Invited research visit (04/05-08/05) to the Jussieu Institute of Mathematics  - PRG, Sorbonne Université and Université Paris Cité, CNRS, France.

I also gave a talk in the framework of the working group "Combinatorics, Arithmetic and Geometry". Seminar's webpage

Title: Toric ideals of graphs minimally generated by a Gröbner basis.

(joint work with Apostolos Thoma)

Abstract: Let G be a connected, undirected, finite and simple graph. We study the complete intersection property on the toric ideal I_G. In general, the toric ideal I_G is complete intersection if and only if it can be generated by h binomials, where h=m-n+1 if G is a bipartite graph or h=m-n if G is not a bipartite graph, where by m we denote the number of the edges of G and by n the number of its vertices. The answer is known in the case of bipartite graphs, i.e. graphs with no odd cycles. In the last years, several useful partial results have been proved and they provide key properties of complete intersection toric ideals of graphs. 

We focus on the general case, where G is a random graph and we present a structural theorem which gives us necessary and sufficient conditions in which the toric ideal I_G is complete intersection. Moreover, we characterize with sufficient and necessary conditions the complete intersection graphs which are planar.