Toric ideals of graphs minimally generated by a Gröbner basis

TL;DR

This paper investigates when toric ideals of graphs are MG-ideals (minimally generated by a Gröbner basis) or UMG-ideals (every reduced Gröbner basis is a minimal generating set). It proves that for toric ideals of graphs, being a UMG-graph is equivalent to generalized robustness ($\mathcal{U}_G = \mathcal{M}_G$), and shows MG-graphs are hereditary under induced subgraphs, with two bipartite families—ring graphs (complete intersections) and graphs where all chordless cycles share the same length—forming MG-graphs. A key result extends to bipartite graphs: if all minimal generators of $I_G$ have the same degree, then $I_G$ is MG; the main theorem characterizes graphs with all chordless cycles of length $2k$ as 2-clique sums of $\varTheta_r^k$ graphs, leading to MG for these graphs. The work connects combinatorial graph structure with Gröbner-basis properties and outlines open questions for non-bipartite cases, including limitations of extending these results beyond bipartite graphs.

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 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 $I_G$ 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 $4$.

Figures (12)

• Figure 1: Bipartite graph with three even cycles. The cycles $c_1$ and $c_2$ are chordless, while $c_3$ is not; indeed, the edge $e_9$ is a chord of $c_3$.
• Figure 2: The even closed walks $w_1, w_2$ yield non-minimal binomials $B_{w_1}, B_{w_2}$ by Lemma \ref{['lm:technical']}, while $B_{w_3}$ is minimal.
• Figure 3: $Q_3$ is a non MG-graph
• Figure 4: Example of an MG-graph.
• Figure 5: A 2-clique sum of two graphs $G_1$ and $G_2$

Theorems & Definitions (35)

• Theorem 1.1
• Proposition 2.1
• Example 2.2
• Example 3.1
• Theorem 3.2
• Lemma 3.3
• Example 3.4
• proof : Proof of Theorem \ref{['thm:UMGgenrobust']}.
• Corollary 3.5
• Example 4.1