On Complete Intersection toric ideals of graphs

TL;DR

This paper characterizes graphs whose toric ideals are complete intersections, revealing structural properties such as bipartite blocks and specific cycle conditions, and classifies when these ideals are normal.

Contribution

It provides a complete characterization of graphs with complete intersection toric ideals, including their block structure, cycle conditions, and normality criteria.

Findings

Graphs with complete intersection toric ideals have bipartite blocks except for at most two.

Generators correspond to even cycles, with at most one generator from odd cycles joined at a vertex.

Complete intersection toric ideals are circuit ideals and satisfy the odd cycle condition.

Abstract

We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. The generators of the toric ideal correspond to even cycles of $G$ except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of the graph satisfy the odd cycle condition. Finally we characterize all complete intersection toric ideals of graphs which are normal.