Invited research visit (04/05-08/05) to the Jussieu Institute of Mathematics  - PRG, Sorbonne Université and Université Paris Cité, CNRS, France.

I will also give a talk in the framework of the working group "Combinatorics, Arithmetic and Geometry". My talk will take place on 06 of May of 2025. Seminar's webpage

Title: Toric ideals of graphs minimally generated by a Gröbner basis.

(joint work with Ignacio García-Marco and Irene Marquez-Corbella)

Abstract:  Describing families of ideals that are minimally generated by at least one, or by all, of their reduced Gröbner bases is a central topic in commutative algebra.  In this talk, we address this problem in the context of toric ideals of graphs. We say that a graph G is an MG-graph if its toric ideal I_G is minimally generated by some Gröbner basis, and a UMG-graph if every reduced Gröbner basis of I_G forms a minimal generating set. We prove that a graph G is a UMG-graph if and only if its toric ideal I_G is a generalized robust ideal (that is, its universal  Gröbner basis coincides with its universal Markov basis). Although the class of MG-graphs is not closed under taking subgraphs, we prove that it is hereditary, that is, closed under taking induced subgraphs. In addition, we describe two families of bipartite MG-graphs: ring graphs (which correspond to complete intersection toric ideals, as shown by Gitler, Reyes, and Villarreal) and graphs in which all chordless cycles have the same length. The latter extends a result of Ohsugi and Hibi, which corresponds to graphs whose chordless cycles are all of length 4.