On a regularity-conjecture of generalized binomial edge ideals

TL;DR

This paper proves a conjecture on the Castelnuovo-Mumford regularity of generalized binomial edge ideals, providing a tighter combinatorial upper bound and demonstrating its tightness with an infinite class of graphs.

Contribution

It establishes a new, improved upper bound for the regularity of generalized binomial edge ideals and confirms its optimality through infinite graph classes.

Findings

Proved the upper bound conjecture for regularity.

Provided a tighter combinatorial upper bound.

Showed the bound is tight with infinite graph classes.

Abstract

In this paper, we prove the upper bound conjecture proposed by Saeedi Madani \& Kiani on the Castelnuovo-Mumford regularity of generalized binomial edge ideals. We give a combinatorial upper bound of regularity for generalized binomial edge ideals, which is better than the bound claimed in that conjecture. Also, we show that the bound is tight by providing an infinite class of graphs.