On vanishing patterns in $j$-strands of edge ideals

TL;DR

This paper investigates vanishing patterns in the Betti tables of edge ideals, proving connectivity of certain strands and applying these results to confirm the subadditivity conjecture for specific cases.

Contribution

It establishes the connectivity of the 3-strand in Betti tables and uses this to prove the subadditivity conjecture for edge ideals when b=2,3.

Findings

The 3-strand in Betti tables is connected.

Examples show j-strand may not be connected for j>3.

Subadditivity conjecture holds for b=2,3 in edge ideals.

Abstract

We consider two problems regarding vanishing patterns in the Betti table of edge ideals $I$ in polynomial algebra $S$. First, we show that the $j$-strand is connected if $j=3$ (for $j=2$ this is easy and known), and give examples where the $j$-strand is not connected for any $j>3$. Next, we apply our result on strand connectivity to establish the subadditivity conjecture for edge ideals, $t_{a+b}\leq t_a+t_b$, in case $b=2,3$ (the case $b=1$ is known). Here $t_i$ stands for the maximal shifts in the minimal free $S$-resolution of $S/I$