On toric ideals arising from signed graphs

Abstract

A signed graph is a pair pG, τ q of a graph G and its sign τ , where a sign τ is a function from tpe, vq | e P EpGq, v P V pGq, v P eu to t1, ´1u. Note that graphs or digraphs are special cases of signed graphs. In this paper, we study the toric ideal I pG,τ q associated with a signed graph pG, τ q, 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 pG,τ q , and then focus on the complete intersection property. More precisely, we find a complete list of graphs G such that I pG,τ q is a complete intersection for every sign τ .