On the extremal Betti numbers of binomial edge ideals of block graphs (Q1753032)

Let \(G\) be a finite simple graph with vertex set \([n]\) and edge set \(E(G)\). Also, let \(S=\mathbb{K}[x_1,\dots,x_n, y_1, \ldots, y_n]\) be the polynomial ring over a field \(\mathbb{K}\). Then the binomial edge ideal of \(G\) in \(S\), denoted by \(J_G\), is generated by binomials \(f_{ij} = x_iy_j-x_jy_i\), where \(i < j\) and \(\{i, j\}\in E(G)\). Also, one could see this ideal as an ideal generated by a collection of \(2\)-minors of a \((2\times n)\)-matrix whose entries are all indeterminate. A graph \(G\) is called a block graph if each block of \(G\) is a clique. A vertex \(v\) of \(G\) is a free vertex if it belong to exactly one maximal clique of \(G\). Otherwise, \(v\) is an inner vertex. The number of free vertices and inner vertices of \(G\) are denoted by \(f(G)\) and \(i(G)\), respectively. A graph \(G\) is decomposable, if there exist two subgraphs \(G_1\) and \(G_2\) of \(G\), and a decomposition \(G=G_1\cup G_2\) with \(\{v\}=V(G_1)\cap V(G_2)\), where \(v\) is a free vertex of \(G_1\) and \(G_2\). If \(G\) is not decomposable, then it is said to be indecomposable. Let \(M\) be a finitely graded \(S\)-module. A graded Betti number \(\beta_{i,i+j}(M)\neq 0\) of \(M\) is called extremal, if \(\beta_{k,k+l}(M)=0\) for all pairs \((k,l)\neq (i,j)\) with \(k\geq i\) and \(l\geq j\). The first main result of the paper under review, states that if \(G\) is an indecomposable block graph with \(n\) vertices and if \(<\) is the lexicographic order induced by \(x_1 > x_2 > \cdots > x_n > y_1 > y_2 > \cdots > y_n\), then \(\beta_{n-1,n-1+i(G)+1}(S/J_G)\) and \(\beta_{n-1,n-1+i(G)+1}(S/\text{in}_<(J_G))\) are extremal Betti numbers, and \(\beta_{n-1,n-1+i(G)+1}(S/\text{in}_<(J_G)) = \beta_{n-1,n-1+i(G)+1}(S/J_G) = f(G) - 1\). This result partially confirms the conjecture of J. Herzog, which asserts that if the initial ideal of a graded ideal \(I\) is a squarefree monomial ideal, then the extremal Betti numbers of \(I\) and \(\text{in}_<(I)\) coincide in their positions and values. The authors use the above mentioned result to show that for any indecomposable block graph \(G\), we have \(\text{reg}(J_G)\geq i(G)\), with equality if and only if \(S/J_G\) has exactly one extremal Betti number, namely \(\beta_{n-1,n-1+i(G)+1}(S/J_G)\). As the second main result, the authors 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 they also give an explicit description of the block graphs with precisely one extremal Betti number.