2025/04/08 by García-Marco, Ignacio, Márquez-Corbella, Irene, Tatakis, Christos
#05C25 #13C05 #14M25 #20M14 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.06216
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 paper, 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 IG is minimally generated by some Gröbner basis, and a UMG-graph if every reduced Gröbner basis of IG forms a minimal generating set. We prove that a graph G is a UMG-graph if and only if its toric ideal IG 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.