# Green-Lazarsfeld Condition for Toric Edge Ideals of Bipartite Graphs

## TL;DR

This paper characterizes the higher syzygies of toric edge ideals of bipartite graphs, linking algebraic properties to combinatorial structures, and extends the Green-Lazarsfeld conditions beyond quadrics.

## Contribution

It provides combinatorial criteria for all Green-Lazarsfeld conditions  N_p  for bipartite graph toric edge ideals, including a characterization of linear presentation.

## Key findings

- I_G is linearly presented iff the bipartite complement of G is a tree of diameter at most 3.
- Provides combinatorial descriptions for Green-Lazarsfeld conditions  N_p for all p  1.
- Establishes criteria for polyomino ideals to satisfy Green-Lazarsfeld conditions.

## Abstract

Previously, Ohsugi and Hibi gave a combinatorial description of bipartite graphs $G$ whose toric edge ideal $I_G$ is generated by quadrics, showing that every cycle of $G$ of length at least $6$ must have a chord. This corresponds to the Green-Lazarsfeld condition $\mathbf{N}_1$. In this paper, we investigate the higher syzygies of $I_G$ and give combinatorial descriptions of the Green-Lazarsfeld conditions $\mathbf{N}_p$ of toric edge ideals of bipartite graphs for all $p \ge 1$. In particular, we show that $I_G$ is linearly presented (i.e. satisfies condition $\mathbf{N}_2$) if and only if the bipartite complement of $G$ is a tree of diameter at most $3$. We also investigate the regularity of linearly presented toric edge ideals and give criteria for polyomino ideals to satisfy the Green-Lazarsfeld conditions.