Toric Ideals of Finite Simple Graphs

Abstract

This thesis deals with toric ideals associated with finite simple graphs. In particular we establish some results pertaining to the nature of the generators and syzygies of toric ideals associated with finite simple graphs. The first result dealt with in this thesis expands upon work by Favacchio, Hofscheier, Keiper, and Van Tuyl which states that for G, a graph obtained by "gluing" a graph H1 to a graph H2 along an induced subgraph, we can obtain the toric ideal associated to G from the toric ideals associated to H1 and H2 by taking their sum as ideals in the larger ring and saturating by a particular monomial f. Our contribution is to sharpen the result and show that instead of a saturation by f, we need only examine the colon ideal with f^2. The second result treated by this thesis pertains to graded Betti numbers of toric ideals of complete bipartite graphs. We show that by counting specific subgraphs one can explicitly compute a minimal set of generators for the corresponding toric ideals as well as minimal generating sets for the first two syzygy modules. Additionally we provide formulas for some of the graded Betti numbers. The final topic treated pertains to a relationship between the fundamental group the finite simple graph G and the associated toric ideal to G. It was shown by Villareal as well as Hibi and Ohsugi that the generators of a toric ideal associated to a finite simple graph correspond to the closed even walks of the graph G, thus linking algebraic properties to combinatorial ones. Therefore it is a natural question whether there is a relationship between the toric ideal associated to the graph G and the fundamental group of the graph G. We show, under the assumption that G is a bipartite graph with some additional assumptions, one can conceive of the set of binomials in the toric ideal with coprime terms, B(IG), as a group with an appropriately chosen operation ⋆ and establish a group isomorphism (B(IG), ⋆) ∼= π1(G)/H where H is a normal subgroup. We exploit this relationship further to obtain information about the generators of IG as well as bounds on the Betti numbers. We are also able to characterise all regular sequences and hence compute the depth of the toric ideal of G. We also use the framework to prove that IG = (⟨G⟩ : (e1 · · · em)^∞) where G is a set of binomials which correspond to a generating set of π1(G).