On the Betti numbers of edge ideal of skew Ferrers graphs

TL;DR

This paper proves a specific equality for Betti numbers of edge ideals of skew Ferrers graphs, confirming a conjecture for the last column of Betti tables and providing formulas for extremal Betti numbers in certain cases.

Contribution

It establishes a key equality for Betti numbers of skew Ferrers graphs and confirms a conjecture, also deriving explicit formulas for extremal Betti numbers of binomial edge ideals.

Findings

Proves $eta_p(I(G)) = eta_{p,p+r}(I(G))$ for skew Ferrers graphs.

Confirms Ene, Herzog, and Hibi's conjecture for the last Betti table column.

Provides explicit formulas for extremal Betti numbers of some closed graphs.

Abstract

We prove that $\beta_p(I(G)) = \beta_{p,p+r}(I(G))$ for skew Ferrers graph $G$, where $p:=\pd(I(G))$ and $r:=\reg(I(G))$. As a consequence, we confirm that Ene, Herzog and Hibi's conjecture is true for the Betti numbers in the last columm of Betti table. We also give an explicit formula for the unique extremal Betti number of binomial edge ideal for some closed graphs.