On the extremal Betti numbers of binomial edge ideals of block graphs

We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.


Introduction
Let K be a field and I a graded ideal in the polynomial ring S = K[x 1 , . . . , x n ]. The most important invariants of I, which are provided by its graded finite free resolution, are the regularity and the projective dimension of I. In general these invariants are hard to compute. One strategy to bound them is to consider for some monomial order the initial ideal in < (I) of I. It is known that for the graded Betti numbers one has β i,j (I) ≤ β i,j (in < (I)). This fact implies in particular that reg(I) ≤ reg(in < (I)), and proj dim(I) ≤ proj dim(in < (I)). In general however these inequalities may be strict. On the other hand, it is known that if I is the defining binomial ideal of a toric ring, then proj dim(I) = proj dim in < (I), provided in < (I) is a squarefree monomial ideal. This is a consequence of a theorem of Sturmfels [14]. The first author of this paper conjectures that whenever the initial ideal of a graded ideal I ⊂ S is a squarefree monomial ideal, then the extremal Betti numbers of I and in < (I) coincide in their positions and values. This conjecture implies that reg(I) = reg(in < (I)) and proj dim(I) = proj dim(in < (I)) for any ideal I whose initial ideal is squarefree.
An interesting class of binomial ideals having the property that all of its initial ideals are squarefree monomial ideals are the so-called binomial edge ideals, see [9], [5], [1]. Thus it is natural to test the above conjectures for binomial edge ideals. A positive answer to this conjecture was given in [7] for Cohen-Macaulay binomial edge ideals of PI graphs (proper interval graphs). In that case actually all the graded Betti numbers of the binomial edge ideal and its initial ideal coincide. It is an open question wether this happens to be true for any binomial edge ideal of a PI graph. Recently this has been confirmed to be true, if the PI graph consists of at most two cliques [2]. In general the graded Betti numbers are known only for very special classes of graphs including cycles [18].
Let J G denote the binomial edge ideal of a graph G. The first result showing that reg(J G ) = reg(in < (J G )) without computing all graded Betti numbers was obtained for PI graphs by Ene and Zarojanu [8]. Later Chaudhry, Dokuyucu and Irfan [3] showed that proj dim(J G ) = proj dim(in < (J G )) for any block graph G, and reg(J G ) = reg(in < (J G )) for a special class of block graphs. Roughly speaking, block graphs are trees whose edges are replaced by cliques. The blocks of a graph are the biconnected components of the graph, which for a block graph are all cliques. In particular trees are block graphs. It is still an open problem to determine the regularity of the binomial edge ideal for block graphs (and even for trees) in terms of the combinatorics of the graph. However, strong lower and upper bounds for the regularity of edge ideals are known by Matsuda and Murai [11] and Kiani and Saeedi Madani [15]. Furthermore, Kiani and Saeedi Madani characterized all graphs are whose binomial edge ideal have regularity 2 and regularity 3, see [12] and [13].
In this note we determine the position and value of one of the distinguished extremal Betti number of the binomial edge ideal of a block graph. Let M be a finitely graded S-module. Recall that a graded Betti number In order to describe our result in detail, we introduce the following concepts. Let G be finite simple graph. Let V (G) be the vertex set and E(G) the edge set of G. The clique degree of a vertex v ∈ V (G), denoted cdeg(v), is the number of cliques to which it belongs. For a tree the clique degree of a vertex is just the ordinary degree. A vertex v ∈ V (G) is a called a free vertex, if cdeg(v) = 1 and an inner vertex if cdeg(v) > 1. Suppose v ∈ V (G) is a vertex of clique degree 2. Then G can be decomposed as a union of subgraphs G 1 ∪ G 2 with V (G 1 ) ∩ V (G 2 ) = {v} and where v is a free vertex of G 1 and G 2 . If this is the case, we say that G is decomposable. In Proposition 1.3 we show that if G decomposable with G = G 1 ∪G 2 , then the graded Poincaré series of G is just the product of the graded Poincaré series of S/J G 1 and S/J G 2 . This result, with a simplified proof, generalizes a theorem of the second author which he obtained in a joint paper with Rauf [17]. As a consequence one obtains that the position and value of the distinguished extremal Betti numbers of S/J G are obtained by adding the positions and multiplying the values of the corresponding distinguished extremal Betti numbers of S/J G 1 and S/J G 2 . The other extremal Betti numbers of S/J G are not obtained in this simple way from those of S/J G 1 and S/J G 2 . But the result shows that if we want to determine the distinguished extremal Betti numbers of S/J G for a graph G (which also give us the regularity and projective dimension of S/J G ), it suffices to assume that G is indecomposable.
Let f (G) be the number of free vertices and i(G) the number of inner vertices of G. In Theorem 2.2 we show: let G be an indecomposable block graph with n vertices. Furthermore let < be the lexicographic order induced by x 1 > x 2 > . . . > x n > y 1 > y 2 > · · · > y n . Then β n−1,n−1+i(G)+1 (S/J G ) and β n−1,n−1+i(G)+1 (S/ in < (J G )) are extremal Betti numbers, and β n−1,n−1+i( . It also implies that reg(J G ) = i(G) if and only if S/J G has exactly one extremal Betti number, namely the Betti number β n−1,n−1+i(G)+1 (S/J G ). In Theorem 2.4 we classify all block graphs with the property that they admit precisely one extremal Betti number, by listing the forbidden induced subgraphs (which are 4 in total), and we also give an explicit description of the block graphs with precisely one extremal Betti number. Carla Mascia informed us that Jananthan et al in an yet unpublished paper and revised version of [10] obtained a related result for trees.
For indecomposable block graphs G the most challenging open problem is to obtain a combinatorial formula for the regularity of J G , or even better, a description of both distinguished extremal Betti numbers of S/J G .

Decomposable graphs and binomial edge ideals
Let G be a graph with vertex set V (G) = [n] and edge set E(G). Throughout this paper, unless otherwise stated, we will assume that G is connected. A Note that any graph has a unique decomposition (up to ordering) The following proposition generalizes a result due to Rinaldo and Rauf [17].

Proposition 1.3. Let G be a decomposable graph, and let
Proof. We may assume that V (G) = [n] and V (G 1 ) = [1, m] and V (G 2 ) = [m, n] . We claim that for the lexicographic order < induced by x 1 > x 2 > · · · > x n > y 1 > y 2 > · · · > y n , we have We recall the notion of admissible path, introduced in [9] in order to compute Gröbner bases of binomial edge ideals. A path π : (2) for each k = 1, . . . , r − 1 one has either i k < i or i k > j; (3) for any proper subset {j 1 , . . . , j s } of {i 1 , . . . , i r−1 }, the sequence i, j 1 , . . . , j s , j is not a path. Given an admissible path π : i = i 0 , i 1 , . . . , i r = j from i to j with i < j we associate the monomial u π = ( i k >j x i k )( i ℓ <i y i ℓ ). In [9] it is shown that in < (J G ) = (x i y j u π : π is an admissible path).
The claim follows by observing that the only admissible paths passing through the vertex m are the ones inducing the set of monomials We have the following cases to study We may assume that V (π) ⊂ V (G 1 ). Assume m is not an endpoint of π. Then π : i = i 0 , . . . , i r = j with m = i k , 0 < k < r. Since i k−1 and i k+1 belong to the maximal clique in G 1 containing m, it follows that {i k−1 , i k+1 } ∈ E(G 1 ) and condition (3) is not satisfied. Therefore π : i = i 0 , . . . , i r = m and x i y m u π is the corresponding monomial.
(b) In this case we observe that m is not an endpoint of the path π : i = i 0 , . . . , i r = j. Since i < m < j this path is not admissible by (2). Now the claim implies that Tor i (S/ in < (J G 1 ), S/ in < (J G 2 )) = 0 for i > 0. Therefore, we also have Tor i (S/J G 1 , S/J G 2 ) = 0 for i > 0. This yields the desired conclusion.

Extremal Betti numbers of Block graphs
Let G be a graph with vertex set V (G) and edge set E(G). A vertex of a graph is called a cutpoint if the removal of the vertex increases the number of connected components. A connected subgraph of G that has no cutpoint and is maximal with respect to this property is called a block.

Theorem 2.2.
Let G be an indecomposable block graph with n vertices. Furthermore let < be the lexicographic order induced by x 1 > x 2 > . . . > x n > y 1 > y 2 > · · · > y n . Then β n−1,n−1+i(G)+1 (S/J G ) and β n−1,n−1+i(G)+1 (S/ in < (J G )) are extremal Betti numbers of S/J G and S/ in < (J G ), respectively. Moreover, Proof. We prove the theorem by induction on i(G). If i(G) = 0, then G is a clique and J G is the ideal of 2-minors of the 2 × n matrix The desired conclusion follows by the Eagon-Northcott resolution [6]. Let us now assume that the above equation holds for i(G) > 0. Let C 1 , . . . , C t be the blocks of G and assume that C t is a leaf of G. Since i(G) > 0, it follows that t > 1. Let i be the vertex of C t of cdeg(i) > 1, and let G ′ be the graph which is obtained from G by replacing C t by the clique whose vertex set is the union of the vertices of the C i which have a non-trivial intersection with C t . Furthermore, let G ′′ be the graph which is obtained from G by removing the vertex i, and H be the graph obtained by removing the vertex i from G ′ .
Note that G ′ and H are indecomposable block graphs for which i(G ′ ) = i(H) = i(G) − 1.
The following exact sequence [7] is used for our induction step. By the proof of [7, Theorem 1.1] we know that proj dim S/J G = proj dim S/J G ′ = n − 1, proj dim S/((x i , y i ) + J H ) = n, and proj dim S/((x i , y i )+J G ′′ ) = n−q, where q+1 is the number of connected components of G ′′ . Since cdeg(i) ≥ 3 it follows that q ≥ 2. Therefore, Tor n−1 (S/((x i , y i ) + J G ′′ ), K) = 0, and hence for each j, the exact sequence (4) yields the long exact sequence where for any finitely generated graded S-module, T S k,l (M) stands for Tor S k,l (M, K). Note that Our induction hypothesis implies that (5) and (6) imply that T n−1,n−1+j (S/J G ) = 0 for j > i(G) + 1, and Next assume that i(G) > 0. As noted in [3], one also has the exact sequence . Therefore, by using the induction hypothesis, one deduces as before that ). This concludes the proof.
The next corollary is an immediate consequence of Proposition 1.3 and Theorem 2.2.

Corollary 2.3.
Let G be a block graph for which G = G 1 ∪ · · · ∪ G s is the decomposition of G into indecomposable graphs. Then each G i is a block graph, β n−1,n−1+i(G)+s (S/J G ) is an extremal Betti number of S/J G and In the following theorem we classify all block graphs which admit precisely one extremal Betti number. Suppose that G contains one of the induced subgraphs T 0 , T 1 , T 2 , T 3 . We will show that reg(S/J G ) > i(G) + 1. By Corollary 2.3 this is equivalent to saying that S/J G admits at least two extremal Betti numbers. To proceed in our proof we shall need the following result [11, Corollay 2.2] of Matsuda and Murai which says that for W ⊂ V (G), one has β ij (J G |W | ) ≤ β ij (J G )) for all i and j.
It can be checked by CoCoA that reg(S/J T j ) > i(T j ) + 1 for each T j . Now assume that G properly contains one of the T j as induced subgraph. Since G is connected, there exists a clique C of G and subgraph G ′ of G such that (1) G ′ contains one of the T j as induced subgraph, (ii) ⇒ (iii): Suppose condition (iii) is not satisfied. Let C 1 , . . . , C r with r ≥ 3 be maximal cliques of G |P that meet in the same cutpoint i. After a suitable relabeling of the cliques C i we may assume that one of the following cases occurs: (α) C 1 , C 2 , C 3 have cardinality ≥ 3; (β) C 1 , C 2 have cardinality ≥ 3, the others have cardinality 2; (γ) C 1 has cardinality ≥ 3, the others have cardinality 2; (δ) C 1 , . . . , C r have cardinality 2. In case (α) observe that G contains C 1 , C 2 and C 3 , too. But this contradicts the fact that G does not contain T 0 as an induced subgraph. Similarly in case (β), G contains C 1 , C 2 . Let C 3 = {i, j}. Since C 3 is an edge in G |P , it cannot be a leaf of G. Therefore, since G is indecomposable, there exist at least two maximal cliques in G for which j is a cut point. It follows that T 1 is an induced subgraph, a contradiction. (γ) and (δ) are discussed in a similar way.
(iii) ⇒ (i): We use induction on i(G). If i(G) = 0, then G is a clique and the assertion is obvious. Now let us assume that i(G) > 0. By (a) it is sufficient to prove that reg S/J G ≤ i(G) + 1. If i(G) = 0, then G is a clique and the assertion is obvious. We choose a leaf of G. Let j be the unique cut point of this leaf, and let G ′ , G ′′ and H be the subgraphs of G, as defined with respect to j in the proof of