Edge rings with $q$-linear resolutions

TL;DR

This paper classifies connected simple graphs whose edge rings have q-linear resolutions, proving that for q ≥ 3, such rings are hypersurfaces, confirming a conjecture by Hibi, Matsuda, and Tsuchiya.

Contribution

It provides a complete classification of graphs with q-linear edge ring resolutions and confirms a conjecture regarding their hypersurface nature for q ≥ 3.

Findings

Edge rings of graphs with q-linear resolutions are hypersurfaces for q ≥ 3.

Complete classification of graphs with q-linear edge ring resolutions.

Confirmation of a conjecture by Hibi, Matsuda, and Tsuchiya.

Abstract

In the present paper, we give a complete classification of connected simple graphs whose edge rings have a $q$-linear resolution with $q \geq 2$. In particular, we show that the edge ring of a finite connected simple graph with a $q$-linear resolution, where $q \geq 3$, is a hypersurface, which was conjectured by Hibi, Matsuda, and Tsuchiya.