Regularity of Initial Ideal of Binomial Edge Ideals in Degree 2 and Their Powers

: In this paper, we study the Castelnuovo-Mumford regularity of the initial ideal of binomial edge ideals in degree 2 ( ) 2 , [ ( )] G in J < and their powers for some classes of graphs. Also, we obtain a lower bound for the regularity of powers of binomial edge ideal associated to complete bipartite graphs.


Introduction
In the present paper, we are interested in the Castelnuovo-Mumford regularity of powers of homogeneous ideals in the polynomial rings.Let R = K[x 1 , ..., x n ] be such a standard graded polynomial ring over a field K and I be a homogeneous ideal in R. It is well-known that the regularity (R/I s ), as a function in s is asymptotically linear for 0 s  (cf.[1][2]).In general, it is difficult to decide when this function starts to be linear.Finding the linear function's exact form is also not easy (cf.[3]).In the following, we will call them the linearization-of-regularity problems, by abuse of terminology.
For these problems, the simplest case is when I is a quadratic squarefree monomial ideal.Whence, it can be recognized as the edge ideal of suitable graphs.Researchers have obtained some results for a few classes of simple graphs (e.g., forest graphs, cycle graphs, bipartite graphs) (cf.[4][5]).In contrast, little is known about the related binomial ideals.Recently, Jayanthan et al. in [6] provided the exact formulas for the regularity of powers of binomial edge ideals of several simple graphs such as cycle graphs, star graphs and balloon graphs.The main result of Jayanthan et al. in [6] is an upper bound for the regularity of powers of an ideal generated by a quadratic sequence.Inspired by their work, in this paper, we will provide the exact formulas for the regularity of powers of I, when I is an initial ideal of a binomial edge ideal of some graphs.As applications, we will study the linearization-of-regularity problems when the ideal I is an initial ideal of binomial edge ideal of some undirected simple graph G.
Given such a graph G, suppose that it has the vertex set V(G) = [n] = {1, •••, n} and edge set E(G).The binomial edge ideal of G is defined as where K denotes an arbitrary field.Herzog et al. in [7] and Ohtani in [8] independently introduced the notion of binomial edge ideal, and after that, the research on "binomial edge ideal" becomes a trend topic in combinatorial commutative algebra.Recently, there have been many results relating to the combinatorial data of graphs with the algebraic properties of the corresponding binomial edge ideals, see [5][6][9][10][11].In particular, there has been active research connecting algebraic invariants of the binomial edge ideals such as Castelnuovo-Mumford regularity, depth, and Betti numbers, with combinatorial invariants associated with graphs such as length of maximal induced path, number of maximal cliques, matching number.Let the polynomial ring [ , : ( )] endowed with the lexicographic order induced by 1 .
The initial ideal of binomial edge ideal in degree 2 is defined as x y i j E G i j = ∈ < i < j, and it is the monomial edge ideal of a bipartite graph on 2n vertices.Herzog et al. showed in [12], that powers of edge ideals with linear resolution have linear resolution themselves.A classic result by Fröberg [13] says that an edge ideal has linear resolution if and only if there is no induced cycle of length greater than three in its complement.Similar attempts have been made for powers of edge ideals of bipartite graphs in [5,14].
In [10], the authors conjectured that the extremal Betti numbers; a Betti number β i, j = 0 will be called extremal if coincide for any graph G that it is still open.In [9], the author gave a positive answer to this conjecture when graph G is a complete bipartite graph or a cycle graph.Also, the conjecture had been stated for every closed graph.
In this paper, we study the regularity of the initial ideal of binomial edge ideal in degree 2 ( ) for complete bipartite, cycle, and tree graphs.Furthermore, we obtain a lower bound for the regularity of powers of the binomial edge ideal of complete bipartite graphs.

Preliminaries
In this section, we review notation and fundamental results on the regularity of edge ideals and power of edge ideals that will be used in the next sections.
All graphs discussed in this paper are simple graphs.We assume basic familiarity with standard graph theory definition as in, e.g., [15], but in finding combinatorial bounds for the regularity of edge ideals, matching numbers and induced matching numbers have been used often.We recall their definitions.
Let G be a simple graph with the vertex set ( ) [ ] V G n = and edge set E(G).A complete graph on [n], denoted by K n , is the graph with the edge set ( ) { , :1 }.
said to be a simplicial vertex if v is contained in only one maximal clique otherwise it is called an internal vertex.We denote the number of internal vertices of G by iv(G).
A collection of edges {e 1 , •••, e s } is said to be a matching if i j e e ∩ = ∅ for all i j ≠ and this is said to be an induced matching if the induced subgraph on the vertices of {e 1 , •••, e s } has an edge set, {e 1 , •••, e s }.For a graph G, the induced matching number of G is the largest size of an induced matching, and it is denoted by v(G).
The complement of a graph G, denoted by G c is the graph on the same vertex set in which {u, v} is an edge of G c if and only if it is not an edge in G.
Definition 1 [5] A graph G is chordal graph if every induced cycle in G of a length greater than three has a chord, and it is called co-chordal if the complement of the graph G c is chordal.The co-chordal cover number, denoted co-chord (G), is the minimum number m such that there exist co-chordal subgraphs be a standard graded polynomial ring over a field K and M be a finitely generated graded R-module.Let be the minimal graded free resolution M, where R(-j) is the free R-module of rank 1 generated in degree j.The numbers ( ) ij M β are called the (i, j)-th graded Betti numbers of M. The regularity of M, denoted by reg(M) is defined as, When edge ideal I(G) has a linear resolution, all the nonzero entries in its Betti diagram are located on the first row.The next theorem, due to Fröberg is used repeatedly throughout this paper.
Theorem 1 [13] The minimal free resolution of edge ideal I(G) is linear if and only if the complement graph G c is chordal, that is, does not have induced cycles of length ≥ 4.
In the following theorem, a lower bound for the regularity of powers of edge ideals is given.Theorem 2 Theorem 4.5 in [4].Let G be a graph with edge ideal I = I(G) and let v(G) denote its induced matching number.Then for all s ≥ 1, we have ( ) 2 ( ) 1.

Regularity of powers of initial ideal of binomial edge ideal in degree 2 associate to cycle graphs
The following theorem was proved by Jayanthan et al. in [6].In this section, we prove it for initial ideals of binomial edge ideals in degree 2 associated to cycle graphs.A connected simple graph G of n vertices whose vertices are all of degree two is called a n-cycle and denoted by C n .
Theorem 3 Theorem 3.6 in [6].Let C n be a cycle graph on vertex set [n] and Therefore, [in < (J C n )] 2 = (x 1 y 2 , ..., x n − 1 y n , x 1 y n ) and the graph associated with [in < (J C n )] 2 is a bipartite graph of the form as shown in Figure 1.Proof.Let C n be a cycle graph.Then [in < (J C n )] 2 is an edge ideal and by Theorem 2 we have: ( ) where v(C n ) is the maximum size of an induced matching in C n , and by Observation 1, v(C n ) = n -2.Hence, ( ) .

Regularity of binomial edge ideal and initial of binomial edge ideal with their powers associated with bipartite graphs
In this section, we give a lower bound for the regularity of powers of binomial edge ideals of complete bipartite graphs and we give the exact value for the regularity of initial degree 2 of binomial edge ideals and their powers.Also, we obtain the regularity of certain bipartite graphs, and, in particular trees.
We proceed by recalling a few well-known results.
Lemma 1 [10] (Lemma 3.3).Let G be a finite bipartite graph on {x 1 , •••, x n } ∪ {y 1 , •••, y n } with the edges {x i , y j } with 1 .i j n ≤ ≤ ≤ Then, the complementary graph G c of G is a chordal graph.Proposition 1 [17], (Proposition 2.18).If G is a bipartite graph, then its complement does not have any induced cycle of length > 4. In particular, edge ideal of a bipartite graph has linear presentation, if and only if all its powers have linear resolution.
Corollary 2 Let G be a bipartite graph and I(G) be an edge ideal of the graph G.If Corollary 4 Let G = K m, n be a complete bipartite graph and [in < (J G )] 2 be the initial ideal of binomial edge ideal in degree 2 associated with G.Then, the following are equivalent:  3) shows that: In the following proposition, we obtain the regularity of powers of [ ] 2 ( ) , associate to the complete bipartite graphs.
Corollary 5 Let G be a complete bipartite graph and J G be the binomial edge ideal of G. Then Example 2 Let G = K 2, 3 be a complete bipartite graph with binomial edge ideal ( )

G J x y x y x y x y x y x y x y x y x y x y x y x y
shows the K 2, 3 graph.Schenzel et al. [14] (Theorem 4.1), proved that reg(J K 1, n ) = 3, where K 1, n is a special class of complete bipartite graphs that called star graphs.Also, Jayanthan et al. [6] computed the regularity of their powers [6] (Theorem 3.5) and show that ( ) where ι is the length of the longest induced path of G.The length of the longest induced path of a complete bipartite graph is 2, that is 2.

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At present, we are unable to prove, but computational evidence indicates that the answer to the following question is affirmative: Question 1 Let G be a complete bipartite graph.Is reg( ) 2 1, + for all s ≥ 1? Now, we deal with tree graphs.The tree is a graph in which any two vertices are connected by exactly one path.For the initial ideal of binomial edge ideal in degree 2 of a tree graph, we fix a labeling of the tree that we call grapes labeling.The labeling is as follows: First, we hang the tree just like a bunch of grapes by any of the vertex having a degree greater than 1.We labeled this vertex by 1 and named this vertex as a father vertex and all the vertices connected with the father vertex by an edge are called its children and labeled by 2, 3 and so on.Now in the next step consider all the children's vertices as father vertices and label their children's vertices as before and continue this process until the whole tree is labeled.
Example 3 Let's apply grapes labeling on the following tree shown in Figure 4.The father vertex is labeled as v 1 who has three children labeled as v 2 , v 3 , and v 4 .Now in the next step, the father vertex labeled with v 2 has three children labeled as v 5 , v 6 , and v 7 .The father vertex labeled with v 3 has no child and v 4 has one child labeled with v 8 as shown in Figure 4.In the rest of this section, we study [in < (J T )] 2 with the above mentioned labeling.We compute the regularity of initial ideal of binomial edge ideal in degree two for an arbitrary tree graph.
Theorem 6 Let T be a tree graph on vertex set [n] and J T be the binomial edge ideal of graph T. Let [in < (J T )] 2 be the initial ideal of binomial edge ideal in degree two associated with graph T.Then, reg([in < (J T )] 2 ) = iv(T ).
Proof.The graph associated to the monomial edge ideal [in < (J T )] 2 is a bipartite graph on [2n] vertices and each connected component of this bipartite graph is K i, 1 -star graph.It can easily be seen that the number of internal vertices iv(T ) or (father vertices) of graph T equals the number of components of K i, 1 -star graph associated to [in < (J T )] 2 .Based on Corollary 2 in [3], we get that: Proof.The upper bound is obtained from [11] (Theorem 5.3 (c)) and Theorem 6.The lower bound deduces from Theorem 2 and regarding that, the number of internal vertices of graph T is equal to the induced matching number of a bipartite graph associated with the monomial ideal [in < (J T )] 2 .

Discussion
Since the early 1990s, binomial ideals have become gradually fashionable.They now appear in various areas of commutative algebra and combinatorics as well as statistics.A comprehensive analysis of the algebraic properties of binomial ideals, including their primary decomposition, regularity, and depth, was given.
The binomial edge ideal of a graph could be seen as the ideal generated by a collection of 2-minors of the generic (2 × n)-matrix X.Therefore, the binomial edge ideal of a complete graph with n vertices is nothing but the well-studied determinantal ideal I 2 (X ).
A big effort concerning the binomial edge ideals has been understanding their minimal graded free resolutions.In general, it is quite difficult to describe the resolution of a binomial ideal.The graded Betti numbers give the numerical data of the resolution and determine the regularity and projective dimension of the ideal.In [19], the graphs whose binomial edge ideals have a linear resolution were characterized.Indeed, it was shown that a graph with no isolated vertices J G has a linear resolution if and only if G is a complete graph.Moreover, it was shown that the equivalent conditions hold if and only if J G has linear relations.Another approach in studying the graded minimal free resolution of J G has been determining its Castelnuovo-Mumford regularity.There have been two conjectures regarding some upper bounds for the regularity of J G , one by Matsuda et al. [20], and the other one by Kiani et al. [21].The first one was proved in [21], while the second one is still widely open.This conjecture was verified in [22] for a class of chordal graphs, namely block graphs.Several other algebraic properties and invariants of binomial edge ideals have also been studied in [7,[9][10].
In this work, we obtain an upper bound for the regularity of powers of the binomial edge ideal of complete bipartite graphs.Also, we compute the regularity of the initial ideal powers of binomial edge ideal associated with the cycle graphs, complete bipartite graphs, and trees.

Conclusion
The initial binomial edge ideal is the monomial edge ideal of a bipartite graph.In [10], the authors conjectured that the extremal Betti numbers of J G and in < (J G ) coincide for any graph G that it is still open.In [9], the author gave a positive answer to this conjecture when the graph G is a complete bipartite graph or a cycle graph.Also, the conjecture had been stated for every closed graph.
In [23], authors obtained the regularity of binomial edge ideal powers for cycle graphs and in the present work, we compute the regularity of powers of initial ideals of binomial edge ideals in degree 2 associated with cycle graphs.This gives a partial answer to the conjecture of Ene et al. [10].Schenzel et al. in [14] (Theorem 4.1), proved that reg(J K 1, n ) = 3, where K 1, n is a special class of complete bipartite graphs that called star graphs.Also, Jayanthan et al. [6] (Theorem 3.5) computed the regularity of their powers and show that ( )

J
be the binomial edge ideal of C n .Then, reg( ) all s ≥ 1. Observation 1 Let C n be a cycle graph on vertex set [n] and n C J be the binomial edge ideal of C n .Dokuyucu in [9] (Corollary 1.4), gave a following presentation for ( ) n C in J < : in < (J C n ) = (x 1 y 2 , ..., x n − 1 y n , x 1 y n , {x i x j + 1 ... x n y 1 ...

Figure 1 .
Figure 1.Graph associated to [in < (J Cn )] 2 I G = then by Theorem 1, G c does not have an induced cycle of length ≥ 4 this means that G c is a chordal graph, and by[18] (Proposition 1.3), we get that I(G) has a linear presentation.Corollary 3 Let G be a finite bipartite graph on {x 1 , •••, x n } ∪ {y 1 , •••, y n } with the edges { , } i j x y with 1 ≤ i ≤ j ≤ n.Then ( )reg( ) 2, I G = where I(G) is the edge ideal of graph G. Proof.Let I(G) be an edge ideal of a finite bipartite graph on {x 1 , •••, x n } ∪ {y 1 , •••, y n }, by Lemma 1, G c is a chordal graph, this means that G c does not have induced cycles of length ≥ 4, and by using Theorem 1, we get that The next corollary immediately follows from Theorem 1, [18] (Proposition 1.3 Theorem 1.8), and [12] (Theorem 3.2).

3 Figure 2 . F 3 graph 1
Figure 2. F 3 graph Remark 1 Let G = K m, n be the complete bipartite graph on the vertex set {1, •••, m} ∪ {m + 1, •••, m + n} with m ≥ n ≥ 1 and let J G be its binomial edge ideal.J G generated by all the binomials ij i j j i f x y x y = − where 1 ≤ i ≤ m and m + 1 ≤ j ≤ m + n.Dokuyucu in [9] (Corollary 1.3) shows that:

=
Since the graph G is complete bipartite, then by Remark 1, the graph related to [ ] bipartite graph.So, it is a co-chordal graph and by[3] (Theorem 5.7), for all s ≥ 1.

Figure 3 .
Figure 3. K 2, 3 graph In the following theorem, we extend this result for binomial edge ideal of complete bipartite graphs.Theorem 5 Let G be a complete bipartite graph K m, n Then reg( ) 2 1,s G J s ≥ + for all s ≥ 1.Proof.Since G is a complete bipartite graph then it is connected and based on[6] (Corollary 3.4), we have reg(

Figure 4 .
Figure 4. Tree graph In Theorem 5, we extend this result for the