Ideals of Graph Homomorphisms

TL;DR

This paper introduces a new class of ideals called ideals of graph homomorphisms, exploring their algebraic properties and Grobner bases, with applications to optimization problems like stable set polytopes.

Contribution

It generalizes previous results by framing graph ideals functorially and analyzing how graph topology affects algebraic properties, using toric fiber products.

Findings

Explicit Grobner bases for several classes of graph homomorphism ideals

Connection between graph topology and algebraic properties of the ideals

Application to stable set polytopes in optimization

Abstract

In combinatorial commutative algebra and algebraic statistics many toric ideals are constructed from graphs. Keeping the categorical structure of graphs in mind we give previous results a more functorial context and generalize them by introducing the ideals of graph homomorphisms. For this new class of ideals we investigate how the topology of the graphs influence the algebraic properties. We describe explicit Grobner bases for several classes, generalizing results by Hibi, Sturmfels and Sullivant. One of our main tools is the toric fiber product, and we employ results by Engstrom, Kahle and Sullivant. The lattice polytopes defined by our ideals include important classes in optimization theory, as the stable set polytopes.