The binomial edge ideal of a pair of graphs

TL;DR

This paper introduces a new class of ideals generated by 2-minors of matrices indexed by pairs of graphs, generalizing binomial edge ideals, and explores their algebraic properties including prime ideals, nilpotency, and Cohen–Macaulayness.

Contribution

It defines a novel class of ideals linked to pairs of graphs and analyzes their algebraic structure, including prime decomposition and special cases for Gr"obner bases.

Findings

Determined minimal prime ideals of the new class of ideals.

Provided lower bounds for their degree of nilpotency.

Characterized conditions for Cohen–Macaulayness in special cases.

Abstract

We introduce a class of ideals generated by a set of 2-minors of $m\times n$-matrix of indeterminates indexed by a pair of graphs. This class of ideals is a natural common generalization of binomial edge ideals and ideals generated by adjacent minors. We determine the minimal prime ideals of such ideals and give a lower bound for their degree of nilpotency. In some special cases we compute their Gr\"obner basis and characterize unmixedness and Cohen--Macaulayness.