On toric ideals arising from signed graphs

TL;DR

This paper investigates the algebraic structure of toric ideals derived from signed graphs, providing a unified framework that encompasses known results for graphs and digraphs, and characterizes when these ideals are complete intersections.

Contribution

It characterizes all primitive binomials of toric ideals from signed graphs and identifies all graphs where these ideals are complete intersections regardless of the sign assignment.

Findings

Characterization of primitive binomials in toric ideals of signed graphs.

Complete list of graphs with universally complete intersection toric ideals.

Unified explanation of known results for graphs and digraphs.

Abstract

A signed graph is a pair $(G,\tau)$ of a graph $G$ and its sign $\tau$, where a \textit{sign} $\tau$ is a function from $\{ (e,v)\mid e\in E(G),v\in V(G), v\in e\}$ to $\{1,-1\}$. Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal $I_{(G,\tau)}$ associated with a signed graph $(G,\tau)$, and the results of the paper give a unified idea to explain some known results on the toric ideals of a graph or a digraph. We characterize all primitive binomials of $I_{(G,\tau)}$, and then focus on the complete intersection property. More precisely, we find a complete list of graphs $G$ such that $I_{(G,\tau)}$ is a complete intersection for every sign $\tau$.