A special case of Postnikov-Shapiro conjecture

TL;DR

This paper proves the conjecture that the graded Betti numbers of two related ideals are equal for a specific class of graphs when the number of vertices is three, confirming a special case of the Postnikov-Shapiro conjecture.

Contribution

It establishes the equality of graded Betti numbers for the ideals associated with a class of complete graphs in the case of four vertices, a previously unproven special case.

Findings

Confirmed the conjecture for n=3 (four vertices).

Validated the equality of Betti numbers for the specific graph class.

Supported the broader Postnikov-Shapiro conjecture in this case.

Abstract

For a graph $G$, Postnikov-Shapiro \cite{PS04} construct two ideals $I_G$ and $J_G.$ $I_G$ is a monomial ideal and $J_G$ is generated by powers of linear forms. They proved the equality of their Hilbert series and conjectured that the graded Betti numbers are equal. When $G=K_{n+1}^{l,k}$ is the complete graph on the vertices $\{0,1,\cdots, n\}$ with the edges $e_{i, j},$ $i, j\neq 0,$ of multiplicity $k$ and the edges $e_{0, i}$ of multiplicity $l,$ for two non-negative integers $k$ and $l,$ they gave an explicit formula for the graded Betti numbers of $I_G,$ which are conjecturally the same for $J_G.$ We prove this conjecture in the case $n=3,$ which was also conjectured by Schenck \cite{S04}.