An upper bound for the regularity of symbolic powers of edge ideals of chordal graphs

Assume that G is a chordal graph with edge ideal I ( G ) and ordered matching number ord-match( G ). For every integer s (cid:62) 1, we denote the s -th symbolic power of I ( G ) by I ( G ) ( s ) . It is shown that reg( I ( G ) ( s ) ) (cid:54) 2 s + ord-match( G ) − 1. As a consequence, we determine the regularity of symbolic powers of edge ideals of chordal Cameron-Walker graphs


Introduction
Let K be a field and S = K[x 1 , . . ., x n ] be the polynomial ring in n variables over K. Suppose that M is a nonzero graded S-module with minimal free resolution The Castelnuovo-Mumford regularity (or simply, regularity) of M , denoted by reg S (M ), is defined as reg S (M ) = max{j − i | β i,j (M ) = 0}.
As a convention, we set reg S (M ) = ∞, when M is the zero module.When there is no fear of confusion, we delete the subscript S in the above notation.We mention that the Castelnuovo-Mumford regularity is an important invariant in commutative algebra and algebraic geometry.
Katzman [16], proved that for any graph G, where ind-match(G) denotes the induced matching number of G. Beyarslan, Hà and Trung [5], generalized Katzman's inequality by showing that reg(I(G) s ) 2s + ind-match(G) − 1, for every integer s 1.Recently, Gu, Hà, O'Rourke and Skelton [9] proved the same inequality for symbolic powers.More explicit, they proved that for any graph G and any integer s 1.
Our approach is to determine an upper bound for the regularity of symbolic powers of edge ideals.Indeed, Hà and Van Tuyl [11,Theorem 6.7] proved that for every graph G, where match(G) denotes the matching number of G.This inequality was strengthen by Constantinescu and Varbaro [7,Remark 4.8] (see also [19,Corollary 2.5]).To be more precise, let G be a graph with ordered matching number ord-match(G) (see Definition 1).Then reg(I(G)) ord-match(G) + 1.
It is natural to ask whether the inequalities and are true.To the best of our knowledge, it is not known whether inequality (1) is true.In this paper, we investigate inequality (2) and as the main result, we show that inequality (2) holds for any chordal graph G and for any integer s 1 (see Theorem 3. As a consequence, we determine the regularity of symbolic powers of edge ideals of graphs belonging to the following classes.
3. K n − e, where e is an arbitrary edge of K n .

Preliminaries
In this section, we provide the definitions and basic facts which will be used in the next section.
Let G be a simple graph with vertex set V (G) = x 1 , . . ., x n and edge set E(G).For a vertex x i , the neighbor set A vertex of degree one is a leaf and the unique edge incident to a leaf is called a pendant edge.A pendant triangle of G is a triangle T of G, with the property that exactly two vertices of T have degree two in G.For every subset every edge of G is incident to at least one vertex of C. A vertex cover C is a minimal vertex cover if no proper subset of C is a vertex cover of G.The set of minimal vertex covers of G will be denoted by C(G).
For every subset C of x 1 , . . ., x n , we denote by p C , the monomial prime ideal which is generated by the variables belong to C. It is well-known that for every graph G with edge ideal I(G), Let G be a graph.A subset M ⊆ E(G) is a matching if e ∩ e = ∅, for every pair of edges e, e ∈ M .The cardinality of the largest matching of G is called the matching number of G and is denoted by match(G).A matching M of G is an induced matching of G if for every pair of edges e, e ∈ M , there is no edge f ∈ E(G) \ M with f ⊂ e ∪ e .The cardinality of the largest induced matching of G is the induced matching number of G and is denoted by ind-match(G).Definition 1.Let G be a graph, and let M = {a i , b i } | 1 i r be a nonempty matching of G.We say that M is an ordered matching of G if the following hold: (1) A := {a 1 , . . ., a r } ⊆ V (G) is a set of independent vertices of G; and the electronic journal of combinatorics 26(2) (2019), #P2.10 (2) {a i , b j } ∈ E(G) implies that i j.
The ordered matching number of G, denoted by ord-match(G), is defined to be ord-match(G) = max{|M | | M ⊆ E(G) is an ordered matching of G}.
We close this section by recalling the definition of symbolic powers.Let I be an ideal of S and let Min(I) denote the set of minimal primes of I.For every integer s 1, the s-th symbolic power of I, denoted by I (s) , is defined to be Let I be a squarefree monomial ideal in S and suppose that I has the irredundant primary decomposition where every p i is an ideal generated by a subset of the variables of S. It follows from [12,Proposition 1.4.4] that for every integer s 1, We set I (s) = S, for any integer s 0.

Main results
In this section, we prove the main result of this paper, Theorem 3. The proof is based on an inductive argument and the following lemma has a crucial role in our induction.
Lemma 2. Let G be a graph and assume that x 1 is a simplicial vertex of G, with N G (x 1 ) = x 2 , . . .x d , for some integer d 1.Then for every integer s 1, We know that I(G) = C∈C(G) p C .Thus, It follows that the electronic journal of combinatorics 26(2) (2019), #P2.10 and the induction hypothesis implies that where the last inequality follows from [19,Lemma 2.1].
Hence, in all cases, we have There, using inequality (5), we conclude that reg(S/I 1 ) 2s + ord-match(G) − 3 and this completes the proof.
As an immediate consequence of Theorem 3, we obtain the following corollary.The class of Cameron-Walker graphs is an interesting class of graphs, introduced in [6].It consists of graphs for which ind-match(G) = match(G).Algebraic properties of Cameron-Walker graphs have been studied in [13,14] and [21] The structure of Cameron-Walker graph has been determined in [6].Indeed, a connected graph G is a Cameron-Walker graph if and only if it is • a star graph, or • a star triangle, or • consisting of a connected bipartite graph H with vertex partition V (H) = X ∪ Y such that there is at least one pendant edge attached to each vertex of X and that there may be possibly some pendant triangles attached to each vertex of Y .
Thus, Corollary 5, essentially determines the regularity of symbolic powers of edge ideal of G, if either G is a star triangle, or the graph H above is a tree.
Let n 3 be a positive integer and suppose that e is an edge of the complete graph K n .Obviously, the graphs K n and K n − e are chordal graphs and their ordered matching number is one.Thus, by using Theorem 3 and [9, Theorem 4.6], we conclude the following corollary.
for any edge e ∈ E(G), we done by G − e, the graph obtained from G by deleting the edge e.A subgraph H of G is called induced provided that two vertices of H are adjacent if and only if they are adjacent in G.The induced subgraph of G on the vertex set U