# Koszul binomial edge ideals of pairs of graphs

## TL;DR

This paper investigates when the algebra defined by the binomial edge ideal of a pair of graphs is Koszul, establishing equivalences involving quadratic Gr"obner bases and linear quotients.

## Contribution

It characterizes the Koszul property of the algebra in terms of Gr"obner bases and linear quotients for binomial edge ideals of graph pairs.

## Key findings

- Koszul property equivalent to quadratic Gr"obner basis
- Linear quotients characterize Koszulness
- Provides criteria for algebraic properties based on graph pairs

## Abstract

We study the Koszul property of a standard graded $K$-algebra $R$ defined by the binomial edge ideal of a pair of graphs $(G_1,G_2)$. We show that the following statements are equivalent: (i) $R$ is Koszul; (ii) the defining ideal $J_{G_1,G_2}$ of $R$ has a quadratic Gr\"obner basis; (iii) the graded maximal ideal of $R$ has linear quotients with respect to a suitable order of its generators