# Binomial edge ideals of cographs

## TL;DR

This paper investigates the algebraic properties of binomial edge ideals in cographs, establishing bounds on their regularity and providing counterexamples to existing conjectures.

## Contribution

It determines the Castelnuovo-Mumford regularity of binomial edge ideals for cographs and constructs counterexamples to a conjecture by Hibi and Matsuda.

## Key findings

- Maximum regularity of cographs grows as 2n/3
- Regularity bounds are related to graph invariants
- Counterexamples to Hibi and Matsuda's conjecture are provided

## Abstract

We determine the Castelnuovo-Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.