Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Groebner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Groebner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.Comment: Discrete Math., to appea

Gluing And Splitting of Homogeneous Toric Ideals

We show that any two homogeneous affine semigroups can be glued by embedding them suitably in a higher dimensional space. As a consequence, we show that the sum of their homogeneous toric ideals is again a homogeneous toric ideal, and that the minimal graded free resolution of the associated semigroup ring is the tensor product of the minimal resolutions of the two smaller parts. We apply our results to toric ideals associated to graphs to show how two of them can be a splitting of a toric ideal associated to a graph or an hypergraph.Comment: 21 pages, 3 figure

Many toric ideals generated by quadratic binomials possess no quadratic Gr\"obner bases

Let $G$ be a finite connected simple graph and $I_{G}$ the toric ideal of the edge ring $K[G]$ of $G$. In the present paper, we study finite graphs $G$ with the property that $I_{G}$ is generated by quadratic binomials and $I_{G}$ possesses no quadratic Gr\"obner basis. First, we give a nontrivial infinite series of finite graphs with the above property. Second, we implement a combinatorial characterization for $I_{G}$ to be generated by quadratic binomials and, by means of the computer search, we classify the finite graphs $G$ with the above property, up to 8 vertices.Comment: 11 pages, 17 figures, Typos corrected, Reference adde

Minimal generators of toric ideals of graphs

Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal $I_G$