On $m$-Closed Graphs

A graph is closed when its vertices have a labeling by $[n]$ such that the binomial edge ideal $J_G$ has a quadratic Gr\"{o}bner basis with respect to the lexicographic order induced by $x_1>\cdots>x_n>y_1>\cdots>y_n$. In this paper, we generalize this notion and study the so called $m-$closed graphs. We find equivalent condition to $3-$closed property of an arbitrary tree $T$. Using it, we classify a class of $3-$closed trees. The primary decomposition of this class of graphs is also studied.


Introduction and Preliminaries
Suppose G is a simple graph on the vertex set [n] and R = k[x 1 , . . . , x n , y 1 , . . . , y n ] is the polynomial ring over the field k. The binomial edge ideal of G is the ideal where f ij = x i y j − x j y i . This notion was first introduced in [9] and independently in [13].
Note that any ideal generated by a set of 2-minors of a 2 × n-matrix X of indeterminates may be viewed as the binomial edge ideal of a graph. In [9], the authors compute the reduced Gröbner basis of the binomial edge ideal with respect to the lexicographic order induced by x 1 > · · · > x n > y 1 > · · · > y n (we show this order by ≺). In particular, they find the necessary and sufficient conditions in which J G has a quadratic Gröbner basis. Graphs whose binomial edge ideal has a quadratic Gröbner basis are called closed graphs and the Cohen-Macaulay property of these graphs is studied in [6]. Recently, many authors studied the algebraic properties of some classes of binomial edge ideals. In particular the regularity and the depth are studied in [1,6,7,10,12,15]. But the reduced Gröbner basis obtained in [9] has not been studied in more details.
In this paper, we study the Gröbner basis of J G where G is a simple graph. We call G an m−closed graph when its vertices can be labeled by [n] such that the elements of the reduced Gröbner basis of J G have degree at most m, and m is the least integer with this property for G. Note that by this definition, a closed graph is a 2−closed graph.
In Section 2 we study some basic properties of m−closed graphs. In particular, we show that a cycle C n (n > 3) is m− closed where m = n 2 + 1 n is even; n+1 2 + 1 n is odd.
(see Theorem 2.5). Using it we conclude that in each m−closed graph, any cycle with at least 2m − 1 vertices has a chord. The notion of weakly closed graphs has been introduced in [11] as a generalization of closed graphs. The final result of section 2 shows that each weakly closed graph is m−closed for some m ≤ 4 (see Theorem 2.9).
In Section 3 we study 3−closed property of trees and we show that a tree T with n vertices is 3−closed if and only if it is not a path and there exists a labeling of its vertices such that d(i, i + 1) ≤ 2 for each i < n (see Theorem 3.1). The class of 3−closed trees and the number of elements of the reduced Gröbner basis of J T for a 3−closed labeling is also studied by means of the bipartite graph G * attached to a simple graph G corresponding to the generators of J G (see Definition 3.4, Theorem 3.5 and Corollary 3.6).
In Section 4, we study a class of trees constructed from caterpillar trees. We characterize the minimal primary decomposition of this class of trees (see Theorem 4.2). Also, we show that they are 3−closed. For some other trees constructed by caterpillar trees we show 3−closed property (see Theorem 4.3). To prove Theorem 4.3, we need an algorithm to give a 3−closed labeling to the vertices of a caterpillar tree such that 1 is assigned to an arbitrary vertex. This is provided in Algorithm 1 presented in the Appendix section.
In the following, we review some definitions and results from [9] which we need in the next sections.
Definition 1.1. Let G be a simple graph on [n], and let i and j be two vertices of G with i < j.
Given an admissible path π : i = i 0 , i 1 , . . . , i r = j from i to j, where i < j, we associate the monomial By [3, Chapter 2, Proposition 6], the reduced Gröbner basis of J G with respect to ≺ is unique. We have: where G 1 , . . . , G c(S) are the connected components of the induced subgraph on the vertices [n] \ S, andG ℓ is the complete graph on the vertices of G ℓ for all ℓ. Then is the number of the connected components of G. Equation (1) also shows that J G is a radical ideal. If G is a connected graph then P ∅ (G) = J Kn is a minimal prime ideal of J G . Note that if S is an arbitrary subset of [n] the prime ideal P S (G) is not necessary a minimal prime ideal of J G . The next lemma detects the minimal prime ideals of J G when G is a connected graph. Note that for S ⊂

m−closed graphs
In this section we study the reduced Gröbner basis of J G . As Theorem 1.2 shows the reduced Gröbner basis depends on the labeling of the vertices of G. We recall that a labeling of G is a bijection V (G) ≃ [n] = {1, . . . , n}, and given a labeling, we typically assume V (G) = [n].
The graph G is called closed with respect to the given labeling if J G has a quadratic Gröbner basis with respect to ≺. By [ Let G be a graph, we recall that the clique complex of G, denoted ∆(G), is the simplicial complex on [n] whose faces are the cliques of G. The graph G is closed if and only if there exists a labeling of G such that all facets of ∆(G) are intervals [a, b] ⊂ [n] (see [6,Theorem 2.2]). Closed graphs are studied in more details in [2,4].
Following the definition of closed graph we introduced m−closed graphs. By the above definition a closed graph is a 2−closed graph. the cycle C 4 (cycle with 4 vertices) is 3−closed and C 5 is 4−closed.
By Theorem 1.2, a graph G is m−closed if and only if, there exists a labeling for its vertices such that each admissible path in G has at most m vertices and in each labeling of the vertices, there exists an admissible path of length ℓ where ℓ ≥ m − 1.
We recall that a bridge is an edge whose removal from a graph increases the number of components. If e is a bridge of a connected graph G, and H 1 and H 2 are the connected components of G \ e, we write G \ e = H 1 ⊔ H 2 .
In the following we find some information about m−closed graphs. Proof. Part (i) and (ii) are followed from the definition of an admissible path and m−closed property.
For part (iii), assume that H 1 is an m−closed graph on [n 1 ], H 2 is an ℓ−closed graph on [n 2 ] and 1 is the label of the end points of e in each H i (i = 1, 2). We give a labeling to G by assigning to each vertex i of H 1 the new label n 1 − i + 1 and to each vertex i of H 2 the new label n 1 + i. So, by this labeling e = {n 1 , n 1 + 1}. It is easy to see that the graph A natural question to ask is that if the reduced Gröbner basis of J G has an element of degree m, can we conclude that it also has an element of degree ℓ for each 1 < ℓ < m. This is not true in general, as the following example shows: two edges {i, j} and {k, j} in E(G) with i < j, k < j and {i, k} / ∈ E(G). So j, i, ℓ or i, j, k is an admissible path of length 2. So, G has an element of degree 3.
Therefore, if G is an m−closed graph, in each labeling of its vertices, there exists an admissible path of length 2. But as we have seen in the above example, we can not extend Theorem 2.1 to check if a labeling is a 3−closed labeling or not.
We recall that if I is an ideal of R, the leading term ideal of I with respect to ≺ is the monomial ideal of R which is generated by ( where LT ≺ (f ) is the leading term of f with respect to ≺ . We write LT ≺ (I) for the leading term ideal of I.
If G is a graph, it is clear that for any arbitrary labeling of the vertices of G, It is well known by [9, Proposition 1.2], that a closed graph is chordal. In the following we are going to find a generalization of this necessary condition for m−closed property. For this we need the following theorem about cycles: n is even; n+1 2 + 1 n is odd. Proof. Let C n be the cycle on n vertices and m be as defined in the theorem. To show the result, we first prove that in any labeling of the vertices of C n , one can find an admissible path with at least m vertices.
In an arbitrary labeling of the vertices of C n , one of the following situation happens: , P 1 has ℓ + 1 vertices and P 2 has n − ℓ + 1 vertices.
In the case that n is even, if ℓ + 1 < n 2 + 1 and n − ℓ + 1 < n 2 + 1, then n + 2 < n + 2 which is a contradiction. So, one of the paths P 1 and P 2 has at least m vertices. Now assume that n is odd. Since d(i, i + 1) = ℓ, we have ℓ ≤ n − ℓ. Moreover, ℓ = n − ℓ if and only if n = 2ℓ which is a contradiction. So, ℓ < n − ℓ.
So in each labeling of the vertices of C n , we have an admissible path with at least m vertices. Now, if we find a labeling of the vertices of C n such that each admissible path has at most m vertices, the conclusion follows.
Suppose that: If n is even, we do as follows: label v 1 as 1, If n is odd, we do as follows: For each 1 ≤ i < m, label v i as 2i − 1 and for each m ≤ i ≤ n, label v i as 2(i − m + 1). By this labeling, for each i, d(i, i + 1) = m − 2 and for each i there is a unique admissible path with m vertices between i and i + 1 . Now assume that P : j 1 , . . . , j t (t > m) is an admissible path in C n . So, j t > j 1 + 1.
If n is odd, by the fact that d(i, i + 1) = m − 2 for each i, we conclude j 1 + 1 ∈ V (P ) which is a contradiction.
The next corollaries are the generalization of the fact that a closed graph is chordal. These results are immediate consequences of Proposition 2.3 and Theorem 2.5. (1) G is weakly closed.
(2) There exists a labeling which satisfies the following condition: for alli, j such that {i, j} ∈ E(G) and j > i + 1, the following assertion holds: for all i < k < j, {i, k} ∈ E(G) or {k, j} ∈ E(G).
In the following we relate the m−closed graphs to weakly closed graphs.
Theorem 2.9. Let G be a weakly closed graph. Then G is m−closed for some m ≤ 4.
Proof. Suppose that G is a weakly closed graph on [n]. Then by Theorem 2.8, for all i, j such that {i, j} ∈ E(G) and j > i + 1, the following assertion holds: for all i < k < j, {i, k} ∈ E(G) or {k, j} ∈ E(G). We prove that each admissible path of G has at most 4 vertices. Assume to the contrary that there exists an admissible path P : i = i 1 , i 2 , . . . , i m−1 , i m = j with m ≥ 5 vertices. Note that i < j. If i 2 > j, then i < j < i 2 and {i, i 2 } ∈ E(G). So {i, j} ∈ E(G) or {i 2 , j} ∈ E(G) which is a contradiction. If i m−1 < i, then i m−1 < i < j and {i m−1 , j} ∈ E(G). So {i m−1 , i} ∈ E(G) or {i, j} ∈ E(G). Again, it is a contradiction. Therefore i 2 < j and i m−1 > i. Since P is an admissible path, we have i 2 < i and i m−1 > j.
Let t = min{r |2 < r ≤ m − 1, i r > j}. So i t−1 < i < j < i t and {i t−1 , i t } ∈ E(G). If t = 3, then {i 2 , j} ∈ E(G) or {j, i 3 } ∈ E(G) which is impossible because m ≥ 5 and P is an admissible path. If t > 3, then {i t−1 , i} ∈ E(G) or {i, i t } ∈ E(G). This case also is impossible since P is an admissible path.
So, in any case we get a contradiction. Thus m ≤ 4 and the result follows.
Note that the converse of Theorem 2.9 is not true since C 5 is 4−closed and not weakly closed.

3−closed trees
In the following we are going to characterize 3−closed trees. Let G be a simple graph on the vertex set [n] and G has no element of degree more than 3, then d(i, i + 1) ≤ 2 for each i. But the converse is not true in general. For example, let C be the cycle on the vertex set [n] and with the edge set {{1, 3}, {3, 4}, {2, 4}, {2, 5}, {1, 5}}. Then for each i, d(i, i + 1) ≤ 2 but C is 4−closed.
We recall that by [9, Corollary 1.3], a tree is a closed graph if and only if it is a path. Next result shows that a 3−closed labeling for a tree T is a labeling in which d(i, i + 1) ≤ 2 for each i.  Case c: There exists j ∈ [n] \ {i 2 , . . . , i m } such that i 1 + 1, j, i 1 is a path. In this case, consider the path P ′ : i 1 +1, j, i 1 , i 2 , . . . , i m . Since ℓ(P ′ ) = m+1, by our choice of m, P ′ is not an admissible path. So, i 1 < i 1 + 1 < j < i m . It is easy to see that P ′′ : j, i 1 , i 2 , . . . , i m is an admissible path of length m which is again a contradiction by our choice of m.   (Figure 1).
Again, by (2) and (3) we can easily see that i7 Figure 1 and i 16 = i 10 − 1 or i 16 = i 10 + 2, and i 16 = i 13 − 1 or i 16 = i 13 + 2. So, i 7 = i 10 or i 7 = i 13 or i 10 = i 13 which is a contradiction. Note that if T is a tree, then T * is also a tree. In the following, we give a characterization of 3−closed trees by means of Definition 3.4. ( (2) for all i ∈ {1, . . . , n − 1} one of the following conditions holds: Proof. If T is a 3−closed graph on [n], then, by Theorem 3.1, d(i, i + 1) ≤ 2, ∀1 ≤ i < n. So T * satisfies condition 2. Since |E(T )| = |E(T * )| = n − 1, the conclusion follows from the fact that T = (T * ) * .
Conversely, if H satisfies condition 1 then H * is defined and is a graph on [n]. By condition 2, in H * , d(i, i+1) ≤ 2 for each i and moreover H * is connected. Now since |E(H * )| = n−1 = |V (H * )|−1, H * is a tree. So, by Theorem 3.1, H * is a 3−closed tree.
In the next corollary, we find the number of elements of the reduced Gröbner basis of a 3−closed tree. Moreover, by [14] the graded Betti numbers of J T is obtained from the graded Betti numbers of LT ≺ (J T ) by a sequence of consecutive cancelations. So β 03 (LT ≺ (J T )) = β 13 (LT ≺ (J T )) = β 13 (I(T * )) and the conclusion follows.
We remark that if G is an arbitrary 3−closed graph, for a 3−closed labeling, the same argument as the proof of Corollary 3.6 shows that |G| = |E(G)| + β 13 (I(G * )) − 2K 3 (G).

BINOMIAL EDGE IDEALS OF CATERPILLAR TREES
In this section, we study the binomial edge ideals of caterpillar trees and some trees constructed from this kind of trees. First we recall its definition.
Definition 4.1. A caterpillar tree is a tree T with the property that it contains a path P such that any vertex of T is either a vertex of P or it is adjacent to a vertex of P . Note that the path P in the definition of a caterpillar tree is a longest induced path of T and we call it the central path of T . Figure 2 is an example of a caterpillar tree with the central path P : v 1 , v 2 , . . . , v 7 .
Caterpillar trees were first studied by Harary and Schwenk [8]. These graphs have some applications in chemistry and physics [5].
Let T be a caterpillar tree and ℓ be the length of its longest induced path. By [6, Theorem 1.1] depth(R/J T ) = |V (T )| + 1 and by [1, Theorem 4.1] reg(R/J T ) = ℓ. In the following we describe the minimal primary decomposition of J T . We recall that since J T is a radical ideal, to know the minimal primary decomposition of J T , it is enough to characterize its minimal prime ideals. . . , v l } where 1 < i 1 < · · · < i k < l satisfy the following conditions: Proof. We prove that each prime ideal corresponding to a set S, where S is satisfying in the mentioned conditions, is a minimal prime ideal by induction on the number of vertices in the set S. For k = 1 the statement is obvious. Now assume theorem is true for each S with |S| = m and S ′ = {v i1 , . . . , v im+1 } has the mentioned conditions. If S = {v i1 , . . . , v im }, by induction hypothesis, P S (T ) is a minimal prime ideal of J T . Let d = deg(v im+1 ) and d ′ = deg(v im ).
Depending on d(v im , v im+1 ), we distinguish the following cases: In this case it is easy to see that c(S ′ ) = c(S) + d − 1 and for all It is obvious that in all of the above situations, c(S ′ \ {v ij }) < c(S ′ ) for all j ∈ {1, · · · , m + 1}. For example, if T is the caterpillar tree described in Figure 2, then by Theorem 4.2, it is easy to find all minimal prime ideals of J T and see that dim(R/J T ) = 19.
Finally, we prove that caterpillar trees and some trees constructed by caterpillar trees are 3−closed. (b) Let T = T 1 ∪ B ∪ T 2 where T 1 and T 2 are two caterpillar trees and B is a bridge between T 1 and T 2 , and the endpoints of B are chosen from the vertices of the central paths of T 1 and T 2 respectively. Then T is 3−closed.
More generally, (c) Let T be a tree and T = T 1 ∪ B ∪ T 2 where T 1 , T 2 and B are caterpillar trees, and the endpoints of the central path of B are chosen from the vertices of T 1 and T 2 respectively. Then T is 3−closed.
Remark 5.1. If one wants to give a 3−closed labeling to a caterpillar tree T in such a way that 1 is assigned to v ∈ N ′ T (v i0 ) for some 1 < i 0 < ℓ, it is enough to label v as 1, v i0 as 2, set N ′ T (v i0 ) = N ′ T (v i0 ) \ {v} and start with t := 3 instead of t := 2. Moreover, if one wants to give a 3−closed labeling to a caterpillar tree T in such a way that n = |V (T )| is assigned to an arbitrary vertex v, it is enough to apply Algorithm 1, by labeling v as 1 and at the end changing the label i of each vertex to n − i + 1.
Example 5.2. Here, we give an example of a labeled caterpillar tree using Algorithm 1. Note that 12 is the label of v 1 , 11 is the label of v 2 , 1 is the label of v 3 and so on . Finally, we give an example of a 3−closed tree described in Theorem 4.3(part b). Note that the labeling is given by Algorithm 1, and Proposition 2.3.