Toric ideals of graphs minimally generated by a Grοbner basis

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 IG is minimally generated by a Grobner basis, while we say that G is an UMG-graph if every reduced Grobner basis of IG is a minimal generating set. We prove that G is an UMG-graph if and only if IG 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 under taking 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 (joint work with Ignacio Garcia-Marco and Irene Marquez-Corbella).

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar