\(d\)-sequence edge binomials, and regularity of powers of binomial edge ideals of trees (Q6607127)

Let \(S = k[x_1,\ldots,x_n,y_1,\ldots,y_n]\) denote the polynomial ring in \(2n\) variables over the field \(k\). Let \(G\) denote a finite graph on \(n\) vertices. Let \(J_G \subset S\) denote the edge ideal generated by the edge polynomials \(f_{ij} = x_iy_j-x_jy_i\) for all edges \(\{i,j\}\) in \(G\). The authors' main results are: Let \(G\) denote a tree of \(n\) vertices. Then they describe exactly the degree sequence of \(G\) whenever the edge binomials form a \(d\)-sequence. Then \(J_G\) has a linear resolution answering partially a conjceture posed by \textit{A. V. Jayanthan} et al. [J. Pure Appl. Algebra 225, No. 6, Article ID 106628, 20 p. (2021; Zbl 1460.13041)]. Moreover, they study the regularity of powers of binomial edge ideals of trees generated by \(d\)-sequences in terms of the number of internal vertices of \(G\).