Binomial edge ideals and bounds for their regularity

TL;DR

This paper investigates bounds on the regularity of binomial edge ideals for various classes of graphs, proving new bounds for quasi-block and semi-block graphs, and confirming conjectures for chordal graphs and Jahangir graphs.

Contribution

It introduces new upper bounds for the regularity of binomial edge ideals in specific graph classes and provides proofs for existing conjectures in the field.

Findings

Bound regularity by c(G)+1 for quasi-block and semi-block graphs

Confirmed Saeedi Madani-Kiani conjecture for chordal graphs

Determined regularity for binomial edge ideals of Jahangir graphs

Abstract

Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in $S = K[x_1, \ldots, x_n, y_1,\ldots, y_n].$ We prove that the Castelnuovo-Mumford regularity of $J_G$ is bounded above by $c(G)+1$ when $G$ is a quasi-block graph or semi-block graph. We give another proof of Saeedi Madani-Kiani regularity upper bound conjecture for chordal graphs. We obtain the regularity of binomial edge ideals of Jahangir graphs. Later, we establish a sufficient condition for Hibi-Matsuda conjecture to be true.