# Edge rings of bipartite graphs with linear resolutions

## TL;DR

This paper investigates the algebraic structure of edge rings of bipartite graphs with linear resolutions, confirming a conjecture that such rings are hypersurfaces for higher linearity degrees.

## Contribution

It proves the conjecture that edge rings of bipartite graphs with q-linear resolutions are hypersurfaces for all q ≥ 3.

## Key findings

- Edge rings of bipartite graphs with q-linear resolutions are hypersurfaces for q ≥ 3
- Confirmed the conjecture for bipartite graphs in the case q ≥ 3
- Extended understanding of the algebraic properties of graph edge rings

## Abstract

Ohsugi and Hibi characterized the edge ring of a finite connected simple graph with a $2$-linear resolution. On the other hand, Hibi, Matsuda and the author conjectured that the edge ring of a finite connected simple graph with a $q$-linear resolution, where $q \geq 3$, is a hypersurface and proved the case $q=3$. In the present paper, we solve this conjecture for the case of finite connected simple bipartite graphs.