Solution to a conjecture on edge rings with 2-linear resolutions

TL;DR

This paper investigates a conjecture relating the projective dimension of edge rings with 2-linear resolutions to the maximum degree of vertices in a graph, providing characterizations for when the conjecture holds.

Contribution

It characterizes graphs for which the Eliahou-Villarreal conjecture on edge rings with 2-linear resolutions is true, using Stanley-Reisner ring interpretation.

Findings

Identifies conditions under which the conjecture holds.

Provides a characterization of graphs satisfying the conjecture.

Clarifies the relationship between projective dimension and vertex degree.

Abstract

For a graph $G=(V,E)$ the edge ring $k[G]$ is $k[x_1,\ldots,x_n]/I(G)$, where $n=|V|$ and $I(G)$ is generated by $\{ x_ix_j;\{ i,j\}\in E\}$. The conjecture we treat is the following. If $k[G]$ has a 2-linear resolution, then the projective dimension of $K[G]$, pd$(k[G])$, equals the maximal degree of a vertex in $G$. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia, and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq. That there are counterexamples was noted already by Moradi and Kiani. By interpreting $k[G]$ as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds.