On Conjecture of Binomial Edge Ideals of Linear Type

TL;DR

This paper investigates the conjecture that binomial edge ideals of certain graphs are of linear type, confirming it for trees but finding counterexamples among unicyclic graphs.

Contribution

It proves the conjecture for trees and shows that it does not hold for all unicyclic graphs, providing new insights into the structure of binomial edge ideals.

Findings

Confirmed the conjecture for trees

Disproved the conjecture for some unicyclic graphs

Enhanced understanding of binomial edge ideals in graph theory

Abstract

An ideal $I$ of a commutative ring $R$ is said to be of linear type when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if $G$ is a tree or a unicyclic graph, then the binomial edge ideal of $G$ is of linear type. Our investigation validates this conjecture for trees. However, our study reveals that not all unicyclic graphs adhere to this conjecture.