Binomial edge ideals of crown graphs

TL;DR

This paper studies algebraic properties of binomial edge ideals in specific graphs, establishing new results on their projective dimension, Vasconcelos number, and proving a related conjecture for cycles and crown graphs.

Contribution

It introduces new characterizations of binomial edge ideals in crown graphs and proves a conjecture regarding their Vasconcelos number for cycles.

Findings

Projective dimension matches big height in certain graphs

Vasconcelos number computed for cycles and crown graphs

Proof of a conjecture related to Vasconcelos number in cycles

Abstract

In this article, we explore the class of graphs for which the projective dimension of the quotient of the binomial edge ideals matches the big height of that ideal. Additionally, we investigate the Vasconcelos number of binomial edge ideals for cycles and crown graphs. We also provide proof for [Conjecture 4.13, 3], which is related to the Vasconcelos number of binomial edge ideals for cycles.