On toric ideals arising from signed graphs

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 $\tau$.