On toric ideals arising from signed graphs

Abstract

A signed graph is a pair (G, τ) of a graph G and its sign τ, where a signτ is a function from { (e, v) ∣ e∈ E(G) , v∈ V(G) , v∈ 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,τ) associated with a signed graph (G, τ) , 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,τ) and then focus on the complete intersection property. More precisely, we find a complete list of graphs G such that I(G,τ) is a complete intersection for every sign τ.