {w 1 , w n } ∈ E(G ), b = w 2 , w n . Consider the edges {a, b} and {w 2 , w n }. These do not form a gap (and a is not connected to neither w 2 nor w n ), and so either {b, w 2 } ∈ E(G) or {b, w n } ∈ E(G). If {b, w 2 } ∈ E(G) then w 2 and w 3 are even-connected with respect to e 1 = ab, which implies that {w 2 , w 3 } ∈ E(G ), a contradiction. If {b, w n } ∈ E(G) then w n−1 and w n are evenconnected with respect to e 1 = ab, which implies that w n−1 w n ∈ E(G ), also a contradiction.
If a ∈ W , say a = w 1 , and b ∈ W (similar to the case where a ∈ W and b ∈ W ) then by considering the edges {a, b} and {w 2 , w n } again, the same arguments as above would lead to a contradiction.
If a, b ∈ W and either a or b is not connected to any vertices in W , then G [W ] (being also an anticycle in G) is an anticycle in either , which is a contradiction to the local linearity of G.
It remains to consider the case that a, b ∈ W , and both a and b are connected to W . Assume that aw 1 ∈ E(G). Consider the pair of edges {a, b} and {w 2 , w n }. If either {b, w 2 } ∈ E(G) or {b, w n } ∈ E(G) then, as before, we would have either {w 2 , w 3 } ∈ E(G) or {w n−1 , w n } ∈ E(G), which is a contradiction. Thus, we must have either {a, w 2 } ∈ E(G) or {a, w n } ∈ E(G). Without loss of generality, we may assume that {a, w 2 } ∈ E(G). We continue by considering the pair of edges {a, b} and {w 3 , w n }. A similar argument shows that {a, w 3 } ∈ E(G). We can keep going in this fashion to get {a, w i } ∈ E(G) for all i = 1, . . . , n − 2. Now, it can be seen that b cannot be connected to any of the w i without creating an even-connection that gives {w i , w i+1 } ∈ E(G), for some i, which is a contradiction.
We have shown that such a collection of the vertices W cannot exists. That is, G is a co-chordal graph. The theorem is proved. Theorem 4.5 immediately recovers the following result of Banerjee [4]. Example 4.7. Let 2K 2 denote a gap and let K 6 denote the complete graph on 6 vertices. Let G = 2K 2 + K 6 be the join of these two graphs (the join of two graphs H and K is obtained by taking the disjoint union of H and K and connecting each vertex in H with every vertex in K). Then, it can be seen G is locally linear but not gap-free. Particularly, it follows that reg I(G) s = 2s for all s ∈ N. This gives an example of a locally linear graph G for which reg I(G) s = 2s for all s ∈ N.

Regularity of Second Powers of Edge Ideals
We end the paper with a flavor of Conjecture A when s = 2. We also take a look at the symbolic square of edge ideals.
Theorem 5.1. Let G be a graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1. Then, for any edge e ∈ E(G), reg(I 2 : e) ≤ r. Particularly, this implies that reg(I 2 ) ≤ r + 2.
Proof. The second statement follows from the first statement and Theorem 2.14. To prove the first statement, we shall use induction on |V (G)|. Let J = I 2 : e and let G be the graph associated to J.
If there are no even-connected vertices in G with respect to e, then I 2 : e = I, and the conclusion follows from [8,Proposition 4.9].
If there are edges in G which are not initially in G, then these edges are of the form xy where x ∈ N (a), y ∈ N (b) or xx where x ∈ N (a) ∩ N (b) and x is a new whisker vertex.
Suppose that there exists at least one new edge of the form xy for x = y. Observe that J : x = I : x + (u | u ∈ N (b)). Thus reg(J : x) ≤ reg(I : x) ≤ r − 1. Furthermore, (J, x) = I(G \ x) 2 : e. Therefore, by induction on |V (G)|, we have reg(J, x) ≤ r. Hence, by Lemma 2.11, we have reg J ≤ r.
Suppose that the only new edges are of the form xx , where x is a new whisker vertex. Observe that, in this case, J : x = I : x + (u | u ∈ N (a) ∪ N (b)) + (u | u is a whisker in the new edges ) (J, x) = I(G \ x) 2 : e Thus, we also have reg(J : x) ≤ reg(I : x) ≤ r − 1 and reg(J, x) ≤ r by induction. Hence, by Lemma 2.11 again, we have reg J ≤ r. This completes the proof.
Symbolic powers in general are much harder to handle than ordinary powers. The symbolic square of an edge ideal appears to be more tractable. We recall and rephrase a result from [32]. The last result of our paper is stated as follows.
Theorem 5.3. Let G be a graph with edge ideal I = I(G). Suppose that G is locally of regularity at most r − 1. Then reg(I (2) ) ≤ r + 2.
Observe that K l = I, and for all i we have the following short exact sequence.
This, particularly, implies that reg(I (2) ) ≤ max Note that if e is an edge in the triangle {x i , x j , x k }, then (x i x j x k : e) is a variable. If e shares a vertex with the triangle, then the colon ideal is generated by an edge and (x i x j x k : e) ∈ I. If e and {x i , x j , x k } have no common vertices, then (x i x j x k : e) = x i x j x k ∈ I. Then, by Theorem 2.16 we have J i = I 2 : e i+1 + (variables) and hence, reg J i ≤ reg(I 2 : e). The conclusion now follows from Theorem 5.1 and the use of [8,Proposition 4.9].