Invited speaker (23/05-24/05) to the Department of Mathematics, University of Crete/Heraklion in the framework of the 10th Greek Algebra and Number Theory Conference, Conference's webpage.

Title: Toric ideals of graphs minimally generated by a Grobner basis

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

Abstract: The problem of describing families of ideals minimally generated by either one or all of its Grobner bases is a central topic in commutative algebra. This work tackles this problem in the context of toric ideals of graphs. We call a graph G an MG-graph if its toric ideal I_G is minimally generated by a Grobner basis, while we say that G is an UMG-graph if every reduced Grobner basis of I_G is a minimal generating set. We prove that G is an UMG-graph if and only if I_G is a generalized robust ideal, i.e. ideal whose universal Grobner basis and universal Markov basis coincide. We observe that the class of MG-graphs is not closed under taking subgraphs, and we prove that it is hereditary (i.e., closed undertaking induced subgraphs). Also, we describe two families of bipartite MG-graphs: ring graphs and graphs whose induced cycles have the same length. The latter extends a result of Ohsugi and Hibi, which corresponds to graphs whose induced cycles have all length 4.